Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 20

There are several methods to determine the particular solutions, like Laplace and Fourier transforms. Here, complex algebra will be used¹. Amplitude and phase are given by a complex pointer denoted by bold italic type as depicted in Figure 1.5.

(1.25)

Figure 1.5 Complex pointer, amplitude and phase relationship. Source: Alexander Peiffer.

F_x is the complex amplitude of the force, and the Re(⋅) expression is usually omitted. The displacement and velocity response is then given by

(1.26)

with u and v_x as complex amplitudes of the displacement and velocity, respectively. Introducing this into Equation (1.23).

(1.27)

and solving this for u gives:

(1.28)

The magnitude u^ and the phase ϕ of u are:

(1.29)

(1.30)

At ω = 0 the static displacement amplitude is u^0=F^x/ks. Using the definitions from (1.6) and dividing u^ by u^0 gives the normalized amplitude

(1.31)

and phase

(1.32)

It can be shown that the maximum of u^ is at

(1.33)

and the maximum value is

(1.34)

with the corresponding phase

(1.35)

The evolution of u^/u^0 and ϕ₀ is shown in Figures 1.6 and 1.7 for different ζ. One can see the resonance amplification at ω_r that would be infinite in case of ζ = 0 and the decrease of the amplitude with increasing damping. For ζ>1/2 the maximum value occurs at ω = 0, so the displacement is just a forced movement without any resonance effect.

Figure 1.6 Normalized amplitude of forced harmonic oscillator. Source: Alexander Peiffer.

Figure 1.7 Phase of forced harmonic oscillator. Source: Alexander Peiffer.

The frequency of highest amplitude is called the amplitude resonance and it is different from the so called phase resonance with ϕ=−π2, which corresponds to the resonance of the undamped oscillator.