Vibroacoustic Simulation - Alexander Peiffer

Скачать книгу

to the overdamped oscillator the equilibrium is reached as can be seen in Figure 1.4.

      Figure 1.4 Motion of the critically damped oscillator. Source: Alexander Peiffer.

      Let us summarize some facts and observations about free damped oscillators:

      1 Oscillation occurs only if the system is underdamped.

      2 ωd is always less than ω0.

      3 The motion will decay.

      4 The frequency ωd and the decay rate are properties of the system and independent from the initial conditions.

      5 The amplitude of the damped oscillator is u^(t)=u^0e−βt with β=ζω0. β is called the decay rate of the damped oscillator.

      The decay rate is related to the decay time τ. This is the time interval where the amplitude decreases to e−1 of the initial amplitude. Thus, the decay time is:

      1.2 Forced Harmonic Oscillator

      When an external force F^xcos⁡(ωt) is exciting the damped oscillator as shown in Figure 1.1 b), applying Newton’s second law we get for the equation of motion:

      This is an inhomogeneous, linear, second-order equation for u. The solution of this equation is given by a particular solution uP(t) and the solutions of the homogeneous Equation (1.1) uH(t).

       u left-parenthesis t right-parenthesis equals u Subscript upper H Baseline left-parenthesis t right-parenthesis plus u Subscript upper P Baseline left-parenthesis t right-parenthesis (1.24)

      1.2.1 Frequency Response

       ModifyingAbove upper F With caret Subscript x Baseline cosine left-parenthesis omega t plus phi right-parenthesis equals upper R e left-parenthesis bold-italic upper F Subscript x Baseline e Superscript j omega t Baseline right-parenthesis (1.25)

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

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

       StartLayout 1st Row 1st Column u left-parenthesis t right-parenthesis 2nd Column equals bold-italic u e Superscript j omega t Baseline 3rd Column v Subscript x Baseline left-parenthesis t right-parenthesis 4th Column equals j omega bold-italic u e Superscript j omega t Baseline equals bold-italic v Subscript x Baseline e Superscript j omega t EndLayout (1.26)

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

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

       StartLayout 1st Row ModifyingAbove u With caret equals StartFraction ModifyingAbove upper F With caret Subscript x Baseline Over StartRoot left-parenthesis k Subscript s Baseline minus m omega squared right-parenthesis squared plus left-parenthesis c Subscript v Baseline omega right-parenthesis Subscript 2 Baseline EndRoot EndFraction EndLayout (1.29)

      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

      and phase

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

       StartFraction omega Subscript r Baseline Over omega 0 EndFraction equals StartRoot 1 minus 2 zeta squared EndRoot (1.33)

      and the maximum value is

      with the corresponding phase

       phi Subscript r Baseline equals arc tangent left-parenthesis minus StartFraction StartRoot 1 minus 2 zeta squared EndRoot Over zeta EndFraction right-parenthesis (1.35)

Скачать книгу