48

Physics Factsheet January 2003

Number 48

Damping and Resonance This Factsheet will explain: • the distiction between free and forced vibrations; • the nature of resonance; • the effects of damping

Resonance Free vibrations occur when a system is given energy and allowed to oscillate without any external forces being applied. The system will oscillate at its natural frequency

Free Vibrations

Forced vibrations occur when a system is oscillating because of some externally applied, oscillating force. The system will be made to vibrate with the same frequency as the external force.

Mechanical systems, no matter how complex or how simple, will vibrate, when they are set in motion. Vibrations are sometimes called oscillations. Examples of simple systems include a mass suspended on a spring, moving up and down or a pendulum swinging to and fro. More complex examples include the swaying of a tall building in the wind or the bouncing up and down of a car chassis on a suspension system.

Resonance occurs when the driving frequency of the externally applied force is equal to the natural frequency of the system being made to oscillate. The amplitude of the vibrations is a maximum.

When a mechanical system is allowed to vibrate freely, without any external forces being applied, it will oscillate at its natural frequency. These vibrations are called free vibrations.

Exam Hint: Questions on vibrations and resonance often start with simple explanations of free vibrations, forced vibrations and resonance. The three paragraphs in this last key concept section would be good explanations of these terms.

Forced Vibrations In some circumstances, an external force may act on a mechanical system. This force may have its own frequency of vibration, which may affect the motion of the mechanical system. An example of this might be a child being pushed on a swing by a parent. The driving force of the pushing of the parent is affecting the oscillations of the child on the swing.

Resonance, free and forced vibrations - Barton’s Pendulums

Since a driving force is forcing the system to vibrate at the frequency of the driving force, such vibration are called forced vibrations.

string connecting all pendulums

Resonance When a mechanical system is forced to oscillate by a driving force that has the same frequency as the natural frequency of the mechanical system, it will vibrate with maximum amplitude. This is called resonance. In practice, the closer the driving frequency is to the natural frequency of the object being forced to vibrate, then the greater its amplitude of vibration.

paper cone pendulums

This is shown on the graph below. The amplitude of the vibrations of the object being forced to oscillator have been plotted against the frequency of the driving frequency. The graph peaks when the driving frequency is equal to the natural frequency, f0, of the forced oscillator.

As the diagram shows the set up consists of a number of paper cone pendulums of varying lengths. All are suspended from the same string as a ‘driver’ pendulum that can act as a driving force when it is set in motion. The driver pendulum is much heavier, usually a metal weight.

Resonance occurs when driving frequency equals natural frequency

amplitude of oscillation

driver pendulum with heavy bob

When the driver pendulum is set going (this would be swinging to and fro perpendicular to plane of this paper) it forces all of the paper cone pendulums to oscillate with the same frequency but with different amplitudes. This is a case of forced oscillation. The paper cone pendulum that has the same length as the driver pendulum has the greatest amplitude as it has the same natural frequency as the driver pendulum. This is an example of resonance.

f0

Careful observation shows that • a resonant pendulum is always a quarter cycle behind the driver pendulum. • the shorter pendulums are almost in phase • the longer pendulums are almost in antiphase (180o out of phase) with the driver pendulum.

driving frequency

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Physics Factsheet

Damping and Resonance

Heavy damping may stop a system from oscillating altogether. Instead, when the system is displaced, it simply returns back to its equilibrium position without moving past the equilibrium position. A system that returns back to its equilibrium position in the shortest possible time is said to be critically damped. A graph of displacement against time for a critically damped oscillator would look like the following:

Typical Exam Question The diagram below shows four pendulums, all suspended from the same string. Pendulum 1 is much heavier than the other pendulums and is made to oscillate perpendicular to the plane of the paper.

displacement

time (a) By completing the table below, describe the motion of the pendulums after pendulum 1 is set in motion. [3] Frequency compared to to frequency of A

It is useful for oscillators to be critically damped. A typical example used in exam questons is the suspension system of a car, so that the ride is not too ‘bouncy’ and uncomfortable. Swinging fire doors are also critically damped to prevent further accidents as people pass through them in a hurry.

Amplitude

Pendulum 2 Pendulum 3

Critical damping When a damped oscillator returns back to the equilibrium position in the shortest possible time, without crossing it, it is said to be critically damped.

Pendulum 4 (b) State what is meant by the term resonance. How is it demonstrated by this experiment? [3]

Typical Exam Question One of the reasons that earthquakes can cause such devastation is that they can act as a driving frequency for high rise buildings. The natural period of oscillation for a multi-storey building is 0.10 multiplied by the number of storeys. (a) A particular earthquake has a frequency of 2.0 Hz. How many storeys would a building have to have so that it is in resonance with this earthquake? [2] (b) Sketch a graph of displacement against time to show how an undamaged building in resonance would oscillate immediately after the earthquake had finished. [3] (c) Why might a taller building be less likely to suffer damage? [2] (d) Why might two high rise buildings that are built close to each other, but with a different number of storeys, be more likely to suffer damage than if they were the same height? [2]

(a) Pendulum 1 acts as a driving force for the other, much lighter pendulums, and so all the other pendulums will be forced to oscillate at the same frequency as pendulum 1. Pendulum 3 is the same length as pendulum 1 and so they will share the same natural frequency. This means that pendulum 3 is in resonance and will oscillate with the largest amplitude. Frequency compared to to frequency of A

Amplitude

Pendulum 2

same as pendulum

1 small

Pendulum 3

same as pendulum

1 large

Pendulum 4

same as pendulum

1 small

Answer (a) If we calculate the period of oscillation of this earthquake, we can use the relationship given in the question to calculate the resonant number of storeys. Period of oscillation of earthquake = 1/f = 1/2 = 0.5s Resonant number of storeys = 0.5/0.1 = 5 storeys (b) This graph should show that the period of oscillation is constant at 0.5 seconds as already calculated, and also that the amplitude of oscillation will decrease over time as the oscillations will be damped once the driving frequency has stopped. The oscillations would also be smooth.

(b) Resonance occurs when the driving frequency of the externally applied force is equal to the natural frequency of the system being made to oscillate. The amplitude of the vibrations is a maximum. This experiment demonstrates resonance as pendulum 1 is forcing pendulum 3 to oscillate at its natural frequency. Pendulum 3 therefore has the largest amplitude of vibration.

displacement

Oscillations and Damping The amplitude of all oscillations also depends on how much the system is damped. Damping is when an oscillator experiences a resistive force meaning that the oscillation gradually dies away. A graph of displacement against time for an oscillator that only experiences light damping or a small resistive force looks as follows: displacement

T

2T

3T

0.5

1.0

1.5

time

(c) A taller building will have a natural frequency that is different to the driving frequency of the earthquake. It will not be in resonance and so will not have as large an amplitude of oscillation. (d) Each of the two buildings will have a different natural frequency. There are now two possible resonant frequencies. If one building is in resonance it may fall onto the other building if it is built close by.

time

Although the amplitude decreases with time, the frequency and time period of the oscillator remain constant.

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Physics Factsheet

Damping and Resonance Exam Hint: Questions on vibrations and resonance often involve calculating a periodic time when you have been given a frequency and vice versa. This can be done with the simple relationship f = frequency (Hz) f = T1 T = periodic time (s)

Exam Workshop This is a typical weak student’s answer to an exam question. The comments explain what is wrong with the answers and how they can be improved. The examiner’s answer is given below. (a) (i) State what is meant by a free vibration. [1] How an object vibrates on its own. 0/1

amplitude of forced oscillations

Damping and Resonance The level of damping on a system also has an effect on the magnitude of the amplitude at the resonant frequency of a system. This is shown in the graph below.

Although the candidate has an idea of what free vibration is, there is no reference to the absence of an external force acting. (ii) State what is meant by a forced vibration. When an object is forced to vibrate by another object. Reference needs to be made to the frequency of oscillation to secure full marks.

light damping

(b) A suspension system of a particular car oscillates with a natural frequency of oscillation of 2.0Hz. What is the natural period of oscillation of the car? [2] T = 1/2 = 0.5 ! ! Although the calculation is correct and the correct answer has been determined marks would be lost as no units have been quoted and the final answer has only been given to one significant figure when two significant figures have been quoted in data in the question.

moderate damping heavy damping

resonant frequency

frequency of driving force

(c) The same car is travelling at a speed of 24 m/s along a road with speed bumps placed 12 m apart. (i) Calculate the time of travel between bumps. [2] ! time = distance/velocity = 12/24 = 0.5 s ! 2/2

Damping and resonance Heavier damping leads to a smaller amplitude of oscillation at the resonant frequency. Damping also makes the resonant peak broader.

(ii) How will this answer effect the suspension system of the car? [2] The periods of oscillation are the same. ?/2 No attempt has been made to explain the consequences of the two periods being the same and how this will effect the oscillations of the car.

Qualitative (Concept) Test 1. 2. 3. 4. 5. 6.

What are free vibrations? What are forced vibrations? What is resonance? What is damping? How does damping effect the amplitude and the period of oscillation? How does damping effect the resonance of a forced oscillator?

(d) The car manufacturer has designed the suspension system to decrease the oscillations of the car by critically damping the suspension system. (i) What is meant by critical damping? [2] The car will not oscillate very much. 0/2

Quantitative (Calculation) Test There are very few calculations associated with this topic and testing tends to concentrate around the qualitative understanding of the topic. The majority of calculations are concerned with determining frequency from periodic time and vice versa. 1. What is the periodic time of an oscillator that has a natural frequency of 2.5Hz? 2. What is the natural frequency of an oscillator with a natural periodic time of 0.0040s?

This answer lacks the detail required about how the car oscillates and the time that it oscillates for. (ii) Give two reasons how this minimises the oscillations of the suspension system of the car travelling down this road. [2] The amplitude of the oscillations are less.

Examiner’s Answer (a) (i) A free vibration is when a system oscillates at its natural frequency and no external forces act.! (ii) A forced vibration is when a system is forced to vibrate by an external driving force.! The system will oscillate at the same frequency as the driving force.! (b) (i) T = 1/f = 1/2 = 0.50s ! ! (c) (i) time = distance / velocity = 12/24 = 0.50s ! (ii) The car will be in resonance(since the periods are equal).! It will oscillate with maximum amplitude.! (d) (i) This is when a damped oscillator returns back to its equilibrium position !without crossing the equilibrium position.! in the shortest possible time (ii) Heavier damping decreases the amplitude of vibrations at resonance.! ! The car returns to its equilibrium position in the shortest possible time, making the ride far less bumpy than if the motion was undamped.

Answers can be found in the text

Quantitative Test Answers

2. frequency =

1 1 = = 0.40s frequency 2.5

1 1 = = 250Hz periodic time 0.0025

Acknowledgements: This Physics Factsheet was researched and written by Jason Slack. The Curriculum Press,Unit 305B, The Big Peg,120 Vyse Street, Birmingham, B18 6NF. Physics Factsheets may be copied free of charge by teaching staff or students, provided that their school is a registered subscriber. No part of these Factsheets may be reproduced, stored in a retrieval system, or transmitted, in any other form or by any other means, without the prior permission of the publisher. ISSN 1351-5136

0/2

No detail or reason has been given to explain this answer.

Qualitative Test Answers

1. Periodic time =

[2] 1/2

3