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Forced vibrations and resonance
Resonance
Forces and Motion
Forced vibrations and resonance
for 14-16
Forced vibrations and resonance
Class experiment
Something like a blade can oscillate on its own at a definite frequency. Here students investigate what happens when it is driven by an oscillating force at a rate which may or may not be the same as the natural blade frequency.
Apparatus and Materials
- Hacksaw blade oscillator
- Hacksaw blade
- Meccano strips (No. 1 and No. 2a) (if not with item 1024)
- Stopwatch or stopclock
- G-clamp (large) The following items should also be available, if required:
- Metre rule
- Postcard, cork, needle, rubber band
- Retort stand base, rod, boss, and clamp 2
Health & Safety and Technical Notes
Assemble the oscillator as shown in the diagram. You may need to substitute Meccano parts and the hacksaw clamping device with something similar.
The amplitude of the motion of the driver pendulum can be maintained by gently tapping the pendulum strip a little below its support with one finger.
Read our standard health & safety guidance
Procedure
- Investigate factors affecting this oscillating system, consisting of a hacksaw blade driven by the heavy pendulum.
- Before coupling the blade to the 1 kg pendulum, find out how to change the natural frequency of each of them.
- Try changing the degree of coupling in the system, by using different rubber bands (or springs).
- Damping may be changed by turning the postcard so that it is at right angles to the direction of motion of the blade.
Teaching Notes
- Students could be given a very loose brief, to investigate how this system behaves. An open brief can stimulate students to think for themselves and to closely observe what happens in this oscillating system.
- Alternatively, students could be divided into pairs or small groups, with each one investigating a specific factor, for example how the system responds with standard coupling, with different degrees of coupling or with damping. In each of these cases, they might collect data showing how the amplitude of hacksaw blade oscillations varies across a range of driving frequencies, concluding by drawing a resonance curve.
- Whatever approach you use, set up one sample apparatus beforehand so that students do not waste time trying to figure out how to set the apparatus up. You may also want to suggest that they observe not only blade oscillation amplitudes but also any phase difference between the blade and pendulum oscillations.
- The natural frequency of the hacksaw on its own can be found by displacing it from its equilibrium position and releasing it to oscillate freely. The frequency is determined by counting, say, the time for 10 complete oscillations. Students should find that the natural frequency of the hacksaw blade depends on its length. A similar procedure can be used to find the natural frequency for the 1 kg mass, which will depend on the length from pivot to the centre of the 1kg mass.
- These investigations provide a good opportunity to practice what may be new vocabulary, using terms such as ‘frequency’, ‘amplitude’, ‘phase’ and ‘resonance’.
- Adequate time is essential. A single long practical session may not be enough.
This experiment has yet to undergo a health and safety check.
Up next
Resonance of a pendulum
Class experiment
In this experiment the amplitude of forced vibrations is measured as the driving frequency varies. It could be used as part of a circus of experiments for students.
Apparatus and Materials
- Resonance kit, as shown in the diagram
- Metre rule
- Retort stand base, rod, boss, and clamp
- Stopclock
- Card
- Adhesive tape
Health & Safety and Technical Notes
Set the apparatus up in advance. The two strips are lock-nutted together about 20 cm from the top of the vertical strip. The solder is bent as shown, so that it is supported by the horizontal strip but free to swing. (Coiled solder is easily bent onto the required shape.) The horizontal strip rests on the open jaws of a clamp, which acts as a pivot.
Read our standard health & safety guidance
Procedure
- The pendulum of solder is the driven oscillator. The driving force is provided by the horizontal strip. Its frequency can be varied by moving the adjustable mass up or down. The amplitude of vibration can be measured using the mm scale fixed on the bench.
- A damping vane made of paper may be attached to the solder with adhesive tape.
- Vary the frequency of the driving force while measuring the amplitude of vibration of the V-shaped solder. Keep the oscillations of the driver gentle.
- Plot a graph of amplitude of the forced vibrations v driving force.
- Fix the card to the solder and then repeat the same experiment.
Teaching Notes
- It should be possible for students to measure the amplitude of forced vibrations over a range of frequencies for both lightly damped and heavily damped vibrations. In systems that are too lightly damped, transient vibrations are persistent and can cause confusion.
- Students can also be asked to closely observe the phase relationship between the oscillations of the driver and the solder, across the range of driving frequencies, with and without damping.
- Adequate time is essential. A single long practical session may not be enough.
This experiment has yet to undergo a health and safety check.
Up next
Energy in forced oscillation
During each cycle of its oscillation the driver transfers some energy to the driven oscillator. Some of this energy may be returned to the driver later in the cycle (for example, a spring which has been forced to stretch can pull back, helping the driver return in its oscillation) but, at resonance, the phase difference p /2 prevents this. Some energy is dissipated in overcoming damping. The rest is stored in the forced oscillator, increasing its amplitude.
The amounts of energy lost to damping in each cycle increases with the amplitude of the oscillation. The final amplitude of oscillation is that for which the energy transferred by the driver in each cycle is equal to the energy used to overcome damping. With heavy damping this happens at small amplitude, and even the resonant oscillations are not violent. With light damping large amplitudes of the driven oscillator are achieved. A lot of energy transferred from the driver is then stored in the driven oscillator, particularly at resonance.
The quality factor, Q
The quality factor, Q, of an oscillator can be formally defined like this:
Q = 2π × energy storedenergy dissipated per cycle
However, there is a much more useful, though non-rigorous, description of Q: it is approximately equal to the number of free oscillations which occur before all the oscillator's energy is dissipated.
Q is related to the degree of damping of the oscillator, and to the sharpness of its resonance peak. Low values of Q are associated with heavily damped oscillations which do not resonate violently and which die away quickly if they are not forced. High values of Q are associated with light damping and sharp resonance.
Some typical values of Q are:
| Car suspension | 1 |
| Tethered trolley | 10 |
| Simple pendulum | 1,000 |
| Guitar string | 1,000 |
| Quartz crystal of watch | 105 |
| Excited atom | 107 |
| Excited nucleus | 1,012 |
Consider the guitar string, for example. The sound waves with a fundamental frequency of, say, 512 Hz (the C above middle C) transfer energy. If Q=1000, then roughly 1000 oscillations occur before all the energy is dissipated. Thus the plucked string will cease to oscillate after 1000/512 approx. 2 seconds: which agrees roughly with experience.
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Resonance
Resonance has wide ranging practical applications. Any machine or structure is likely to be subjected to periodic forces, either as a result of its own operation (e.g., the motor in any vehicle imposes an oscillation or vibration on every part of the vehicle) or through the action of some external agent (e.g., wind exerts a periodic force on buildings and structures through vortex shedding). If you keep your eyes and ears open you will notice countless examples of forced oscillations.
Forced oscillations can prevent machines operating efficiently, as when an unevenly loaded spin drier cannot achieve its normal working speed because much of its energy is being diverted into a violent wobbling. More seriously, forced oscillations can result in fatigue failure of metal components at stresses well below the tensile strength of the metal, simply as a result of repeated flexing (like breaking a piece of wire by bending it to and fro). If resonance occurs, forced oscillations can be violent and may have catastrophic results (as in the Tacoma Narrows bridge collapse). An understanding of forced oscillations is clearly essential to engineering.
Forced oscillations are not always destructive; sometimes engineers and scientists can make positive use of them. Nor is the phenomenon confined to mechanical oscillations. Microwave ovens heat food as a result of a forced oscillation of the molecules within the food, particularly water molecules, which are polar (they are permanently charged positive at one end and negative at the other, see figure D21). Infra-red absorption spectroscopy, which is an important technique for chemists, involves the forced oscillation of atoms or groups of atoms within a molecule. The conversion of radio waves to electric currents in an aerial is an example of a forced oscillation, and the operation of a tuning circuit in a radio relies on resonance.