ME 411
Class No. 17942
S. F. Felszeghy

Winter 2009
Text: Theory of Vibration with Applications, Thomson, W. T.,
and Dahleh, M. D., 5th. Ed., Prentice-Hall, 1998.
Errata Sheet

1Jan. 51.1, 2.1, 2.21.3, 2.4, 2.5
Jan. 72.2, 2.32.13, 2.16, 2.20, FEM #1
2Jan. 122.4, 2.62.22, 2.30
Jan. 142.7, 2.82.40, 2.42
3Jan. 213.12.48, 2.54, FEM #2
4Jan. 263.2, 3.63.3, 3.4
Jan. 283.5, 3.7, 3.10, 3.113.8, 3.13, FEM #3
5Feb. 23.8, 3.93.26, 3.27
6Feb. 91.2, Handout1.9, 1.12, Handout
Feb. 114.1, 4.24.1, 4.2, 4.5
7Feb. 16Appendix B
Feb. 184.34.7, 4.9, 4.10, 4.12
8Feb. 234.4, 4.5, 4.64.13, 4.20, 4.21, 4.31
Feb. 255.1, 5.2, 6.6
Mar. 45.3, 6.7 5.1, 5.7, 5.11
10Mar. 96.8, 5.4, 5.65.29, 5.31, 5.57, FEM #4
Mar. 116.9, 6.10

FINAL EXAM: Monday, Mar. 16, 4:30 - 7:00 p.m.

Notebook containing solutions to problems is available in CSULA Library. Call No. is 2468 .

Concepts and Key Words

Lecture 1


Classical mechanics --> dynamics --> vibration (oscillation): back and forth motion about equilibrium position.

Mechanical vibration examples.

Classification of vibration: free or forced; discrete-parameter or lumped system (ordinary differential equations), or distributed-parameter or continuous system (partial differential equations); degrees of freedom (DOF); undamped or damped; linear (superposition applies) or nonlinear.

Free Vibration of a Single DOF, Undamped, Linear System

Massless and linearly elastic spring. Spring constant or stiffness. Hooke's law. Newton's second law of motion. Equation of motion. General solution (rotating vector interpretation). Initial conditions. Amplitude. Phase angle. Natural (circular) frequency. Period.

Lecture 2

Velocity "leads" displacement by pi/2 radians. Acceleration "leads" displacement by pi radians.
Example: Finding mass moment of inertia from measurements of a compound pendulum.
Spring equivalent to springs connected in parallel or springs in series.
Conservation of energy; derivation of equation of motion from energy.
Rayleigh's method for determining natural frequency.

Lecture 3

Application of Rayleigh's method: effect of spring mass on natural frequency.

Free Vibration of a Single-Degree-of-Freedom (SDOF), Viscously Damped, Linear System

Dashpot. Viscous damping.
Equation of motion. Trial exponential solution. Characteristic equation. Three types of pairs of roots possible: (a) conjugate complex, (b) real and equal, (c) real and distinct.
Critical damping constant.

Lecture 4

Damping ratio. Three possible pairs of roots, (a), (b) and (c), stated in terms of natural frequency and damping ratio. Graphical representation of roots in complex plane for various values of damping ratio.
Case (a): damping ratio less than 1, underdamped case. Oscillatory motion decays exponentially in time. Damped (circular) frequency.
Example: Pivoted massless bar with attached point mass, restrained by spring and dashpot.
Logarithmic decrement.
Case (c): damping ratio greater than 1, overdamped case. Nonoscillatory motion; dies out with increasing time. At most one crossing of time axis.
Case (b): critically damped case. Nonoscillatory motion; dies out with increasing time. At most one crossing of time axis.

Lecture 5

Example of Cases (a), (b) and (c).
Coulomb damping. Dry friction. Oscillatory motion decays linearly in time.

Response of a SDOF, Linear System, to Harmonic Excitation

Forcing (driving) frequency.
General solution to equation of motion: general solution to homogeneous equation plus trial particular solution.
Particular solution: phase angle and amplitude.
Transient part of general solution; steady-state (part of general) solution.
Vectorial representation of excitation and steady-state solution.
Rotating force vector polygon representation of terms in equation of motion for steady-state solution.

Lecture 6

Harmonic excitation and steady-state solution represented by complex exponentials; (complex) frequency response.
Nondimensional amplitude; plot of nondimensional amplitude and phase angle as function of frequency ratio (forcing frequency/natural frequency) for various damping ratios.
The condition of resonance.
Force vector polygon for frequency ratios much less than 1, equal to 1, and much greater than 1.
The undamped forced vibration case. General solution when forcing frequency not equal to natural frequency. General solution when forcing frequency equal to natural frequency. Solution when forcing frequency nearly equal to natural frequency. Beats.

Lecture 7

Application: rotating unbalance. Plots of amplitude ratio vs. frequency ratio for various damping ratios.
Application: transmitted forces and vibration isolation. Plots of transmissibility vs. frequency ratio for various damping ratios.
Application: support motion. Nondimensional amplitude ratio for relative motion, as function of frequency ratio and damping ratio (same nondimensional formula as for rotating unbalance). Nondimensional amplitude ratio for absolute motion, as function of frequency ratio and damping ratio (same nondimensional formula as for transmissibility).

Lecture 8

Vibration measuring instruments: (a) displacement, (b) velocity, and (c) acceleration measuring. Instruments (a) and (b) have very low natural frequencies. Instrument (c) has very high natural frequency. Role of damping in (c).
Energy dissipated in a SDOF, viscously damped system during sinusoidal steady-state vibration. Energy dissipated by viscous damper per cycle.
Average power dissipated as a function of frequency ratio. Half-power points. Quality factor Q.

Structural (solid, hysteretic) damping. Equivalent viscous damping. Structural damping factor. Plots of amplitude ratio and phase angle vs. frequency ratio for various values of structural damping factor. Complex stiffness.

Lecture 9

Coulomb (dry friction) damping. Equivalent viscous damping for assumed steady-state harmonic motion. Plots of amplitude ratio and phase angle vs. frequency ratio for various values of friction force.
The superposition principle.
Response to periodic forces. Fourier series of periodic functions. Pointwise convergence theorem. Even and odd functions.
Fourier cosine series for even function. Fourier sine series for odd function. Amplitude and phase angle representation of Fourier series. Fourier spectrum or frequency spectrum. Steady-state response of SDOF damped system to periodic excitation.

Lecture 10

Response of a SDOF, Linear System, to Aperiodic Forces. Transient Vibration

Impulse response: Dirac delta function (unit impulse) as forcing function. Impulsive force produces instantaneous change in velocity. Equivalent initial value problem. Impulse response.

Lecture 11

Response to aperiodic forces. Convolution integral.
Graphical interpretation of convolution integral.
Step (indicial) response: unit step function as forcing function.
System response by the Laplace transform method. Three steps: (1) subsidiary equation of differential equation in time, (2) subsidiary equation solved by algebraic manipulations, (3) solution to subsidiary equation transformed back to time. Existence of Laplace transform. A short table of Laplace transform pairs of functions. Laplace transformation is linear.

Lecture 12

A short table of Laplace transforms of a number of "operations." Application: SDOF, linear system subject to impulsive force.
Concepts from "systems" analysis: block diagram, input, output, transfer function.
Zero (initial) state response. Zero input response. Concepts from electrical circuits: complex impedance, complex admittance, admittance transform, impedance transform.
Expansion of proper fraction in partial fractions when solution to subsidiary equation has simple or higher order poles.

Lecture 13

Shock response spectrum (SRS). Example: step forcing function with rise time.
Example: rectangular pulse. SRS for sawtooth, triangular, reverse sawtooth, haversine, and half-sine pulses, for undamped and damped systems.

Lecture 14

Two-Degree-of-Freedom (TDOF) Systems

Free vibration of an undamped, TDOF system. Matrix notation. Is synchronous motion possible, that is, motion which is separable in "space" and time? Or, stated differently, is a motion that is represented as a product of an amplitude vector and a function of time possible? Answer: solution of "space" part of synchronous motion involves an eigenvalue problem. Characteristic equation. Eigenvalues. Eigenvectors, and their normalization. Solution for time part of synchronous motion. Synchronous solutions are necessarily simple harmonic motions. Natural frequencies. Normal modes. Natural modes of vibration.
Mode shapes.

Lecture 15

General free vibration is a linear combination of natural modes of vibration. Determination of constants in equation for general motion from initial conditions.
Orthogonality of normal modes with respect to mass and stiffness matrices.
Coupled coordinates: static coupling, dynamic coupling. To guarantee getting symmetric mass and stiffness matrices in equations of motion, use energy principles.

Lecture 16

Coordinate transformation, using normal modes, to uncouple equations of motion. Principal or normal coordinates.
Steady-state response of an undamped, TDOF system, to sinusoidal fores.

Lecture 17

Vibration absorber.
Design considerations for vibration absorber.
General response of an undamped, TDOF system. Modal analysis (mode summation) method. Solution of uncoupled equations of motion. Case of separable forcing function; mode participation factor.

Lecture 18

General response of a viscously damped, TDOF system. Potential energy and kinetic energy written as quadratic forms. Rayleigh dissipation function. Equation of motion from energy balance equation. Modal analysis applied to damped system. Proportional (Rayleigh) damping. Modal damping factor. Example: case of separable forcing function.

The final exam will be closed book. But, you will be allowed to refer to two 8 1/2 x 11 in. sheets of notes during the exam.


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