Consider the initial value problem m y\'\' + c y\' + k y = F(t),y(0) = 0, y\'(0)

Question

Consider the initial value problem m y'' + c y' + k y = F(t),y(0) = 0, y'(0) = 0 modelling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and F (t) = 30 e^-t Newtons. Solve the initial value problem. Y (t) = Determine the long-term behaviour of the system. Is lim_t right arrow infinite y (t) = 0? If it is, enter zero. If not, enter a function that approximates y (t) for very large positive values of t. For very large positive values of t, y (t)

Explanation / Answer

Here ,

m y" + c y' + k * y = F(t)

m = 2 Kg

F(t) = 30 e^(-t)

c = 8 Kg/s

k = 80 N/m

a) for the equation

2 y'' + 8 y' + 80 y = 30 e^(-t)

y'' + 4 y' + 40 y = 15 e^(-t)

solving for y

y = -5/74 * e^(-2t) * (-6 e^t + sin(6t) + 6 * cos(6t))

the function y is -5/74 * e^(-2t) * (-6 e^t + sin(6t) + 6 * cos(6t))

b)

for y(t) lim t --> infinte

as e^(-2t) --> 0

y will approach zero

hence , y = 0 for large t

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