Question

a weight is suspended from the ceiling by a spring (k = 20 ln/in) and is connected to the floor by a dashpot producing viscous damping. The damping force is 10 lb when the velocity of the dashpot plunger is 20 in/sec. The weight and plunger have W = 12 lb. What will be the frequency of the damped vibrations?

Answer 1

The frequency of the damped vibrations is 3.82 Hz.

Step-by-step explanation:

Given that,

Spring constant = 20 lb/in

Damping force = 10 lb

Velocity = 20 in/sec

Weight = 12 lb

We need to calculate the damping constant

Using formula of damping force

`b* v=F_(d)`

`b=(F_(d))/(v)`

Put the value into the formula

`b =(10)/(20)`

`b=0.5\ lb-sec/in`

`b=0.5*12 =6\ lb-sec/ft`

We need to calculate the frequency

Using formula of angular frequency

`\omga=\sqrt{\omega_(0)^2-((b)/(2m))^2}`

`\omega=\sqrt{(k)/(m)-((b)/(2m))^2}`

Put the value into the formula

`\omega=\sqrt{(20*12*32)/(12)-((6*32)/(2*12))^2}`

`\omega=24\ rad/s`

We need to calculate the frequency of the damped vibrations

Using formula of frequency

`f=(\omega)/(2\pi)`

Put the value into the formula

`f=(24)/(2\pi)`

`f=3.82\ Hz`

Hence, The frequency of the damped vibrations is 3.82 Hz.