Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 59

If there is damping present (as there always is in real systems) the homogenous solution of a harmonically forced vibration system decays away with time. It has to be noted that when damping is included in the mathematical model, the eigenvalues and eigenvectors can be complex numbers, unlike in the undamped case. Although in practice the damping of a structural system is often small, its effect on the system response at or near resonance may be significant. If the damping matrix is a linear combination of the mass and the stiffness matrix (proportional damping), the system of differential Eq. (2.22) can be uncoupled using the modal matrix method [13]. This method is based on calculating the eigenvalues and eigenvectors of the system and the application of a modal transformation in a new set of coordinates called modal coordinates. This technique is not possible to apply if the damping matrix is arbitrary. In this case, a state‐space representation is often used to uncouple the system [10]. This technique reduces the order of the differential equations at the expense of doubling the number of degrees of freedom.

For the case of an n‐degree of freedom system with viscous damping and subject to a single‐frequency harmonic excitation, we can assume harmonic solutions in the form of Eq. (2.36) and use the same arguments employed to obtain Eq. (2.39). Thus, the amplitudes of Eq. (2.36) are now expressed as [13]

(2.50)

Several examples are discussed in textbooks on vibration theory [10–13].

Equation (2.50) shows that if the forcing frequency is very low in comparison to the lowest natural frequency, the term [K] is dominant and the vibration amplitudes are controlled mainly by the system's stiffness. If the system is excited significantly above their resonance frequency region, the term −ω²[M] dominates and the system is mass‐controlled. Damping only has an appreciable effect around the resonance frequencies. The effects of these frequency regions on the sound transmitted through a forced vibrating panel are discussed in Chapter 12.