Question

A spring is stretched 6 in by a mass that weighs 8 lb. The mass is attached to a dashpot mechanism that has a damping constant of 0.25 lb·s/ft and is acted on by an external force of 4 cos 2t lb. (a) Determine the steady state response of this system.

Answer 1

Answer:

y= 240/901 cos 2t+ 8/901 sin 2t

Explanation:

To find mass m=weighs/g

  m=8/32=0.25

To find the spring constant

Kx=mg    (given that c=6 in and mg=8 lb)

K(0.5)=8               (6 in=0.5 ft)

K=16 lb/ft

We know that equation for spring mass system

my''+Cy'+Ky=F  

now by putting the values

0.25 y"+0.25 y'+16 y=4 cos 20 t  ----(1) (given that C=0.25 lb.s/ft)

Lets assume that at steady state the equation of y will be

y=A cos 2t+ B sin 2t

To find the constant A and B we have to compare this equation with equation 1.

Now find y' and y" (by differentiate with respect to t)

y'= -2A sin 2t+2B cos 2t

y"=-4A cos 2t-4B sin 2t

Now put the values of y" , y' and y in equation 1

0.25 (-4A cos 2t-4B sin 2t)+0.25(-2A sin 2t+2B cos 2t)+16(A cos 2t+ B sin 2t)=4 cos 20 t

So by comparing the coefficient both sides

30 A+ B=8

A-30 B=0

So we get

A=240/901 and B=8/901

So the steady state response

y= 240/901 cos 2t+ 8/901 sin 2t

Answer 2

A steady-state response is the behavior of a circuit over a lengthy period of time when stable conditions have been achieved. The steady-state response of this system will be y= 240/901 cos 2t+ 8/901 sin 2t.

What is a steady-state response?

A steady-state response is the behavior of a circuit over a lengthy period of time when stable conditions have been achieved following an external stimulus.

The given data in the problem will bge;

C=0.25 lb.s/ft

Weight is defined as the product of mass and gravity.

$\rm{m=\frac{W}{g} }\\\\\rm{m=\frac{8}{32}$

$\rm m=0.25$

Spring constant is defined as the ratio of force per unit displaced length.

The spring force is balanced by the weight;

$\rm Kx=mg\\\\ \rm x= \frac{mg}{K} \\\\ \rm x=\frac{8}{0.5} lb/ft$

The equation for the spring-mass system is given by;

$\rm {my''+Cy'+Ky=F }$

$\rm 0.25 y"+0.25 y'+16 y=4 cos 20 t$

Steady-state equation;

$\rm y=A cos 2t+ B sin 2t$

For finding the value of A and B

$\rm y'= -2A sin 2t+2B cos 2ty"=-4A cos 2t-4B sin 2t$

By putting the value we got

$\rm 0.25 (-4A cos 2t-4B sin 2t)+0.25(-2A sin 2t+2B cos 2t)+16(A cos 2t+ B sin 2t)=4 cos 20 t$

The value of cofficient obtained from the equation

$30 A+ B=8$

Getting the value as

$A= \frac{240/901}\\\\ B=\frac{8}{901}$

The steady-state response got

$y= 240/901 cos 2t+ 8/901 sin 2t$

Hence the steady-state response of this system.y= 240/901 cos 2t+ 8/901 sin 2t

To learn about the steady-state response refer to the link;

https://brainly.com/question/14960844