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Ch2-1(1-2)c

Ch2-1(1-2)c - 1 MAE351 Mechanical Vibrations Lecture 6(Chap...

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1 1 MAE351 Mechanical Vibrations Lecture 6 (Chap. 2.1-2.2) 2 Chapter 2. Response to Harmonic Excitation Introduces the important concept of resonance

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2 3 External Forces and Vibration * (adapted from B&K Technical Note BA767412) 4 2.1 Harmonic Excitation of Undamped Systems Displacement Consider the usual spring mass damper system with applied force F )= F cos M k x F=F 0 cos t system with applied force ( t ) 0 t is the driving frequency F 0 is the magnitude of the applied force We take c = 0 to start with
3 5 Equations of motion x x t F t kx t x m cos ) cos( ) ( ) ( 2 0 M k x Displacement F=F 0 cos t m k m F f t f t t n n where , ) cos( ) ( ) ( 0 0 0 Solution is the sum of homogenous and particular solution () () 0 mx t kx t  0 cos( ) Ft The particular solution assumes a form of forcing function (physically, the input wins) 6 Substitute particular solution into the equation of motion: 2 2 2 0 0 2 2 n x n x f X t f t X t X p n p : yields solving cos cos cos       Thus the particular solution has the form Thus, the particular solution has the form:

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4 7 Add particular and homogeneous solutions to get general solution: x ( t ) A 1 sin n t A 2 cos n t f 0 2 2 cos t particular  homogeneous  n  A 1 and A 2 are constants of integration. 8 00 12 2 0 22 (0) sin0 cos0 nn ff x AA A x   Apply the initial conditions to evaluate the constants 0 2022 0 1 0 (0) ( sin0) n n f Ax f xA A A v   0 1 0 () n v A v xt 0 sin cos cos n tx t t     
5 9 Comparison of free and forced response Sum of two harmonic terms of different Sum of two harmonic terms of different frequency Free response has amplitude and phase affected by forcing function Our solution is not defined for n =  because it produces division by 0 because it produces division by 0. If forcing frequency is close to natural frequency, the amplitude of particular solution is very large 10 Response for m =100 kg, k =1000 N/m, F =100 N, = n +5, v 0 =0.1m/s and x 0 = -0.02 m.

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Ch2-1(1-2)c - 1 MAE351 Mechanical Vibrations Lecture 6(Chap...

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