The nonlinear transfer of energy among modes of different frequencies on a vibrating string is investigated both theoretically and experimentally. The nonlinearity is associated with the wellknown variation of string tension caused by the vibration modes, but it is essential that at least one of the end supports has finite mechanical admittance if there is to be any mode coupling. If the nonrigid bridge support has zero admittance in a direction parallel to the string, the coupling is of third order in the mode amplitudes. For a more realistic model in which the string changes direction as it passes over a bridge of finite admittance there are additional coupling terms of second order. The first mechanism gives driving terms of frequency 2On _____ COm where COn and COm are, respectively, the angular frequencies of the nth and mth modes present on the string, while the second mechanism gives driving terms of angular frequencies 2On and 2Om. Analysis shows that modes absent from the initial excitation of the string can be driven to appreciable amplitude by these mechanisms, reaching their maximum amplitude after a time typically of order 0.1 s. Modes that are in nearly harmonic frequency relationship behave simply but coupling of modes that are appreciably inharmonic may give rise to rapid amplitude fluctuations. A simple experiment with a wire deflected by a bridge of elastic cord and plucked so as to eliminate a particular mode from the initial excitation provided general semiquantitative confirmation of the theoretical predictions.

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Experimental studies on several orchestral gongs of the tamtam and cymbal families suggest that two separate nonlinear mechanisms contribute to the evolution of the sound. The first mechanism is an upward cascade of energy from the low-frequency modes initially excited into high-frequency modes, caused by coupling between tension and shear stresses at regions of sharp change in shape of the gong. The second is a transition from simple periodic nonlinear modal motion to multiple fractional subharmonics, or even chaotic motion, which fills out the radiated spectrum at frequencies between those of the normal linear modes. Each of these mechanisms has considerable hysteresis, so that the spectrum of the radiated sound evolves over a period of several seconds. Measurements using high-level sinusoidal excitation have elucidated some of the features of this behavior.

A theoreticaland experimental study of energy transfer between the vibrational modes of a symmetrically kinked bar with clamped ends is described. Two non-linear mechanisms responsible for energy transfer from one mode to another at twice the frequency are identified. The first arises from the interaction between shear and tensional forces at the kink and the second from unbalanced moments across the kink. In the system studied, the first of these mechanisms is dominant. A further related mechanism is responsible for energy transfer to modes at three times the base frequency. When a kinked bar with mode frequencies of a few hundred hertz is shaped so that two modes have the desired 2 to 1 frequency relation and is excited by striking it at a node of the higher mode, then the amplitude of that mode rises from zero to a maximum in a time of order 0.1 s and then decays. Theory and experiment are in quite good agreement in relation to the time delay to the maximum amplitude and the magnitude of this maximum. The results contribute to an understanding of the vibrational behaviour of certain musical gongs and are also relevant to other systems.

The paper investigates the use of optimization techniques for the design of percussive beams. Profiles were generated to satisfy predetermined overtone tunings and specified optimization criteria. It is shown that there is no unique solution to the problem, and that the generated profiles need not be the same as traditionally used profiles. The vibration of the beams was modeled using a thick beam theory to take into account the possible effects of shear deformation on the natural frequencies. A number of beams were manufactured with the generated profiles, and their frequency characteristics were obtained experimentally. The experimental results confirm the validity of the numerically generated profiles.