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| Preface | |
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| Disclaimer of warranty | |
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| Selected topics for a first course on vibration analysis and computation | |
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| Acknowledgements | |
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| Fundamental concepts | |
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| General solution for one degree of freedom | |
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| Steady-state harmonic response | |
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| Expansion of the frequency-response function in partial fractions | |
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| Negative frequencies | |
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| Root locus diagram | |
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| Impulse response | |
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| Special case of repeated eigenvalues | |
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| Frequency response of linear systems | |
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| General form of the frequency-response function | |
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| Example of vibration isolation | |
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| Logarithmic and polar plots | |
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| General expansion in partial fractions | |
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| Expansion for complex eigenvalues | |
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| Numerical examples | |
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| Undamped response | |
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| Undamped mode shapes | |
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| Damped response | |
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| Logarithmic and polar plots of the damped response | |
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| Partial-fraction expansion when there are repeated eigenvalues | |
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| Frequency response of composite systems | |
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| Natural frequencies of composite systems | |
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| General response properties | |
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| Terminology | |
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| Properties of logarithmic response diagrams | |
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| Receptance graphs | |
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| Properties of the skeleton | |
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| Mobility graphs | |
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| Reciprocity relations | |
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| Measures of damping | |
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| Logarithmic decrement | |
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| Bandwidth | |
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| Energy dissipation | |
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| Modal energy | |
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| Proportional energy loss per cycle | |
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| Loss angle of a resilient element | |
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| Forced harmonic vibration with hysteretic damping | |
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| Numerical example | |
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| Time for resonant oscillations to build up | |
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| Acceleration through resonance | |
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| Matrix analysis | |
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| First-order formulation of the equation of motion | |
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| Eigenvalues of the characteristic equation | |
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| Finding the A-matrix and its eigenvalues | |
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| Calculating eigenvalues | |
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| Eigenvectors | |
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| Normal coordinates | |
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| Uncoupling the equations of motion | |
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| General solution for arbitrary excitation | |
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| Application to a single-degree-of-freedom system | |
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| Solution for the harmonic response | |
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| Comparison with the general expansion in partial fractions | |
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| Case of coupled second-order equations | |
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| Transforming to nth-order form | |
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| Reduction of M second-order equations to 2M first-order equations | |
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| General solution of M coupled second-order equations | |
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| General response calculation | |
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| Natural frequencies and mode shapes | |
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| Introduction | |
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| Conservative systems | |
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| Example calculations for undamped free vibration | |
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| Systems with three degrees of freedom | |
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| Bending vibrations of a tall chimney | |
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| Torsional vibrations of a diesel-electric generator system | |
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| Non-conservative systems | |
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| Example calculations for damped free vibration | |
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| Systems with three degrees of freedom | |
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| Interpretation of complex eigenvalues and eigenvectors | |
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| Damped vibrations of a tall chimney | |
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| Stability of a railway bogie | |
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| Checks on accuracy | |
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| Singular and defective matrices | |
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| Singular mass matrix | |
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| Three-degree-of-freedom system with a singular mass matrix | |
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| System with a zero mass coordinate | |
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| General case when one degree of freedom has zero mass | |
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| Multiple eigenvalues | |
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| Calculating the Jordan matrix and principal vectors | |
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| Example of a torsional system with multiple eigenvalues | |
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| Interpretation of principal vectors | |
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| Numerical methods for modal analysis | |
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| Calculation of eigenvalues | |
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| Eigenvalues of a triangular matrix | |
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| Step (i) Transformation to Hessenberg form | |
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| Step (ii) Transformation from Hessenberg to triangular form | |
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| Eigenvalues of a nearly triangular matrix | |
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| Choice of the transformation matrices for the QR method | |
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| Practical eigenvalue calculation procedure | |
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| Calculation to find the eigenvalues of a 5 x 5 matrix | |
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| Calculation of the determinant of a Hessenberg matrix | |
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| Calculation of eigenvectors | |
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| Inversion of a complex matrix | |
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| Discussion | |
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| Response functions | |
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| General response of M coupled second-order equations | |
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| Properties of the partitioned eigenvector matrix | |
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| Frequency-response functions matrix | |
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| Computation of frequency-response functions | |
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| Frequency-response functions of the torsional system in Fig. 8.2 | |
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| Frequency-response functions when the eigenvector matrix is defective | |
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| Frequency-response function of a system with repeated eigenvalues | |
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| Alternative method of computing the frequency-response function matrix | |
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| Impulse-response function matrix | |
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| Computation of impulse-response functions | |
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| Impulse-response functions of the torsional system of Fig. 8.2 | |
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| Impulse-response functions when the eigenvector matrix is defective | |
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| Use of the matrix exponential function | |
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| Application to the general response equation | |
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| Application of response functions | |
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| Fourier transforms | |
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| Delta functions | |
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| The convolution integral | |
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| Unit step and unit pulse responses | |
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| Step response of the torsional system in Fig. 8.2 | |
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| Time-domain to frequency-domain transformations | |
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| General input-output relations | |
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| Case of periodic excitation | |
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| Example calculation for the torsional vibration of a diesel engine | |
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| Discrete response calculations | |
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| Discrete Fourier transforms | |
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| Properties of the DFT | |
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| Relationship between the discrete and continuous Fourier transforms | |
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| Discrete calculations in the frequency domain | |
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| Discrete calculations in the time domain | |
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| Discrete finite-difference calculations | |
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| Numerical integration | |
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| Stability | |
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| Fourth-order Runge-Kutta method | |
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| Variable stepsize | |
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| Systems with symmetric matrices | |
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| Introduction | |
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| Lagrange's equations | |
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| Application of Lagrange's equation | |
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| Potential energy of a linear elastic system | |
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| Kinetic energy for small-amplitude vibrations | |
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| General equations of small-amplitude vibration | |
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| Special properties of systems with symmetric matices | |
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| Three important theorems | |
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| Standard forms of the equations of motion | |
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| Proof that a positive-definite matrix always has an inverse | |
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| Similarity transformation to find a symmetric matrix that is similar to m[superscript -1]k | |
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| Eigenvectors of m[superscript -1]k | |
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| Proof that the eigenvalues of m[superscript -1]k cannot be negative | |
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| Other orthogonality conditions | |
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| Alternative proof of orthogonality when the eigenvalues are distinct | |
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| Time response of lightly-damped symmetric systems | |
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| Frequency response of lightly-damped symmetric systems | |
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| Impulse-response and frequency-response matrices | |
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| Reciprocity relations | |
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| Modal truncation | |
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| Computational aspects | |
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| Scaling of eigenvectors | |
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| Damping assumptions | |
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| Causality conditions | |
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| Continuous systems I | |
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| Normal mode functions | |
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| Equations of motion | |
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| Longitudinal vibration of an elastic bar | |
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| Impulse-response and frequency-response functions | |
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| Application to an elastic bar I | |
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| Alternative closed-form solution for frequency response | |
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| Application to an elastic bar II | |
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| Frequency-response functions for general damping | |
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| Frequency-response functions for moving supports | |
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| Application to an elastic column with a moving support | |
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| Example of a flexible column on a resilient foundation | |
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| Response at the top of the column | |
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| Alternative damping model | |
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| Discussion of damping models | |
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| General response equations for continuous systems | |
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| Continuous systems II | |
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| Properties of Euler beams | |
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| Simply-supported beam | |
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| Beams with other boundary conditions | |
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| Simply-supported rectangular plates | |
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| Timoshenko beam | |
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| Effect of rotary inertia alone | |
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| Effect of rotary inertia and shear together | |
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| Beam with a travelling load | |
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| Approximate natural frequencies | |
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| Rayleigh's method | |
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| Example of the whirling of a shaft subjected to external pressure | |
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| The Rayleigh-Ritz method | |
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| Corollaries of Rayleigh's principle | |
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| Parametric and nonlinear effects | |
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| Introduction | |
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| Parametric stiffness excitation | |
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| Solutions of the Mathieu equation | |
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| Stability regions | |
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| Approximate stability boundaries | |
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| Effect of damping on stability | |
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| Autoparametric systems | |
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| Internal resonance | |
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| Nonlinear jump phenomena | |
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| Stability of forced vibration with nonlinear stiffness | |
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| Numerical integration: chaotic response | |
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| Methods for finding the periodic response of weakly nonlinear systems | |
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| Galerkin's method | |
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| Ritz's method | |
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| Krylov and Bogoliubov's method | |
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| Comparison between the methods of Galerkin and Krylov-Bogoliubov for steady-state vibrations | |
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| Nonlinear response of a centrifugal pendulum vibration absorber | |
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| Logical flow diagrams | |
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| Upper Hessenberg form of a real, unsymmetric matrix A(N, N) using Gaussian elimination with interchanges | |
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| One iteration of the QR transform | |
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| Eigenvalues of a real unsymmetric matrix A(N, N) by using the QR transform of Appendix 2 | |
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| Determinant of an upper-Hessenberg matrix by Hyman's method | |
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| Eigenvectors of a real matrix A(N, N) whose eigenvalues are known | |
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| Inverse of a complex matrix | |
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| One Runge-Kutta fourth-order step | |
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| Problems | |
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| Answers to selected problems | |
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| List of references | |
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| Summary of main formulae | |
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| Index | |