Vibroacoustic Simulation. Alexander Peiffer
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left-bracket Start 1 By 1 Matrix 1st Row upper K EndMatrix minus omega squared Start 1 By 1 Matrix 1st Row upper M EndMatrix right-bracket Start 1 By 1 Matrix 1st Row bold-italic u EndMatrix 2nd Column equals StartBinomialOrMatrix 0 Choose 0 EndBinomialOrMatrix EndLayout"/> (1.75)
The non trivial solutions of this are given by:
This leads to the characteristic equation with λ=ω2
With ω12=ks1/m1, ω12=ks1/m1 and ωc2=ksc(m1+m2)m1m2 the solutions are:
The eigenvalues shall be entered into the equations to solve for {Ψi}.
leading to the surprisingly simple eigenvalues after some painful math
The results present the modes of the system or the shape of movement for this natural frequency. Let us simplify the above expression by additional conditions: ks1=ks2=ks and m1=m2=m.
So the modal frequencies read with ω02=ks/m
with the eigenvectors:
Thus, the first mode represents a uniform motion of both masses without any relative motion and thus no effect of the centre spring. The second mode is a symmetric resonance and the centre spring adds some extra stiffness leading to higher frequencies. See Figure 1.12 for both modes.
Figure 1.12 Mode shapes of the 2DOF example. Source: Alexander Peiffer.
When we enter numerical figures with m = 0.1 kg, ks=10 N/m and ksc=2 N/m we get the modal frequencies ωn1=10.0 s−1(fn1=1.59 Hz) and ωn2=11.83 s−1(fn2=1.88 s−1).
1.3.1.1 Forced Vibration of the 2DOF System
If the 2DOF system is excited by a force or a combination of several forces, the system of equations (1.73) is solved for every frequency. Equation (1.73) can by written in a more generic way
The solution can be written using the inverse of the stiffness matrix
This frequency response can be received for example by using numerical packages like MATLABTM, Python (with NumPy) for a set of frequencies. In Figure 1.13 the response of the system for unit excitation of Fx1=1 N and Fx2=0 N is shown.
Figure 1.13 Magnitude and phase of response to unit force at mass 1. Source: Alexander Peiffer.
1.3.1.2 Dynamic Vibration Absorber
The harmonic oscillator can be an anti-vibration device. This is called a tuned vibration absorber (TVA) or dynamic vibration absorber (DVA) and is a kind of multi-purpose tool whenever you have to combat resonance issues. It is a very useful device if vibration at a particular frequency must be reduced. Many applications are single frequency cases as for example propeller harmonics. In addition DVAs are used for reducing the resonance effects under broadband excitation. Usually real technical systems have multiple resonances but the principle can be shown with a SDOF system as master system. In Figure 1.14 such a setup is shown. The exciting force can be for example a rotating or vibrating machinery.
Figure 1.14 DVA mounted on resonant master system. Source: Alexander Peiffer.
The equation of motion is