Engineering Acoustics. Malcolm J. Crocker

Figure 2.7 Motion of a damped mass–spring system, R < (4MK)^1/2.

      The amplitude of the motion decreases with time unlike that for undamped motion (see Figure 2.3). If the damping is increased until R equals (4MK)^1/2, the damping is then called critical, R_crit = (4MK)^1/2. In this case, if the mass in Figure 2.6 is displaced, it gradually returns to its equilibrium position and the displacement never becomes negative. In other words, there is no oscillation or vibration. If R > (4MK)^1/2, the system is said to be overdamped.

      The ratio of the damping constant R to the critical damping constant R_crit is called the damping ratio δ:

      (2.16)

      In most engineering cases, the damping ratio, δ, in a structure is hard to predict and is of the order of 0.01–0.1. There are, however, several ways to measure damping experimentally [8, 9].

      Example 2.3

      A 600‐kg machine is mounted on springs such that its static deflection is 2 mm. Determine the damping constant of a viscous damper to be added to the system in parallel with the springs, such that the system is critically damped.

      Solution

The static deflection is given by Eq. (2.8a) as d = Mg/K. Therefore K = Mg/d = 600(9.8)/2 × 10⁻³ = 294 × 10⁴ N/m. The system is critically damped when the damped constant R_crit = (4MK)^1/2 = (4 × 600 × 294 × 10⁴)^1/2 = 84 000 Ns/m.

       (c) Forced Vibration – Damped

If a damped spring–mass system is excited by a simple harmonic force at some arbitrary angular forcing frequency ω (see Figure 2.8), we now obtain the equation of motion Eq. (2.17):

(2.17)

Figure 2.8 Forced vibration of damped simple system.

      The force F is normally written in the complex form for mathematical convenience. The real force acting is, of course, the real part of F or |F| cos(ωt), where |F| is the force amplitude.

      If we assume a solution of the form y = A e^jωt then we obtain from Eq. (2.17):

(2.18)

      We can write A = |A| e^jα, where α is the phase angle between force and displacement. The phase, α, is not normally of much interest, but the amplitude of motion |A| of the mass is. The amplitude of the displacement is

      (2.19)

      This can be expressed in alternative form:

(2.20)

Equation (2.20) is plotted in Figure 2.9. It is observed that if the forcing frequency ω is equal to the natural frequency of the structure, ω_n, or equivalently f = f_n, a condition called resonance, then the amplitude of the motion is proportional to 1/(2δ). The ratio |A|/(| F|/K) is sometimes called the dynamic magnification factor (DMF). The number |F|/K is the static deflection the mass would assume if exposed to a constant nonfluctuating force |F|. If the damping ratio, δ, is small, the displacement amplitude A of a structure excited at its natural or resonance frequency is very high. For example, if a simple system has a damping ratio, δ, of 0.01, then its dynamic displacement amplitude is 50 times (when exposed to an oscillating force of |F| N) its static deflection (when exposed to a static force of amplitude |F| N), that is, DMF = 50.