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