---
title: "Transmissibility and isolator design"
subject: "Theory of Mechanism and Machine II"
chapter: 6
key: "semester_7/theory_of_mechanism_and_machine_ii:vibration-isolation-transmissibility"
source: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/vibration-isolation-transmissibility
site: BCT Engineering
---

# Transmissibility and isolator design

Theory of Mechanism and Machine II · Chapter 6: Vibration of Single Degree of Freedom Systems

- Asked in: year not given, 3+5 marks
- Video: [Vibration Isolation And Transmissibility | Theory Of Machine](https://www.youtube.com/watch?v=vRahpSpgyMA) — LEARN AND GROW
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/vibration-isolation-transmissibility
- Whole chapter: https://www.bctengineer.com/md/semester_7/theory_of_mechanism_and_machine_ii/chapter-6

## Question

(a) Define transmissibility and derive an expression for the force transmissibility of a damped spring-mass system under a harmonic exciting force. State the frequency ratio above which isolation begins.
(b) A machine of mass 400 kg is mounted on undamped springs and has a vertical exciting force of frequency 1200 rpm. The transmitted force must not exceed 20% of the exciting force. Find the required stiffness of the springs and the static deflection.

## Short answer

Transmissibility is the ratio of transmitted force to applied force. Its expression is derived from damped forced vibration and isolation begins when frequency ratio r > √2. Required total spring stiffness is 1053 kN/m, giving a static deflection of 3.73 mm.

## Answer

#### (a) Transmissibility
**Transmissibility** $T_R$ is the ratio of the amplitude of the force transmitted to the foundation to the amplitude of the force applied on the machine (force transmissibility). The same ratio applies to motion (displacement transmissibility) for base excitation.

For $m\ddot{x} + c\dot{x} + kx = F_0\sin\omega t$, the steady-state amplitude is

$$
X = \frac{F_0/k}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}, \qquad r = \frac{\omega}{\omega_n}
$$

The force reaching the foundation is the sum of spring and damper forces (in quadrature):

$$
F_T = X\sqrt{k^2 + (c\omega)^2} = kX\sqrt{1 + (2\zeta r)^2}
$$

$$
T_R = \frac{F_T}{F_0} = \sqrt{\frac{1 + (2\zeta r)^2}{(1-r^2)^2 + (2\zeta r)^2}}
$$

Key points:
- $T_R > 1$ for $r < \sqrt2$ (the foundation sees more force than applied).
- $T_R = 1$ at $r = \sqrt2$ for every damping ratio.
- Isolation ($T_R < 1$) begins only for $r > \sqrt2$. Above this, more damping increases transmissibility, so light damping is best, while some damping is needed to limit the peak when passing resonance.

#### (b) Isolator design (undamped)
For $\zeta = 0$:
$$
T_R = \frac{1}{r^2 - 1} \;(r > \sqrt{2})
$$

$$
0.2 = \frac{1}{r^2 - 1} \Rightarrow r^2 = 6 \Rightarrow r = 2.4495
$$

$$
\omega = \frac{2\pi \times 1200}{60} = 125.66\ \text{rad/s}, \qquad \omega_n = \frac{\omega}{r} = 51.30\ \text{rad/s}
$$

$$
k = m\omega_n^2 = 400 \times 51.30^2 = 1052758\ \text{N/m} = 1052.8\ \text{kN/m}
$$

$$
\delta_{st} = \frac{mg}{k} = \frac{g}{\omega_n^2} = 0.00373\ \text{m} = 3.73\ \text{mm}
$$

Isolation efficiency $= 1 - T_R = 80\%$.

**Answer:** Spring stiffness $\approx 1052.8$ kN/m (total, all springs); static deflection $\approx 3.73$ mm.

## Check yourself

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

1. Force transmissibility is the ratio of:
   A. force transmitted to the foundation to force applied on the machine
   B. applied force to transmitted force
   C. machine displacement to static deflection
   D. damping force to spring force
   Answer: A. T_R = F_T / F0, the amplitude of force reaching the foundation over the amplitude of the exciting force. (Key point: Definition of transmissibility)

2. Isolation (T_R < 1) begins only when the frequency ratio r exceeds:
   A. 1
   B. sqrt(2)
   C. 2
   D. 0.707
   Answer: B. T_R = 1 at r = sqrt(2) for every damping ratio; above this T_R < 1. (Key point: Isolation threshold)

3. In the isolation region (r > sqrt(2)), increasing the damping ratio:
   A. makes the isolation better
   B. has no effect on transmissibility
   C. makes the system resonate
   D. increases transmissibility, so light damping is best
   Answer: D. Above r = sqrt(2) more damping raises T_R, but some damping is needed to limit the peak when passing through resonance. (Key point: Effect of damping above sqrt(2))

4. To keep T_R = 0.2 for an undamped system, use T_R = 1/(r^2 - 1). The frequency ratio required is:
   A. r = 2.449 (r^2 = 6)
   B. r = 1.5 (r^2 = 2.25)
   C. r = 4 (r^2 = 16)
   D. r = 1.095 (r^2 = 1.2)
   Answer: A. 0.2 = 1/(r^2 - 1) gives r^2 = 6, r = 2.4495, so wn = 125.66/2.4495 = 51.3 rad/s and k = m wn^2 about 1052.8 kN/m. (Key point: Design of undamped isolator)

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