CA-AA242B-Ch2

CA-AA242B-Ch2 - AA242B: MECHANICAL VIBRATIONS AA242B:...

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Unformatted text preview: AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of n-DOF Systems These slides are 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 AA242B: MECHANICAL VIBRATIONS Outline 1 Linear Vibrations 2 Natural Vibration Modes 3 Orthogonality of Natural Vibration Modes 4 Modal Superposition Analysis 5 Spectral Expansions 6 Forced Harmonic Response 7 Response to External Loading 8 Mechanical Systems Excited Through Support Motion AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Linear Vibrations Equilibrium configuration q s ( t ) = q s (0) q s ( t ) = s = 1 , , n (1) Recall the Lagrange equations of motion- d dt T q s + T q s- V q s- D q s + Q s ( t ) = 0 where T = T + T 1 + T 2 Recall the generalized gyroscopic forces f s =- n X r =1 2 T 1 q s q r q r + T 1 q s , s = 1 , , n Definition : the effective potential energy is defined as V ? = V - T The Lagrange equations of motion can be re-written as d dt T 2 q s- T 2 q s = Q s ( t )- V ? q s- D q s + f s- t T 1 q s AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Linear Vibrations Recall that T ( q , t ) = 1 2 N X k =1 3 X i =1 m k U i k t 2 (transport kinetic energy) From the Lagrange equations of motion d dt T 2 q s- T 2 q s = Q s ( t )- V ? q s- D q s + f s- t T 1 q s it follows that an equilibrium condition exists if and only if V = V ( q ) and Q s ( t ) = 0 Hence, at equilibrium Q s ( t ) = 0 and V ? q s = ( V - T ) q s = 0 , s = 1 , , n AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Linear Vibrations Free-Vibrations About a Stable Equilibrium Position Consider a system that does not undergo a transport (or overall motion) T = T 2 ( q ) The equilibrium position is then given by Q s ( t ) = 0 and V q s = 0 , s = 1 , , n Consider now a conservative system E = T + V = cst Usually, the potential energy is defined only up to a constant shift the origin of the generalized coordinates to have equilibrium at q s = 0 , s = 1 , , n Now, suppose that a certain energy E (0) is initially given to the system in equilibrium AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Linear Vibrations Free-Vibrations About a Stable Equilibrium Position Definition : the equilibrium position ( q s = 0 , s = 1 , , n ) is said to be stable if E ?...
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CA-AA242B-Ch2 - AA242B: MECHANICAL VIBRATIONS AA242B:...

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