Ch6-1(1-3)c - 2009-04-22 1 MAE351 Mechanical Vibrations...

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2009-04-22 1 1 MAE351 Mechanical Vibrations Lecture 25 (Chap. 6.1-6.3) 2 Chapter 6 Distributed Parameter Systems Extending the first 5 chapters to systems with distributed mass and stiffness properties: strings, rods, beams, membranes, and plates (“ continuous ” systems)
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2009-04-22 2 3 Continuous systems Discrete or lumped parameter system: ODE Distributed or continuous system: PDE ( DOF) String Rod, Beam Membrane Plate 4 6.1 Vibration of a String or Cable Start by considering a uniform string stretched between two fixed boundaries Assume constant axial tension in string Let a distributed force f ( x , t ) act along the string   , wxt
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2009-04-22 3 5 Movement of a small element of the string 2 2 (,) y wxt F Ax 11 2 2 sin sin fx t x    t x x ds Force balance on an infinitesimally small element Linearization of the equation by approximating the sine with small angles: 6 If w(x,t) is small, tension remains constant along ds x  ds N t ti l f ti 2 12 , tan , x x For small displacement ww x x       21 (sin sin ) z F  x xx x Net vertical force acting on an element z F f x   
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2009-04-22 4 7 21 2 2 (,) xx wxt f xt x A x t       Recall the Tayor series of about x w 1 2 () x ww w xO x x x   2 2 xf xt x A x    Recall the Tayor series of about 1 : x 1 x x xt 2 2 fx t A t  8 For no external forces and constant , the equation of motion becomes: 22 2 1( , ) ( , ) , c t x Second order in time and second order in space; therefore, 4 constants of integration are required. Two from i nitial c onditions: 00 (,0 ) , ) 0 t wx w x a t t  And two from (fixed) b oundary c onditions:
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2009-04-22 5 9 Physical quantities at any point x along the string at time t Deflection occurs only in the y-direction and defined by w ( x , t ) - The slop of the string is w x ( x , t ) - The restoring force per unit length is - The velocity is w t ( x , t ) The acceleration is w x 2 (,) wxt A     - The acceleration is tt ( , t ) Note that the above applies to cables as well! 2 x xt  10 Cables in the suspension bridge The world-first bungee jumping site near Queenstown in New Zealand
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2009-04-22 6 11 6.2 Modes and Natural Frequencies Solve the eq’n of motion by the method of separation of variables: () ( ) ( ) wxt X xTt  22 1( , ) ( , ) c   ( , )( Substitute into the equation of motion to get: " dd where and dx dt 2 "( ) ( ) "( ) Xx T t Xx acons t    2 , ct x A 2 .
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This note was uploaded on 02/11/2010 for the course MECHANICAL ms316 taught by Professor Abduljaba during the Spring '09 term at Kalamazoo.

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Ch6-1(1-3)c - 2009-04-22 1 MAE351 Mechanical Vibrations...

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