Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 45
Solution
ОглавлениеWe model the system as a hanging spring‐mass system (see Figure 2.5). Equation (2.9) governs the displacement of the machine from its static‐equilibrium position. Since we have four equal springs, the equivalent stiffness is 4 × 2 × 105 = 8 × 105 N/m. The natural frequency is then determined using Eq. (2.10) as
We have seen that a solution to Eq. (2.9) is y = A sin(ωt + ϕ) or the same as Eq. (2.3). Hence we know that any system that has a restoring force that is proportional to the displacement will have a displacement that is simple harmonic. This is an alternative definition to that given in Section 2.2 for simple harmonic motion.
b) Free Vibration – Damped
Many mechanical systems can be adequately described by the simple mass–spring system just discussed above. However, for some purposes it is necessary to include the effects of losses (sometimes called damping). This is normally done by including a viscous damper in the system (see Figure 2.6). See Refs. [8, 9] for further discussion on passive damping. With viscous or “coulomb” damping the friction or damping force Fd is assumed to be proportional to the velocity, dy/dt. If the constant of proportionality is R, then the damping force Fd on the mass is
(2.11)
and Eq. (2.9) becomes
(2.12)
or equivalently
where the dots represent single and double differentiation with respect to time.
Figure 2.6 Movement of damped simple system.
The solution of Eq. (2.13) is most conveniently found by assuming a solution of the form: y is the real part of A ejλt where A is a complex number and λ is an arbitrary constant to be determined. By substituting y = A ejλt into Eq. (2.13) and assuming that the damping constant R is small, R < (4MK)1/2 (which is true in most engineering applications), the solution is found that:
Here ωd is known as the damped “natural” angular frequency:
(2.15)
where ωn is the undamped natural frequency . The motion described by Eq. (2.14) is plotted in Figure 2.7.
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, Rcrit = (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 Rcrit 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].
