dynamic analysis using mode superposition

dynamic analysis using mode superposition - D YNAMIC...

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DYNAMIC ANALYSIS USING MODE SUPERPOSITION The Mode Shapes Used To Uncouple The Dynamic Equilibrium Equations Need Not Be The Exact Free-Vibration Mode Shapes 13.1 EQUATIONS TO BE SOLVED The dynamic force equilibrium Equation (12.4) can be rewritten in the following form as a set of N second order differential equations: Mu Cu Ku F f g && & (t) (t) (t) (t) (t) j j j + = =1 J (13.1) All possible types of time-dependent loading, including wind, wave and seismic, can be represented by a sum of “J” space vectors j f , which are not a function of time, and J time functions g (t) j , where J cannot be greater than the number of displacements N. The number of dynamic degrees-of-freedom is equal to the number of lumped masses in the system. Many publications advocate the elimination of all massless displacements by static condensation prior to the solution of Equation (13.1). The static condensation method reduces the number of dynamic equilibrium equations to solve; however, it can significantly increase the density and the bandwidth of the condensed stiffness matrix. In building type structures, in which each diaphragm
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STATIC AND DYNAMIC ANALYSIS 2 has only three lumped masses, this approach is effective and is automatically used in building analysis programs. For the dynamic solution of arbitrary structural systems, however, the elimination of the massless displacement is, in general, not numerically efficient. Therefore, the modern versions of the SAP program do not use static condensation in order to retain the sparseness of the stiffness matrix. 13.2 TRANSFORMATION TO MODAL EQUATIONS The fundamental mathematical method that is used to solve Equation (13.1) is the separation of variables. This approach assumes the solution can be expressed in the following form: uY (t) (t) = Φ (13.2a) Where Φ is an “N by L” matrix containing L spatial vectors which are not a function of time, and Y (t) is a vector containing L functions of time. From Equation (13.2a) it follows that & & && (t) (t) (t) (t) and ΦΦ (13.2b) and (13.2c) Prior to solution, we require that the space functions satisfy the following mass and stiffness orthogonality conditions: TT 2 M = I a n d K = Φ Φ Φ Φ (13.3) where I is a diagonal unit matrix and 2 is a diagonal matrix which may or may not contain the free vibration frequencies. It should be noted that the fundamentals of mathematics place no restrictions on these vectors, other than the orthogonality properties. In this book all space function vectors are normalized so that the Generalized Mass φφ n T n M = 1 . After substitution of Equations (13.2) into Equation (13.1) and the pre-multiplication by T Φ , the following matrix of L equations are produced:
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MODE SUPERPOSITION METHOD 3 IY dY p g && & (t) (t) (t) j j j + = 2 =1 J (13.4) where j j p f T Φ and are defined as the modal participation factors for time function j. The term nj p is associated with the n th mode. For all real structures the “L x L” matrix d is not diagonal; however, in order to uncouple the modal equations it is necessary to assume that there is no coupling between the modes.
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This note was uploaded on 06/18/2009 for the course CE xxx taught by Professor Tuken during the Spring '09 term at Middle East Technical University.

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dynamic analysis using mode superposition - D YNAMIC...

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