Give step-by-step solution with explanation and final answer:2.1 Modeling a two degree of freedom mass-spring-damper system
‘The system in Figure 1 consists of two masses (m1 and 712) connected by li

Question

Give step-by-step solution with explanation and final answer:2.1 Modeling a two degree of freedom mass-spring-damper system
‘The system in Figure 1 consists of two masses (m1 and 712) connected by linear springs (hr. /» and.
3) and viscous dampers (c1. ¢2 and ¢3). Two external forces fi and f are applied to m1 and mn.
respectively.
0(0./iC) 00.5)
h kB ks
EN ==
a 0 0 O_O =
Figure 1. Two degrees of freedom system: two masses with springs and dampers.

Drarw the free body diagram. derive the equations of motion and write the equations in the following
format:
i=
=

VIPSolutions · Accepted Answer

Let's solve the problem **step by step**:

---

# **Step 1: Free Body Diagram**

- **Mass $ m_1 $** is acted upon by:
  - Spring $ k_1 $ and damper $ c_1 $ (to the left, connected to ground)
  - Spring $ k_2 $ and damper $ c_2 $ (to the right, connecting $ m_1 $ and $ m_2 $)
  - External force $ f_1(t) $

- **Mass $ m_2 $** is acted upon by:
  - Spring $ k_3 $ and damper $ c_3 $ (to the right, connected to ground)
  - Spring $ k_2 $ and damper $ c_2 $ (to the left, connecting $ m_1 $ and $ m_2 $)
  - External force $ f_2(t) $

Let:
- $ x_1(t) $: displacement of $ m_1 $
- $ x_2(t) $: displacement of $ m_2 $

---

# **Step 2: Write the Equations of Motion**

## **For Mass $ m_1 $:**
Sum forces acting on $ m_1 $:

- Left spring and damper (to ground): 
  - Spring: $ -k_1 x_1 $
  - Damper: $ -c_1 \dot{x}_1 $
- Right spring and damper (to $ m_2 $): 
  - Spring: $ -k_2 (x_1 - x_2) $
  - Damper: $ -c_2 (\dot{x}_1 - \dot{x}_2) $
- External force: $ +f_1(t) $

Apply Newton's second law:
$$
m_1 \ddot{x}_1 = -k_1 x_1 - c_1 \dot{x}_1 - k_2 (x_1 - x_2) - c_2 (\dot{x}_1 - \dot{x}_2) + f_1(t)
$$

---

## **For Mass $ m_2 $:**
Sum forces acting on $ m_2 $:

- Right spring and damper (to ground): 
  - Spring: $ -k_3 x_2 $
  - Damper: $ -c_3 \dot{x}_2 $
- Left spring and damper (to $ m_1 $): 
  - Spring: $ -k_2 (x_2 - x_1) $
  - Damper: $ -c_2 (\dot{x}_2 - \dot{x}_1) $
- External force: $ +f_2(t) $

Apply Newton's second law:
$$
m_2 \ddot{x}_2 = -k_3 x_2 - c_3 \dot{x}_2 - k_2 (x_2 - x_1) - c_2 (\dot{x}_2 - \dot{x}_1) + f_2(t)
$$

---

# **Step 3: Rearranging and Writing State-Space Form**

Let’s write the equations as:

$$
\begin{align*}
m_1 \ddot{x}_1 + (c_1 + c_2)\dot{x}_1 - c_2\dot{x}_2 + (k_1 + k_2)x_1 - k_2 x_2 &= f_1(t) \
m_2 \ddot{x}_2 + (c_2 + c_3)\dot{x}_2 - c_2\dot{x}_1 + (k_2 + k_3)x_2 - k_2 x_1 &= f_2(t)
\end{align*}
$$

---

# **Step 4: Write in Given Format**

Let:
- $\dot{x}_1 = v_1$
- $\dot{x}_2 = v_2$

Then, the equations for $\dot{v}_1$ and $\dot{v}_2$ (i.e., $\ddot{x}_1$ and $\ddot{x}_2$) are:

$$
\boxed{
\begin{aligned}
\ddot{x}_1 &= \frac{1}{m_1} \left[ - (c_1 + c_2)\dot{x}_1 + c_2\dot{x}_2 - (k_1 + k_2)x_1 + k_2 x_2 + f_1(t) \right] \
\ddot{x}_2 &= \frac{1}{m_2} \left[ - (c_2 + c_3)\dot{x}_2 + c_2\dot{x}_1 - (k_2 + k_3)x_2 + k_2 x_1 + f_2(t) \right]
\end{aligned}
}
$$

Or, in the requested format:

$$
\boxed{
\begin{aligned}
\dot{x}_1 &= v_1 \
\dot{x}_2 &= v_2 \
\dot{v}_1 &= \frac{1}{m_1} \left[ - (c_1 + c_2)v_1 + c_2v_2 - (k_1 + k_2)x_1 + k_2 x_2 + f_1(t) \right] \
\dot{v}_2 &= \frac{1}{m_2} \left[ - (c_2 + c_3)v_2 + c_2v_1 - (k_2 + k_3)x_2 + k_2 x_1 + f_2(t) \right]
\end{aligned}
}
$$

---

## **Summary**

- **Free body diagrams** are drawn for each mass.
- **Equations of motion** are derived using Newton's second law.
- **Final answer** is written in state-space (first-order ODE) format.

If you need the free body diagram sketch, let me know!

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