---
title: "Transfer function from symbolic mass-spring-damper state model"
subject: "Control Systems"
chapter: 8
key: "semester_3/control_systems:ss-tf-mbk-symbolic"
source: https://www.bctengineer.com/q/semester_3/control_systems/ss-tf-mbk-symbolic
site: BCT Engineering
---

# Transfer function from symbolic mass-spring-damper state model

Control Systems · Chapter 8: State Space Analysis

- Asked in: 2076 Chaitra, 4 marks
- Video: [ME 357 09 L Converting State Space to Transfer Function Form 1DOF System](https://www.youtube.com/watch?v=VF7IE40cc_U) — Joseph Mahoney
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/ss-tf-mbk-symbolic
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-8

## Question

Obtain the transfer function from the following system equations given by [ẋ1; ẋ2] = [0 1; −K/M −B/M][x1; x2] + [0; 1/M]u, y = [1 0][x1; x2].

## Short answer

Apply the transfer function formula C(sI-A)^{-1}B with the given state matrices. The result is Y(s)/U(s) = 1/(Ms^2 + Bs + K).

## Answer

$$
\frac{Y(s)}{U(s)} = C(sI - A)^{-1}B
$$

with

$$
A = \begin{bmatrix} 0 & 1 \\ -\dfrac{K}{M} & -\dfrac{B}{M} \end{bmatrix},\quad B_m = \begin{bmatrix} 0 \\ \dfrac{1}{M} \end{bmatrix},\quad C = \begin{bmatrix} 1 & 0 \end{bmatrix}
$$

(The input matrix is written $B_m$ to avoid confusion with the damping coefficient $B$.)

#### Step 1: $(sI - A)$ and determinant

$$
sI - A = \begin{bmatrix} s & -1 \\ \dfrac{K}{M} & s + \dfrac{B}{M} \end{bmatrix}
$$

$$
|sI - A| = s\left(s + \frac{B}{M}\right) + \frac{K}{M} = s^2 + \frac{B}{M}s + \frac{K}{M}
$$

#### Step 2: Inverse

$$
(sI - A)^{-1} = \frac{1}{s^2 + \frac{B}{M}s + \frac{K}{M}}\begin{bmatrix} s + \dfrac{B}{M} & 1 \\ -\dfrac{K}{M} & s \end{bmatrix}
$$

#### Step 3: Transfer function

$$
(sI-A)^{-1}B_m = \frac{1}{s^2 + \frac{B}{M}s + \frac{K}{M}}\begin{bmatrix} \dfrac{1}{M} \\ \dfrac{s}{M} \end{bmatrix}
$$

$$
\frac{Y(s)}{U(s)} = \begin{bmatrix} 1 & 0 \end{bmatrix}(sI-A)^{-1}B_m = \frac{1/M}{s^2 + \frac{B}{M}s + \frac{K}{M}} = \frac{1}{Ms^2 + Bs + K}
$$

This is the transfer function of the mass–spring–damper system $M\ddot{y} + B\dot{y} + Ky = u$, with $\omega_n = \sqrt{K/M}$ and $\zeta = \dfrac{B}{2\sqrt{KM}}$.

**Answer:** $\dfrac{Y(s)}{U(s)} = \dfrac{1}{Ms^2 + Bs + K}$.

## Check yourself

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

1. What is |sI - A| for A = [0 1; -K/M -B/M]?
   A. s^2 - (B/M)s + K/M
   B. s^2 + (B/M)s + K/M
   C. s^2 + (K/M)s + B/M
   D. s^2 + BMs + K
   Answer: B. s(s + B/M) + K/M. (Key point: Determinant)

2. What is Y/U?
   A. M/(Ms^2 + Bs + K)
   B. s/(Ms^2 + Bs + K)
   C. 1/(Ms^2 + Bs + K)
   D. 1/(s^2 + Bs + K)
   Answer: C. (1/M)/(s^2 + (B/M)s + K/M) simplifies to 1/(Ms^2 + Bs + K). (Key point: Final result)

3. Why is the input matrix called B_m?
   A. Because it is a mechanical matrix with different rules
   B. Because it is multiplied by M
   C. Because B is already used for the output
   D. To avoid confusion with the damping coefficient B
   Answer: D. The letter B is used both for the damper and the input matrix. (Key point: Why B_m)

4. What is the damping ratio of this system?
   A. B/(2 sqrt(KM))
   B. B/(2KM)
   C. 2 sqrt(KM)/B
   D. sqrt(K/M)
   Answer: A. Comparing with s^2 + 2 zeta wn s + wn^2 gives wn = sqrt(K/M) and zeta = B/(2 sqrt(KM)). (Key point: Damping ratio)

## Related in Control Systems

- [Vertical two-mass system transfer function X1/F](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-vertical-m1-m2-k1-b1) — Control Systems, chapter 2
- [State model of a mass-spring-damper system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-model-mass-spring-damper) — Control Systems, chapter 8
- [Two-mass system X1/F, X2/F with F-V analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afv-tf-m1-m2-k1b1-k2b2) — Control Systems, chapter 2, asked 3×
- [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
- [Two-mass system equations, X2/F, F-I and F-V analogies](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ade-tf-fi-fv-m1-m2) — Control Systems, chapter 2
- [State model of two-mass spring system, output x2](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-m1-m2-k1-k2-x2) — Control Systems, chapter 8

## Related in other subjects

- [Spring-mass-damper instrument parameters](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Aspring-mass-damper-instrument-numerical) — Instrumentation and Measurement, chapter 4