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SD-Lecture04-2D0fFreedom

SD-Lecture04-2D0fFreedom - Soil Dynamics Lecture 04 Systems...

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Unformatted text preview: Soil Dynamics Lecture 04 Systems with Two-Degrees of Freedom © Luis A. Prieto-Portar, August 2006. The figure at right shows a simple non-damped mass and spring system with two-degrees of freedom. This simple system can be excited into vibration in two different ways: (1) A sinusoidal force is applied to the mass m 1 , thereby resulting in a forced vibration of the system, or (2) The system is set to vibrate by an impact force on the mass m 2 . The calculation of the system’s natural frequency. Consider the free-body diagram on the previous slide. The differential equations of motion are, 1 1 1 1 2 1 2 2 2 2 2 1 1 2 2 1 2 1 2 2 2 2 2 n n n n m z k z k ( z z ) and m z k ( z z ) Let z A sin t and z B sin t Backsubstituting these solution s int o the basic differential equations, A( k k m ) k B and Ak ( k m )B Since A and B are not zero, the n ω ω ω ω ω ω ω ω ω ω + + − = + − = = = + − − = − + − = ɺɺ ɺɺ 2 2 2 1 2 1 2 2 2 4 2 1 1 2 2 2 1 1 2 1 2 1 2 n n n n on trivial solution is, ( k k m )( k m ) k or , k m k m k m k k m m m m ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω − + − − = + + − + = 1 2 1 2 1 2 2 1 1 2 1 2 2 4 2 2 2 2 2 1 1 n nl nl n nl nl n nl nl The equation for the natural frequency of the system can be simplified by m setting m k k and and m m m which yields, ( )( ) ( )( )( ) ω η ω ω ω ω ω ω ω ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω ω η ω ω = = = + − + + + + = Case 1: The amplitude of vibration for a force on mass m 1 . Consider the case when a vibration is induced on the system through a force acting upon the mass m 1 . The differential equations of motion are now, 1 1 1 1 2 1 2 2 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 2 2 2 2 1 2 m z k z k ( z z ) Q sin t and m z k ( z z ) Let z A sin t and z A sin t Substituting back int o the d .e.s, A ( m k k ) A k Q A ( k m ) A k ω ω ω ω ω ω ω ω ω ω ω + + − = + − = = = − + + − = − − = ɺɺ ɺɺ ( )...
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SD-Lecture04-2D0fFreedom - Soil Dynamics Lecture 04 Systems...

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