UB CIE 619 - “Exact” Dynamic Analysis of Linear MDOF Structures (27 pages)

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“Exact” Dynamic Analysis of Linear MDOF Structures



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“Exact” Dynamic Analysis of Linear MDOF Structures

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Lecture Notes


Pages:
27
School:
University at Buffalo, The State University of New York
Course:
Cie 619 - Structural Dynamics and Earthquake Engineering II

Unformatted text preview:

3 Exact Dynamic Analysis of Linear MDOF Structures Free Vibrations of Undamped Systems Example p Rayleigh s y g Method m 0 m 0 m k 6EI 2 3 7 l 3 3 8 1 0 806 2 2 815 CIE 619 Chapter 4 Seismic Analysis EI ml 3 EI ml 3 103 3 Exact Dynamic Analysis of Linear MDOF Structures Free Vibrations of Undamped Systems Example p Rayleigh s y g Method We use a static load vector corresponding to the weight associated to the degrees offreedom mg F mg The fundamental mode of vibration corresponds to the static deformations under this static load vector A 1 k 1 F f F 3 3 l 8 3 mg mg l 11 6 EI 3 2 mg 6 EI 5 CIE 619 Chapter 4 Seismic Analysis 104 1 3 Exact Dynamic Analysis of Linear MDOF Structures Free Vibrations of Undamped Systems Example p Rayleigh s y g Method N Substituting Rayleigh s formula it yields T 1 2 2 T 1 2 mgl 3 2 2 mg 11 5 6EI mgl 3 g mg 11 5 6EI T 1 7 749 3 ml EI W j 2j j 1 N g W j j j 1 3 T 1 7 796 ml EI We notice that the difference between the estimate of Rayleigh s formula and the exact fundamental period calculated in equation 4 176 is only 0 6 CIE 619 Chapter 4 Seismic Analysis 105 3 Exact Dynamic Analysis of Linear MDOF Structures Modal Analysis The Modal Superposition Method Strategy The modal analysis or the modal superposition method is a very useful technique to determine the dynamic response of a linear MDOF system subjected to an arbitrary dynamic load or a base acceleration The governing system of differential equations is written as m x c x k x F t 4 239 where m c k F t x x et x global mass matrix usually diagonal global damping matrix hard to evaluate global stiffness matrix x s t dynamic load vector m r x relative displacement velocity and acceleration vectors In general the differential equations can be coupled by m c or k In other words these matrices are not necessarily diagonal matrices If mij exists for i j the system is said to be dynamically coupled The basic strategy of the modal analysis is to introduce a linear transformation of the



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