Mechanical Vibrations (differential equations)A mass weighing 4 pounds is attached to a sping whose constant is 2lb/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point one foot above the equilibrium position with a downward velocity of 8ft/s. Determine the time at which the mass passes through the equilibrium position.

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Asked Jun 24, 2019
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Mechanical Vibrations (differential equations)

A mass weighing 4 pounds is attached to a sping whose constant is 2lb/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point one foot above the equilibrium position with a downward velocity of 8ft/s. Determine the time at which the mass passes through the equilibrium position.

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Expert Answer

Step 1

Let m be the mass attached, k be the spring constant and let b be a positive damping constant.

Then, by Newton’s second law for the system is:

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d2x dxc m -b- dt dt or а?x, b dx, k +x0 m dt m (1) dt

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Step 2

Where x(t) is the displacement from the equilibrium position.

Now, determine the equation of motion.

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W = mg 4 lb 1 m _ g 32 ft/sslug 8 g

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Step 3

Now, put m=1/8 slugs, k=2 lb/ft and b=1 into t...

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d2x ,dx 8 16x=0 dt dt

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