Active control of flexural waves in in-vacuo pipes

Abstract

In pipe vibration, the n = 2 (ovalling mode) flexural wave causes a large increase of strain in the pipe wall at the cut-on frequency of the wave when it starts to propagate. Hence from the fatigue perspective, this wave is a major concern. In this thesis, an active control system is designed to suppress this wave. Before the control strategy can be considered, however, the dynamic behaviour of the pipe must first be understood. For this reason, the wavenumbers of the pipe and the mobilities of an infinite and semi-infinite pipe based on Flugge's shell theory are derived. Once the dynamic behaviour is known, the active control system can be modelled in a straightforward manner. An n = 2 PVDF (polyvinylidene fluoride) modal sensor and PZT (lead zirconate titanate) modal actuator are designed to selectively sense and control this wave. Other types of control systems, passive and adaptive-passive control, have also been investigated for comparison with the active control system.

For the infinite pipe, the mobility is derived using the method of residues and an analytical method with eight boundary conditions. Both methods are shown to give the same result. Analytical and wave methods are used to derive the mobility of the semi-infinite pipe. Like the infinite pipe, numerical evaluation shows that both methods are identical. In addition, the mobilities of infinite and semi-infinite pipes are simplified for low frequencies, which facilitates physical insight, and allows explicit expressions for the circumferential wave amplitudes to be derived.

For the modal sensor, PVDF elements are shaped in the form of sine and cosine functions similar to the n = 2 mode shape of the pipe. Since the orientation of the n = 2 mode shape at some arbitrary point on the pipe is unknown, both of these elements are required to sense the wave. The relationship between the charge generated on the sensor to the combination of the axial and circumferential bending strains is established for the case of in-extensional deformation. It is found that a practical modal sensor is sensitive to higher order modes as well as the n = 2 mode, and this cross-sensitivity is dependent upon the width of the modal sensor. However, provided that the width is small enough to keep the axial strain constant over its surface; the sensor will only be dominantly sensitive to the circumferential bending strain of the pipe and hence to the n = 2 mode. For this to occur, the width of the modal sensor has to be less than one third of the wavelength of the flexural wave at the ring frequency.

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