CA-AA242B-Ch5

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

This preview shows pages 1–8. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
AA242B: MECHANICAL VIBRATIONS Outline 1 The Rayleigh-Ritz Method 2 The Finite Element Method AA242B: MECHANICAL VIBRATIONS
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
AA242B: MECHANICAL VIBRATIONS The Rayleigh-Ritz Method

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern