SD-Lecture10-2DOF-Systems

SD-Lecture10-2DOF-Systems - CEG CEG-5065C Soil Dynamics...

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Unformatted text preview: CEG CEG-5065C Soil Dynamics 5065C Soil Dynamics Lecture 10 Lecture 10 Systems with Two-Degrees of Freedom Luis A. Prieto-Portar 2009 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 (1) The system is set to vibrate by applying an impact force on the mass m 2 . 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 and Backsubstituting these solutions into the basic differential equations, + +- = +- = = = && && o o m z k z k ( z z ) and m z k ( z z ) Let z Asin t z B sin t ω ω ω ω ω ω ω ω 2 1 2 1 2 2 2 2 2 and Since A and B are not zero, the n +-- =- +- = o o A( k k m ) k B Ak ( k m )B ω ω 2 2 2 1 2 1 2 2 2 4 2 1 1 2 2 2 1 1 2 1 2 1 2 on-trivial solution is, +-- = & ¡ + + ∴- + = ¢ £ ¤ ¥ o o o o ( k k m )( k m ) k 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 2 1 1 2 2 4 2 2 2 2 2 The equation for the natural frequency of the system can be simplified by setting and and which yields, 1 1 = = = +- + + + + = o ol ol o ol ol o ol ol w m k k m m m m ( )( ) ( )( )( ) η ω ω η ω ω η 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 ω η ω ω Case 1: The Amplitude of vibration for a force on the mass m 1 . Consider the case when a vibration is induced on the system through a force Q o sin & t 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 and + +- = +- = && && m z k z k ( z z ) Q sin t m z k ( z z ) ω 1 1 2 2 2 1 1 1 2 2 2 2 2 2 2 1 2 Let and Substituting backinto the differential equation, = =- + +- =-- = z A sin t z A sin t A ( m k k ) A k Q A ( k m ) A k ω ω ω ω ω ω ω ω ω ω ( )...
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This note was uploaded on 09/11/2011 for the course CEG 5065c taught by Professor Staff during the Fall '09 term at FIU.

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SD-Lecture10-2DOF-Systems - CEG CEG-5065C Soil Dynamics...

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