Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

ζ η Q Δω τ Viscous damping c _v 1 ζcvc Critical damping ratio ζ cv/cvc 1 η/2 12Q Δω/2ω0 1/ω0τ Critical damping c _vc 4mks 2ζmω0 Damping loss η 2cv/cvc 2ζ 1 1/Q Δω/ω0 2/ω0τ Qualtity factor Q ccv2cv 12ζ 1/η 1 ω0Δω ω0τ2 3dB bandwidth Δω 2ζω0 ηω0 ω0/Q 1 12τ Decay time τ 1/ζω0 2/ω0η 2Q/ω0 2/Δω0 1

In tools and software for vibroacoustic simulations many different quantities are used. The overview of all those different criteria shall help to avoid mistakes and confusion.

1.3 Two Degrees of Freedom Systems (2DOF)

The harmonic oscillator is also named as single degree of freedom (SDOF) system. Realistic systems consist of multiple degrees of freedom (MDOF). In order to keep things manageable we stay in a first step with two degrees of freedom. As for the SDOF case several phenomena can be treated exemplarily, especially the coupling effects. For presenting the idea of MDOF systems the start is done by two degrees of freedom (2DOF). The equations of motion for such a system as shown in Figure 1.11 are

StartLayout 1st Row m 1 ModifyingAbove u 1 With two-dots plus c Subscript v Baseline 1 Baseline ModifyingAbove u 1 With dot plus k Subscript s Baseline 1 Baseline u 1 plus k Subscript s c Baseline left-parenthesis u 1 minus u 2 right-parenthesis equals upper F Subscript x Baseline 1 EndLayout (1.69)

StartLayout 1st Row m 2 ModifyingAbove u 2 With two-dots plus c Subscript v Baseline 2 Baseline ModifyingAbove u 1 With dot plus k Subscript s Baseline 2 Baseline u 2 plus k Subscript s c Baseline left-parenthesis u 2 minus u 1 right-parenthesis equals upper F Subscript x Baseline 2 EndLayout (1.70)

Figure 1.11 Two degrees of freedom system. Source: Alexander Peiffer.

By introducing harmonic motion for u1=u1ejωt and u2=u2ejωt we get

StartLayout 1st Row 1st Column left-parenthesis minus omega squared m 1 plus j omega c Subscript v Baseline 1 Baseline plus k Subscript s Baseline 1 Baseline plus k Subscript s c Baseline right-parenthesis bold-italic u 1 minus k Subscript s c Baseline bold-italic u 2 2nd Column equals 3rd Column bold-italic upper F Subscript x Baseline 1 EndLayout (1.71)

StartLayout 1st Row 1st Column minus k Subscript s c Baseline bold-italic u 1 plus left-parenthesis minus omega squared m 2 plus j omega c Subscript v Baseline 2 Baseline plus k Subscript s Baseline 2 Baseline plus k Subscript s c Baseline right-parenthesis bold-italic u 2 2nd Column equals 3rd Column bold-italic upper F Subscript x Baseline 2 EndLayout (1.72)

neglecting the time dependence ejωt. It is practical to write this in matrix form:

(1.73)

In the following, the square brackets and the curly brackets denote a coefficient matrix and vector, respectively.

1.3.1 Natural Frequencies of the 2DOF System

We start with a simplified system without damping and external forces in order to get the natural frequencies of the system.

(1.74)

This equation can be rearranged so that it corresponds to the general eigenvalue problem

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