---
title: "Free body diagram of masses between walls"
subject: "Control Systems"
chapter: 2
key: "semester_3/control_systems:fbd-tf-m1-m2-k1-b"
source: https://www.bctengineer.com/q/semester_3/control_systems/fbd-tf-m1-m2-k1-b
site: BCT Engineering
---

# Free body diagram of masses between walls

Control Systems · Chapter 2: Component Modeling

- Asked in: 2066 Jestha (old course), 8 marks
- Video: [Mechanical Vibrations 8 - Newton 2 - Double Mass-spring-damper system](https://www.youtube.com/watch?v=NcvTrl_fzAM) — Jurnan Schilder
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/fbd-tf-m1-m2-k1-b
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-2

## Question

Draw free body diagram, write complete differential equations and find the transfer function X1(s)/F(s) for the dynamic system shown below. [Figure: wall – spring K1 – mass M1 (displacement x1, force f(t) applied) – damper B – mass M2 (displacement x2) – spring K2 – wall]

## Short answer

Equations: M1x1''+K1x1+B(x1'-x2')=f, M2x2''+K2x2+B(x2'-x1')=0. Transfer function X1(s)/F(s)=(M2s²+Bs+K2) / [M1M2s⁴+B(M1+M2)s³+(M1K2+M2K1)s²+B(K1+K2)s+K1K2].

## Answer

Displacements $x_1$ ($M_1$, force $f$) and $x_2$ ($M_2$); no ground friction. $K_1$ ties $M_1$ to the left wall, $B$ couples the masses and $K_2$ ties $M_2$ to the right wall.

#### Free body diagrams
```text
          M1                            M2
 K1x1  ┌──────┐ f(t)    B(x2'-x1') ┌──────┐
 ◄─────┤      ├──►      ◄──────────┤      │
 M1x1''│  M1  │         M2x2''     │  M2  │
 ◄─────┤      │         ◄──────────┤      │
 B(x1'-x2')   │         K2x2       │      │
 ◄─────┤      │         ◄──────────┤      │
       └──────┘                    └──────┘
```

#### Differential equations
$$
\begin{aligned}
M_1\ddot x_1 + K_1x_1 + B(\dot x_1 - \dot x_2) &= f(t) \\
M_2\ddot x_2 + K_2x_2 + B(\dot x_2 - \dot x_1) &= 0
\end{aligned}
$$

#### Laplace form
$$
\begin{aligned}
(M_1s^2 + Bs + K_1)X_1 - BsX_2 &= F \\
-BsX_1 + (M_2s^2 + Bs + K_2)X_2 &= 0
\end{aligned}
$$
From the second equation $X_2 = \dfrac{Bs}{M_2s^2 + Bs + K_2}X_1$. Substituting:
$$
\left[(M_1s^2 + Bs + K_1) - \frac{B^2s^2}{M_2s^2 + Bs + K_2}\right]X_1 = F
$$

#### Transfer function
$$
\frac{X_1(s)}{F(s)} = \frac{M_2s^2 + Bs + K_2}{(M_1s^2 + Bs + K_1)(M_2s^2 + Bs + K_2) - B^2s^2}
$$
Expanding the denominator:
$$
\frac{X_1(s)}{F(s)} = \frac{M_2s^2 + Bs + K_2}{M_1M_2s^4 + B(M_1 + M_2)s^3 + (M_1K_2 + M_2K_1)s^2 + B(K_1 + K_2)s + K_1K_2}
$$
Check: for a steady force ($s\to 0$), $X_1 = F/K_1$; the damper carries no static force, so $K_2$ does not share the load.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What couples M1 and M2 in this system?
   A. Spring K2
   B. Damper B
   C. Spring K1
   D. Friction with ground
   Answer: B. Only the damper B links the masses; K1 and K2 tie each mass to its own wall. (Key point: Coupling element)

2. Which equation describes M2?
   A. M2 x2'' + K2 x2 + B(x2' - x1') = 0
   B. M2 x2'' + K2 x2 = f
   C. M2 x2'' + B x2' = 0
   D. M2 x2'' + K2(x2 - x1) = 0
   Answer: A. M2 feels K2 and the damper force that depends on the relative velocity. (Key point: Equation for M2)

3. From the second Laplace equation, X2 in terms of X1 is
   A. B s / (M2 s^2 + B s + K2)
   B. K2 / (M2 s^2)
   C. B / (M2 s + K2)
   D. (M2 s^2 + B s + K2)/B s
   Answer: A. Setting -B s X1 + (M2 s^2 + B s + K2) X2 = 0 gives this ratio. (Key point: Elimination)

4. Under a steady force, why is X1 = F/K1?
   A. The damper carries no static force so K2 does not share the load
   B. K2 is zero
   C. K1 and K2 are in parallel
   D. M2 does not move
   Answer: A. At s = 0 the damper term vanishes, so only K1 resists F. (Key point: Steady-state check)

## Related in Control Systems

- [Free body diagram and X1/F1 of two-mass system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afbd-tf-m1-m2-k12-b12) — Control Systems, chapter 2
- [Free body diagram and X1/F, X2/F of two-mass system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afbd-m1-m2-k1k2-b1b2) — Control Systems, chapter 2
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