A significant portion of your grade will be assigned based on a multi-week long project involving the modeling, analysis, and design of a real vibrating system. You will be paired with a partner and together you must select a reasonably complex system that can be modeled using planar kinematics and dynamics.

You will start by suggesting a well-posed question about and/or design criteria for the system which the simulation and analysis can help you to address. You will derive the differential equations of motion, study the system by simulating the system dynamics, and present you results in a well formatted written report in the form of a Jupyter notebook (or set of) including whatever charts, tables, figures, and visualizations you may need to clearly communicate your results.

Many students believe that once the equations are written and the simulation is completed, the job is finished. Actually, dynamic simulations are merely a tool to learn about the possible dynamic behaviors that a system can exhibit and how these dynamics depend on other parameters. They must be used cleverly and resourcefully to elucidate the behavior of the system. This involves asking the right questions about the system and understanding which parametric dependencies are of interest. Usually this will come from some engineering job requirements and specifications or from some scientific question which one would like answered (e.g. "Design an antenna deployment system for a communication satellite which works in the near earth orbit, deploys in less that 10 sec, ..., etc." or "What are the important parameters which limit the maximum range of achievable in a throw of the discus and how should it be launched optimally?").

In this project you will be responsible for generating this project requirement before writing the simulation to help you answer the question you have asked. A good, precise statement of an interesting question is every bit as important as good answer. Indeed, poorly defined or worded questions are often impossible to answer in a satisfying and fulfilling way.

Basic Project Tasks

  • Develop a research question and/or design criteria for the system
  • Perform a literature review and report on at least two relevant and useful references
  • Develop a simple planar free body diagram along with an associated mathematical and computational model
  • Calculate fundamental modal frequencies and frequency response of the system
  • Design inputs to the systems that mimic reality
  • Identify parameters (mass, geometry, stiffness, damping, etc.) to investigate
  • Simulate the system for the inputs and design changes
  • Design your system to answer your question or meet the design criteria

Proposal [Due Friday February 7th]

Report to the instructors in writing the system of your choice, including a motivation for the problem, background on how the system works heuristically, a literature search to identify previous work on this problem, and a relatively complete discussion of the way you hope to use the vibration simulation to learn what you have chosen to learn. The proposal should be no more than 3 pages and address these things:

  • What do you want to figure out or solve? Propose either a research question or some specifications a design should meet.
  • Background and literature review of prior related work others have done.
  • What are inputs/outputs of the system?
  • What are relevant masses/inertias in the system?
  • What are flexible elements of the system?
  • What dissipates energy in the system?
  • Include a basic sketch or schematic of the system with descriptive annotation.

Proposal Rubric

Report Rubric
Item Exceed expectations [10 points] Meets expectations [5 points] Does not meet expectations [0 points]
Research question/Design Criteria Defined a research question or design criteria Research question or design criteria is present but not articulated clearly Did not define a research question or design criteria
Literature review Identified more than two relevant work or extensive discussion of at least two Identified and discussed at least two relevant works Did not look into the literature
System description System described from a vibrational and dynamics perspective System described but lacking in description of dynamics/vibration System not described
Methods description Clear plan on how analysis and simulation may be used to answer research question Plan on how to answer question is murky No discussion of how analysis and simulation may be used to answer research question or address design criteria
Writing Clear, coherent, and well organized Writing and organization needs improvement Not clear, coherent, or well organized
Contributions Very clear that each partner contributed equitably to all aspects of the project. Need to improve the contributions of one or more members Clear that everyone is not contributing equitably

Modeling Report [Due Friday, February 28th]

This report will extend your proposal with a detailed description of your model and a demonstration that it behaves like the real system. The report should be no longer than 6 pages in total.

The report should include:

  • Everything from the proposal (you will receive full points if the feedback is addressed, otherwise the grade from the proposal will be used).
  • Description of the modeling assumptions, number of degrees of freedom, inertial, restorative & dissipative elements and any forcing functions.
  • A free body diagram (or diagrams) of your system that indicates the generalized coordinates & speeds, important velocities, and applied forces & torques.
  • A Lagrange derivation of the nonlinear equations of motion using SymPy (start with kinetic and potential energy definitions and show the equations of motion in explicit nonlinear first order form).
  • A demonstration through numerical simulation that the model behaves like the real system.
  • A short paragraph describing each team members' contributions to this report.

Submit your written report as a PDF alongside a zip file that contains your functioning Jupyter notebooks (.ipynb), Python (.py), and/or data files. Make sure that "Kernel > Restart Kernel and Run All Cells" runs without error on any notebooks before submitting. The instructors should be able to run and inspect the notebooks. Make use of Markdown cells with section headings and text to describe what you are doing in each section of the notebook(s).

Modeling Report Rubric

Report Rubric
Item Exceed expectations [10 points] Meets expectations [5 points] Does not meet expectations [0 points]
Proposal Proposal included and feedback addressed Proposal grade if not present or feedback not addressed Proposal grade if not present or feedback not addressed
Model description Model fully described Model partially described Model not described
Free body diagram Complete & fully descriptive free body diagram(s) Partially descriptive free body diagram(s) No free body diagram
Equations of motion Correct Lagrange derivation and resulting nonlinear equations of motion in explicit first order form Partially correct derivation and resulting nonlinear equations of motion No derivation and equations of motion
Demonstration of model Simulation demonstrates that the model behaves like the real system Simulation present but does not necessarily demonstrate the model behaves as expected No simulation
Writing Clear, coherent, and well organized Writing and organization needs improvement Not clear, coherent, or well organized
Contributions Very clear that each partner contributed equitably to all aspects of the project. Need to improve the contributions of one or more members Clear that everyone is not contributing equitably

Final Report [Due Tuesday March 17th]

This report will cover the entirety of the project with the primary goal of showing how you utilized your model to answer your research question or meet your design requirements. You should make use of any relevant topics learned in class or homeworks and apply those to your system to characterize the system's vibrational dynamics in context of the research question or design requirements. The topics are expected to be covered in the report:

  • introduction with physical system description and background literature for context
  • describe your model of the system (assumptions, free body diagram, forcing, etc.)
  • show the resulting symbolic linear equations in canonical form
  • characterize the free response by identifying, showing, and explaining the eigenvalues and the associated mode shapes (modal analysis)
  • characterize the forced response
  • exercise the non-linear and/or linear model to answer your research question or meet your design requirements (show that you used your understanding of the free and forced dynamics in this process)
  • show and explain the results of your analysis
  • conclusion and discussion of the results

You should have an accompanying Jupyter notebook (or notebooks) that demonstrate your analyses with working code. These notebooks should be executable, with no errors, by the instructors so make sure to "Restart Kernel and Run All Cells" before saving and submitting. Include all necessary ancillary files in a zip file with the notebook(s). The notebook(s) should include:

  • nonlinear equation of motion deviation via Lagrange's method
  • linearization of the equations of motion
  • modal analysis of the unforced system
  • analysis of the effects of forcing (if relevant)
  • simulation(s) of the model to answer your research question or design criteria
  • code to produce any figures used in your report
  • animation of your system (optional)


  • The report should be not more than 12 pages long in total!
  • Make liberal use of figures to describe your system and the analysis results.
  • Thoroughly explain your figures and results with accompanying text (if you include a figure, it needs explanation).
  • Address any feedback from the prior reports.

Final Report Rubric

Report Rubric
Item Exceed expectations [10 points] Meets expectations [5 points] Does not meet expectations [0 points]
Introduction Physical system thoroughly introduced in context of external information. Physical system introduced but lacks clarity and/or context. Physical system not introduced.
Model description Model fully described and final linear equations presented. Model partially described and final linear equations presented. Model not described and final linear equations not presented.
Modal analysis Modal analysis complete with clear explanations of modes. Modal analysis present but not correct and/or clear. No modal analysis.
Forced response Forced response of the system explored and explained thoroughly. Basic forced response explanation. No forced response analysis.
Results Model exercised and clear, relevant results present. Model exercised but results are not clear. Model not exercised, no results.
Research question/Design criteria Resarch question or design criteria fully addressed. Resarch question or design criteria partially addressed. Reearch question or design criteria not addressed.
Writing Clear, coherent, and well organized Writing and organization needs improvement Not clear, coherent, or well organized
Notebooks Functioning notebooks with correct analysis. Function notebooks with partially correct analysis and/or partially functioning. No notebooks.
Contributions Very clear that each partner contributed equitably to all aspects of the project. Need to improve the contributions of one or more members Clear that everyone is not contributing equitably

Project Idea Prompts

You may propose your own project idea if you'd like. Each team must choose a unique project topic with respect to the other teams. Here are some possible ideas to choose from or to use as inspiration:

Utensil/Tool Design for People with Parkinson's Disease

Parkinson's disease often causes uncontrollable shaking. This prevents people with the disease from performing many daily tasks. For example, it is difficult to eat with utensils because the vibration in the hand causes the food to fall from the utensil or not make it into the mouth. There are products that damp the vibrations in the utensil, for example the Liftware Steady Spoon. The goal of this project would be to design a utensil or tool that could allow those with Parkinson's to continue performing the selected task.

You will need to characterize the typical motion and vibrations that occur in the task. The task should be one that can be modeled with a planar model of the arm, hand, and utensil/tool. The idea would be do design a passive mechanism with appropriate damping that causes the effector of the utensil to move more smoothly than that of the shaking input.

Record Player Needle

Record players produce sound by vibrating a thin structure, the needle, across a dimpled surface. The vibration of the needle then has to be transformed into vibrations of the air to produce sound. The simplest setup can be created by attaching a vibrating needle to a paper cone that amplifies the air vibration magnitude. Electronic record players use a voice coil that transforms mechanical motion into voltage changes in a coil via the Lorentz effect which is then amplified via the transformation back into the motion of the speaker diaphragm. This project could explore the design geometry of the needle, the surface shapes of the record dimpling, the transformation into electric energy, fatigue constraints, material selection, and/or resonance. It is even possible to produce sound waves with Python based on our simulations.

Cricket Sound Production

Cricket's and other insects produce sound by vibrating elements of their exoskeletons. This project would involve investing the geometric and material properties of the exoskeleton elements that are used to make their chirp, creating a simple model of the mechanism, and designing the model to produce chirps of frequency and amplitude that match an actual cricket or other insect.

Braking On Cobblestone

A cobblestone road is shaped such that a tire (e.g. bicycle tire) doesn't create a full contact patch between the tire and the road, as it does on a smooth road. This short article gives some initial ideas about the issues:

Here you would develop a model that shows the difference in braking ability and affects of the vehicle due to the cobblestone road. Once the simulations are functioning you can turn to designing the suspension, tire, materials, or other aspects to provide better braking and suspension performance.

Car, Motorcycle, etc. Traversing Periodic Roads with Active Damping

Two and four wheel vehicles are often modeled as a "half car" with a rigid body representing the sprung mass mounted on front and rear suspension elements and an unsprung mass representing the mass of the wheels. Develop a half car model and select realistic parameter values for a real vehicle of your choice. Develop a variety of road inputs for different travel speeds and design a suspension system that provides a comfortable rider to the passengers and sufficiently low forces to the vehicle structure. There is also the concept of the Skyhook damper that could be investigated:

Here is a paper that implements a model that would be of interest:

Bouncy Bus Seat

The driver's seat of buses are typically mounted on special suspension systems that have large travel. This project could investigate why this is the only seat with suspension, how should this suspension be designed, data collection of acceleration of different locations on a bus. You can use a smartphone to collect angular rate and linear acceleration data different locations on a Unitrans bus to characterize inputs to seat locations. You would then need to design a seat suspension system to provide comfortable motion to the driver and/or passengers.

Here is a related paper:

Tuned Mass Damper

Tuned mass dampers are often designed and installed in skyscrapers to damp oscillations due to earthquakes. This project would focus on modeling a multistory building and designing a tune mass damper to suppress motion from earthquake-like input vibrations.

Energy Harvesting From Waves

Ocean waves provide an oscillation input. If designed correctly a machine that floats on the surface or that is attached to the sea floor can harvest energy from the periodic motion of the waves. The moving machine can be coupled to an electric motor to transform rotational or linear motion into electricity. This project would investigate a wave energy harvesting device and design it such that energy can be stored from the "vibrating" ocean waves.

Design of Front Wheel Suspension in an Automobile

There are a variety of non-trivial suspension designs for ground vehicles. This project would select a suspension system that has a reasonably complex mechanism to model and simulation under realistic road conditions.

Here is a paper some Formula SAE students wrote about their suspension design that could be a starting point:

Design and Analysis of a Mountain Bike Suspension

There are a variety of interesting bicycle suspension designs (see for a starting point). This project would model and investigate a non-trivial mountain bike suspension over downhill off-road shapes with a goal to provide comfortable traversal of the rough terrain.

Design of a Tire Balance Machine

Automobile tires need to be "balanced" to minimized vibrations due to asymmetries in the mass distribution of the wheel. Autoshop typically have a machine that spins the wheel and recommends a location and mass size to add to the wheel to ensure minimal vibration when rotating at speed. This project would focus on figuring out how this machine works and designing the machine through a model and simulation.

Estimating of the Inertia of a Sports Implement

It is potentially useful to know the inertia of a sports implement for further dynamic study. For example, tennis rackets, baseball bats, cricket bats, bowling balls, etc. all have moments and products of inertia. This project would be to design a vibrating machine that could automatically estimate the inertia of a sports implement that is place on a vibrating table. You can see how Jason has done this with bicycle parts here:

but this is a labor intensive process. It would be much nicer if the item can be placed in a machine and vibrated in such a way that doesn't require special mounting to arrive at the full set of inertia values.

Piezoelectric Hydraulic Pump

Piezoelectric materials are those which convert applied mechanical stress into electrical signals. These materials are used in a wide array of transducers (sensors and actuators).

In this project, you will model a positive displacement piston pump powered by a piezoelectric stack actuator. The piezoelectric actuator will be driven by a sinusoidal voltage at a frequency of approximately 1kHz. The pump will consist of a single piston moving axially in a frictionless bore. Your simulation will include the mass and stiffness of the pump housing, piston, and fluid, as well as pressure losses from flow resistance. This study will examine how elements of mechanical design are driven by the properties and limitations of real materials. An effective model will aid in the identification of design criteria that will drive the selection of materials, and the geometry of the final product.