Explanation Given: The arbitrary velocity of body is x ˙ 0 . Concept used: Write the equation of motion a system containing spring, mass and damper. m x ¨ + c x ˙ + k x = 0 ...... (1) Here, m is the mass, k is the spring constant, c is the damping constant, x ˙ is the velocity of damper and x ¨ is the acceleration of mass. For heavy damping the solution for equation (1) is. x = c 1 e λ 1 t + c 2 e λ 2 t ...... (2) Here, x is the displacement of system, λ 1 and λ 2 are the roots of equation (1) and t is time. The above solution corresponds to a non-vibratory motion, as 1 and 2 are both negative, x approaches to zero as t increases indefinitely. Differentiate equation (2) with respect to time t . x ˙ = c 1 λ 1 e λ 1 t + c 2 λ 2 e λ 2 t ...... (3) Here, x ˙ is the velocity. Calculation: Substitute x 0 for x and 0 for t in equation (2). x 0 = c 1 + c 2 ...... (4) Here, x 0 is any arbitrary position. Substitute x ˙ 0 for x ˙ and 0 for t in equation (3). x ˙ 0 = c 1 λ 1 + c 2 λ 2 ...... (5) Substitute x 0 − c 1 for c 2 in equation (5) and solve for c 1 . c 1 = − x ˙ 0 + λ 2 x 0 λ 2 − λ 1 Substitute − x ˙ 0 + λ 2 x 0 λ 2 − λ 1 for c 1 in equation (4)

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Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the init…

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Chapter 19.5, Problem 19.128P

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To show for heavy damping a body released from an arbitrary position with an 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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To show for heavy damping a body released from an arbitrary position with an initial velocity cannot pass more than once through its equilibrium position.

Solution Summary: The author explains that a body released from an arbitrary position with an initial velocity cannot pass more than once through its equilibrium position.

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To show for heavy damping a body released from an arbitrary position with an initial velocity cannot pass more than once through its equilibrium position.

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Chapter 19 Solutions