Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

s Baseline 4 Baseline 3rd Column Blank 4th Column Blank 5th Column 0 6th Column minus k Subscript s Baseline 1 Baseline 7th Column Blank 8th Column Blank 3rd Row 1st Column minus k Subscript s Baseline 2 Baseline 2nd Column 0 3rd Column k Subscript s Baseline 2 Baseline plus k Subscript s Baseline 3 Baseline 4th Column 0 5th Column Blank 6th Column Blank 7th Column Blank 8th Column Blank 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column k Subscript s Baseline 5 Baseline 5th Column Blank 6th Column Blank 7th Column Blank 8th Column minus k Subscript s Baseline 5 Baseline 5th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column k Subscript s Baseline 6 Baseline 6th Column Blank 7th Column minus k Subscript s Baseline 6 Baseline 8th Column Blank 6th Row 1st Column 0 2nd Column minus k Subscript s Baseline 4 Baseline 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column k Subscript s Baseline 4 Baseline 7th Column Blank 8th Column Blank 7th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column minus k 6 6th Column Blank 7th Column k Subscript s Baseline 6 Baseline plus k Subscript s Baseline 7 Baseline 8th Column Blank 8th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column minus k Subscript s Baseline 5 Baseline 5th Column Blank 6th Column Blank 7th Column Blank 8th Column k Subscript s Baseline 5 EndMatrix"/> (1.98)

With this procedure the matrix formulation of the equation of motion can be created. A similar approach can be used if other elements like dampers are involved. Generally the local elements can be everything that can be expressed by a dynamic stiffness matrix [D(ω)] and can be added into a global matrix, independent from the fact if it comes from other models, simulation or test.

1.4.3 Power Input into MDOF Systems

The power introduced into the MDOF system is calculated according to (1.49).

(1.99)

The input power can be reconstructed from the dynamic stiffness matrix (1.89). We know that

Start 1 By 1 Matrix 1st Row bold-italic upper F EndMatrix equals Start 1 By 1 Matrix 1st Row bold-italic upper D EndMatrix Start 1 By 1 Matrix 1st Row bold-italic q EndMatrix (1.100)

hence

normal upper Pi equals StartFraction omega Over 2 EndFraction sigma-summation Underscript i comma j Endscripts upper I m left-bracket bold-italic q Subscript i Superscript asterisk Baseline bold-italic upper D Subscript i j Baseline bold-italic q Subscript j Baseline right-bracket (1.101)

or in matrix notation

normal upper Pi equals StartFraction omega Over 2 EndFraction upper I m left-bracket Start 1 By 1 Matrix 1st Row bold-italic q EndMatrix Superscript upper H Baseline Start 1 By 1 Matrix 1st Row bold-italic upper D EndMatrix Start 1 By 1 Matrix 1st Row bold-italic q EndMatrix right-bracket (1.102)

1.4.4 Normal Modes

Modes are natural shapes of vibration for a dynamic system. For a given excitation it would be of interest to see how well each mode is excited. In addition, these considerations lead to a coordinate transformation that simplifies the equations of motion.

We start with the discrete equation of motion in the frequency domain (1.89) and set the damping matrices [C] and [B] to zero

left-bracket Start 1 By 1 Matrix 1st Row upper K EndMatrix minus omega squared Start 1 By 1 Matrix 1st Row upper M EndMatrix right-bracket Start 1 By 1 Matrix 1st Row bold-italic q EndMatrix equals Start 1 By 1 Matrix 1st Row bold-italic upper F EndMatrix (1.103)

Without external forces we get the equation for free vibrations, and we get the generalized eigenvalue problem

left-bracket Start 1 By 1 Matrix 1st Row upper K EndMatrix minus omega Subscript n Superscript 2 Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix right-bracket Start 1 By 1 Matrix 1st Row normal upper Psi Subscript n Baseline EndMatrix equals 0 (1.104)

The non-trivial solutions of this are determined by zero determinants

det left-brace Start 1 By 1 Matrix 1st Row upper K EndMatrix minus omega Subscript n Superscript 2 Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix right-brace equals 0 (1.105)

providing the modal frequencies ω_n. Entering these frequencies and solving for Ψn provides the mode shape of the dynamic system. These are the natural modes (shapes) of vibration that occur at the modal frequencies.

The mode shapes are orthogonal as can be derived by assuming two different solutions m,n

StartLayout 1st Row 1st Column left-bracket Start 1 By 1 Matrix 1st Row upper K EndMatrix minus omega Subscript m Superscript 2 Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix right-bracket Start 1 By 1 Matrix 1st Row normal upper Psi Subscript m EndMatrix 2nd Column equals 3rd Column 0 EndLayout (1.106)

StartLayout 1st Row 1st Column left-bracket Start 1 By 1 Matrix 1st Row upper K EndMatrix minus omega Subscript n Superscript 2 Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix right-bracket Start 1 By 1 Matrix 1st Row normal upper Psi Subscript n EndMatrix 2nd Column equals 3rd Column 0 EndLayout (1.107)

Multiplying (1.107) from the left with the transposed {Ψm}T gives

Start 1 By 1 Matrix 1st Row normal upper Psi Subscript m EndMatrix Superscript upper T Baseline left-bracket Start 1 By 1 Matrix 1st Row upper K EndMatrix minus omega Subscript n Superscript 2 Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix right-bracket Start 1 By 1 Matrix 1st Row normal upper Psi Subscript n Baseline EndMatrix equals 0 (1.108)

Transposing (1.106) and multiplying from the right with {Ψ}n reads as

Start 1 By 1 Matrix 1st Row normal upper Psi Subscript m EndMatrix Superscript upper T Baseline left-bracket Start 1 By 1 Matrix 1st Row upper K EndMatrix minus omega Subscript m Superscript 2 Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix right-bracket Superscript upper T Baseline Start 1 By 1 Matrix 1st Row normal upper Psi Subscript n Baseline EndMatrix equals 0 (1.109)

The difference between (1.108) and (1.109) leads to

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