alt="equation"/>
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.
2.4.2 Forced Vibration – Undamped
By forced vibration, we mean that the system is vibrating under the influence of continuous (external) forces that do not cease. The total response of a multi‐degree of freedom system due to a force excitation is the sum of a homogeneous solution and a particular solution. The homogenous solution depends upon the system properties while the particular solution is the response due to the particular form of excitation. The homogenous solution is often ignored for a system subjected to a periodic vibration for being of lesser practical importance than the particular solution. For a general form of excitation, a closed‐form solution of a multi‐degree of freedom system can be very difficult to obtain and numerical methods are often used.
The equations of motion of an n‐degree‐of‐freedom undamped linear system excited by simple harmonic forces at some arbitrary angular forcing frequency ω (all excitation terms at the same phase) can be expressed in matrix form as
where F is an n‐dimensional complex column vector of dynamic amplitude forces. We assume harmonic solutions of the form
where A is a vector of undetermined amplitudes. Substituting Eq. (2.36) into (2.35) leads to
A unique solution of Eq. (2.37) exists unless
which has the same form as Eq. (2.26). Equation (2.38) is satisfied only when the forcing frequency coincides with one of the system's natural frequencies. In this condition, called resonance, the response of the system grows linearly with time and thus use of the solution Eq. (2.36) is unsuitable. When a solution of Eq. (2.37) exists, the amplitudes can be determined as [13]
If we consider the two‐degree of freedom system discussed in Example 2.5 but now harmonic force excitations of frequency ω and amplitude F1 and F2 are applied to the masses m1 and m2, respectively (see Figure 2.13), the equations of motion are
(2.40a)
and
(2.40b)
