ME 511
Class No. 14998
S. F. Felszeghy

Spring 2008

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

1Mar. 251.0, 1.1; 2.0, 2.1, 2.2, 2.62.3, 2.10
Mar. 272.6; 3.0, 3.12.35, 2.56
2Apr. 13.7, 3.10; 4.0, 4.13.7, 3.24
Apr. 34.2, 4.4, 4.54.36
3Apr. 84.6, 5.0, 5.15.5
Apr. 105.2, 5.35.6, FEM #1
4Apr. 155.3, 5.45.32
Apr. 176.1 thru 6.46.1, 6.5, 6.7
Apr. 246.5, 6.6, 6.76.8, 6.12, 6.17
6Apr. 296.7 thru 6.116.34, 6.37, 6.38, 6.39, FEM #2
May 19.0, 9.19.1, 9.3, FEM #3
7May 69.2 thru 9.59.5, 9.6
May 87.1, 7.2, 7.39.10, 9.11, FEM #4
8May 137.3, 7.47.10, 7.18
9May 2012.1, 12.312.2, 12.11, FEM #5
May 228.3 thru 8.512.19
10May 2710.1, 10.2
May 2911.1

FINAL EXAM: Tuesday, June 3, 4:30 - 7:00 p.m.

Concepts and Key Words

Lecture 1


Vibration (oscillation): back and forth motion about equilibrium position.

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-Degree-of-Freeedom (SDOF), Undamped, Linear System

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

Free Vibration of a SDOF, Viscously Damped, Linear System

Massless dashpot. Viscous damping. Dashpot placed in parallel with spring.
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 2

Damping ratio. Three possible pairs of roots, (a), (b) and (c), stated in terms of natural frequency and damping ratio.
Case (a): damping ratio less than 1, underdamped case. Oscillatory motion decays exponentially in time. Damped (circular) frequency.
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.

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 solution.
Vectorial representation of excitation and steady-state solution.
Rotating force vector polygon representation of terms in equation of motion for steady-state solution.
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.

Lecture 3

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.
The superposition principle.

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

Impulse response: Dirac delta function (unit impulse) as forcing function.

Lecture 4

Impulsive force produces instantaneous change in velocity. Equivalent initial value problem. Impulse response.
Response to aperiodic forces. Convolution integral.
Graphical interpretation of convolution integral.
Step (indicial) response: unit step function as forcing function.

Pseudo spectra. Strong ground motion. Velocity Spectra. Spectral displacement. Pseudo-relative spectral velocity. Pseudo-absolute spectral acceleration. Tripartite plot.

Lecture 5

Steady-State Response of an Undamped SDOF System to a Periodic Train of Impulses

Homework. Application: find steady-state response of SDOF system to alternating square wave forcing function. Develop suitable convolution integral. Apply to finding closed form solution.

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 as a product of an amplitude vector and a scalar function of time? Answer: Yes, solution for amplitude vector is necessarily an eigenvalue problem.
Characteristic equation. Eigenvalues. Eigenvectors, and their normalization.

Lecture 6

For each eigenvalue, scalar function of time-part of synchronous motion is governed by a second order ordinary differential equation. Synchronous solutions are necessarily harmonic motions. Natural frequencies. Normal modes. Synchronous motions are called natural modes of vibration. Mode shapes. 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.

Lecture 7

To guarantee getting symmetric mass and stiffness matrices in equations of motion use energy principles. Lagrange's equations of motion.
Coordinate transformation, using normal modes, to uncouple equations of motion. Principal or normal or modal coordinates.

Steady-state response of a TDOF, undamped system, to sinusoidal forces. Solution for unknown steady-state vibration amplitude by expansion of amplitude as a linear combination of normal modes.

Lecture 8

Vibration absorber.

Properties of Vibrating Multi-Degree-of-Freedom (MDOF) Systems

Influence coefficients. General axial spring-node system. Displacements and external forces at nodes. Relations between these: flexibility influence coefficients, stiffness influence coefficients. Flexibility matrix. Stiffness matrix. Flexibility matrix is inverse of stiffness matrix whenever inverse exists.

Lecture 9

Elastic potential energy of a general linear discrete system, as quadratic form in terms of external forces, and in terms of displacements. Symmetry of flexibility matrix: Maxwell's reciprocity theorem.

Quadratic forms for elastic potential energy and kinetic energy in terms of displacements and velocities, respectively. Stiffness influence coefficients and mass coefficients. Matrix notation. Application of Lagrange's equations. Existence of synchronous motions. The eigenvalue problem associated with finding the amplitude vector. Properties of eigenvalues and eigenvectors derived from symmetry and positive properties of energy quadratic forms associated with stiffness and mass matrices.

Lecture 10

Normalization of eigenvectors with respect to mass matrix: normal modes.
Formation of modal matrix.
To each eigenvalue, there corresponds a sinusoidal solution to the differential equation governing the time part of the synchronous motion. The general solution to the equations of motion in the physical coordinates is the sum of all the harmonic synchronous motions. This sum represents a linear transformation from the physical coordinates to the modal coordinates.
Response of an undamped system. Modal analysis (mode summation method). Transformation of initial conditions in physical coordinates to modal coordinates. Case when forcing function is separable as a product of an amplitude vector and function of time. Response for each modal coordinate expressed as a sum of zero input response and zero state response. Mode participation factor.
General response of a viscously damped system. Rayleigh dissipation function. Damping matrix. Solution by modal analysis. Rayleigh (proportional damping) leads to uncoupled equations of motion in modal coordinates. Response for each modal coordinate expressed as a sum of zero input response and zero state response.
Application: response to support motion of an n-story building.

Lecture 11

The SRSS (square root of the sum of the squares) method for finding the maximum response using seismic data presented as pseudo relative velocity spectra.

Vibration of Continuous Systems

Free vibration of a string in tension. The wave equation. Solution by D'Alembert's method. General solution for a horizontal string represents a wave travelling to the right summed with a wave travelling to the left.
Case of a string with one end fixed to a wall. A wave travelling towards the wall is reflected antisymmetrically as a wave travelling away from the wall. A wave travelling in a string of finite length produces periodic motion. Can a finite string undergo synchronous motion? Yes, write motion as a product of a spatial function and a function of time. Separation of variables leads to eigenvalue problem (Sturm-Liouville problem) for spatial function. General solution is sum of sinusoids. Application of boundary conditions leads to transcendental characteristic equation. Eigenvalues. Eigenfunctions are sinusoids. Normalization. Normal modes.
Solution for time part of synchronous motion corresponding to each eigenvalue. Solutions are necessarily harmonic. Natural frequencies. General solution to string motion problem is the infinite sum of the synchronous motions combined with undetermined constants. Orthogonality of normal modes.
Lecture 12

Determination of constants in general solution from initial conditions.
The decomposition of a natural mode into sinusoidal travelling waves. Wavelength number. Wavelength.

Mathematically analogous vibration problems. Longitudinal vibration of a bar (or rod). Bar velocity. Torsional vibration of circular bars (rods). Shear velocity.

Free vibration of a beam. Equation of motion neglecting shear deformation and rotatory inertia: the Bernoulli-Euler beam. Examine if travelling sinusoidal waves can propagate in beam. Wave speed and wave number. Sinusoidal wave speed is a function of the wavelength. This phenomenon called dispersion.

Lecture 13

Free vibration of a finite-length simply-supported beam. Boundary conditions. Seek synchronous solutions.
Spatial part of trial solution governed by eigenvalue problem. General solution. Application of boundary conditions leads to transcendental characteristic equation. Infinite number of eigenvalues. Normalized eigenfunctions.

Lagrange's equations

Concepts from analytical mechanics. Constrained particle. Smooth surface: (holonomic) constraint. Given (applied) force. Reaction (constraint) force is normal to constraint surface. Virtual displacement.
Resultant force on particle. Particle in equilibrium. Principle of virtual work.
Conservative given forces. Variation of potential function. The fundamental equation. Extension of virtual work principle to dynamics.

Lecture 14

Generalized coordinates. Least number of coordinates necessary and sufficient to specify the configuration of a system. Lagrange's equations. Transform fundamental equation for a system of particles from physical coordinates to generalized coordinates.
Introduce kinetic energy and generalized force. Extract from fundamental equation Lagrange's equations. The case when generalized forces are conservative. The Lagrangian function. Inclusion of conservative and nonconserbative generalized forces in Lagrange's equations.
Application to linear systems. Motion of MDOF system in neighborhood of stable equilibrium position. Small displacements and velocities. Kinetic and potential energies expressesd as quadratic forms with generalized mass and generalized stiffness coefficients, respectively. Viscous damping forces introduced through Rayleigh's dissipation function. Lagrange's equations.

Lecture 15

Approximate Numerical Methods.

Rayleigh's Quotient. Rayleigh's Principle. Rayleigh's quotient constructed from quadratic forms formed from eigenvalue problem for MDOF system. Rayleigh's principle: fundamental frequency squared is minimum of Rayleigh's quotient. Example: estimating fundamental frequency by using static deflections due to gravity.
Extension to continuous systems. Example: fixed-free bar in axial motion. Rayleigh's quotient constructed from eigenvalue problem through integration; result is a functional. Fundamental frequency of bar estimated by using static deflection of bar due to gravity.

Rayleigh's (Energy) Method. Assume an MDOF system is vibrating harmonically about its equilibrium position. Form ratio of maximum potential energy and maximum kinetic energy divided by the square of the circular frequency. Ratio same as Rayleigh's quotient. A continuous system: cantilevered Bernoulli-Euler beam. Assume harmonic motion. Form ratio of maximum potential energy and maximum kinetic energy divided by the square of the circular frequency. Compare to Rayleigh's quotient formed from differential equation associated with eigenvalue problem. Integration by parts of numerator shows that this ratio same as that from Rayleigh's method. Geometric and natural boundary conditions. Admissible amplitude functions need to satisfy geometric boundary conditions only.

Lecture 16

Rayleigh-Ritz Method. Extension of Rayleigh's method. Amplitude function assumed to be a linear combination of functions that satisfy geometric boundary conditions. Rayleigh's quotient is ratio of quadratic forms in the linear combination constants. Choose constants to minimize quotient. This leads to Ritz eigenvalue problem. Get upper bounds for exact eigenvalues of structure. Example: axial motion of fixed-spring grounded bar.

Approximate Numerical Methods (Continued). Matrix Iteration (by Power Method). Eigenvalue problem defined with dynamical matrix. The iteration scheme. Process produces a vector proportional to first normal mode. Ratio of the same nonzero component from consecutive iterated vectors converges to fundamental frequency.

Lecture 17

To compute other eigenvalues and modes, define and iterate with deflated matrix. Three story building example.

Introduction to the Finite Element Method

The bar (or axial) element. Formulation by Rayleigh-Ritz method using shape (interpolation) functions over portions of a long bar. Divide bar into segments (finite elements) connected at ends to material points or particles (nodes). Define displacement field in each element, with shape (interpolation) functions, expressed as linear functions in the nodal displacements. Apply Rayleigh's quotient to generate element stiffness and (consistent) mass matrices, and system stiffness and mass matrices. Kinetic and potential energies in terms of nodal displacements and velocities. Application of Lagrange's equations in the nodal displacements. External nodal forces. The beam element. Beam element connected to end nodes. Nodal tranverse translational and rotational displacements, and corresponding external nodal forces and couples. Develop deflected shape function, with shape (interpolation) functions, expressed as a linear function in the nodal displacements.

Lecture 18

Potential (strain) energy in terms of nodal displacements. The element stiffness matrix. Kinetic energy in terms of nodal velocities. The element mass matrix. Application of Lagrange's equations in the nodal displacements.

Modal Analysis of Continuous Systems

Free vibration of beam. Show eigenfunctions are orthogonal. Forced vibration of a pinned-pinned beam with initial conditions. Solve by modal analysis: expand motion in an eigenfunction series. Coefficients of eigenfunctions are normal coordinates. Substitute series in equation of motion of beam. Use orthogonality property of eigenfunctions to resolve equation of motion into system of SDOF equations of motion in the normal coordinates. Solve these normal coordinate equations of motion as sums of zero input response and zero state response. Relate initial values of normal coordinates to initial displacements and velocities of beam using orthogonality property of eigenfunctions. Form general response. Example: case when forcing function is separable in space and time, and time part is a square pulse.


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