Vibroacoustic Simulation - Alexander Peiffer

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is shown in Figures 1.6 and 1.7 for different ζ. One can see the resonance amplification at ωr 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.

      1.2.2 Energy, Power and Impedance

       StartLayout 1st Row 1st Column upper E Subscript kin Baseline plus upper E Subscript pot 2nd Column equals one-half m ModifyingAbove u With dot squared plus one-half k Subscript s Baseline u squared 2nd Row 1st Column Blank 2nd Column equals one-half m omega squared ModifyingAbove u With caret squared sine squared left-parenthesis omega t right-parenthesis plus one-half k Subscript s Baseline ModifyingAbove u With caret cosine squared left-parenthesis omega t right-parenthesis 3rd Row 1st Column Blank 2nd Column equals one-half m omega squared ModifyingAbove u With caret squared left-bracket sine squared left-parenthesis omega t right-parenthesis plus cosine squared left-parenthesis omega t right-parenthesis right-bracket equals one-half m omega squared ModifyingAbove u With caret squared EndLayout (1.36)

      and is constant, but spring and mass exchange energy twice over one period T0.

      Figure 1.8 Kinetic and potential energy of the harmonic oscillator. Source: Alexander Peiffer.

       StartLayout 1st Row 1st Column u Subscript ms Superscript 2 Baseline equals mathematical left-angle u squared mathematical right-angle Subscript upper T Baseline 2nd Column equals StartFraction 1 Over upper T EndFraction integral Subscript 0 Superscript upper T Baseline u squared left-parenthesis t right-parenthesis d t 3rd Column u Subscript rms Baseline equals StartRoot mathematical left-angle u squared mathematical right-angle Subscript upper T Baseline EndRoot EndLayout (1.37)

      In the following ⟨⋅⟩T=1T∫0T⋅dt denotes a time average. If the signal is harmonic with u(t)=u^cos⁡(ω0t) then

       StartLayout 1st Row 1st Column u Subscript rms 2nd Column equals StartFraction ModifyingAbove u With caret Over StartRoot 2 EndRoot EndFraction 3rd Column u Subscript rms Superscript 2 4th Column equals StartFraction ModifyingAbove u With caret squared Over 2 EndFraction 5th Column Blank EndLayout (1.38)

      1.2.3 Impedance and Response Functions

      So far the frequency response of the oscillator was expressed as the relationship between displacement and force. The ratios u/Fx and D=Fx/u are called mechanical receptance and, respectively. Often used force response relationships are the mechanical impedance impedance ! mechanical (force/velocity=Fx/vx) and the mobility (velocity/force=vx/Fx). The symbols and definitions are:

       Impedance colon bold-italic upper Z equals StartFraction bold-italic upper F Subscript x Baseline Over bold-italic v Subscript x Baseline EndFraction Mobility colon bold-italic upper Y equals StartFraction bold-italic v Subscript x Baseline Over bold-italic upper F Subscript x Baseline EndFraction (1.39)

      Considering the solution of the damped oscillator and vx=jωu both quantities become:

      The real and imaginary part have specific names resistance reactance

       bold-italic upper Z equals upper R plus j upper X Subscript normal upper Z Baseline resistance plus j reactance (1.41)

      Figure 1.9 Magnitude and phase of oscillator impedance. Source: Alexander Peiffer.

      1.2.3.1 Power Balance

      We multiply Equation (1.23) by u˙

      The first and third term can be integrated

       StartFraction d Over d t EndFraction left-parenthesis one-half m ModifyingAbove u With dot squared plus one-half k Subscript s Baseline u squared right-parenthesis plus c Subscript v Baseline ModifyingAbove u With dot squared equals upper F Subscript x Baseline ModifyingAbove u With dot period (1.43)

      The terms in the parenthesis are kinetic and potential energy and known as constant. The expression cvu˙2 is the dissipated power, because it is the damping force times velocity.

       normal upper Pi Subscript diss Baseline equals upper F Subscript x Baseline ModifyingAbove u With dot equals c Subscript v Baseline ModifyingAbove u With dot squared (1.44)

      And Fxu˙ is the introduced power,

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