Consider the initial value problem my′'+ c y′+ k y=F(t),y(0)=0,y,(0)=0 modeling 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)=20sin(6t)Newtons. a. Solve the initial value problem. b. Determine the long-term behavior of the system. c. Is limt→[infinity]y(t)=0?If no, enter a function that approximates y(t)for very large positive values of t.

A)      

B)      

Step-by-step explanation:

- The following initial value problem is given as follows:

- The above equation is the Newtonian mathematical model of a spring-mass-dashpot system. The displacement ( y ) and velocity ( y' ) are zeroed at the initial value t = 0.

- The equivalent mass ( m ) , damping constant ( c ) and the equivalent spring stiffness ( k ) are given as follows:

- The system is subjected to a sinusoidal force F ( t ) given. We will plug in the constants ( m , c, and k ) and applied force F ( t ) into the given second order ODE.

- The solution to a second order ODE is comprised of a complementary function ( yc ) and particular function ( yp ).

- To determine the complementary function ( yc ) we will solve the homogeneous part of the given second order ODE. We will assume the independent solution to the homogeneous ODE takes the form:

Where,

                          a: The root of the following characteristic equation

- Substitute ( y ) into the given ODE as follows:

- Solve the above characteristic quadratic equation:

- The complementary solution for the complex solution to the characteristic equation is of the form:

Where,

                          a = α ± β

Therefore,

- To determine the particular solution we will scrutinized on the non-homogeneous part of the given ODE. The forcing function F ( t ) the applied force governs the form of the particular solution. For sinusoidal wave-form the particular solution takes form as following:

Where,

                           C & D are constants to be evaluated.

- Determine the first and second derivatives of the particular solution (yp) as follows:

- Plug in the particular solution ( yp ) and its derivatives ( first and second ) into the given ODE.

- The particular solution can be written as follows:

- Now we use the principle of super-position and combine the complementary and particular solution and form a function of general solution as follows:

- To determine the complete solution of the given ODE we have to calculate the constants ( A and B ) using the given initial conditions as follows:

- The complete solution to the initial value problem is:

- To determine the long term behavior of the system we will apply the following limit on our complete solution derived above:

- We see that the complementary part of the solution decays as t gets large and the particular solution that models the applied force F ( t ) is still present in the system response when t gets large.

part a: 1.08

part b: 27.59