# User:Egm4313.s12.team6.hickey

## Contents

## Report 1[edit]

### Question R1.2[edit]

#### Question[edit]

Derive the equation of motion of the spring - mass - dashpot in Fig. 53, in K 2011 p. 85, with an applied force r(t) on the ball.

#### Solution[edit]

There are 2 possible cases that can occur in this problem, depending on the direction of the applied force.

In both cases,

##### Case 1[edit]

The applied force is in the positive direction, and therefore the displacement is in the positive direction.

From the Free Body Diagram, we get the equation

Rearranging the equation, we get

Replacing Force variables, we get

##### Case 2[edit]

The applied force is in the negative direction, and therefore the displacement is in the negative direction.

From the Free Body Diagram, we get the equation

Rearranging the equation, we get

Replacing Force variables, we get

#### Conclusion[edit]

Since both cases return the same solution, the equation of motion is derived as:

## Report 2[edit]

### Question R2.9[edit]

#### Question[edit]

Find and plot the solution for the Linear Second order ODE with constant coefficients corresponding to equation (1).

Equation (1):

Initial Conditions:

No excitation:

In another Figure, superpose 3 Figures: (a) This figure, (b)The figure in R2.6 p. 5-6, (c) The Figure in R2.1 p. 3-7.

#### Solution[edit]

The question implied by the problem statement is that we are solving the following ODE:

Where we know that

and the corresponding characteristic equation is

To find the roots of this equation, we must use the Quadratic Equation, giving us

Inserting our corresponding values for these variables, we have

Knowing that we have complex roots, we can now insert the roots of our characteristic equation into the proper form for the homogeneous solution:

And taking the first derivative of this equation, we get

Now, we can use our initial conditions to find the values of our C coefficients:

Therefore, we now know that our final solution is:

The plot for this equation is given as shown in plot 1:

Comparing the graphs between problems R2.1, R2.6, and R2.9, we get the following plot:

Where the solution for R2.1 is given in red, the solution for R2.6 is given in green, and the solution for R2.9 is given in blue.