# Driven Pendulums

• L. P. Pook
Chapter
Part of the History of Mechanism and Machine Science book series (HMMS, volume 12)

## Abstract

One of the functions of the escapement in a pendulum clock is to transfer energy to the pendulum in order to replace energy lost due to friction etc. It does this by impulses whose timing is determined by the pendulum. By contrast, in a driven pendulum, energy is transferred to a pendulum by periodic forces whose period is not controlled by the pendulum. A driven pendulum is sometimes called a forced pendulum. There are three distinct ways in which a simple rod pendulum can be driven. In rotary driving the pendulum is subjected to prescribed varying torques about the suspension point. Theoretical investigation of a damped simple rod pendulum under sinusoidally varying torque has shown that a wide range of chaotic behaviour sometimes occurs, especially for large amplitudes and low values of the pendulum quality, Q. Experimental results, using a specially designed real pendulum with an electromagnetic drive have confirmed the existence of both periodic and chaotic behaviour. A simple rod pendulum can be started from the rest position by using rotary driving. A simple string pendulum cannot be driven by rotary driving. In horizontal driving the suspension point of a simple rod pendulum is subjected to prescribed varying horizontal displacements. In vertical driving, sometimes called lifting, the suspension point is subjected to prescribed varying vertical displacements. In both cases there are two forces on the simple rod pendulum. One is that due to gravity on the point mass, m, and is constant. The other is that needed to impose the prescribed force or displacement, and this force varies with time. A simple rod pendulum can be started from the rest position by using horizontal driving, but it cannot be started from the rest position by using vertical driving. A simple string pendulum can be driven by horizontal driving, and by vertical driving. A prescribed force or a prescribed displacement can be either periodic, such as a sine wave or random. In the latter case random process theory is needed for understanding the effects of driving. Driven damped simple harmonic motion provides a useful approximation to some aspects of the behaviour of driven pendulums. Some aspects of horizontal driving are described.

## Keywords

Root Mean Square Chaotic Behaviour Significant Wave Height Spectral Density Function Rest Position
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Baker GL, Gollub JP (1996) Chaotic dynamics. An introduction, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
2. Bateman D (1977) Vibration theory and clocks. Part 2. Forced harmonic motion. Horological J 120(2):48–52Google Scholar
3. Bendat JS, Piersol AG (2000) Random data: analysis and measurement procedures, 3rd edn. Wiley, New York
4. Bishop SR, Xu DL, Clifford J (1996) Flexible control of the parametrically excited pendulum. Proc R Soc Lond 452:1789–1806
5. Butikov EI (2008) Extraordinary oscillations of an ordinary forced pendulum. Eur J Phys 29(2):215–233
6. D’Humieres D, Beasey MR, Huberman BA, Libchaber A (1982) Chaotic pendulum states and routes to chaos in the forced pendulum. Phys Rev A 26(6):3483–3496
7. Frost NE, Marsh KJ, Pook LP (1974) Metal fatigue. Clarendon, OxfordGoogle Scholar
8. Jurriaanse D (1987) The practical pendulum book, with instructions for use and thirty-eight pendulum charts. The Aquarium Press, WellingboroughGoogle Scholar
9. Lamb H (1923) Dynamics, 2nd edn. Cambridge University Press, Cambridge
10. Macbeth N (1943) About pendulums. Pendulum play. A scientific pastime needing no knowledge of science etc. Michael Houghton, LondonGoogle Scholar
11. Papoulis A (1965) Probability, random variables, and stochastic processes. McGraw-Hill Book Company, New York
12. Pippard AB (1988) The parametrically maintained Foucault pendulum and its perturbations. Proc R Soc Lond A 420:81–91
13. Pook LP (1978) An approach to practical load histories for fatigue testing relevant to offshore structures. J Soc Env Eng 17–1(76): 22–23, 25–28, 31–35Google Scholar
14. Pook LP (1984) Approximation of two parameter Weibull distributions by Rayleigh distributions for fatigue testing. NEL Report 694. National Engineering Laboratory, East KilbrideGoogle Scholar
15. Pook LP (1987) Random load fatigue and r. m. s. NEL Report 711. National Engineering Laboratory, East KilbrideGoogle Scholar
16. Pook LP (1989) Spectral density functions and the development of Wave Action Standard History (WASH) load histories. Int J Fatigue 11(4):221–232
17. Pook LP (2007) Metal Fatigue: What it is, why it matters. Springer, Dordrecht
18. Rõssler OE, Stewart HB, Wiesenfeld K (1990) Unfolding a chaotic bifurcation. Proc R Soc Lond 431(1882):371–383Google Scholar
19. Tritton DJ (1986) Ordered and chaotic motion of a forced spherical pendulum. Eur J Phys 7(3):162–169
20. Tritton DJ (1992) Chaos in the swing of a pendulum. In: Hall N (ed) The new scientist guide to chaos. Penguin, London, pp 22–32Google Scholar