Chapter 4, Problem 4.52P

To determine

(a)

The equation of motion of the system.

Answer to Problem 4.52P

The Equation of motion of the system are $m_1{x}_1=-c{x}_1+k\left(x_2-x_1\right)$ and $m_2{x}_2=f-k\left(x_2-x_1\right)$.

Explanation of Solution

Given information:

The damping coefficient is $c$ and friction is negligible.

The figure below shows the free body diagram.

Figure-(1)

Write the equation of motion for mass 1.

$\begin{array}{l}{m}_1{x}_1+c{x}_1+k\left(x_1-x_2\right)=0\\ m_1{x}_1=-c{x}_1-k\left(x_1-x_2\right)\\ m_1{x}_1=-c{x}_1+k\left(x_2-x_1\right)\end{array}$ ...... (I)

Here, mass of body 1 is $m_1$, damping coefficient is $c$, spring constant is $k$, displacement of mass1 is $x_1$, and the displacement of mass 2 is $x_2$.

Write the Equation of motion for mass 2.

$\begin{array}{l}{m}_2{x}_2+k\left(x_2-x_1\right)=f\\ m_2{x}_2=f-k\left(x_2-x_1\right)\end{array}$ ...... (II)

Here, mass of body 2 is $m_2$ and the input force is $f$.

Conclusion:

The equation of motion of the system is $m_1{x}_1=-c{x}_1+k\left(x_2-x_1\right)$ and $m_2{x}_2=f-k\left(x_2-x_1\right)$.

To determine

(b)

The equation of motion of the system.

Answer to Problem 4.52P

The equation of motion of the system is $m_1{x}_1=-k{x}_1+c\left(x_2-x_1\right)=0$ and $m_2{x}_2=f-c\left(x_2-x_1\right)$.

Explanation of Solution

Given information:

The damping coefficient is $c$ and friction is negligible.

The figure below shows the free body diagram of the block 1 and 2.

Figure-(2)

Write the Equation of motion for mass 1.

$\begin{array}{l}{m}_1{x}_1+k{x}_1+c\left(x_1-x_2\right)=0\\ m_1{x}_1=-k{x}_1-c\left(x_1-x_2\right)\\ m_1{x}_1=-k{x}_1+c\left(x_2-x_1\right)=0\end{array}$ ........ (III)

Here, mass of body 1 is $m_1$, damping coefficient is $c$, spring constant is $k$, displacement of mass1 is $x_1$, and the displacement of mass 2 is $x_2$.

Write the equation of motion for mass 2.

$\begin{array}{l}{m}_2{x}_2+c\left(x_2-x_1\right)=f\\ m_2{x}_2=f-c\left(x_2-x_1\right)\end{array}$ ...... (IV)

Here, mass of body 2 is $m_2$ and the input force is $f$.

Conclusion:

The equation of motion of the system is $m_1{x}_1=-k{x}_1+c\left(x_2-x_1\right)=0$ and $m_2{x}_2=f-c\left(x_2-x_1\right)$.

To determine

(c)

The equation of motion.

Answer to Problem 4.52P

The equation of motion of the system is $m_1{x}_1=-c{x}_1-k_1{x}_1+k_2\left(x_2-x_1\right)=0$ and $m_2{x}_2=f-k_2\left(x_2-x_1\right)$.

Explanation of Solution

Given information:

The damping coefficient is $c$ and friction is negligible.

The Figure below shows the free body diagram of the system.

Figure-(3)

Write the equation of motion for mass 1.

$\begin{array}{l}{m}_1{x}_1+c{x}_1+k_1{x}_1+k_2\left(x_1-x_2\right)=0\\ m_1{x}_1=-c{x}_1-k_1{x}_1-k_2\left(x_1-x_2\right)\\ m_1{x}_1=-c{x}_1-k_1{x}_1+k_2\left(x_2-x_1\right)=0\end{array}$ ........ (V)

Here, mass of body 1 is $m_1$, damping coefficient is $c$, spring constant for first spring is $k_1$, the spring constant for spring 2 is $k_2$ displacement of mass1 is $x_1$, and the displacement of mass 2 is $x_2$.

Write the Equation of motion for mass 2.

$\begin{array}{l}{m}_2{x}_2+k_2\left(x_2-x_1\right)=f\\ m_2{x}_2=f-k_2\left(x_2-x_1\right)\end{array}$ ...... (VI)

Here, mass of body 2 is $m_2$ and the input force is $f$.

Conclusion:

The equation of motion of the system is $m_1{x}_1=-c{x}_1-k_1{x}_1+k_2\left(x_2-x_1\right)=0$ and $m_2{x}_2=f-k_2\left(x_2-x_1\right)$.

To determine

(d)

The Equation of motion.

Answer to Problem 4.52P

The Equation of motion of the system is $m_1{x}_1=-c_1{x}_1-k_1{x}_1+c_2\left(x_2-x_1\right)+k_2\left(x_2-x_1\right)$ and $m_2{x}_2=f-c_2\left(x_2-x_1\right)-k_2\left(x_2-x_1\right)$.

Explanation of Solution

Given information:

The damping coefficient of damper 1 is $c_1$, the damping coefficient of damper 2 is $c_2$.

The figure below shows the free diagram of the system.

Figure-(4)

Write the Equation of motion for mass 1.

$\begin{array}{l}{m}_1{x}_1+c_1{x}_1+k_1{x}_1+c_2\left(x_1-x_2\right)+k_2\left(x_1-x_2\right)=0\\ m_1{x}_1=-c_1{x}_1-k_1{x}_1-c_2\left(x_1-x_2\right)-k_2\left(x_1-x_2\right)\\ m_1{x}_1=-c_1{x}_1-k_1{x}_1+c_2\left(x_2-x_1\right)+k_2\left(x_2-x_1\right)\end{array}$ ........ (VII)

Here, mass of body 1 is $m_1$, damping coefficient of first damper is $c_1$, damping coefficient of second damper is $c_2$ spring constant for first spring is $k_1$, the spring constant for spring 2 is $k_2$ displacement of mass1 is $x_1$, displacement of mass 2 is $x_2$.

Write the Equation of motion for mass 2.

$\begin{array}{l}{m}_2{x}_2+c_2\left(x_2-x_1\right)+k_2\left(x_2-x_1\right)=f\\ m_2{x}_2=f-c_2\left(x_2-x_1\right)-k_2\left(x_2-x_1\right)\end{array}$ ...... (VIII)

Here, mass of body 2 is $m_2$ and the input force is $f$.

Conclusion:

The equation of motion of the system is $m_1{x}_1=-c_1{x}_1-k_1{x}_1+c_2\left(x_2-x_1\right)+k_2\left(x_2-x_1\right)$ and $m_2{x}_2=f-c_2\left(x_2-x_1\right)-k_2\left(x_2-x_1\right)$.