Aspring 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.

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.

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.

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

The spring force is balanced by the weight;

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

Steady-state equation;

For finding the value of A and B

By putting the value we got

The value of cofficient obtained from the equation

Getting the value as

The steady-state response got

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

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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

which three statements are true about all alloys?