Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 35

Modes are natural shapes of vibration for a dynamic system. For a given excitation it would be of interest to see how well each mode is excited. In addition, these considerations lead to a coordinate transformation that simplifies the equations of motion.

We start with the discrete equation of motion in the frequency domain (1.89) and set the damping matrices [C] and [B] to zero

(1.103)

Without external forces we get the equation for free vibrations, and we get the generalized eigenvalue problem

(1.104)

The non-trivial solutions of this are determined by zero determinants

(1.105)

providing the modal frequencies ω_n. Entering these frequencies and solving for Ψn provides the mode shape of the dynamic system. These are the natural modes (shapes) of vibration that occur at the modal frequencies.

The mode shapes are orthogonal as can be derived by assuming two different solutions m,n

(1.106)

(1.107)

Multiplying (1.107) from the left with the transposed {Ψm}T gives

(1.108)

Transposing (1.106) and multiplying from the right with {Ψ}n reads as

(1.109)

The difference between (1.108) and (1.109) leads to

(1.110)

Since ωn2≠ωm2 this requires

(1.111)

Using this in Equation (1.109) gives also

(1.112)

Thus, the mode shapes are orthogonal to each other with respect to [K] and [M]. For normalisation we multiply (1.106) from the left with {Ψ}m

(1.113)

and get

(1.114)

(1.115)

for the modal mass m_n and stiffness k_n with the following relation to the modal frequency

(1.116)