Lecture 1 - Welcome and Modeling of Mechanical Systems
Monday, January 4, 2021
6:22 PM
Lecture Outline:
• Welcome and expectations
• Introduction to modeling
• Basic mechanical system elements
ME 6210 – Mechanical Vibrations
Department of Mechanical Engineering, School of Engineering and Computer Science,
Oakland University
12017
Winter 2023
T Th 5:30 pm – 7:17 pm in Dodge Hall 135
Course Description:
Linear vibration of mechanical systems; System modeling; Derivation of governing equations using Newtonian
mechanics and Lagrange’s equations; Free and forced response of single degree of freedom systems; Multi-degree of
freedom systems; Modal analysis for response calculations; Application of vibration principles to applications like
vibration isolation, vibration absorbers, and resonator devices; Gyroscopic effects for spinning rotors; and
Introduction to vibration of continuous systems.
Prerequisites: Graduate standing.
Student Hours:
Tuesday and Thursday from 4-5 pm, after lectures, or by appointment. You are welcome to join me in person. I will also
have a virtual option using Zoom. Links to Zoom office hours will be posted each week on Moodle.
Textbook:
Meirovitch, L., 2010, Fundamentals of Vibrations, Waveland Press, 2010.
Course Objectives:
By the end of the course, the successful student will be able to:
• Derive the equation(s) of motion of lumped-parameter mechanical systems using vector (Newton’s laws) and
analytical (Lagrange’s equations) methods.
• Derive the free response of linear single degree of freedom systems and use its response to solve vibration problems.
• Determine the frequency response for single degree of freedom systems and identify its important features.
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Determine the frequency response for single degree of freedom systems and identify its important features.
Solve for the response of single degree of freedom systems due to nonperiodic excitations.
Determine the natural frequencies and vibration modes for systems with multiple degrees of freedom.
Determine the forced response of systems with multiple degrees of freedom using modal analysis.
Apply vibration control techniques to solve engineering problems.
Formulate and analyze the vibration of distributed-parameter systems.
Course Materials:
a) System modeling and kinematics of vibration
Modeling of mechanical systems in terms of discrete elements. Characteristics of discrete stiffness and damping
elements. Nonlinear element behavior and linearization. Amplitude, frequency, and phase of harmonic signals. Periodic
signals and their Fourier series representation. Spectrum of periodic signals and the fast Fourier transform.
b) Dynamics fundamentals
Newtonian mechanics for particles and rigid bodies in planar motion. Free body diagrams. Power, work, kinetic energy,
and potential energy. Virtual work. Lagrange’s equations for deriving equations of motion.
c) Single degree of freedom system vibration
Equations of motion, free and forced response, damping, and response to transient and harmonic excitations.
Oscillations excited by direct forcing, mass imbalance, and base excitation. Transmissibility and vibration isolation.
d) Multi-degree of freedom system vibration
Equations of motion, free and forced response, and modal analysis using the expansion theorem. Beating phenomenon,
vibration absorbers, centrifugal pendulum vibration absorbers, and untuned viscous vibration damper. Introduction to
gyroscopic (spinning) system vibration.
e) Continuous system vibration
Introduction to vibration of elastic continuum, for example, strings, rods, shafts, and beams. Equations of motion,
natural frequencies, and vibration modes. Free and forced response calculations. Approximate methods.
Grading:
Homework – 20%, Exam I – 25%, Exam II – 25%, and Final Exam – 30%.
The final grades for the course will be assigned using the weighting above. The grade distribution is A: 93% and above,
A-: 90-92%, B+: 86-89%, B: 80-85%, B-: 77-79%, C+: 70-76, C: 65-69%, D: 60-64%, and F: 59% and below.
Homework:
To assess your understanding of the course material, there will be eight homework assignments within the semester.
The course schedule gives tentative dates when each assignment is assigned and due. These dates may be adjusted
based on the actual progression of the course.
Homework should be submitted electronically as a single pdf file that is uploaded to Moodle. Handwritten
homework solutions can be scanned using a smartphone with free scanning apps like Genius Scan. Please help me
stay organized by naming your pdf file in the following format:
“LastName_FirstName_ME6210_HW#.pdf”
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Submissions without this naming convention will be given a total grade of zero points. Please submit your work for
all the assigned problems. Not all problems will be graded, however. In addition to your result, your process,
organization, and legibility will be evaluated. Points will be deducted for unsupported work. Points will also be
deducted for unclear or illegible solutions. All homework grading will be done in Moodle. All grades will be recorded
online in Moodle’s gradebook.
Late homework will only be accepted at the discretion of the instructor.
Exams:
To assess your understanding of the course material, there will be two exams within the semester and one final exam at
the end of the semester. All exams are open notes. Hardcopy only please. No electronic devices are allowed on exams. All
other materials (including the textbook and homework solutions) are prohibited. Consultation with other students,
colleagues, or the instructor is not allowed. Use of unauthorized materials is not allowed. Violation of these rules, or any
other attempt to subvert the academic process, will result in a grade of “F” on the exam. The university may apply
additional penalties.
Exam I – Thursday, February 2
Exam II – Tuesday, March 14
Final Exam – Tuesday, April 25 from 7-10 pm
Adjustments to these dates, if they occur, will be announced one week prior to any exam.
No make-up exams will be given without prior approval from the instructor.
Models
A model is a mathematical representation of a mechanical system (or subsystem) that can be used to
interpret, predict, or design the dynamics of the system.
Modeling mechanical systems is an ____________________________________________.
A given mechanical system can have ___________________________________________.
Modeling can be an _______________________________________, where a model is developed then
analyzed. The results are scrutinized, ideally using experiments or other measured data, and the model
is revised.
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All models are ______________________________________. They should be developed with their
use in mind.
Types of Models
Lumped-parameter models (discrete element models) represent the dynamics of the
system using ____________________________________________________________
• ___________________________________________________
• ___________________________________________________
• ___________________________________________________
Examples:
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Lumped-parameter models result in ____________________________________________.
Distributed-parameter models (continuous element models) represent the dynamics of the
system using _____________________________________________________________.
• ___________________________________________________
• ___________________________________________________
• ___________________________________________________
Examples:
Distributed-parameter models result in _________________________________________.
Hybrid lumped- and distributed-parameter models (hybrid discrete-continuous element models)
represent the dynamics of the system using
__________________________________________________________________________.
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Examples:
Hybrid lumped- and distributed-parameter models result in
__________________________________________________________________________.
Example 1
Given the washing machine shown in Figure 1.28(a) (from Meirovitch) and a possible model shown in
Figure 1.28(b).
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Figure from Meirovitch
Briefly describe what each element represents.
Solution
Example 2
Given the gear pair and a possible dynamic model shown below.
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Briefly describe what each element represents.
Solution
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Example 3
Given an automobile shown below.
Figure from Meirovitch
Propose three different vibration models. Briefly describe what each element represents.
Solution
Model 1
Model 2
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Model 3
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Vehicle Acceleration Data
From Modern Automotive Structural Analysis
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