Understanding Pendulums pp 107-127 | Cite as

# Entertainment

## Abstract

Pendulums are an essential part of some structures used for entertainment. Three of these uses are described. These are child’s swings, a child’s rocking horse, and pendulum harmonographs. Child’s swings are ubiquitous in both public playgrounds and private gardens. One of the attractions is that pumping is possible so they can be ridden without external assistance. Details vary widely. One form consists of a horse mounted on two wooden bars. These are suspended by pivoted metal links from a fixed wooden bar, which is at the top of the supporting frame. The attraction of the rocking horse is that it can be pumped by the rider to give an exhilarating ride. The rocking horse is, in essence, a four bar linkage with a coupler. Lissajous figures are plane curves produced when two orthogonal simple harmonic motions are combined. This can be done either theoretically or experimentally. They are named after Jules Antoine Lissajous, who demonstrated them experimentally in 1857. He used an optical method involving mirrors mounted on two tuning forks arranged at right angles. Some authors describe Lissajous figures without saying that this is what they are. They were first described by Nathaniel Bowditch in 1815, and they are sometimes called Bowditch curves. Lissajous figures are sometimes called rectangular curves or rectilinear harmony curves because they are produced by compounding two rectilinear motions. Lissajous figures are aesthetically satisfying, and they are closely related to the mathematics of music. Lissajous figures and some related curves are described. A mechanical device for the production of Lissajous figures and some related curves is called a harmonograph. This term was in use by 1877. There are two main types of harmonograph. A circular harmonograph is based on the rotation of interconnected wheels or gears, and numerous versions have been described. The geometric chuck and lathe described by Bazley in 1875 is, in effect, a circular harmonograph. Pendulum harmonographs are based on the oscillations of pendulums. Curves are close approximations to theoretical ideals provided that pendulum amplitudes are small. Because of friction etc. the amplitude of oscillation decays with time. Hence, the curves produced become progressively smaller. There are many different types of pendulum harmonograph. The earliest pendulum harmonographs were based on the Blackburn pendulum, which was first discussed by James Dean in 1815. It appears to have been re-discovered independently by Blackburn, who first described it in 1844. Twin pendulum harmonographs, which use two pendulums suspended by gimbals so that they are free to swing in any direction, appeared in the 1870s. The first of these was Tilley’s compound pendulum, ca 1873 (for ‘compound’ read ‘twin’). Commercial versions of twin pendulum harmonographs started to become available at about this time. In 1906 Charles Benham introduced his twin-elliptic harmonograph. Both parts of the pendulum of a twin elliptic harmonograph are free to swing in all directions. A harmonograph always has a recording device. This is usually a sheet of paper fastened to a moving table. A pen is mounted on a pivoted pen arm so that it remains in contact with the paper. The figures produced by harmonographs are sometimes called harmonograms. The aesthetic quality of harmonograms produced by pendulum harmonographs became appreciated towards the end of the nineteenth century, and their production was a popular pastime. At about the same time the relationship between Lissajous figures and the theory of music became of interest.

## Keywords

Rest Position Double Pendulum Negative Impulse Private Garden Positive Impulse## References

- Anonymous (1930a) Meccano instructions for outfits nos. 4 to 7. Meccano Ltd., LiverpoolGoogle Scholar
- Anonymous (1930b) Meccano twin-elliptic harmonograph. Special instructions for building Meccano super models. No. 26. Meccano Ltd., LiverpoolGoogle Scholar
- Anonymous (1955) A fascinating designing machine. Meccano Mag 40(7): 384–385Google Scholar
- Anonymous (2010a) Lissajous figures. http://www.youtube.com/results?search_query=Lissajous+ figures&search_type=&aq=f Accessed 4 Feb 2010
- Anonymous (2010b) Lissajous figures. http://physci.kennesaw.edu/javamirror/explrsci/dswmedia/ lisajous.htm Accessed 7 Feb 2010
- Ashton A (2003) Harmonograph. A visual guide to the mathematics of music. Walker & Company, New YorkGoogle Scholar
- Bazley TS (1875) Index to the geometric chuck: a treatise upon the description, in the lathe, of simple and compound epitrochoidal or “geometric” curves (with plates). Waterlow & Sons, London (Reprinted by Nabu Public Domain Reprints)Google Scholar
- Benham CE (1909) Descriptive and practical details as to harmonographs. In: Newton HC (ed) Harmonic vibrations and vibration figures. Newton & Co., London, pp 26–86Google Scholar
- Case WB (1996) The pumping of a swing from the standing position. Am J Phys 64(3):215–220CrossRefGoogle Scholar
- Case WB, Swanson MA (1990) The pumping of a swing from the seated position. Am J Phys 58(5):463–467CrossRefGoogle Scholar
- Coxeter HSM (1961) Introduction to geometry. Wiley, New YorkMATHGoogle Scholar
- Cundy HM, Rollett AR (1981) Mathematical models, 3rd edn. Tarquin Publications, StradbrookeGoogle Scholar
- Falconer K (1990) Fractal geometry. Mathematical foundations and applications. Wiley, ChichesterMATHGoogle Scholar
- Goold J (1909) Vibration figures. In: Newton HC (ed) Harmonic vibrations and vibration figures. Newton & Co., London, pp 89–162Google Scholar
- Greenslade TB (2010) Instruments for natural philosophy. http://physics.kenyon.edu/ EarlyApparatus/index.html Accessed 5 Feb 2010
- Kœnig R (1865) Catalogue des appareils d’acoustique construits par Rudolph Kœnig. Rudolph Kœnig, ParisGoogle Scholar
- Lamb H (1923) Dynamics, 2nd edn. Cambridge University Press, CambridgeMATHGoogle Scholar
- Loney SL (1913) An elementary treatise on the dynamics of a particle and of rigid bodies. Cambridge University Press, CambridgeGoogle Scholar
- Newton HC (1909a) Simple harmonographs. In: Newton HC (ed) Harmonic vibrations and vibration figures. Newton & Co., London, pp 1–25Google Scholar
- Newton HC (ed) (1909b) Harmonic vibrations and vibration figures. Newton & Co., LondonGoogle Scholar
- Roura P, González JA (2010) Towards a more realistic description of swing pumping due to the exchange of angular momentum. Eur J Phys 31(5):1195–1207MATHCrossRefGoogle Scholar
- Stong CL (1965) Zany mechanical devices that draw figures known as harmonograms. Sci Am 212(5):128–130, 132, 134, 136Google Scholar
- Tea PL, Falk H (1968) Pumping on a swing. Am J Phys 36:1165–1166CrossRefGoogle Scholar
- van den Berg H (1997) A modern Meccano harmonograph. Constructor Q 36:18–25Google Scholar
- von Seggern DH (1990) CRC handbook of mathematical curves and surfaces. CRC Press, Boca RatonMATHGoogle Scholar
- Whitaker RJ (1991) A note on the Blackburn pendulum. Am J Phys 59(4):330–333MathSciNetCrossRefGoogle Scholar
- Whitaker RJ (2001a) Harmonographs I. Pendulum design. Am J Phy 69(2):162–173MathSciNetGoogle Scholar
- Whitaker RJ (2001b) Harmonographs II. Circular design. Am J Phy 69(2):174–183MathSciNetGoogle Scholar
- Whitaker RJ (2005) Types of two-dimensional pendulums and their uses in education. In: Matthews MR, Gauld CF, Stinner A (eds) The pendulum. Scientific, historical and educational perspectives. Springer, Dordrecht, pp 377–391Google Scholar
- Whitty HI (1893) The harmonograph. Jarrrold & Sons, NorwichGoogle Scholar
- Yang T, Fang B, Li S, Huang W (2010) Explicit analytical solution of a pendulum with periodically varying length. Eur J Phys 31(5):1089–1096MATHCrossRefGoogle Scholar