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hwk12_soln - Problem 12.1 SOLUTION Consider the undamped...

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Problem 12.1 SOLUTION Consider the undamped, four-DOF system shown below. This system has 4x4 symmetric and positive definite mass and stiffness matrices M ! " # $ and K ! " # $ , respectively. However, we are not given the details of the problem in terms of masses and individual spring stiffnesses. The fundamental natural frequency for this system is ! 1A with the corresponding modal vector ! X A 1 ( ) . a) Consider now System B shown below. This system is the same as System A above except a particle of mass m has been rigidly attached to one of the original particles. Use the Rayleigh’s method to prove that the fundamental natural frequency is decreased by this addition of mass to the system; that is, show that ! 1A > ! 1 B . Consider the following suggestions for your proof: If the mass matrix for System B is written as M ! " # $ B = M ! " # $ + ! M ! " # $ , determine ! M ! " # $ . Note that the stiffness matrix for System B is the same as for System A. Write down the Rayleigh quotient for System B in terms of an arbitrary trial vector !
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