Engineering Acoustics. Malcolm J. Crocker

      which leads to the following results

      (2.48)

      (2.49)

      where and .

      2.4.3 Effect of Damping

      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.

      Example

As an illustrative example consider the forced two‐degree of freedom system of Example 2.6, where k₁ = k₂ = k and m₁ = m₂ = m. In addition, two equal dampers of damping constant R are connected in parallel to the springs. The displacement amplitudes A₁ and A₂ can be determined from Eq. (2.50). Figure 2.14 shows the response of ∣A₁∣ and ∣A₂∣ with the forcing frequency. Note that both ∣A₁∣ and ∣A₂∣ reach maximum values at the same frequencies given by Eq. (2.33). It is also noted that the mass m₁ theoretically does not move when the excitation frequency is .

      Example 2.9

A small electric motor is fixed on a rigid rectangular plate resting on springs. The total mass of the motor and the plate is 45.5 kg. The system is found to have a natural frequency of 15.9 Hz. It is proposed to suppress the vibration when the motor operates at 764 rpm by attaching an undamped vibration absorber underneath the motor, as shown in Figure 2.15. Determine the necessary stiffness of the absorber if m₂ = 4.5 kg.

      Solution

      The natural frequency of the original system is 15.9 Hz = 100 rad/s. Then, the stiffness k₁ = m₁(ω)² = 45.5(100)² = 455 000 N/m. Now, the operating frequency of the motor is 764/60 = 12.7 Hz = 80 rad/s, so the absorber should have the natural frequency 80 rad/s. Then, the total stiffness of the absorber is

Figure 2.14 Forced response spectra of a damped two‐degree of freedom system.

Figure 2.15 Undamped dynamic vibration absorber defined in Example 2.9.

2.5 Continuous Systems

      All structural systems such as beams, columns, and plates are continuous systems with an infinite number of degrees of freedom. Consequently, a continuous system has an infinite number of natural frequencies and corresponding mode shapes. Although easier, modeling a structure using a finite number of degrees of freedom provides just an approximation of the behavior of the system. The analysis of continuous systems requires the solution of partial differential equations. However, analytical solutions to partial differential equations are often difficult to obtain and numerical or approximate methods are usually employed to analyze continuous systems in particular at high frequencies. However, flexural vibration of some common structural elements can be analytically studied. Sound radiation can be produced by the vibration of these structural elements. Such is the case of the vibration of thin beams, thin plates and thin cylindrical shells that will be discussed in the following sections.

      2.5.1 Vibration of Beams

      If we ignore the effects of axial loads, rotary inertia, and shear deformation,