Explanation Calculation: Since ( c > c c ) which is over damped, the wavelength λ 1 and λ 2 is less than zero. The expression for the differential equation of over damping as follows: x = C 1 e λ 1 t + C 2 e λ 2 t (1) Differentiate the above equation with respect to ‘ t ’. v = d x d t = C 1 λ 1 e λ 1 t + C 2 λ 2 e λ 2 t (2) Since the body released from an arbitrary position with an arbitrary initial velocity cannot pass more than once through its equilibrium position. Substitute 0 for t , x 0 for x , and v 0 for v in equation (1). x 0 = C 1 e λ 1 ( 0 ) + C 2 e λ 2 ( 0 ) x 0 = C 1 e 0 + C 2 e 0 x 0 = C 1 ( 1 ) + C 2 ( 1 ) x 0 = C 1 + C 2 C 1 = x 0 − C 2 (3) Substitute 0 for t , x 0 for x , and v 0 for v in equation (2). d x d t = C 1 λ 1 e λ 1 t + C 2 λ 2 e λ 2 t v 0 = C 1 λ 1 e λ 1 ( 0 ) + C 2 λ 2 e λ 2 ( 0 ) v 0 = C 1 λ 1 ( 1 ) + C 2 λ 2 ( 1 ) v 0 = C 1 λ 1 + C 2 λ 2 Substitute x 0 − C 2 for C 1 . C 1 λ 1 + C 2 λ 2 = v 0 ( x 0 − C 2 ) λ 1 + C 2 λ 2 = v 0 x 0 λ 1 − C 2 λ 1 + C 2 λ 2 = v 0 x 0 λ 1 − C 2 ( λ 1 − λ 2 ) = v 0 x 0 λ 1 = v 0 + C 2 ( λ 1 − λ 2 ) − ( x 0 λ 1 − v 0 ) = C 2 ( λ 2 − λ 1 ) C 2 = v 0 − x 0 λ 1 λ 2 − λ 1 Substitute v 0 − x 0 λ 1 λ 2 − λ 1 for C 2 in equation (3). C 1 = x 0 − v 0 − x 0 λ 1 λ 2 − λ 1 = x 0 ( λ 2 − λ 1 ) − v 0 + x 0 λ 1 λ 2 − λ 1 = x 0 λ 2 − x 0 λ 1 − v 0 + x 0 λ 1 λ 2 − λ 1 = x 0 λ 2 − v 0 λ 2 − λ 1 = − ( v 0 − x 0 λ 2 ) λ 2 − λ 1 Substitute v 0 − x 0 λ 1 λ 2 − λ 1 for C 2 and − ( v 0 − x 0 λ 2 ) λ 2 − λ 1 for C 1 in equation (1)

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Free Damped Vibrations

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Chapter 19.5, Problem 19.128P

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Show that in case of heavy damping (c>cc) a body released from an arbitrary position with an arbitrary initial velocity cannot pass more than once through its equilibrium position.

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Chapter 19.5, Problem 19.127P

Chapter 19.5, Problem 19.129P

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Show that in case of heavy damping ( c &gt; c c ) a body released from an arbitrary position with an arbitrary initial velocity cannot pass more than once through its equilibrium position.

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Show that in case of heavy damping (c>cc) a body released from an arbitrary position with an arbitrary initial velocity cannot pass more than once through its equilibrium position.

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Show that in case of heavy damping ( c > c c ) a body released from an arbitrary position with an arbitrary initial velocity cannot pass more than once through its equilibrium position.