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A recently proposed concept for a ring-based multi-axis rate sensor requires that two in-plane modes of vibration and two out-of-plane modes of vibration of a ring each have identical natural frequencies. This work proposes a practical mass trimming approach for eliminating the frequency splits between these four natural frequencies for an initially imperfect ring. Given that the modes of an imperfect ring are unlikely to consist of pure in-plane and pure out-of-plane modes, the proposed trimming approach takes account of the cross-coupling that exists between the in-plane and out-of-plane modes. The proposed approach consists of three stages: (i) eliminate the frequency split between a pair of predominantly in-plane modes and a pair of predominantly out-of-plane modes; (ii) eliminate the cross-coupling between the in-plane and out-of-plane modes; (iii) match the in-plane and out-of-plane natural frequencies. Numerical examples demonstrating and validating the approach are provided. [Copyright &y& Elsevier]

This paper considers the statistical distribution of natural frequency splits for an initially perfect ring with different types of random mass imperfection. The analysis used to derive analytical expressions for the natural frequency splits is based on a Rayleigh–Ritz approach, in which it is assumed that the mode shapes of the imperfect rings are identical to those of a perfect ring. The types of random mass imperfection investigated are: (i) random harmonic variations in the mass per unit length around the circumference of the ring; (ii) the attachment of random point masses at random locations on the ring; and (iii) the attachment of random point masses at uniformly spaced positions on the ring. For case (i) it is found that the frequency splits always have a Rayleigh distribution. For case (ii) an expression for the statistical distribution is deduced which tends to a Rayleigh distribution as the number of attached masses increases. For case (iii) it is found that the frequency split distribution is dependent upon the mode considered and the number of attached masses, and that in some situations the frequency splits have a “half Gaussian” (i.e., non-Rayleigh) distribution. [Copyright &y& Elsevier]

This paper presents a practical method for trimming the natural frequencies of an initially imperfect ring to simultaneously eliminate certain of the frequency splits present. Compared with previous work, the novel feature of this method is that the trimming masses are positioned at pre-selected locations on the ring. The basis for the proposed method is the concept of equivalent imperfection mass, which allows any imperfect ring to be considered as a perfect ring with equivalent imperfection masses attached. By considering this trimming problem it is deduced that it is possible to trim N pairs of modes simultaneously by removing (a minimum of) 2 N trimming masses at particular locations around the ring. By positioning the trimming masses at pre-selected locations, it is shown that a simple set of trimming masses can be calculated easily, and from this set an infinite number of solution sets can be found. Methods for generating these sets are outlined for the trimming of both a single and a dual pair of modes. In practice, it is likely that the trimming masses will be spaced regularly. For this special case, it is found that it is not possible to trim all single- and dual-mode pairs with any arrangement of masses. Validation of the derived simple solution set and the proposed procedure to generate further sets is achieved by studying a number of theoretical examples. [Copyright &y& Elsevier]

This paper addresses limitations to existing analytical models for piezoelectric energy harvesters. The presented model is targeted at predicting behaviours of highly flexible piezoelectric devices (FPEDs) and includes high orders of substrate and piezoelectric material nonlinearity, geometric nonlinearity, and additionally the effects of both self-weight and pre-stress. Validation through experimental testing is provided. The influence of self-weight on vibratory dynamics becomes important in FPEDs due to both material composition and dimension. The developed model facilitates the simulation of FPED performance mounted at specified angles to the horizontal. In one study, for a FPED of 120 mm in length, the resonant frequency changed by over 30 percent with mounting angle. Consideration of mounting orientation is advised as self-weight increases damping and significantly lowers FPED performance – over a 50 percent reduction in one presented case. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a systematic, wave propagation approach for the free vibration analysis of networks consisting of slender, straight and curved beam elements and complete rings. The analysis is based on a ray tracing method and a procedure to predict the natural frequencies and mode shapes of complex ring/beam networks is demonstrated. In the wave approach, the elements are coupled using reflection and transmission coefficients, and these are derived for discontinuities encountered in a MEMS rate sensor consisting of a ring supported on an array of folded beams. These are combined, taking into account wave propagation and decay, to provide a concise and efficient method for studying the free vibration problem. A simplification of the analysis that exploits cyclic symmetry in the structure is also presented. The effects of decaying near-field wave components are included in the formulation, ensuring that the solutions are exact. To illustrate the effectiveness of the approach, several numerical examples are presented. The predictions made using the proposed approach are shown to be in excellent agreement with a conventional FE analysis. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers thermoelastic damping of the in-plane vibration of rings. The work was motivated by the need to gain improved understanding of energy-dissipation effects in silicon MEMS resonators, for which a high Q-factor is often a key design objective. The presented analysis is based on Zener''s classical work on the modelling of thermoelastic loss in uniform beams, and on a recent refinement of Zener''s analysis by Lifshitz and Roukes. A review of Zener''s and Lifshitz and Roukes’ analysis is given. The paper then extends the above work by applying the thermoelastic models to the in-plane vibration of uniform rings of rectangular cross-section. Using both approaches, numerical predictions of modal Q-factors are developed and compared. The relationships between ring geometry, scale and Q-factor are explored and the ability to choose resonator dimensions to control Q-factor due to thermoelastic loss is illustrated. [Copyright &y& Elsevier]