Mechanical Vibration Analysis And Computation

by D. E. Newland

• ISBN13: 9780486445175
• ISBN10: 0486445178
• Format: Paperback
• Copyright: 2006-01-04
• Publisher: Dover Publications

Summary

Focusing on applications rather than rigorous proofs, this volume is suitable as a text for upper-level undergraduates and graduate students concerned with vibration problems. It also serves as a practical handbook for performing vibration calculations and features extensive appendices and answers to selected problems. 128 figures. 1989 edition.

Table of Contents

Preface	xi
Disclaimer of warranty	xv
Selected topics for a first course on vibration analysis and computation	xvi
Acknowledgements	xx
1 Fundamental concepts	1	(16)
General solution for one degree of freedom	1	(2)
Steady-state harmonic response	3	(3)
Expansion of the frequency-response function in partial fractions	6	(3)
Negative frequencies	9	(2)
Root locus diagram	11	(1)
Impulse response	12	(2)
Special case of repeated eigenvalues	14	(3)
2 Frequency response of linear systems	17	(39)
General form of the frequency-response function	17	(2)
Example of vibration isolation	19	(3)
Logarithmic and polar plots	22	(4)
General expansion in partial fractions	26	(5)
Expansion for complex eigenvalues	31	(2)
Numerical examples	33	(11)
Example 2.1: Undamped response	34	(3)
Example 2.2: Undamped mode shapes	37	(2)
Example 2.3: Damped response	39	(4)
Example 2.4: Logarithmic and polar plots of the damped response	43	(1)
Partial-fraction expansion when there are repeated eigenvalues	44	(3)
Frequency response of composite systems	47	(5)
Natural frequencies of composite systems	52	(4)
3 General response properties	56	(41)
Terminology	56	(2)
Properties of logarithmic response diagrams	58	(1)
Receptance graphs	59	(3)
Properties of the skeleton	62	(3)
Mobility graphs	65	(2)
Reciprocity relations	67	(1)
Measures of damping	68	(10)
Logarithmic decrement	68	(3)
Bandwidth	71	(2)
Energy dissipation	73	(1)
Modal energy	74	(1)
Proportional energy loss per cycle	75	(1)
Loss angle of a resilient element	76	(2)
Forced harmonic vibration with hysteretic damping	78	(5)
Numerical example	83	(6)
Time for resonant oscillations to build up	89	(4)
Acceleration through resonance	93	(4)
4 Matrix analysis	97	(25)
First-order formulation of the equation of motion	97	(2)
Eigenvalues of the characteristic equation	99	(2)
Example 4.1: Finding the A -matrix and its eigenvalues	99	(1)
Example 4.2: Calculating eigenvalues	100	(1)
Eigenvectors	101	(1)
Normal coordinates	102	(2)
Example 4.3: Uncoupling the equations of motion	103	(1)
General solution for arbitrary excitation	104	(1)
Application to a single-degree-of-freedom system	105	(2)
Solution for the harmonic response	107	(2)
Comparison with the general expansion in partial fractions	109	(1)
Case of coupled second-order equations	110	(3)
Example 4.4: Transforming to nth-order form	111	(2)
Reduction of M second-order equations to 2M first-order equations	113	(1)
General solution of M coupled second-order equations	114	(8)
Example 4.5: General response calculation	115	(7)
5 Natural frequencies and mode shapes	122	(57)
Introduction	122	(1)
Conservative systems	123	(2)
Example calculations for undamped free vibration	125	(20)
Example 5.1: Systems with three degrees of freedom	125	(3)
Example 5.2: Bending vibrations of a tall chimney	128	(8)
Example 5.3: Torsional vibrations of a diesel-electric generator system	136	(9)
Non-conservative systems	145	(3)
Example calculations for damped free vibration	148	(3)
Example 5.4: Systems with three degrees of freedom	148	(3)
Interpretation of complex eigenvalues and eigenvectors	151	(25)
Example 5.5: Damped vibrations of a tall chimney	154	(9)
Example 5.6: Stability of a railway bogie	163	(13)
Checks on accuracy	176	(3)
6 Singular and defective matrices	179	(22)
Singular mass matrix	179	(1)
Three-degree-of-freedom system with a singular mass matrix	180	(3)
Example 6.1: System with a zero mass coordinate	182	(1)
General case when one degree of freedom has zero mass	183	(2)
Multiple eigenvalues	185	(6)
Example 6.2: Calculating the Jordan matrix and principal vectors	188	(3)
Example of a torsional system with multiple eigenvalues	191	(5)
Interpretation of principal vectors	196	(5)
7 Numerical methods for modal analysis	201	(25)
Calculation of eigenvalues	201	(2)
Example 7.1: Eigenvalues of a triangular matrix	202	(1)
Step (i) Transformation to Hessenberg form	203	(2)
Step (ii) Transformation from Hessenberg to triangular form	205	(3)
Example 7.2: Eigenvalues of a nearly triangular matrix	207	(1)
Choice of the transformation matrices for the QR method	208	(4)
Practical eigenvalue calculation procedure	212	(4)
Example 7.3: Calculation to find the eigenvalues of a 5 x 5 matrix	213	(3)
Calculation of the determinant of a Hessenberg matrix	216	(2)
Calculation of eigenvectors	218	(2)
Inversion of a complex matrix	220	(3)
Discussion	223	(3)
8 Response functions	226	(32)
General response of M coupled second-order equations	226	(1)
Properties of the partitioned eigenvector matrix	227	(2)
Frequency-response functions matrix	229	(2)
Computation of frequency-response functions	231	(7)
Example 8.1: Frequency-response functions of the torsional system in Fig. 8.2	232	(6)
Frequency-response functions when the eigenvector matrix is defective	238	(4)
Example 8.2: Frequency-response function of a system with repeated eigenvalues	240	(2)
Alternative method of computing the frequency-response function matrix	242	(2)
Impulse-response function matrix	244	(1)
Computation of impulse-response functions	245	(2)
Example 8.3: Impulse-response functions of the torsional system of Fig. 8.2	246	(1)
Impulse-response functions when the eigenvector matrix is defective	247	(4)
Use of the matrix exponential function	251	(3)
Application to the general response equation	254	(4)
9 Application of response functions	258	(25)
Fourier transforms	258	(1)
Delta functions	259	(3)
The convolution integral	262	(2)
Unit step and unit pulse responses	264	(4)
Example 9.1: Step response of the torsional system in Fig. 8.2	265	(3)
Time-domain to frequency-domain transformations	268	(2)
General input-output relations	270	(2)
Case of periodic excitation	272	(2)
Example calculation for the torsional vibration of a diesel engine	274	(9)
10 Discrete response calculations	283	(24)
Discrete Fourier transforms	283	(2)
Properties of the DFT	285	(2)
Relationship between the discrete and continuous Fourier transforms	287	(2)
Discrete calculations in the frequency domain	289	(2)
Discrete calculations in the time domain	291	(2)
Discrete finite-difference calculations	293	(6)
Numerical integration	299	(1)
Stability	299	(2)
Fourth-order Runge-Kutta method	301	(2)
Variable stepsize	303	(4)
11 Systems with symmetric matrices	307	(34)
Introduction	307	(1)
Lagrange's equations	308	(3)
Example 11.1: Application of Lagrange's equation	309	(2)
Potential energy of a linear elastic system	311	(3)
Kinetic energy for small-amplitude vibrations	314	(2)
General equations of small-amplitude vibration	316	(2)
Special properties of systems with symmetric matices	318	(7)
Three important theorems	318	(1)
Standard forms of the equations of motion	319	(1)
Proof that a positive-definite matrix always has an inverse	320	(1)
Similarity transformation to find a symmetric matrix that is similar to m-1 k	321	(1)
Eigenvectors of m-1k	322	(1)
Proof that the eigenvalues of m-1k cannot be negative	323	(1)
Other orthogonality conditions	324	(1)
Alternative proof of orthogonality when the eigenvalues are distinct	325	(1)
Time response of lightly-damped symmetric systems	326	(3)
Frequency response of lightly-damped symmetric systems	329	(1)
Impulse-response and frequency-response matrices	330	(2)
Reciprocity relations	332	(1)
Modal truncation	332	(1)
Computational aspects	333	(2)
Scaling of eigenvectors	335	(1)
Damping assumptions	336	(2)
Causality conditions	338	(3)
12 Continuous systems I	341	(42)
Normal mode functions	341	(4)
Equations of motion	345	(4)
Longitudinal vibration of an elastic bar	346	(3)
Impulse-response and frequency-response functions	349	(4)
Application to an elastic bar I	351	(2)
Alternative closed-form solution for frequency response	353	(3)
Application to an elastic bar II	354	(2)
Frequency-response functions for general damping	356	(1)
Frequency-response functions for moving supports	357	(10)
Application to an elastic column with a moving support	358	(9)
Example of a flexible column on a resilient foundation	367	(13)
Response at the top of the column	371	(2)
Alternative damping model	373	(3)
Discussion of damping models	376	(4)
General response equations for continuous systems	380	(3)
13 Continuous systems H	383	(37)
Properties of Euler beams	383	(1)
Simply-supported beam	384	(2)
Beams with other boundary conditions	386	(3)
Simply-supported rectangular plates	389	(3)
Timoshenko beam	392	(4)
Effect of rotary inertia alone	393	(2)
Effect of rotary inertia and shear together	395	(1)
Beam with a travelling load	396	(3)
Approximate natural frequencies	399	(4)
Rayleigh's method 400 Example of the whirling of a shaft subjected to external pressure	403	(7)
The Rayleigh–Ritz method	410	(4)
Corollaries of Rayleigh's principle	414	(6)
14 Parametric and nonlinear effects	420	(54)
Introduction	420	(1)
Parametric stiffness excitation	421	(1)
Solutions of the Mathieu equation	422	(7)
Stability regions	429	(5)
Approximate stability boundaries	434	(3)
Effect of damping on stability	437	(2)
Autoparametric systems	439	(4)
Internal resonance	443	(2)
Nonlinear jump phenomena	445	(5)
Stability of forced vibration with nonlinear stiffness	450	(4)
Numerical integration: chaotic response	454	(2)
Methods for finding the periodic response of weakly nonlinear systems	456	(11)
Galerkin's method	456	(1)
Ritz's method	457	(3)
Krylov and Bogoliubov's method	460	(4)
Comparison between the methods of Galerkin and Krylov–Bogoliubov for steady-state vibrations	464	(3)
Nonlinear response of a centrifugal pendulum vibration absorber	467	(7)
Appendices: Logical flow diagrams	474	(29)
Appendix 1 Upper Hessenberg form of a real, unsymmetric matrix A (N, N) using Gaussian elimination with interchanges	474	(3)
Appendix 2 One iteration of the QR transform	477	(5)
Appendix 3 Eigenvalues of a real unsymmetric matrix A (N, N) by using the QR transform of Appendix 2	482	(5)
Appendix 4 Determinant of an upper-Hessenberg matrix by Hyman's method	487	(3)
Appendix 5 Eigenvectors of a real matrix A (N, N) whose eigenvalues are known	490	(7)
Appendix 6 Inverse of a complex matrix	497	(3)
Appendix 7 One Runge–Kutta fourth-order step	500	(3)
Problems	503	(51)
Answers to selected problems	554	(14)
List of references	568	(7)
Summary of main formulae	575	(3)
Index	578