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Characteristics of Free and Forced Vibrations of Elementary Systems
Published in Franklin Y. Cheng, Matrix Analysis of Structural Dynamics, 2017
When ρ = 0, as in the case of undamped forced vibration, Eq. (1.121a,b) becomes Eq. (1.95a or b). Equations (1.95) and (1.121) are in function of forcing frequency, natural frequency, magnitude of the applied force, and damping ratio, if any. Force magnitude and stiffness can be combined as a static displacement, xs1 = F/Κ. The Duhamel’s integral can be used to find the maximum response for a given forcing function, natural period, and damping ratio during and after the force applied. Maximum responses may be plotted in a shock spectrum as discussed in Section 1.4.3. The spectra of two typical forces are shown in Fig. 1.28. For the force rising from zero to F at ζ, if ζ is less than one-quarter of T, then the effect is essentially the same as for a suddenly applied force. In practical design, the rise can be ignored if it is small. When ζ is a whole multiple of T, the response is the same as though F had been applied statically.
A time domain method for wheel-rail force identification of rail vehicles
Published in Vehicle System Dynamics, 2022
Tao Zhu, Xiao-rui Wang, Yi-wei Fan, Ming-meng Wang, Jing-ke Zhang, Shou-ne Xiao, Guang-wu Yang, Bing Yang
According to the theory of structural dynamics, the displacement, velocity and acceleration response of the vibration system can be obtained by Duhamel integral (i.e. a method to solve the response of linear system under any external excitation) under the action of dynamic load [15–17]. where is the damping ratio; is the natural frequency; is the damped natural frequency; , are the displacement and speed of the system at initial time.
High efficient dynamic analysis of vehicle–track–subgrade vertical interaction based on Green function method
Published in Vehicle System Dynamics, 2020
Mei Chen, Yu Sun, Wanming Zhai
For the fasteners along the x-direction, (xj, yp, zp) is the coordinate of the jth fastener, and Frsj(t) is the jth fastener force at the t moment. It is assumed that the fastener forces are the external excitation of the slab–subgrade system which are applied to the fastener positions in the form of a concentrated dynamic force. The Green function method is employed to solve the dynamic responses of the slab–subgrade system under the excitation of the fastener forces. The system responses W(x, y, z, t) (such as vertical displacement, velocity, acceleration and stress) at the position (x, y, z) and the time t induced by the fastener forces can be represented by the Duhamel integral as where G(x, y, z, xj, yp, zp, t − τ) as well as Gj(x, y, z, t − τ) denotes the Green function, which indicates the response at the position (x, y, z) and the time t − τ, when a unit impulse excitation is applied to the jth fastener at time t = 0.