Forced vibrations, damping and resonance

Part of a machine to generate electrical energy using waves is a cylindrical drum whose cross-section is shown below. The height of the drum is 2.0 m and its cross-sectional area is 0.79 m².

The mass of the drum is 1200 kg.

The centre of mass of the drum is 0.50 m from one of the circular faces, on a line joining the centres of the circular faces.

The density of water is 1000 kg m^–3.

The drum floats in flat water with its long axis vertical. The base of the drum is a distance d below the surface, where d is measured in m.

Explain why the weight of water displaced must be equal to the weight of the drum.

Calculate d.

Upthrust forces act at the centre of mass of the fluid displaced.

Explain why the drum cannot float in equilibrium on its side with the long axis horizontal.

When the vertical drum is displaced a distance x deeper into the water from the equilibrium position, the magnitude of the resultant force on the drum, F, is given by

F = –kx

where k is a constant and d is measured in m.

By considering the magnitude of the additional upthrust on the drum, show that k is about 8000 Nm^–1.

When the drum is released, it oscillates vertically. Explain why the oscillations are simple harmonic motion.

Calculate the natural frequency f of the oscillations.

* Two designs, A and B, of the machine are tested in a large ripple tank.

The ripple tank can produce waves whose amplitude and frequency can be varied over wide ranges.

The natural frequencies of the two machines are identical. The graphs show the results of the tests but are incomplete.

• Describe how the amplitude of the machine could be measured at each frequency tested. You do not need to describe how to vary the amplitude or frequency of the waves.

• Complete the graphs with suitable labels and annotations.

• Discuss which of the designs is the most effective at generating electrical energy.

Additional space if required

The length of an unloaded spring is approximately 4 cm.

The force constant k of the spring is 0.62 N cm^–1.

Describe how you could determine k using an appropriate experiment.

Calculate the magnitude of the block’s initial acceleration at the instant that the force F is removed.

Assume that the spring is not extended beyond its limit of proportionality.

acceleration =........ m s^–2

The block now moves with simple harmonic motion.

Calculate the frequency of this motion.

The block is replaced by a strong magnet L of slightly greater mass.

The oscillation frequency of this new arrangement is 2.5 Hz.

The magnet L is placed inside a coil X of insulated copper wire.

The coil X is connected with long wires to a second, identical coil Y.

A second strong magnet M is placed inside Y and attached to a vibration generator.

The vibration generator is then forced to oscillate with a frequency of approximately 2.5 Hz by

As magnet M oscillates, it moves in and out of coil Y.

........

The frequency of the vibration generator is now varied between 0.5 Hz and 5.0 Hz.

Suggest how the amplitude and frequency of the oscillations of L will change as the frequency of the generator is varied.

You may draw a diagram to support your answer.

An oscillator is forced to oscillate at different frequencies. The graph of amplitude A against driving frequency f for this oscillator is shown.

The damping on the oscillator is now decreased.

Which of the following statements is/are correct?

1 The amplitude of the oscillations at any frequency decreases. 2 The maximum amplitude occurs at a lower frequency. 3 The peak on the graph becomes thinner.

A Only 1

B Only 2

C Only 3

D 1, 2 and 3

Your answer

Use the defining equation for simple harmonic motion to show that the natural frequency f₀ of the mass-spring system is about 1.4 Hz.

Calculate the amplitude A of the oscillations.

The vibrator frequency is varied from 0 Hz to 2.5 Hz. On Fig. 18.3, sketch a graph to show the variation with vibrator frequency of the amplitude of the mass. Label your graph K.

A light disc is now attached to the mass to increase the damping. The vibrator frequency is again varied from 0 Hz to 2.5 Hz. Sketch a second graph on Fig. 18.3 to show the new variation of the amplitude.

Label this graph D.

Explain why the phenomenon demonstrated in this experiment can cause problems for engineers when designing suspended footbridges.

The 500 m tall Taipei 101 tower is shown in Fig. 2.1. The tower has a massive sphere suspended across five floors near the top of the building to dampen down movement of the tower in high winds and earthquakes. The sphere is connected to pistons (not shown) which drive oil through small holes providing damping. The vibration energy of the sphere is converted to thermal energy.

Fig. 2.2 models the damper system as the sphere held between two springs. The movement of the walls of the tower forces the sphere to oscillate in simple harmonic motion.

In the strongest wind, the natural frequency of the oscillations of the tower is 0.15 Hz and the maximum acceleration of the sphere is 0.050 m s⁻².

Calculate the maximum displacement of the sphere in the strongest wind.

Explain why the natural frequency of the damper system must be about 0.15 Hz.

The acceleration a of the sphere is given by the equation

where k is the force constant of the spring combination, x is the displacement of the sphere and m is the mass of the sphere.

The mass of the sphere is 6.6 × 10⁵ kg. The natural frequency of the oscillations of the sphere is 0.15 Hz.

The S-wave of an earthquake causes a sudden movement of the building displacing the sphere 0.71 m from its equilibrium position relative to the building.

Use your answer in (i) to calculate the energy transferred to the springs of the damper system.

This question is about a simple pendulum made from a length of string attached to a mass (bob). For oscillations of small amplitude, the acceleration a of the pendulum bob is related to its displacement x by the expression

Show that the period T of the oscillations is given by the expression

A student notices that the amplitude of each oscillation decreases over time. Explain this observation and state what effect this may have on T.

* Describe with the aid of a labelled diagram how an experiment can be conducted and how

amplitude.

Another student conducts a similar experiment in the laboratory to investigate the small amplitude oscillations of a pendulum of a mechanical clock. Each ‘tick’ of the clock corresponds to half a period.

Show that the length of the pendulum required for a tick of 1.0 s is about 1 m.

If the pendulum clock were to be used on the Moon, explain whether this clock would run on time compared with an identical clock on the Earth.