Engineering Acoustics. Malcolm J. Crocker

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(2.24) into (2.23) yields

(2.25)

Equation (2.25) has a nontrivial solution if and only if the coefficient matrix ([K] − ω²[M]) is singular, that is the determinant of this coefficient matrix is zero,

(2.26)

Equation (2.26) is called the characteristic equation (or characteristic polynomial) which leads to a polynomial of order n in ω². The roots of this polynomial, denoted as (for i = 1, 2, …, n), are called the characteristic values (or eigenvalues). The square root of these numbers, ω_i are called the natural frequencies of the undamped multi‐degree of freedom system and they can be arranged in increasing order of magnitude by ω₁ ≤ ω₂ ≤ … ≤ ω_n. The lowest frequency ω₁ is referred to as the fundamental frequency. The characteristic equation has only real roots due to the symmetry of the mass and stiffness matrices. In general, all the roots are distinct except in degenerate cases.

Note that are the eigenvalues of the matrix [M]⁻¹[K], where [M]⁻¹ is the inverse of [M]. Associated with each characteristic value , there is an n‐dimensional column linearly independent vector called the characteristic vector (or eigenvector) A_i which is referred to as the i‐th natural mode (normal mode, principal mode or mode shape) [10–13]. A_i is obtained from the homogeneous system of equations represented by Eq. (2.27) as

(2.27)

Since the system of equations represented by Eq. (2.27) is homogeneous, the mode shape is not unique. However, if is not a repeated root of the characteristic equation then there is only one linearly independent nontrivial solution of Eq. (2.27). The eigenvector is unique only to an arbitrary multiplicative constant [13]. It can be shown that the mode shapes are orthogonal. This property is important and allows a set of n decoupled differential equations of motion of a multi‐degree of freedom system to be obtained by using a modal transformation [10].

      Solving Eq. (2.27) and replacing it into Eq. (2.24), we obtain a set of n linearly independent solutions q_i = A_i exp{jω_i t} of Eq. (2.23). Thus, the total solution can be expressed as a linear combination of them,

(2.28)

where β_i are arbitrary constants which can be determined from initial conditions [usually with initial displacements and velocities q(t = 0) and ]. Equation (2.28) represents the superposition of all modes of vibration of the multi‐degree of freedom system.

      Example 2.5

It is illustrative to consider an example of a two‐degree‐of‐freedom system, as the one shown in Figure 2.11, because its analysis can easily be extrapolated to systems with many degrees of freedom.

      Solution

      The 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:

(2.29a)

      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

(2.31)

For simplicity, consider the situation where m₁ = m₂ = m and k₁ = k₂ = k. Then, Eq. (2.31) becomes

(2.32)

Solving Eq. (2.32) gives the natural frequencies of the system as

(2.33)

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