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acceleration amplitude angular velocity applied assuming beam bearing boundary conditions cepstrum Chapter components compressor computed critical speed damping constant damping ratio Daniel Bernoulli deflection degrees of freedom diameter differential equation disk displacement dynamic eigenvalue elastic electric element engineering equations of motion equilibrium example demonstrates Fourier Fourier series frequency equation friction function gear harmonic excitation harmonic force harmonic functions harmonic oscillator horizontal inertia influence coefficients initial conditions instability integration Lagrange's equation lateral vibration length linear machine mass matrix maximum measured mechanical method midspan motor natural frequency natural modes nodes nonlinear obtain pendulum plane plot potential energy problem rad/s ratio Rayleigh response resulting rigid rotation rotor shaft shear modulus shown in Figure signal solution spectrum spring constant static stiffness torsional spring torsional vibration turbine unbalance undamped vector vibration amplitude vibration modes yields zero