engineering homework examples

17 Problem Solving & Modeling

Introduction.

Aerospace engineering students soon begin to ask when they can start to solve actual problems relevant to aircraft, rockets, or spacecraft. Of course, it is natural to ask such questions, even early on in an engineering education. However, practical problem-solving in engineering is a serious business that requires engineers-in-training to become well-versed in the fundamental subjects appropriate to their field. Besides the usual engineering disciplines, this process of learning and becoming proficient will inevitably include having a solid background in physics, chemistry, mathematics, numerical methods, and computer programming.

In general, engineering problem-solving is an established and well-proven process in which the bigger problem is first dissected into smaller, more manageable, and perhaps better digestible parts, as illustrated in the figure below for a flight vehicle. Then, each vehicle component can be analyzed separately, at least initially, perhaps with experiments, mathematical models, numerical models, or all these working approaches working hand-in-hand. Finally, the understanding and functionality of the parts can be reassembled using a synthesis approach to understand the flight vehicle as a whole, with interactions considered, too. Of course, such an approach is always flawed, but it forms at least one rational basis for understanding and designing complex engineering systems, including aircraft and spacecraft.

Today, aerospace engineers must also have increasingly multidisciplinary technical skill sets, which means they must follow a broader-based educational path and become more knowledgeable and versatile in using a more comprehensive range of subject matter. It is no longer sufficient to be an aerodynamics specialist or “aerodynamicist,” or a “structural dynamicist” or an “acoustician.” Successful aerospace engineers must be technically broad and increasingly versatile to work with others in interdisciplinary contexts.

Furthermore, using artificial intelligence (or AI), machine learning, and big data analytics is increasingly prevalent in the aerospace industry, for which engineers must know the benefits and limitations. As the industry evolves and solves more challenging problems, it is also crucial that aerospace engineers continue to develop their problem-solving skills throughout their careers to stay current with the latest technological advancements.

Learning Objectives

Begin to understand the fundamental processes involved in engineering problem-solving.
Appreciate the significance of governing equations and how such equations can be reduced so that they apply to specific problems.
Understand the ideas of modeling complexity and the trades between fidelity and cost.
Better understand the importance and expectations of doing homework problems in engineering classes.

Hypothetico-Deductive Method

Many have argued that the engineering design process must follow the hypothetico-deductive method (or H-D method), a primary method for testing hypotheses or conjectures. The “hypothetico” (or hypothetical) part is where a hypothesis or theory is proposed, which needs to be tested, and the “deductive” part is where the consequences are drawn from the hypothesis or hypotheses. [1] The H-D method is sometimes called THE scientific method , but it is not the only method used in scientific and engineering work. The H-D method can be divided into the four stages outlined in the figure below.

1. Identify the theory, the hypothesis, or the conjecture to be tested . This approach does not necessarily need to rely on facts and allows for “imaginative preconceptions, intuition, and even luck.” However, the hypothesis usually relies on prior understanding or awareness of pervasive laws.

2. Generate predictions from the theory. The theories are used to make predictions about what we see, i.e., we proceed to imitate what we perceive as the “real world.” These predictions would also encompass the range of conditions perceived as the theory’s domain of applicability, which cannot necessarily have limitless bounds and will trade cost with theoretical complexity and value with predictive fidelity.

3. Use various types of experiments and measurements to test whether predictions are, in fact, correct. If done correctly, the measurements represent the truth. The data acquired, assuming the quantities needed can be measured to the necessary levels of fidelity, provides the evidence to test the proposed theories. Replication of an experiment by others and repeatability of the data over time are critical to doing good science. The data may often uncover unexpected outcomes and spawn new directions if sufficiently comprehensive and high-quality.

4. Expose the theory to criticism, then reject or modify the theory or declare that the theory has been validated or otherwise proved. This process may take significant time and often depends on specific experimental measurements and other data availability. However, it is the most crucial part of doing scientific research. Alternative hypotheses may be pursued after criticism to see which is more likely to explain the predictions. In this regard, the principle of parsimony, or Ockham’s Razor , is essential in mathematical modeling, i.e., in generating equations and mathematical models to represent any given physical behavior.

The goal is to keep modeling complexity and predictive fidelity in equilibrium. This can be achieved through careful, systematic validation studies and requires a degree of engineering common sense. After all, as the figure below suggests, this is the ultimate foundation on which the scientific method is based.

Other methods are used in scientific and engineering work. For example, descriptive methods observe and describe phenomena, while experimental methods manipulate variables to establish causal relationships. Correlational studies examine the relationships between variables without manipulation. Qualitative research gathers in-depth insights through non-numerical data, while mixed-methods research combines quantitative and qualitative approaches. Longitudinal studies track changes over time, meta-analysis synthesizes findings from multiple studies, and action research collaboratively addresses practical issues. Each method offers unique strengths suited to different research questions and objectives, guiding scientific exploration across diverse disciplines and contexts. Nevertheless, the hypothetico-deductive method remains the primary method for testing scientific and engineering hypotheses.

Starting Out

As students think more about engineering concepts and various types of problem-solving in aerodynamics, structures, flight vehicle performance, and other areas, some words of caution are appropriate. First, it must be appreciated that there are few “handy equations” for solving engineering problems, especially in aerodynamics. Instead, the relevant equations for problem-solving must be selected carefully in terms of the specific equations that most appropriately govern the problem, called the governing equations . Choosing the governing equations is one issue, but solving them using the correct boundary conditions, and perhaps with any appropriate simplifications, involves considerable skill that comes only from much practice, i.e., consistent, purposeful, and focused practice over time. The solutions to some of these problems become homework exemplars of the field. “ Those who study a scientific discipline are expected to know its exemplars .” There is no fixed set of exemplars in aerospace engineering; however, this ebook is filled with a few hundred for starters.

“When you can measure what you are speaking about and express it in numbers, you know something about it; but when you cannot measure it when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science , whatever the matter may be.” Source: S ir William Thompson (Lord Kevin), Popular Lectures and Addresses, V ol. 1 (1889) “Electrical Units of Measurement,” from a presentation delivered on May 3, 1883.

Aerodynamics

Remember that aerodynamics is the underpinning of flight, so some form of aerodynamic analysis, such as shown in the figure below, comes into almost all types of problem-solving with aircraft and even with spacecraft, i.e., launch and re-entry vehicles. Therefore, it is essential that the selected aerodynamic models be sufficiently comprehensive and detailed to predict what is needed and for the right reason. Ultimately, they will provide a more substantial basis for decision-making, resource allocation, risk management, and adaptability. By considering a wide range of factors and understanding the underlying reasons, it becomes possible to navigate options and reduce uncertainties to make informed choices.

Value-Based Analysis

How comprehensive and detailed any mathematical model needs to be will depend on the specific type and complexity of the physical problem, which also affects the time and effort (i.e., cost) to solve the problem. If the model is intended for precise predictions, design optimization, or critical engineering applications, a higher level of detail and accuracy is typically required, and the penalty is time for a solution. In the workplace, time equates to money, i.e., a value-based decision process is needed, so it is essential to balance the technical level of detail in the model and the resulting computational cost.

Mathematical models must consider additional factors such as non-linearities, coupling effects, boundary interactions, or time-dependent behavior. These complexities demand more comprehensive models that incorporate more detail to represent the physical problem and its behavior accurately at the price of longer execution time and higher costs. For example, aerodynamic equations may need to be solved consistently and simultaneously with other sets of equations describing the behavior of different aspects of the aircraft, such as its dynamic flight motion, aeroelasticity, acoustics, etc.

It is soon concluded that problem-solving in engineering, especially concerning flight vehicles, can be very challenging, time-consuming, and quite costly because the forms of the governing equations for different disciplines will inevitably be different. This issue means different solution methods, including the various specific techniques, numerical methods, etc., may be required in each case, e.g., numerically solving together or coupling the governing equations that are applicable and traditionally used in the different engineering fields.

Governing Equations

As students learn more about aerospace engineering disciplines, they will be exposed to more general forms of potentially applicable governing equations in each field. In many cases, the relevant governing equations that apply to a given problem may be subsets of more general governing equations intended to apply to a broader range of problems and conditions. In other cases, the governing equations may need to be developed from first principles, e.g., directly invoking the conservation principles of mechanics and thermodynamics. However, the approach must be systematic when choosing or creating the equations that apply to or govern a specific problem. The approach of picking a few equations and hoping they will apply is called an ad hoc approach. However, such an approach will inevitably prove disastrous; engineering case histories and experience speak for themselves.

Problem-solving also requires that the meaning of the governing equations must be understood, as well as each of the terms that comprise these equations. There may be terms or groups of terms in these equations that have different levels of complexity and then may also have interdependencies, so dropping one term may have unintended consequences on the evaluation of other terms. In many cases, simplified (reduced) or otherwise particular governing equations may be appropriate because they simplify the solution process. Indeed, an approximate (and fast) solution might be adequate in the initial design phases. However, a more accurate (and likely more time-consuming) solution might be needed later. Part of the skill that must be developed in engineering problem-solving is to decide what terms in the equations must be retained and what terms can be eliminated without substantially affecting the outcomes of the final solution.

Setting Up Aerodynamic Models

Solving problems in aerodynamics requires that appropriate mathematical models of the flow are set up correctly. The derivation of the mathematical equations that describe fluid dynamic and aerodynamic flows is a systematic process that has become well-established in engineering practice. However, because air (like all fluids) will continuously deform as it flows, its behavior must naturally be expected to be more difficult to describe than a solid material.

The starting point of any aerodynamic analysis is the statement of the physical problem, a definition of the appropriate boundary conditions, and making justifiable assumptions and approximations about how the flow develops. An example of a boundary condition is one that defines the values of the free-stream flow conditions or how the flow behaves on a boundary or surface, e.g., it flows parallel to the surface. The three fundamental conservation principles of mechanics must then be applied to the aerodynamic problem, namely:

Conservation of mass, i.e., mass is neither created nor destroyed.
Conservation of momentum, i.e., a force applied to a mass equals its time rate of change of momentum.
Conservation of energy, i.e., energy is neither created nor destroyed and can only be converted from one form into another.

The resulting mathematical equations should then describe the aerodynamic behavior of the flow of interest, at least within the bounds of the stated assumptions and approximations. The solution to these equations can proceed analytically, numerically, or both, hopefully giving an engineer the desired results.

All practical problems will inevitably require some assumptions and approximations to obtain solutions. A common assumption is that air behaves as an ideal gas. Other assumptions might include two-dimensional flow, steady flow, incompressible flow, inviscid flow, etc. However, all such assumptions must be justified, and such justifications often take skill and experience. Skill and experience are obtained through solving engineering problems, which first develops from diligently doing homework problems.

For example, the figure below shows a hierarchy of aerodynamic methods that could be used to model the real flow, starting from lifting line theory, passing through lifting surface theory to a panel method that also models the effects of wing thickness, and finally to a full computational fluid dynamics (CFD) model. Each method is accompanied by a commensurate increase in fidelity and an increase in execution time and computational cost. Whether the problem is aerodynamics or otherwise, the idea is often to start with a more straightforward method to get some initial understanding of the problem at a relatively low cost and then progress to a more complex model with higher fidelity for the final calculations. For example, a CFD calculation may take up to five orders of magnitude more time and cost than the lifting line theory.

Verification and Validation

Inevitably, the outcomes from any model (i.e., the computed results) would be checked to make sure that they make sense, such as by comparing them with measurements, i.e., to establish how well the model works, a process called verification and validation or “V & V.” If the outcomes are positive, then the process inevitably also involves checking it for different input scenarios to ensure that the model predicts what it needs to predict and for the right reason. However, the process may conclude that the results do not agree. The V & V of all mathematical models is critically important, especially if such models are to be used in the design process. Confidence in the design tools being used is critical if the design is to prove viable or successful.

An example is shown in the figure below, where predictions from four mathematical models (i.e., different competing methods or “theories”) are compared against measurements. The question is: “Which mathematical model works the best compared to measurements?” All methods work reasonably well over some ranges, but one or more may be better than the others. The exact theory fails for higher values of the independent variable. Subjectively, Method 3 seems to compare best to the measurement points. However, when measurement uncertainty is accounted for, Methods 1 and 2 may be just as good regarding their predicted values. Unfortunately, in this case, the final answer as to the “best” model may be in the eye of the beholder! The other issue to examine is to ensure that the best method predicts the behavior for the right reason, which may take additional measurements and/or analysis to answer fully. Predicting the correct outcome for the wrong reason is not a verification or validation of the method. Indeed, some may argue that such models cannot be validated, only disproved, i.e., the essence of Popper’s falsification principle.

Finally, these equations (i.e., the “model,” in general) can be used to analyze similar problems of interest, such as in a parametric study where one quantity is varied using the model to establish the corresponding effects on the other quantities. Such parametric studies with mathematical and/or numerical models are critical in engineering design. For example, a particular combination of parameters may be sought to optimize the system performance or for another reason. However, it should always be remembered that no mathematical description of a physical problem can be perfect for its behavior. It can only be an approximation whose accuracy depends on how diligently the model is set up, including the nature of the assumptions and approximations.

Multi-Disciplinary Problems

As previously discussed, aircraft and spacecraft structures are lightweight, thin-walled structures of various beams, columns, shafts, plates, shells, etc. All must be modeled in aggregate with aerodynamics using a multi-disciplinary approach, the idea being shown in the figure below. Coupling aerodynamics with the structures and structural dynamics is called aeroelasticity because the action of the aerodynamic loads will elastically deform the structure and so will feed back to the aerodynamic loads. Combining models of aerodynamics and structural loads and deformations usually requires an iterative approach, adding significantly to the computational time and overall effort.

Like aerodynamics, the governing equations of structural elasticity comprise sets of partial differential equations. Computational Fluid Dynamics (CFD) can help predict the detailed aerodynamic loads if sufficient computing power is available, i.e., the need for memory and execution speed. For the structure, the finite element method (FEM) is usually necessary to predict the behavior of complex structural geometries. The FEM is highly developed and sophisticated enough to handle just about any aerospace structure, but like CFD also requires significant computing power. Sometimes, subsets of the governing equations may be adequate to speed up the computational process, depending on the assumptions and the approximations that can be justified, e.g., small displacements or isotropic structural properties.

Flutter is a form of aeroelasticity and is a potentially catastrophic dynamic phenomenon that can happen with the inherently flexible structures of aircraft and spacecraft. Flutter usually occurs when the forces created on an object cause it to displace or deform, elastically return to where it was, but also overshoot and then repeat the process and begin to oscillate, i.e., it is a dynamic process requiring an inherent coupled aerodynamic and structural dynamic solution process. If the forces subsequently increase in magnitude, the oscillations also increase until the object eventually fails structurally, e.g., the tail or a wing may break off during flight. Therefore, flutter must be avoided, and great efforts must be undertaken to ensure a flight vehicle is flutter-free. Unfortunately, flutter can still occur on flight vehicles, and it can also happen with buildings, bridges, and other flexible objects exposed to the wind.

Problem-Solving Process

The figure below indicates that a general procedure for solving a physical problem requires many steps. The steps are not unique and will vary from problem to problem, but the iterative process is typical to all aspects of design, particularly for flight vehicles. The process requires many specialized activities and inevitably takes considerable time.

1. Specify or define and then describe the nature of the physical problem, which can often be done relatively quickly using an appropriately annotated sketch. Typically, for an aerodynamic problem, the annotations could include the size and shape of a control volume, the general flow directions, and the specifications of relevant boundary conditions.

2. Mathematically specify any relevant known boundary conditions, such as upstream free-stream conditions. On the surface of a solid body placed in the flow, the flow would not pass through that body, so another boundary condition is that the flow is parallel to the body surface.

3. Decide on the primary form of the needed model, i.e., whether an integral or differential approach is required. For example, if detailed properties are needed at all points, an integral approach is unlikely to be appropriate, and the problem should be approached using a differential model.

4. For aerodynamic problems, decide whether an Eulerian or Lagrangian flow model is required, i.e., whether the aerodynamic behavior at a fixed point or over a volume in space is needed or whether the identical fluid particles need to be tracked as they move through the flow.

5. Make any justifiable assumptions about the problem. The idea here is to take the actual physical problem and derive a simplified but still relevant mathematical version of the physical problem. By drawing on experience or from experiments on similar problems, it may be possible to make reasonable assumptions. For example, for aerodynamic problems, it may be possible to assume that the flow is steady and/or in predominantly two dimensions, which usually results in considerable simplifications of the mathematics. These assumptions are then used to help develop the appropriate governing equations for the model.

6. Use the conservation principles to establish the model’s mathematical form that describes the physical problem. Because there are three physical principles to invoke (i.e., mass, momentum, and energy), most problems will involve three governing equations. However, auxiliary equations (e.g., an equation of state) may also be needed. These equations then need to be solved consistently and concurrently.

7. Conduct the solution process where the relevant equations are solved for the desired physical quantities, e.g., for flow problems, velocities, pressures, etc., may be needed. In some cases, the equations may be solved analytically in closed form, meaning that the resulting solutions are pure mathematics and the creation of final sets of descriptive equations. The equations will likely need to be solved numerically, i.e., a computer program must be written with numbers as the outputs.

8. Verify and validate the model to determine the model’s accuracy and correctness, a process often called “V & V,” i.e., use predicted outcomes to determine how good the model is in representing the physical behavior that was the desired outcome. The model’s validity can be established by comparing the results against measurements if such measurements are already available or can be conducted, as shown in the figure below. This step is essential in aerodynamic modeling and is one reason wind tunnels are critical in understanding all types of aerodynamic flows. If experimental data are unavailable, sometimes other theories can be used for validation, but validation is rarely conducted without reference to appropriate measurements. In all cases, experience and good judgment must be used to establish that the predictive credibility of the model has been obtained. This is rarely a test that one person can objectively conduct because confirmational bias, also known as confirmation bias, can play a role. [2]

9. Improve upon the capabilities of the model to broaden its capabilities. As experience is gained in the validation process regarding what the model predicts adequately and what it does not, the limitations and assumptions within the model can be progressively removed, or other enhancements can be made. For example, extending the range of validity of an aerodynamic model may be possible by including unsteady effects or with a representation of turbulence. In a structural model, it may be necessary to include non-linear effects, e.g., from large deflections.

10. Finally, balance the model’s complexities and capabilities against the time and cost of obtaining solutions from the model. In this case, questions will have to be asked about how the model will be used and whether the full fidelity of the model is needed. For example, the need to model compressibility effects in the flow might not be required if the problem is restricted to low Mach numbers. Furthermore, if the model is to be used exclusively for research, then computational time and cost will be less crucial than for use in the industry, where a short turnaround time is always needed.

Generalization of Data & Data Fitting

$\Pi$

For example, a pure theoretical model may be of the form

$\begin{equation*} \Pi_1 = \Pi_2^{~2} \end{equation*}$

which may show good qualitative agreement with measurements but not quantitive agreement. However, a semi-empirical model of the form

$\begin{equation*} \Pi_1 = A + B \, \Pi_2^{~2} \end{equation*}$

It is rarely possible to cover the entire domain of the parameter space in one single experiment. In this regard, not all test facilities are created equally. One experiment may cover one limited range of the domain, and another experiment may cover another range, perhaps with some overlap, as suggested in the figure below. Each experiment may be inadequate in establishing any useful, robust, causal correlation. However, if the data is taken collectively and used intelligently in the process of generalization, then a semi-empirical model can show a more robust and positive correlation.

Repetition of experiments is always a good practice, which can also benefit from the advancements in measurement technologies. Indeed, the scientific method requires that any one experiment can only be fully trusted once it has been independently repeated and the results confirmed. Unfortunately, repeating experiments is often difficult to justify based on cost, time, or both. Inherent differences in the testing facilities can also be a consideration, i.e., wind tunnel interference effects, measurement limitations, etc. However, the logical rationalization here from the perspective of the scientific method is that the ability to make better measurements will lead to lower errors and inaccuracies in the acquired data. So, a better and stronger causal correlation is likely to be shown. Therefore, more confidence in prediction can then be gained. Nevertheless, for any given dataset, there is always considerable uncertainty in extrapolation beyond the range of the measured data.

Inevitably, new experiments on a given problem are performed in time, and some experiments may be ambitious enough to extend the range of the domain. Such data may confirm existing trends or be disruptive, suggesting a different correlation, as shown in the figure above. On the one hand, such disruptive data leads to an improved correlation and more confident mathematical models that show enhanced predictive confidence at the system level. On the other hand, predictions at the system level may be worse, in which case one can suspect that one modeling error has previously acted to cancel another, i.e., duo mala faciunt unum bonum , and so point to further modeling deficiencies. The stakes in such outcomes are high in that the validity of the entire system model can be thrown into question until the weaknesses in the other parts of the model are identified and corrected. In this regard, complex models, especially those with significant empiricism, can remain perpetually tentative until more and/or confirmatory measurements are obtained.

Ockham’s Razor – The KIS 2 Principle

One issue with complex engineering models for multidisciplinary aerospace applications is that significant empiricism may be needed. Some physical problems are difficult to model without resorting to substantial empiricism, an unavoidable artifact of representing complex physical processes with parsimonious models with practical levels of computational efficiency. In this regard, there is always a need to balance the complexity of the mathematical model against the model’s predictive accuracy while aiming to minimize the variability and maximize the intelligibility of the resulting simulations. For complex mathematical models, history has proved that predictive accuracy increases with increasing modeling complexity only up to a point where the cumulative uncertainties in the components of the model (particularly those with significant empiricism) begin to increase the “noise” in the predictions. Then, beyond a certain level of complexity, the predictive accuracy decreases again, the system exhibiting a classic “Ockham’s Hill,” as shown in the figure below.

In this regard, it is essential to remember the principle of Ockham’s Razor, i.e., given two sets of solutions from methods of equivalent accuracy, one should side with the simpler or parsimonious method, i.e., Frustra fit per plura quod potest fieri per pauciora. This approach is sometimes called the KISS or KIS2 principle, which means “Keep it Short and simple.” Ernst Mach was also an advocate of a similar principle he called the “Principle of Economy,” stating: “Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses.” The message is evident in that the goal for engineers is to keep modeling complexity and predictive fidelity in balance, something that cannot just be achieved through careful, systematic validation studies but also requires a degree of common sense.

Approaching Homework Problems

The skills and abilities to solve real engineering problems develop from the exemplar problems encountered in the classroom. To this end, budding engineers must first be good at doing homework problems. Results generated in homework problems may be single numbers, tables, or graphs in any or all combinations that must be adequately presented. Below are some general guidelines to help new students tackle homework problems and build up the skills they need to learn as engineers.

Homework is NOT a quiz!

The idea of homework for engineering students is to begin to learn the process of solving real problems, which becomes a lifelong endeavor for an engineer. Homework problems expand upon what can be done in the classroom, which are usually simpler and straightforward. In doing the homework, students can use all the resources available, including exemplar problems and solutions given in the textbook or ebook, for which many are former exam or quiz questions. Students should use these exemplars to help them understand the process of solving similar problems that may come up on homework and future exams.

If you are a student and need help understanding how to go about a specific homework problem, then ask your instructor before or after class, during office hours, or by email. Refrain from guessing! Also, do not copy down former “solutions” to similar problems and present them as your own – use the exemplar solutions to understand the process of solving the current homework questions, then write out your own solutions. Finally, do not assume that the current question(s) is (are) exactly the same as a previous question you may have seen; submitting the correct answer to the wrong homework problem will not have a good outcome.

A student’s homework score may be a significant fraction of their total grade in an engineering course. This fraction reflects the importance instructors place on learning the methods and techniques of solving engineering problems and developing and maintaining other skills, including presentation formats, time management, MATLAB use, drawing graphs, etc. Remember that students may see exam questions very similar to homework questions, which are often easier! So, if students approach homework problems seriously and put in the required time to understand the processes of solving them, they can expect to do well on the exams. History speaks for itself in this regard.

Pointers for Doing Good Homework

Preparations.

Start to review the homework problem set as soon as the professor or course instructor sets it. Remember: Start working on the problem set early! Time management is essential.
Don’t guess! Homework is not a quiz. It is a learning exercise, so don’t consider homework as a quiz or exam.
If you need help with the homework questions, attend the professor’s, instructor’s, or TA’s office hours. Come prepared and ask specific questions; other students may be waiting to ask their questions.
Don’t post your homework problems online, hoping that someone in cyberspace knows better than the person who wrote the question. Ask the professor or whoever wrote the problem(s) for advice.
Check whether a similar problem is solved in the textbook, the ebook, or the notes. Review old problem sets and their solutions – professors usually make them freely available to students.
Meet with other students, your study group, the TA, the grader, or anyone else and try to understand the problem and potential solution method. An effective learning method is for one student in a study group, for example, to try to teach the solution method to the others.

Starting on the Problems

State the problem statement briefly and concisely based on the information given. It is often helpful to restate the information in the question, clearly and straightforwardly laying out what is known and what is not.
If appropriate, draw a sketch or schematic of the problem/approach. In most cases, an annotated sketch of the physical problem will help you decide the nature of the mathematical model that needs to be adopted, e.g., control volume approach or otherwise.
Write down the appropriate mathematical equations necessary to solve the problem. These equations should not be expected to be given in the homework problem, at least not in all cases. Choosing (or deriving) a set of simpler equations from a broader, more general set of governing equations may be necessary.
List and develop any simplifying assumptions appropriate to the problem. Sometimes, the assumptions will be specified; other times, they may be left as part of the problem. For some more challenging problems, it may not be apparent initially what assumptions are needed. Several attempts may be necessary until the correct assumptions can be confirmed and verified.

Working the Problems

Be sure to work carefully and systematically through each of the problems. It is better to work slowly but get the correct answer than to work faster and make silly mistakes.
Complete the analysis in algebraic form (i.e., equations with symbols and variables) before substituting the specific numerical values. Sometimes, the answer will be presented as an equation rather than a numerical value.
Substitute known numerical values (using a consistent set of engineering units ) to obtain a numerical answer or answers. If the problem is given in SI units, it is best to work it entirely in SI units; conversely, if it is given in USC units, then work it entirely in USC units. For example, switching back and forth between USC and SI is inadvisable because this approach is often a source of mistakes and numerical errors.
All numerical answers must have appropriate units unless they are in non-dimensional form. Always double-check the engineering units of the solution (s). Units should be used consistently throughout, ideally in base units.
Check that the number of significant digits in the answer(s) is/are consistent with the given data. For example, if you are given information to 3 significant digits, it would not be appropriate to calculate your final results to 5 significant digits. However, it is good practice to round off numerical values at the end of the problem.
Review the answer(s) for correctness. In some cases, it will be evident if the result is wrong; it may be challenging in other cases. In many cases, the question is: Does that result seem physically correct? Check with someone else if in doubt, such as a course instructor.
Draw a box around the final answer to clarify that this is your definitive answer. The answer will often be an equation (formula) or number, but it could be a table or a graph. You do not need to draw a box around tables or graphs.
Import the results into the appropriate software if a graph needs to be drawn. Never draw graphs freehand!! Use MATLAB, Excel, or your favorite graphing program. Kaleidagraph was used for many of the plots in this ebook. As appropriate, all graphs need to have proper legends, labels, and other annotations.
Your submitted homework must be neat and easy to follow. You will also need to follow the submission rules. For example, you may have to use squared engineering paper, put your name and student identifying number on each page, staple together your pages, etc. If not, you will likely get less credit regardless of the correct solution. In industry, great emphasis is placed on the clarity and presentation of reports and papers; for the same reasons, clarity in homework is a place to start.
Write down clearly and unambiguously the names of the student(s) you worked with on the homework, if any. Working with others is acceptable , but you should submit your OWN answers to the questions. For example, you may state, “I cross-checked my final answers with John and Kevin, and my answers agreed with theirs.”

All those Equations!

In engineering (or any science field), students do not have to memorize hundreds of equations or “formulas.” The questions and concerns from new students of the field inevitably soon start to flow, such as: ” What equation do I use?” or “Where do I find the equation?” or “What is this symbol in this equation?” or “Where do I get the solution to this integral?” Such questions are natural, and guidance from experienced engineers and professors is essential. As taught in many educational contexts, the “plug and chug” paradigm of plugging a numerical value into some equation and chugging out an answer is a surefire recipe for disaster in actual engineering problem-solving. Bona fide engineering students need to learn to do much better.

The number of equations in science and engineering could be as many as the number of stars in the galaxy! No exaggeration. As the figure below suggests, most successful engineers and professors remember the details of the biggest stars and the most general form of the equations or the “governing” equations. They also understand the concepts and have done enough problems to know how to reduce and simplify the equations under certain assumptions and conditions to produce subsets of equations with specific applicability to the problem at hand. The equations they can’t remember can usually be derived! The governing equations can then be adapted and applied successfully to various situations by grasping the underlying concepts without memorizing dozens of problem-specific equations. Science and engineering is not a memory game.

The risk in memorizing without understanding is that the wrong equation (or equations) is (are) applied, so the answer obtained is inevitably wrong too, which is always a disastrous outcome in engineering problem-solving. Therefore, rather than rote memorization or hunting down a specific formula that may or may not be applicable, understanding the fundamental principles allows engineers to derive and apply the relevant equations to particular problems. While some foundational equations may be essential at one’s fingertips, the focus should still be on understanding the underlying physical principles and fundamental concepts expressed by the equations, appreciating the meaning of all the terms in the equations, and rationalizing the best solution strategies for these equations.

Moreover, with access to the Internet, lots of information and resources are readily available for students to find the more general form of the equation(s) that might be needed. Unfortunately, there is a lot of opinionated absurdity on the Internet, especially regarding engineering topics, so using authoritative resources is critical. Utilizing selective online resources, as well as peer-reviewed publications, textbooks, and sanctioned computational tools, can help new students and practicing engineers retrieve or verify specific equations and/or solution methods when needed. Labora sapienter, non strenue.

What is Brainstorming?

Brainstorming is an informal but highly effective approach to engineering problem-solving that works for homework problems. Usually conducted with a small group of engineers or students and a single moderator, the process encourages all participants to think laterally and develop ideas that might initially seem unusual or even sound slightly crazy!

A group of five to seven people is usually the most influential, with a mix of experienced and less experienced engineers. The moderator should be someone other than the chief or lead engineer or an engineer in management, and the group itself should select the moderator. The people in the group should come from several technical disciplines to foster and develop the most effective and productive brainstorming environment. Excellent ideas may ultimately flow from engineers who see an avenue of opportunity in a discipline different from their own, i.e., when they start looking at the problem with a fresh mind. Some of these ideas may be developed into a rational basis for engineering problem-solving, often following a new or innovative path that nobody else had considered previously. The basic idea is to get all participants to think “out of the box,” be creative, and divert their attention away from using their “conventional wisdom” to solve problems.

Brainstorming is usually very effective for solving complex engineering problems requiring multi-disciplinary engineering. The best ideas in a brainstorming session will often flow from the less experienced engineers, who are not so encumbered by conventional wisdom. Brainstorming is best conducted in an informal, relaxed environment away from the typical day-to-day work environment, often at a retreat location. Brainstorming can also be fun, an excellent environment for team building, and a way of getting to know other engineers outside one’s primary technical discipline or organization. It is not unusual for companies to work together on solving complex engineering problems in the aerospace field, and brainstorming sessions can foster more substantial inter-company dialog where everyone works more effectively together.

For brainstorming to be effective, all group members must be active participants, and personal criticism is inappropriate. Often, some quirky idea from one group member makes no sense at first, but after discussion, there can be an “aha!” or “I never thought of that!” moment, and the idea can be built upon by the group after that. Alternatively, the idea may lead to some other revelation and a different path to solving the problem that nobody thought about before.

In preparing for the brainstorming session, a location with no distractions must be found, all phones and computers should be turned off, the doors locked, and a whiteboard should be available to write down the ideas. Everyone should have an opportunity to speak when they want to. The moderator must refrain from allowing any one member of the group to dominate the brainstorming session. At the end of the session, the group decides on the best ideas and then moves forward, as needed, to follow up and pursue them. The history of engineering suggests that many of the best and most innovative ideas can come from brainstorming sessions.

Summary & Closure

Developing mathematical models that can be used to study and solve various problems is an integral part of engineering. However, the selection process must be conducted carefully and systematically to choose or develop the relevant governing equations that apply to the specific problem (or problems) of interest. In some cases, it may be possible to down-select the appropriate governing equations for a particular problem from more general forms of governing equations that are intended to apply to a broader range of conditions. In other cases, the governing equations may need to be developed from first principles.

Once a mathematical model or set of models has/have been developed, it is necessary to solve the equations to make predictions about the system being modeled. The solution process can involve analytical methods, such as closed-form solutions, or numerical methods, such as finite element, finite difference, or boundary element methods. The choice of solution method will depend on the nature of the problem, the available resources, and the desired accuracy of the solution. In all cases, the justification of assumptions and/or approximations is needed. In this regard, any reason may require reliance on outcomes from experiments, i.e., for verification and validation of the modeling. A combination of solution methods may often arrive at an acceptable solution. Also, post-processing and visualization techniques may help interpret the results and behavior of the modeled system.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

The “KISS” or “KIS 2 principle refers to the acronym for “Keep It Short & Simple.” Discuss the meaning of this principle as it might apply to engineering modeling.
What is a parametric study, and why might we conduct one in engineering design?
How might assumptions impact the accuracy of models? Can overly simplifying a problem lead to inaccurate results
What might be the trade-offs between making assumptions to simplify a model and retaining complexity for better accuracy?
Why is it essential to validate and verify models? What are some standard methods for validation, and how do they contribute to model credibility?
How might advances in computational power and simulation techniques influence how we set up and solve flow models in the future?

Additional Online Resources

An excellent video on the use of mathematical and computer models in engineering.
Video on the use of models and simulation in engineering.
View a video on Eulerian and Lagrangian flow models.
Navigate here to watch a video from the National Science Foundation on types of flow models.
Review the KIS 2 Principle here and a video here .
The hypothetico-deductive method is a fundamental approach used in scientific inquiry across various disciplines, guiding the systematic investigation of natural phenomena and developing scientific theories. It emphasizes the importance of empirical evidence, logical reasoning, and rigorous testing in the process of scientific discovery. ↵
Confirmational bias is a cognitive bias where individuals tend to favor information that confirms their existing beliefs or hypotheses while disregarding or downplaying contradictory evidence. In other words, people tend to seek information that supports what they already believe and ignore information that contradicts it. This bias can lead to flawed decision-making and judgment because it can prevent individuals from critically evaluating evidence and considering alternative viewpoints. ↵

Introduction to Aerospace Flight Vehicles Copyright © 2022, 2023, 2024 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License , except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n15

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Calculations and spreadsheets.

Calculations and spreadsheets are a form of technical communication. Often calculations are shared by engineers to other engineers, bosses, even clients. When the calculation is shared it becomes communication! Learn how to create calculations that communicate effectively.

Hand-written Calculations

Engineering calculations are the heart of any design or analysis of laboratory data. They need to have more than just the right answer. They need to be formatted in such a way that a reader can understand the calculation and make sense of the answer. You should already be experienced at communicating effectively using calculations (i.e. homework problems).

Calculation Format

Each sheet of calculations should have identifiers about the calculation. These include:
Your name (or who did the calculation). Sometimes there is also space for who checked the calculation.
The class/section for an engineering course homework or the company name for a professional design calculation
The date (always date your work)
Page number/total pages (let the reader know the total number of pages so they can know if one is missing)

Many engineering calculations (and your homework problems) are hand written on engineering paper and most engineering paper has handy little boxes to put this information in at the top of the paper. If you work in a design company often then the company will often provide paper that will have spaces for this information.

The main box of the engineering paper is for well-written, legible, engineering calculations. Each problem or design should be organized in the following manner:

Problem identifier. This can be the homework problem number, or an identifier for the design (i.e. Beam C2 shear calculation)
Given information or assumptions. It is important that you restate your given information. This prevents the reader from the need to go back and try to find that information. If this is a homework problem then restate the assigned problem. If it is a design problem then you may include the design constraints or other given information. Always end this section with the goal of the calculation, what you are trying to find.
Most engineering is very visual and a sketch of the problem is often very useful. Make sure the sketch is clean and well-drawn (use straight edges to help draw straight lines).
Write out all the calculations required to get to the solution. Make sure that you provide a numerical example for every formula used. Be clear what your variables are. Make sure that you use units and the correct number of significant figures.
Clearly indicate your solution (draw a box around it).
If there is something interesting you want the reader to know about the calculation include it here.

Good Practices

The following suggestions will help with the creation of effective engineering calculations.

There should be no more than one problem on each page. If you start a new problem start it on a new page. If your calculation exceeds a page then that is fine.
Write in pencil not in ink. If you must erase something then make sure it erases completely (use the white drafting erasers). Do not leave smudges or cross-outs in your final calculation. This may mean recopying the calculation for a clean result.
Write in print not cursive
Use units in all your calculations (even went putting numbers into a problem). This greatly helps the reader (or professor) follow the work and helps you not make silly unit conversion errors.
Use the correct number of significant figures for each calculation.
Double check your work to make sure the math is correct.
Provide the name, symbol, and value for all variables used. (i.e. “acceleration due to gravity (g) = 9.81 m/s”)
Use blank space. Paper is cheap. There is no need to cram all your calculations in as small a space as possible. Spread out and make the calculations look good.
For more information see: http://users.rowan.edu/~jahan/fatetransport/rowan_hw_format.doc.

Significant Figures

When presenting data, it is important to have the number formatted correctly as well. Pay attention to the number of significant figures in your data.

Number of significant figures indicates precision. Significant digits of a number are those that can be used with confidence, e.g., the number of certain digits plus one estimated digit. It is very important that numerical results be reported using an appropriate number of significant figures. Using too many digits implies a higher level of precision than was actually obtained in the lab procedure. Often putting a number in scientific notation makes it clearer the number of significant figures.

The number of significant figures (non-zero numbers) used to report results should be appropriate to the sensitivity of the measurement.

Wrong: Young’s modulus, E , for aluminum was found to be 12601.01 ksi. Right: Young’s modulus, E , for aluminum was found to be 12600 ksi.

It is impossible to determine a .01 ksi difference in E ; therefore, 12601.01 contains an inappropriate number of significant figures.

When adding or subtracting numbers, the result must have the least number of decimal figures from the measured data.

65.43 in. 1.245 in. + 0.4652 in.

Wrong: 67.1402 in. Right: 67.14 in.

When multiplying or dividing numbers, the result must have the least number of significant figures that occur in all numbers.

0.156 × 84.24 × 1.7854 = 23.5 (not 23.4655)

Because 0.156 has the smallest number of significant figures (3), the answer is only meaningful to three significant figures.

53,8 00 How many significant figures?

5.38 x 10 4 3 5.380 x 10 4 4 5.3800 x 10 4 5

Zeros are sometimes used to locate the decimal point not significant figures.

0.00001753 4 0.0001753 4 0.001753 4

Error and Percent Difference

For a quantitative comparison you will often need to present the error or the percent difference. Percent error is used when you are comparing an experimental result to a theoretical calculation, or in comparing two values in which you take one as the reference.

Percent Error Formula

Often in the lab reports you are asked to compare two values. For example, the tensile strength of a sample of CFRP was 121 ksi and rebar was 100 ksi. If you say “The tensile strength of the CFRP was 21% higher than the rebar.” Then you are making a percent error calculation where the CFRP is the experimental value and the rebar is the theoretical value. When you say “higher than rebar” then the rebar strength is the reference and becomes the value in the denominator.

Percent Error Example

If you say “The rebar’s tensile strength was 17% less than the CFRP”. Then you are making the percent error calculation using the rebar as the experimental and the CFRP as the theoretical. The “less than CFRP” means that the CFRP strength become the reference point and the value in the denominator.

Percent difference is used when you are comparing two experimental results. For example, if you measure diameter of a cylinder twice and read two different numbers.

Percent Difference

For example, if measurement one is 0.5467 in. and measurement two is 0.4967 in. then the percent difference is 9.584% (notice the correct number of significant figures).

Spreadsheets

More and more spreadsheets are being used to create engineering calculations, but that doesn’t mean that they are a table of a bunch of numbers. Use the spreadsheet to its best advantage and organize the sheet so that you have calculations that communicate effectively.

Many of the same rules apply to spreadsheet calculations as to hand-written calculations. The major benefit of a spreadsheet is that they are calculation tools. Once you have one setup, then you can make changes to inputs (like different length beams) and the spreadsheet can be made to automatically update the solution. This is a major benefit when making repetitive calculations. However, if it is to serve as a design calculation (or you are going to show it to anyone else) you want someone to be able to follow your work. This means that you need to:

Provide a problem identifier. This can be the first couple of rows of the spreadsheet, or the name of the tab (only if the spreadsheet is to be only viewed electronically)
Provide the given information or assumptions. It is important that you restate your given information. This prevents the reader from the need to go back and try to find that information. In a spreadsheet you can use highlighted cells or different color fonts to differentiate between information that is constant and information that the user inputs.
You may think that you can’t use a sketch in a spreadsheet…but you can! More importantly you should use sketches just like you do in hand calculations. Just like in Word, Excel will let you draw and write equations. You can even draw something by hand if needed and insert the picture into the spreadsheet.
Write out all the calculations required to get to the solution. Spreadsheets like to hide the calculation in the cells. If the spreadsheet is to be printed to be shared, make sure that you show the calculations or write the calculation in the cell beside the solution. Be clear what your variables are. Make sure that you use units and the correct number of significant figures.
Clearly indicate your solution.

Spreadsheets provide some unique calculation capabilities. Below are some suggestions for good spreadsheets.

Don’t let your spreadsheet be a page of numbers. Use the formatting features available in Excel such as borders, highlighting cells, and font colors to indicate different types of information or areas of calculation.

Don’t make the spreadsheet too large. Use the tabs in Excel to help organize your workbook, especially if the calculations is long. Use units! The number does not mean anything unless there is a unit.

Watch significant figures. Excel automatically fills up the cell with the result of the calculation, but typically you do not need to many significant figures. Go ahead and tell Excel how many to use in that cell(s).

Provide notes and information throughout the spreadsheet to help the reader follow the calculation.

This example shows a spreadsheet that can be printed and used as a design calculations (or homework solution)

This spreadsheet shows information entered or calculated in a table used to make a graph that would be part of a design or lab report. The spreadsheet could be made available electronically to those wanting the data.

Other software

Beyond Excel, there are several different software programs that can make writing engineering calculations easy. For example, Mathcad is a great tool for creating engineering calculations that calculate themselves. The free Express version is able to do simple calculations such as the one shown here.

Students walking by Engineering Systems Building.

Homework Format Standards

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Need a printed version of the standards?

This document is provided as a guide to the format/style that is expected and required for all homework assignments submitted in CEE courses. The format described below follows standard engineering practice and is what will be expected from you when you enter the profession. By requiring this format across the CEE curriculum, it will become second nature to you and your ability to communicate technical solutions will meet professional standards well before you graduate.

All submitted assignments should include at the top of the first page the statement “I pledge to the ODU Honor Code” followed by your signature . The ODU Honor Code is in force at all time. It is an honor code violation to copy the work of others and submit it as your work.
The homework solutions that you submit should stand on their own (i.e. anyone reviewing your submittal should not need any other source to understand what you are doing). Your solution should begin with a statement of the problem and then be followed by a list of the given data, and then statements of any assumptions, estimations, or approximations that need to be made in solving the problem. The engineering principle or solution technique you use (if there is a single identifiable one) should be stated early in the solution. This information is then followed by calculations that often occur in a series of steps with each step typically providing a result that feeds into a following step. Each solution step should be clearly identified, and this is typically accomplished with headings that are underlined . Ultimately what you submit should be a problem solution that consists of a well-organized series of steps that follow a natural progression leading to the final answer. Your instructor may give you examples of this format to further illustrate what is expected.
Always put a date on the first page of your submittals (typically near your name). This should include the month, day, and year.

When using equations with variables represented by letters (e.g. Q, µ), identify what each variable (letter or symbol) in the equation stands for the first time it appears; e.g. v = Q/A ; v is the velocity, Q is the volumetric flow rate, and A is the area).
Make sure to show the units that are associated with the numbers you are using in your calculations. Pay particular attention to make sure your final answer has the correct units. Remember units give meaning to the numbers.
You should never have a naked decimal point. If a number contains a decimal point there must be a number to the left and the right of the decimal point. For example, if you were reporting the number two-tenths it should be written as 0.2 and not .2 .
Your final answer must also contain the correct number of significant digits (also referred to as significant figures). The number of significant digits indicates the precision of the number you are reporting. In most civil engineering calculations, the number of significant digits is four or less. Surveying is one area where a greater number of significant digits is often appropriate.
Your solution of an engineering problem should be neat and well space so that the individual steps can be easily followed. For problems with engineering calculations, engineering paper should be used. When drawing a Mohr’s circle, a compass must be used.
Use a straight edge to draw lines and a French curve, flexible curve, or other curved objects when drawing curves. Where a circle is required (e.g., Mohr’s circle analysis) a compass should be used. In place of hand drawings, computer programs with graphics capabilities may be used.
When using engineering paper: In the top right, place a figure with the numeric page number/total pages format: "1/2" and "2/2"
In other instances: Use words on the bottom right: "Page 1 of 2" or "Page 2 of 2"
When graphs (usually called figures) are included, each figure should be numbered and accompanied by a figure title that identifies what the figure is (e.g., Figure 1. Relationship Between Flow Rate and Head Loss). Figure titles should be placed below the figures. Axis titles identifying the plotted variables and their units should always be included with each graph. If a graph is plotted in landscape mode on paper (i.e., turned sideways), the bottom of the graph should be on the right-hand side of the paper when viewed normally. Hand-drawn figures must always be on engineering or graph paper.
When a table is included in your work, a table title should be assigned (e.g. Table 1. Student Test Scores on Mathematics Placement Tests) and be directly above the table.
Your homework should be completed in a timely manner. The deadline that is set should be adhered to. If you are late with a submission, either points will be deducted or it will not be accepted (each instructor sets this policy). If an answer key is posted before you hand in your homework solution, it's too late to hand in your assignment.
You are encouraged to utilize computer software packages (or write your own) that will assist you in completing homework assignments. If you use word processing software, you should master the equation writing feature in the software and use it where you would normally be writing out equations in your solution. Anytime you conduct calculations within a computer program and import the results to your homework, you need to explain where these data came from and how they were calculated; e.g. “a linear regression analysis of the stress-strain data for the steel specimen used in Laboratory Exercise #6 was conducted using Excel.” Tables in solutions should be created with the software package or in a spreadsheet and copied into the assignment.
If an assignment submittal is delayed due to sickness, or some other legitimate reason, you should immediately contact your instructor. Don’t wait until the next regularly scheduled class to inform the instructor of your situation.
When writing text, write in a technical writing style, use proper punctuation, capitalization, spelling, and grammar, and do not use jargon or slang.
The emphasis in having you complete homework assignments is for you to learn the proper technical analysis and design techniques through practice and to hone the skill of being able to communicate a technical solution using the proper engineering format.

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Engineering Statics: Open and Interactive

(5 reviews)

Daniel W. Baker, Colorado State University

William Haynes, Massachusetts Maritime Academy

Publisher: Daniel Baker and William Haynes

Language: English

Formats Available

Conditions of use.

Learn more about reviews.

Reviewed by Mohammad Valipour, Assistant Professor, Metropolitan State University of Denver on 8/28/23

Comprehensiveness rating: 4 see less

I found this book very interesting and thorough. Fortunately, the book covers most of the syllabus for Statics courses that I teach. The order of the chapters is also logical and in a standard form including Introduction to Statics, Forces and Other Vectors, Equilibrium of Particles, Moments and Static Equivalence, Rigid Body Equilibrium, Equilibrium of Structures, Centroids and Centers of Gravity, Internal Forces, Friction, and Moments of Inertia. There are interactive problems at the end of each chapter of the book which can be considered as a quick review of the book. Also, I found the trigonometric tables very useful for students who have forgotten basic geometry and trigonometry. One of the interesting parts of this book is using QR codes for which you can have access to dynamic pictures of the book. The only negative point that I can mention is the lack of enough sample problems for each content. Students who take Engineering Statics need to practice many problems/examples to get ready for their exams. Only one sample problem for most of the topics is not enough particularly for those who need time to understand the concepts through solving sample problems. I think students can use this book as a study guide very well but they also need an auxiliary textbook/pamphlet to fill the gap of not solving multiple problems for each topic. Adding an index/glossary is also helpful.

Content Accuracy rating: 5

In the current format, the book looks accurate. I could not find any problem with a wrong answer neither in the solution nor in the final answers.

Relevance/Longevity rating: 5

I found the relevancy aspect of this book pretty high. Since the topics rely on the fundamentals of physics and follow Euclidean geometry, we cannot expect any change in the near future. However, the authors try to stay up to date by using matrix solutions which is highly appreciated. In addition, the users can take advantage of the graphical solutions if they use technology such as GeoGebra or a CAD program to make the diagram, then their answers will be precise.

Clarity rating: 4

Using subsections facilitates the readability aspect of the book. Supporting graphs are also useful. Thinking Deeper sections are also interesting for those students who want to know more about the concepts behind the problems. I like the friendly tone that has been used by the authors and the vocabulary and terminology are understandable for all readers even international students. Using more colorful figures may be attractive for students who are less interested in engineering problems.

Consistency rating: 4

This is my favorite part of this book. It is very consistent and readable like a story from chapter 1 to chapter 10. The overall structure of the book holds together quite well. I like that figures in each chapter have their own numbering system with respect to the chapter number. However, figure captions look too short, and adding some more information may be useful.

Modularity rating: 5

There are 10 chapters in this book. The author has done a nice job by putting sections and subsections in the right place. Different parts of the text are useable to be presented to students in different topics which might not be even the same in the book. The book includes occasional references to other subsections for further information, but such self-references do not look disruptive.

Organization/Structure/Flow rating: 5

I think the material is provided and put together realistically. The organization of this book is logical. The sections and even subsections flow easily together with the previous and following sections/subsections. I like the idea of using key questions at the beginning of each chapter. This book is organized and follows a clear structure. Each chapter starts with a chapter description and a list of sections. I found the online version of the book flow better due to the interactive design. The PDF version is also well formatted. Also, the topics are presented in a short concise fashion.

Interface rating: 5

An advantage of this book is having a user-friendly interface which makes it much easier for students to follow up with the materials discussed in the book. Unlike many other OER books, the PDF version does look like a regular book. In my opinion, the navigation of the book is easy for both the PDF version and the online version. All figures and graphs are clear and readable. The sections and subsections are loaded quickly, and the figures and diagrams can be loaded without any issues.

Grammatical Errors rating: 5

I could not find any grammatical errors in the book.

Cultural Relevance rating: 4

I did not find the book offensive/insensitive in any way. It would be better to cover more examples from different parts of the world to be fairer for international readers/students.

Reviewed by James Book, Assistant Instructional Professor, Pittsburg State University on 12/15/22

This text covers most all topics under the statics umbrella with the notable exception of virtual work. The end of chapter interactive problem sets are excellent and make for a good review of the topics covered. The option to "randomize" each problem is a great help in exam preparation. The back matter covers useful trigonometric functions and provides handy steel sections tables.

I found no obvious errors or biases in the text. The back matter equations and reference tables are accurate.

The basic principles of statics covered by this text will remain relevant for a very long time. The end of chapter problem "randomization" feature will provide fresh challenges to the student and teacher. The prose used in writing the text is modern and up to date. Since it is provided on-line, the text should be relatively easy to keep current.

Clarity rating: 5

The text is written in an easy-to-read style with a fairly basic vocabulary. The technical terms used are well explained. The illustrations are adequate but could be expanded to include some simple video content. A feature to vary the font type and size would make the text more accessible. The use of color in many of the illustrations helps with clarity. The method of concealing the answer and solution guides to the example problems is clever.

Consistency rating: 5

The text follows a consistent, repeated pattern throughout. The presentation of each topic adequately builds on ideas and concepts from previous chapters. The overall structure of the text holds together well.

The text is divided into very manageable segments that can stand on their own. It does not appear to be overly self-referential. There are no enormous blocks of text or sections that run-on unnecessarily.

The text is well organized with deliberate and logical progression through statics concepts. The on-line version of the text seems to flow better due to the interactive design, but both on-line and pdf versions are well organized.

Both the on-line and pdf versions of the text provide easy and clear navigation. A check of several of the text hyperlinks showed accurate navigation. Charts, graphs, and other images were clear and readable.

No obvious grammatical errors were encountered.

Cultural Relevance rating: 5

The topic of statics does not lend itself to much discussion of cultural topics. The examples used in the text were generic and did not seem to be culturally biased, offensive, or insensitive.

Reviewed by Michael Pastor, Assistant Professor, Tidewater Community College on 11/27/22

A fully hyperlinked and intuitive table of contents is available for this text. I could find no index, however, the PDF version is searchable, and a search bar exists in the browser edition. For the most part, this text covers all topics normally associated with a typical engineering undergraduate class in mechanics dealing with the state of bodies at rest. The text begins with Newton's Laws, Forces, and Vectors, then moves on to the analysis of particles and rigid bodies. The text also includes chapters on centroids and moments of inertia, as well as chapters on internal loading and friction. A section on virtual work and energy methods is however, missing.

I have not noticed in inaccuracies and have used this text as an optional online resource in my sophomore level statics class for a number of semesters now. Students have never reported any issues.

The relevance for this text is quite high. The material presented here has not changed in at least the last 4 decades. The online mode of delivery, however, is quite refreshing and a relative new achievement. This text is licensed under CC Attribution-Non Commercial-Share Alike.

Chapters are divided into sections and the information is brief and to the point. Occasionally supporting graphics are presented and these can be easily magnified. Very little time is spent on derivation or formulation of relationships presented. However, occasionally extra "thinking deeper" information is presented at a mouse click. Here optional background information is presented on certain subjects. There are some example problems associated with topics and these are somewhat interactive... usually involving showing an answer then showing a more detailed solution on mouse click . Occasionally, there is an interactive diagram demonstrating some concept visually. However, I did not always find these intuitive, and in some cases did not understand how to effectively manipulate them.

I did not notice any inconsistencies in terminology or framework. The work is authored in PreTeXt and powered by MathJax. I have always found it quite easy to navigate chapters and sections consistently in this text. However, I have never tried using it on a cell phone or pad. In the appendix, there is a notation chapter outlining many if not all symbols used in the text.

Modularity rating: 4

The text is divided into chapters (common to most texts of this type). Each chapter is then broken down into sections. An appendix with math formulas and steel section properties is also included. This helps comply with ABET standards. I, however, could not locate simple shape properties in any tables or diagrams either in the chapters or appendices.

Organization/Structure/Flow rating: 4

The topics are present is a short concise fashion. This in my opinion is appropriate for an online resource. However, I would like to see the availability of more details concerning the derivation and/or development of some concepts and equations. Perhaps this could be added in optional user interactive sections.

Navigation of chapters, sections, and pages is quite easy and intuitive for this text. Some of the interactive diagrams were confusing and not well explained. A search bar is available to help locate specific ideas. However, this material is so consistently organized that navigation with the interactive Contents menu is all that I have ever needed.

I saw no issues here.

I noticed no insensitive or offensive areas in this text.

I have used this text for a number of semesters as a secondary resource for students in my engineering static classes. I think it would also work well as an instructor resource. The license allows for it to be upgraded and specialized to a users needs in a non-commercial and open way.

Reviewed by Anahita Khodadadi, Assistant Professor, Portland State University on 6/22/22

The textbook covers the fundamental concepts of statics including, force and vector analysis, equilibrium, internal reactions, and geometrical properties. The textbook also includes required steel section tables and a review of trigonometry principles. I would suggest this textbook in combination with a textbook on basics of mechanics and material properties to students who seek learning about basic concepts of engineering design.

To the best of my understanding the content of the textbook and the interactive exercises look accurate, unbiased, and thorough.

The textbook contains basic principles of statics which are not expected to be changed unless a groundbreaking theory in physics emerges in the future! The text itself is arranged in a way that can easily be edited and extended. I appreciate the efforts that the author has made to create and embed the interactive diagrams and exercises within the textbook instead of inserting the link of available items across different references. This will maintain the configuration of the textbook in the long term.

The text is written in a friendly tone and even students’ presumptions and concerns are early discussed in first chapters. Technical terms are well explained for those who may not have any background in engineering. In the future, the author may include relevant videos to the textbook as well to enhance the clarity of the materials and better engage audio-visual learners.

The text and even visuals are consistent in terms of terminology, format, and graphics. All figures are numbered and mentioned within the text.

The textbook is appropriately organized in 10 chapters. Each chapter is explained in multiple subsections that allow readers focusing on small chunks of learning materials. The text includes occasional references to other subsections for further information but such self-references do not look disruptive.

Both online and PDF version of the book are presented in a fine, clear and logical fashion. The online version allows easy navigation between different sections and subsections. The PDF version is also clearly formatted. It is helpful that each chapter begins with a series of key questions and ends with a number of exercises.

I reviewed the online version both on a computer and smartphone. The interface looks fine on a computer but on a smartphone some of the interactive diagrams cannot be displayed. However, I think students may rarely use a smartphone as a primary means of accessing the textbook.

The textbook looks well proofed.

The textbook is focused on math and physics and doesn’t discuss culturally sensitive topics. Examples intrinsically do not have the capacity to demonstrate diversity and inclusion matters.

Reviewed by Peter Kazarinoff, Professor, Portland Community College on 12/16/21

Comprehensiveness rating: 5 see less

The topics covered in a typical college Engineering Statics course are present. The chapters follow a common Statics textbook pattern of concepts, starting with forces and particles and ending with friction and moments of inertia. Chapter 6 includes the method of cuts and the method of joints. The only thing that many commercial Statics textbooks have compared to this book is an extensive number of problems at the end of each chapter (the fiction chapter, in particular, had few practice problems) and more reference material at the end of the book such as the centroid of common shapes. What this book has that those commercial books lack are interactive problems.

To the best of my knowledge, the content in the book is accurate. The interactive problems I attempted showed the same auto-generated answer as I recorded using pencil and paper. The equations seem accurate throughout. The reference material at the end of the book which contains things like trig identities and properties of steel sections seems accurate.

The fundamentals of Engineering Statics, like introductory Physics and Chemistry, have not changed in a decade. So the content in the book is relevant to a current Statics course and will be relevant to future Statics classes. The only reason the book could become dated is that the interactive animations and interactive problems are no longer supported by new web browsers or new web browsing tools that I can’t even imagine will be in place in 10 years. The book has a pdf version that can be printed.

The clarity of the writing is high, the font and spacing are easy to read. The book is written in a formal academic style which is clear but can seem terse. The diagrams in the book are easy to read and use a common style to show forces, angles, and geometry.

The book is consistent from chapter to chapter and the formatting is consistent from chapter to chapter. The book has a clear numbering system for chapters, sections, and subsections. Each of the diagrams and pictures in the book follows the same captioning format. Equations in the book are formatted consistently and labeled in the same way. Each section within a chapter in the book contains a set of “key questions” that section addresses.

The book is broken up into chapters and each chapter is broken down into sections. A typical quarter or semester-long Statics course would cover almost all of the book. It would be possible to only cover a few chapters. These chapters would need to start at the beginning of the book. It wouldn’t make sense to try and pull out just the middle or end chapters as the material in the book builds up chapter to chapter. One way the book could be used is to just assign the interactive problems for practice.

This book is organized and follows a clear structure. Each chapter starts with a chapter description and a list of sections. Each section starts with “Key Questions” and then proceeds with the section content. There are interactive problems at the end of each chapter.

Interface rating: 4

The online book interface is easy to navigate. Each chapter and section is clickable and it is easy to determine which part of the book you are reading. The sections load quickly and the images, diagrams, and interactive problems load without issue. In particular, the interactive problems are pretty slick. The only reason I don’t rate the interface as a 5 is that there is no search function. I don’t know how hard it would be to add a search bar to the online version of the book, but I do think a search function would be helpful. On my device, the book only took up the left half of my screen. This may be related to the browser/device I use, but in my reading, it seems like half of the screen real estate is wasted and a lot of scrolling is needed.

No grammatical or structural errors were found. The book seems to be free of typos and seems well-proofed. There also don’t seem to be any formatting inconsistencies chapter to chapter or section to section.

Cultural Relevance rating: 3

From what I read, I didn't notice any insensitive or offensive passages in the book. However, the lens of diversity and cultural relevance is not addressed in this book. Some of the pictures in the book depicting statics topics cover common “male-dominated” examples such as motorcycles, and football training sleds.

This is a well-written, high-quality, and organized book. It is a great resource for both instructors and students in undergraduate courses in Engineering Statics. For our needs, at a community college with a 2-year program in Mechanical Engineering and Civil Engineering, this book is a good alternative to commercial offerings from Pearson or McGraw-Hill. It’s a high-quality and interesting book with fantastic interactive problems. The only knock against it is that there could be more worked examples and problems at the end of each chapter for student practice.

1 Introduction to Statics
2 Forces and Other Vectors
3 Equilibrium of Particles
4 Moments and Static Equivalence
5 Rigid Body Equilibrium
6 Equilibrium of Structures
7 Centroids and Centers of Gravity
8 Internal Loadings
10 Moments of Inertia

Ancillary Material

Daniel Baker and William Haynes

About the Book

Engineering Statics is a free, open-source textbook appropriate for anyone who wishes to learn more about vectors, forces, moments, static equilibrium, and the properties of shapes. Specifically, it has been written to be the textbook for Engineering Mechanics: Statics, the first course in the Engineering Mechanics series offered in most university-level engineering programs.

This book’s content should prepare you for subsequent classes covering Engineering Mechanics: Dynamics and Mechanics of Materials. At its core, Engineering Statics provides the tools to solve static equilibrium problems for rigid bodies. The additional topics of resolving internal loads in rigid bodies and computing area moments of inertia are also included as stepping stones for later courses. We have endeavored to write in an approachable style and provide many questions, examples, and interactives for you to engage with and learn from.

About the Contributors

Daniel W. Baker , Colorado State University

William Haynes , Massachusetts Maritime Academy

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Purdue Online Writing Lab Purdue OWL® College of Liberal Arts

Why include writing in engineering courses?

Welcome to the Purdue OWL

This page is brought to you by the OWL at Purdue University. When printing this page, you must include the entire legal notice.

Copyright ©1995-2018 by The Writing Lab & The OWL at Purdue and Purdue University. All rights reserved. This material may not be published, reproduced, broadcast, rewritten, or redistributed without permission. Use of this site constitutes acceptance of our terms and conditions of fair use.

Writing assignments incorporated into engineering courses allow students to both “write to learn” and “learn to write.” The concepts “writing to learn” and “learning to write” are integral to the study of how writing is used in all disciplines across the university. Writing studies scholars call this “writing across curriculum” because it promotes writing instruction in courses where students may not expect to encounter writing assignments and courses that students take throughout their undergraduate education.

When students write to learn, they are actively engaging with material by thinking through and articulating important concepts and issues addressed within the course. Writing in an engineering course will not only help students learn subject matter, but also enable them to synthesize and organize their thoughts to better retain information learned in the course. Furthermore, writing to learn enables students to make connections and understand the importance of the course beyond the classroom.

Assignments that emphasize “writing to learn” serve several purposes:

Promoting a deeper understanding of course material
Building critical thinking skills in students
Showing students linkages to real-world applications
Building students’ confidence in their ability to utilize technical content

Although the assignments provided in this resource primarily focus on “writing to learn,” students will also gain skills associated with “learning to write,” such as

Developing design skills
Practicing technical writing genres

For more information on “writing to learn” and “learning to write,” consult the OWL’s “Writing Across the Curriculum: An Introduction”

You might also check out the OWL vidcast, "An Introduction to Writing Across the Curriculum" on the Purdue OWL's YouTube Channel.

Using Bloom’s Taxonomy to Create Writing Prompts

This page provides examples for how to modify a standard end-of-chapter homework problem to craft write-to-learn exercises at all levels of Bloom’s Taxonomy, a framework for classifying educational learning objectives.

Types of Writing Assignments

This page provides an overview and description of many types of writing assignments suitable for use in engineering homework and class activities.

Writing Tips for Students

This page provides instructors with quick tips that they can give their students to help them navigate the writing process, from the pre-writing to revising stages.

Assessment and Feedback of Engineering Writing Assignments

These resources describe easy-to-implement grading and feedback schemes for engineering writing assignments. Grading is one of the key obstacles to implementing writing in engineering courses as class sizes may be large, or instructors/TAs may teach multiple sections. Therefore, this section also provides techniques for quicker streamlined grading practices.

This work was supported by a Research Initiation Grant in Engineering Education (RIGEE) grant from the Engineering Education and Centers (EEC) Division of the National Science Foundation (grant no. EEC-1340491). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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Mechanical Engineering

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Mechanical engineering combines engineering physics and mathematics principles with materials science to design, analyze, manufacture, and maintain mechanical systems. The mechanical engineering field requires an understanding of core areas including mechanics, dynamics, thermodynamics, materials science, structural analysis, and electricity.

No image available Engineering Mechanics - Statics (Osgood, Cameron, and Christensen)

Thumbnail: Archimedes' screw was operated by hand and could efficiently raise water, as the animated red ball demonstrates. (CC BY-SA 2.5; Silberwolf via Wikipedia )

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7 Engineering Resume Templates and Examples for Job Seekers

Praburam Srinivasan

Growth Marketing Manager

April 17, 2024

You’ve spent years honing your engineering skills, solving complex problems, and creating innovative solutions. But now, you face the greatest challenge yet—crafting a compelling engineering resume showcasing your talents and expertise.

While your engineering skills and technical prowess are excellent, is your engineering resume doing it justice?

Employers are looking for someone beyond crunching numbers and following protocols. They want a passionate problem-solver, a tireless innovator, and a collaborator who can inspire teams to reach new heights.

All too often, engineering candidates get bogged down listing roles and responsibilities while mapping out their careers , failing to highlight the unique value they bring to the table.

Only 2-3% of resumes ( source ) typically make it to the interview stage, so it’s critical to ensure yours strikes the perfect balance between technical competence and career narrative.

What Makes a Good Engineering Resume Template?

1. engineering resume template by beamjobs, 2. engineer template by resume genius, 3. mechanical engineering manager resume, 4. engineering resume template by novoresume, 5. engineering cv template by myperfectresume, 6. pragmatic engineer’s resume template by pragmatic engineer, 7. civil engineering technician template by template.net, level up with clickup docs, automate complex processes with clickup brain, maximize your job search with the clickup job search template, streamline your engineering workflow with clickup’s tool for software teams, ace project management with clickup engineering and product templates , get that coveted job with engineering resume templates.

An engineering resume example or template is a pre-formatted document that provides a structured outline and design for creating an engineering-specific resume.

It serves as a starting point or framework to help engineering professionals, such as software developers or engineers, effectively organize and present their relevant education, skills, experience, and accomplishments and achieve their professional goals .

A good engineering resume template should:

Have a clean, well-structured format that is easy to scan
Allow you to showcase your relevant technical skills and expertise prominently
Provide clear sections to highlight your education, certifications, and key projects/achievements
Use formatting such as bullets, whitespace, and consistent formatting to improve readability
Be customizable to your specific engineering discipline and career level
Help you create a compelling snapshot of your qualifications to grab the hiring manager’s attention

The best templates let you market your capabilities effectively and maintain a professional, uncluttered design. They are a strong foundation for crafting an impactful engineering resume tailored to your background and target roles.

To help you get started, we’ve curated a selection of seven high-quality engineering resume examples and templates that will allow you to create an engineering resume that truly stands out. These options cater to various engineering disciplines and experience levels, from clean and professional designs to more creative layouts.

7 Best Engineering Resume Examples and Templates

Let’s explore the seven best engineering resume examples and templates that you can use to apply to engineering jobs:

This customizable Engineering Resume Template by BeamJobs features a clean, professional resume layout with dedicated sections. It’s designed to highlight your technical prowess, relevant work experience, and hard and soft skills like a pro.

This template includes the following customizable sections:

Career Objective
Work Experience

Simply input your information manually, or link your LinkedIn profile and let the template work magic. It’ll whip up a polished engineering resume summary tailored to your unique background, making you look like the total package.

You can explore industry-specific templates to match your target company’s job title and description. The outcome of templates like these depends on the information you feed it—the more data you provide, the better your resume gets.

BeamJobs also has a feature that offers an instant resume score when you upload your engineering resume.

As you can tell from its name, the platform Resume Genius is bound to have some compelling engineering resume examples and templates, and our favorite is the Network Engineer Resume Template .

This resume template is designed for a network engineer and has the following dedicated sections:

Professional Experience
Certification
Additional Skills

You can add or remove sections depending on your requirements.

This template allows you to highlight your engineering certifications, programming language proficiencies, and specialized skills to make your engineering resume stand out in the job market.

With Resume Genius, you have the flexibility to build your networking resume from scratch using their clean, modern templates. Or, if you already have an existing CV, simply upload it and let their professionals work their resume magic to transform it into a polished document. You can download the finished product as a Word file for free.

Other than offering excellent templates, Resume Genius also provides industry-specific tips for writing better engineering resumes so you can achieve your preset goals as an engineer . This includes tips on job titles, skills section, technical expertise, and engineering resume summary.

Moreover, if you love Resume Genius but the current template does not meet your requirements, don’t worry. Depending on the job ad, you can customize a functional resume on their website.

To help you search for a job, your resume should include factors such as the companies you have worked with, your experience managing a team of engineers, and your ability to use engineering project management software .

This Mechanical Engineering Manager Resume Template from Template.net strongly emphasizes managerial experience.

The template includes the following sections,

Professional Skills
Career Objective

With this template, you can highlight your hard and soft skills, the ability to lead technical teams, engineering, and strong project management skills per the job description.

Template.net also provides an AI writing tool to help you craft compelling content that effectively highlights your skills and experience.

Once complete, you can download your resume in Word or Apple Pages format.

The Engineering Resume Template by Novoresume has a clean, professional design that’s easy to read and versatile to customize.

The benefits of this template

Utilizes the reverse-chronological format to showcase work experience prominently
Clear and accessible contact details, including email, phone, LinkedIn, and Skype
A compelling summary concisely introduces the candidate’s background and expertise
The achievement-focused work experience section highlights quantifiable accomplishments
Skills targeted specifically to engineering roles like computer-aided design
Blends professional format with strategically highlighted achievements and technical proficiencies

Additionally, NovoResume goes beyond providing a template—it’s a comprehensive resource that covers all aspects of crafting an impressive engineering resume.

From guidance on effectively detailing your work experience to tips for creating a compelling resume, even with minimal prior experience, NovoResume has you covered.

Additionally, it offers insights on writing an engaging cover letter tailored specifically for engineering roles, ensuring you make a solid first impression.

If you’re an engineer looking to create a detailed and comprehensive resume, the Engineering CV Template by MyPerfectResume is an excellent choice.

This template is designed for engineers seeking academic or research positions. The template highlights and emphasizes your educational details so the hiring manager knows you are the perfect person for the job.

Boasting 30% higher chances of getting a job, this resume template provides you with the following sections:

Summary Statement
Core Qualifications

Moreover, MyPerfectResume has templates crafted for various engineering roles, such as automation engineer, data center engineer, senior mechanical engineer, civil engineer, chemical engineer, or marine engineer.

Instead of filling out a standard form, the platform leads you through a series of questions to understand your experience and needs to craft the best engineering resume template.

Pragmatic Engineer's Resume Template by Pragmatic Engineer

The Pragmatic Engineer’s Resume Template by The Pragmatic Engineer prioritizes practical experience and skills. With dedicated sections for education and certifications and technologies and languages, it helps highlight hands-on experience and technical skills.

This might just turn out to be your favorite template, as the theme of this template is practicality and efficiency.

The pragmatic engineer Gergely Orosz, created this template for engineering team leaders to help them land job offers at top companies like Facebook, Google, and Microsoft.

With over 6,500 downloads, this template has become a popular tool for job seekers.

It’s also featured and analyzed in Gergely Orosz’s popular book, The Tech Resume Inside Out: What a Good Software Engineer Resume Looks Like .

As civil engineering is a broad field, you must structure your professional resume based on relevant experience in the civil engineering sub-sector you want to work in.

The Civil Engineering Technician Template by Template.net prioritizes technical skills and features a special skill section for a comprehensive view of abilities and expertise.

It’s a functional resume that emphasizes hard skills and experience over other aspects to showcase your engineering expertise.

This template features the following sections:

It’s a useful template for civil engineers, who often need to showcase their ability to collaborate and exchange ideas effectively with other stakeholders.

This template is the answer if you’ve struggled to find the perfect civil engineering resume.

Other Engineering Tools

A big part of engineering is putting different pieces together. You need technical expertise, continuous learning, soft skills, and tools to back you up.

So explore these tools and resources to find a great role and uplevel your career.

Once you’ve selected the engineering resume template that works best for you, use ClickUp to create your engineering resume objective and content. With ClickUp Docs , crafting your engineering resume is easy.

ClickUp Docs lets you create visually appealing resume pages with multiple formatting options. Its AI-powered advanced features allow you to write, edit, and optimize your resume content.

ClickUp Brain helps you instantly generate templates or resume formats based on your needs.

Moreover, when it comes to sifting through stacks of technical docs, ClickUp Brain steps in like a superhero, pulling out the most helpful insights to help with your engineering research and decision-making as a senior engineer or team leader.

All you need to do is ask a question. It works like your assistant, giving prompt answers to your questions.

After crafting your resume using ClickUp Docs, take the next step. Start looking for engineering jobs!

During the job hunt, staying organized and up-to-date with your applications, openings, and company ratings is crucial. That’s where the ClickUp Job Search Template comes into play.

This template streamlines your job search process, helping you manage tasks, track progress, and maintain a clear overview of your job applications.

This template lets you easily input essential details about each job opportunity, including application deadlines, contact information, salary, and interview dates.

This centralized platform ensures no critical details slip through the cracks, keeping you on track and focused on landing your dream job.

Keep track of your job applications easily with the ClickUp Job Search Template

Staying organized and prioritizing your applications can increase your chances of securing that coveted engineering position.

The ClickUp Job Search Template also provides a space for tracking company ratings and interview feedback, empowering you to make informed decisions about potential employers. With this insight, you can assess which companies align with your values, culture, and career goals.

If you’ve landed a tech job, you can leverage ClickUp’s tool for software teams to streamline your day-to-day work and simplify the development lifecycle for cross-functional teams.

Whether collaborating on designs, tracking project progress, or communicating with team members, ClickUp offers you the right support.

Think of it as your all-in-one hub for managing engineering projects, tasks, and teams.

With its intuitive interface and customizable features, you can easily organize tasks, set priorities, and track deadlines. Plus, its built-in communication tools make it simple to stay connected with your team, whether you’re in the office or working remotely.

It also offers robust integrations with other tools and software commonly used by engineers, such as CAD software, version control systems, and issue-tracking platforms.

This seamless integration ensures that all your essential data and files are synced up and accessible in one place, saving you time and minimizing headaches.

Picture this: you’re knee-deep in agile sprints, bug tracking, and engineering processes. It’s enough to make anyone’s head spin, right? That’s where ClickUp’s engineering and product templates can help.

Whether you’re managing multiple projects, tracking bugs, or managing applicants, ClickUp has a range of engineering templates tailored to your needs. The best part? You can access them right in your workspace, so you’re always equipped to handle whatever comes your way.

Plan sprints easily with the ClickUp Agile Sprint Planning Template

So, whether you’re a seasoned engineer or just starting, trust ClickUp to enhance your workflow.

You’re one step closer to your dream job by accessing these top engineering resume templates and examples and knowing how to leverage ClickUp.

Pick the right resume template based on the job description and the recruiter’s needs. Remember to practice your elevator pitch and prepare for basic interview questions beforehand.

Get a head start with these templates and ClickUp’s features, such as ClickUp Brain and ClickUp Docs.

Remember to sign up for free with ClickUp today.

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PDF Engineering Homework Format
An example is given below. The example was completed in pen in order to scan well; YOU must use a mechanical pencil. Figure 5: Rowan University College of Engineering Homework Format . 1. Headers - The five boxes at the top of each sheet of a homework assignment must contain the following PRINTED information from left to right.
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Problem Solving & Modeling
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PDF A Collection of Engineering Design Problems
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Calculations and Spreadsheets // Mizzou Engineering
For example, if measurement one is 0.5467 in. and measurement two is 0.4967 in. then the percent difference is 9.584% (notice the correct number of significant figures). Examples. Spreadsheets. More and more spreadsheets are being used to create engineering calculations, but that doesn't mean that they are a table of a bunch of numbers.
PDF Required Homework Format
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Homework Format Standards
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Engineering Statics: Open and Interactive
Engineering Statics is a free, open-source textbook appropriate for anyone who wishes to learn more about vectors, forces, moments, static equilibrium, and the properties of shapes. Specifically, it has been written to be the textbook for Engineering Mechanics: Statics, the first course in the Engineering Mechanics series offered in most university-level engineering programs.
PDF CEE Homework Format Standard
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CIVL 1112
CIVL 1112 - Homework Format. All assignments are to be submitted on engineering paper. You may use any engineering paper as long as it has a background grid. The example page below is for the paper available in the bookstore. If an alternate form of paper is used, the headings at the top of the page should be modified to match the printed ...
Why include writing in engineering courses?
This page provides examples for how to modify a standard end-of-chapter homework problem to craft write-to-learn exercises at all levels of Bloom's Taxonomy, a framework for classifying educational learning objectives. ... This page provides an overview and description of many types of writing assignments suitable for use in engineering ...
Mechanical Engineering
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PDF 1.5 Engineering Design Process Part 1
with the Engineering Design Process and may be only taken through the simple prototype phase. If the problem chosen allows, students may continue further in the Engineering Design Process as time and class resources permit. 2. Divide the class into teams of 2-3 students. 3. As a class, write a Problem Statement for the selected problem. In
Homework Format
All writing must be done in pencil and be easily readable (i.e., neatly printed or cursive letters of sufficient darkness). It is suggested a mechanical drafting pencil with grade 2B or softer lead be used. All straight lines are to drawn with a ruler. It is suggested a 6 inch clear plastic ruler be purchased.
Mastering Engineering
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PDF Homework1: GettingStarted 1 Softwareengineering
Homework 1: Getting Started. This homework introduces the environment and tools you will be using to complete your future project assignments. It includes a quick C primer. You should use this assignment to familiarize yourself with the tools you will be using throughout the course.
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ENGR 120. Homework 2. Solution. 1. All students must install SolidWorks to participate in ENGR 120. Click the "helpdesk & events" link on the left sidebar at www.livingwiththelab.com; select "week" on the calendar so you can view the calendar details. Find the software installation date and time for your ENGR 120 section, and go to ...
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7 Engineering Resume Templates & Examples for Job Seekers
Let's explore the seven best engineering resume examples and templates that you can use to apply to engineering jobs: 1. Engineering Resume Template by BeamJobs. via BeamJobs. This customizable Engineering Resume Template by BeamJobs features a clean, professional resume layout with dedicated sections.