Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 53
Solution
ОглавлениеThe two‐coordinates x1 and x2 uniquely define the position of the system illustrated in Figure 2.11 if it is constrained to move in the x‐direction. The equations of motion of the system are:
and
(2.29b)
We observe that the equations of motion are coupled, that is to say the motion x1 is influenced by the motion x2 and vice versa. Equation (2.29) can be written in matrix form as
(2.30)
where q = , and .
Equation (2.26) gives the characteristic equation
For simplicity, consider the situation where m1 = m2 = m and k1 = k2 = k. Then, Eq. (2.31) becomes
Solving Eq. (2.32) gives the natural frequencies of the system as
Note that Eq. (2.32) has four roots, the additional two being −ω1 and −ω2. However, since these negative frequencies have no physical meaning, they can be ignored. For each positive natural frequency there is an associated eigenvector that is obtained from Eq. (2.27). Substitution of Eq. (2.33) into Eq. (2.27) and solving for Ai, yields:
and
where X1 and X2 are the elements of vector Ai. Equations (2.34a) and (2.34b) are homogenous, so that no unique solution is possible. Indeed, a solution with all its components multiplied by the same constant is also a solution [11]. Choosing arbitrarily X1 = 1 and solving Eq. (2.34) we get the eigenvectors
When used to describe the motion of a multi‐degree of freedom system, the mode shape refers to the amplitude ratio. These ratios are possible to obtain because their absolute values are arbitrary [12]. Thus, we express the mode shapes as the ratio of the amplitudes X1/X2. Then, for ω1, X1/X2 = 0.618 and for ω2, X1/X2 = −1.618. These ratios can be represented in the mode plot of Figure 2.12. We note that when this simple two‐degree of freedom system vibrates at the first (fundamental) natural frequency ω1, the two masses vibrate in phase (Figure 2.12a). When the system vibrates at the second natural frequency ω2, the two masses vibrate out of phase (Figure 2.12b).
Figure 2.11 Two‐degree‐of‐freedom system.
Figure 2.12 Mode shapes for the two‐degree of freedom system shown in Figure 2.11; (a) first mode, (b) second mode.
