CA-AA242B-Ch5 - AA242B MECHANICAL VIBRATIONS AA242B...

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AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Approximation of Continuous Systems by Displacement Methods These slides are partially based on the recommended textbook: M. G´ eradin and D. Rixen, “Mechanical Vibrations: Theory and Applications to Structural Dynamics,” Second Edition, Wiley, John & Sons, Incorporated, ISBN-13:9780471975465 AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS Outline 1 The Rayleigh-Ritz Method 2 The Finite Element Method AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS The Rayleigh-Ritz Method Choice of Approximation Functions Rayleigh-Ritz approximation u i ( x 1 , x 2 , x 3 , t ) = n X j =1 n ij ( x 1 , x 2 , x 3 ) q j ( t ) , i = 1 , 2 , 3 q j , j = 1 , · · · , n are the generalized coordinates vector form of “assumed modes” n j ( x 1 , x 2 , x 3 ) = 2 4 n 1 j ( x 1 , x 2 , x 3 ) n 2 j ( x 1 , x 2 , x 3 ) n 3 j ( x 1 , x 2 , x 3 ) 3 5 u ( x 1 , x 2 , x 3 , t ) = n X j =1 n j ( x 1 , x 2 , x 3 ) q j ( t ) admissibility conditions: C 0 continuity and satisfaction of the essential boundary conditions AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS The Rayleigh-Ritz Method Discretization of the Displacement Variational Principle Rayleigh-Ritz approximation in matrix form u ( x 1 , x 2 , x 3 , t ) = N ( x 1 , x 2 , x 3 ) q ( t ) N R 3 × n is the displacement interpolation matrix [ N ( x 1 , x 2 , x 3 )] ij = n ij ( x 1 , x 2 , x 3 ) , i = 1 , 2 , 3 , j = 1 , · · · , n q ( t ) = ˆ q 1 · · · q n ˜ T is the vector of generalized coordinates Strain components recall the spatial differentiation operator D = 2 6 4 x 1 0 0 x 2 0 x 3 0 x 2 0 x 1 x 3 0 0 0 x 3 0 x 2 x 1 3 7 5 continuous strain vector ε ( x , t ) = Du = DN ( x 1 , x 2 , x 3 ) q ( t ) = B ( x 1 , x 2 , x 3 ) q ( t ) where ( B ( x 1 , x 2 , x 3 ) = DN ( x 1 , x 2 , x 3 )) R 6 × n is the strain interpolation matrix AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS The Rayleigh-Ritz Method Discretization of the Displacement Variational Principle Displacement variational principle kinetic energy ˙ u = N ˙ q ⇒ T = 1 2 Z V ρ ( N ˙ q ) T N ˙ q dV = 1 2 ˙ q T M ˙ q where M = Z V ρ N T N dV is the symmetric positive definite mass matrix of the discrete system strain energy V int = 1 2 Z V ε T H ε dV = 1 2 Z V ( Bq ) T HBq dV = 1 2 q T Kq where H is the Hooke matrix of elastic coefficients, and K = Z V B T HB dV is the symmetric positive semi-definite stiffness matrix of the discrete system the rigid body modes u are the solutions of Ku = 0 — they produce no strain energy AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS The Rayleigh-Ritz Method Discretization of the Displacement Variational Principle Displacement variational principle (continue) external potential energy V ext = - Z S σ ( Nq ) T ¯ t dS - Z V ( Nq ) T ¯ X dV = - q T g where g ( t ) = Z S σ N T ¯ t ( t ) dS + Z V N T ¯ X ( t ) dV is called the external load factor discretized displacement variational principle δ Z t 2 t 1 1 2 ˙ q T M ˙ q - 1 2 q T Kq - q T g «ff dt = 0 = h δ q T M ˙ q i t 2 t 1 - Z t 2 t 1 δ q T { M ¨ q + Kq - g } dt = 0 recall that δ q ( t 1) = δ q ( t 2 ) = 0 = M ¨ q + Kq = g ( t ) AA242B: MECHANICAL VIBRATIONS
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AA242B: MECHANICAL VIBRATIONS The Rayleigh-Ritz Method
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