Vibroacoustic Simulation. Alexander Peiffer

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Vibroacoustic Simulation - Alexander Peiffer


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u 1 Choose bold-italic u 2 EndBinomialOrMatrix equals StartBinomialOrMatrix bold-italic upper F Subscript x Baseline 1 Baseline Choose 0 EndBinomialOrMatrix"/> (1.84)

      with the following transfer function

       bold-italic u 1 equals minus StartFraction k left-parenthesis i c Subscript v Baseline omega plus k Subscript s Baseline minus m Subscript s Baseline omega squared right-parenthesis Over left-parenthesis i c Subscript v Baseline omega plus k Subscript s Baseline right-parenthesis squared minus left-parenthesis i c Subscript v Baseline omega plus k Subscript s Baseline minus m Subscript s Baseline omega squared right-parenthesis left-parenthesis i c Subscript v Baseline omega plus k Subscript s b Baseline plus k Subscript s Baseline minus m omega squared right-parenthesis EndFraction bold-italic upper F Subscript x Baseline 1 (1.85)

      Assuming zero damping gives the characteristic equation for the combined resonances

       left-parenthesis k Subscript s Baseline minus omega Subscript i Superscript 2 Baseline m Subscript s Baseline right-parenthesis left-parenthesis k Subscript s b Baseline plus k Subscript s Baseline minus omega Subscript i Superscript 2 Baseline m right-parenthesis minus k Subscript s Superscript 2 Baseline equals 0 (1.86)

      With the resonance frequencies of each single system ω02=ksb/m, ωs2=ks/ms and the mass ratio μ=ms/m the resonance frequencies of the combined undamped system are given by

       omega Subscript 1 slash 2 Baseline equals StartFraction omega Subscript s Superscript 2 Baseline left-parenthesis 1 plus mu right-parenthesis plus omega 0 Over 2 EndFraction minus-or-plus StartRoot left-parenthesis StartFraction omega Subscript s Superscript 2 Baseline left-parenthesis 1 plus mu right-parenthesis plus omega 0 Over 2 EndFraction right-parenthesis plus omega Subscript s Superscript 2 Baseline omega 0 squared EndRoot (1.87)

      Figure 1.15 1DOF system with and without DVA. m = 0.1 kg, ms=0.02 kg, ksb=10 N/m, ks=2 N/m. Source: Alexander Peiffer.

      Figure 1.16 Frequency spread for DVA tuned to the same frequency depending on mass ratio. Source: Alexander Peiffer.

      1.4 Multiple Degrees of Freedom Systems MDOF

       Start 1 By 1 Matrix 1st Row upper M EndMatrix Start 1 By 1 Matrix 1st Row ModifyingAbove q With two-dots EndMatrix plus Start 1 By 1 Matrix 1st Row upper C EndMatrix Start 1 By 1 Matrix 1st Row ModifyingAbove q With dot EndMatrix plus Start 1 By 1 Matrix 1st Row upper K EndMatrix Start 1 By 1 Matrix 1st Row q EndMatrix equals Start 1 By 1 Matrix 1st Row upper F left-parenthesis t right-parenthesis EndMatrix (1.88)

      or in the frequency domain as

      The coefficients qi of {q} are generic displacement degrees of freedom, for example the displacements u,v,w in x-, y- and z-directions at different positions. The first matrix [D] is called the dynamic stiffness matrix. The matrices in the parentheses are called mass matrix, damping matrix, stiffness matrix and proportional damping matrix. The solution of equation 1.89 with regard to {q} is called frequency response.

      Figure 1.17 Multiple degrees of freedom network Source: Alexander Peiffer.

      We speak about nodes for the different locations in space running over index i. Each node may have different degrees of freedom(DOF). In our case, there are two translational coordinates u and v. So the displacement and force vector is running over all DOF of all N nodes:

       Start 1 By 1 Matrix 1st Row bold-italic q EndMatrix equals Start 7 By 1 Matrix 1st Row bold-italic u 1 2nd Row bold-italic v 1 3rd Row bold-italic u 2 4th Row bold-italic v 2 5th Row vertical-ellipsis 6th Row bold-italic u Subscript upper N Baseline 7th Row bold-italic v Subscript upper N Baseline EndMatrix Start 1 By 1 Matrix 1st Row bold-italic upper F EndMatrix equals Start 7 By 1 Matrix 1st Row bold-italic upper F Subscript x Baseline Subscript 1 Baseline 2nd Row bold-italic upper F y 1 3rd Row bold-italic upper F Subscript x Baseline Subscript 2 Baseline 4th Row bold-italic upper F y 2 5th Row vertical-ellipsis 6th Row bold-italic upper F Subscript x Baseline Subscript upper N Baseline 7th Row bold-italic upper F y Subscript upper N Baseline EndMatrix (1.90)

      1.4.1 Assembling the Mass Matrix

      The mass matrix in multidimensional space follows from Newton’s law of point masses in free space.

      


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