Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

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.

1.2.2 Energy, Power and Impedance

It is helpful to investigate the ongoing processes from the energy perspective. For an undamped system cv=0 that oscillates with u(t)=u^cos⁡(ω0t) and absence of external forces the energy remains constant. The energy is the sum of kinetic E_kin and potential energy E_pot

(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.

As energy quantities are not always constant, this motivates the definition of time average values. average ! time We introduce the mean-square (ms) and root-mean-square (rms) value for the time average for an arbitrary signal x(t). The mean-square value over a time T is

(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:

bold-italic upper Z equals StartFraction bold-italic upper F Subscript x Baseline Over bold-italic v Subscript x Baseline EndFraction equals c Subscript v Baseline plus j left-parenthesis minus StartFraction k Subscript s Baseline Over omega EndFraction plus m omega right-parenthesis (1.40)

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)

The meaning of this convention becomes clear when inspecting Figure 1.9. When the oscillator is excited in the mass- or stiffness-controlled regime below or above the resonance, the force introduces a reactive movement. The energy can be taken back from the system and it is called reactive. Whether the system response is mass or stiffness controlled can be identified from the slope of the impedance magnitude and the phase. In the resonance regime the system is driven at resonance and the damper dissipates energy. The resistance of the system becomes important and is thus called resistive.

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˙

m ModifyingAbove u With two-dots ModifyingAbove u With dot plus c Subscript v Baseline ModifyingAbove u With dot squared plus k Subscript s Baseline ModifyingAbove u With dot x equals upper F Subscript x Baseline ModifyingAbove u With dot (1.42)

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,

Скачать книгу