ارتعاشات توأم و تجزیه و تحلیل حساسیت پارامتر از ژیروسکوپ ارتعاشی تکان انبوه

Vibrating beam gyroscopes are widely used to measure the angle or the rate of rotation of many mechanical systems. The vibration and parameters sensitivity analyses of a specific type of vibrating beam gyroscope namely rocking-mass gyroscopes are presented in this paper. These types of gyroscopes by far have a better performance than the conventional single-beam gyroscopes. The system comprises of four slender beams attached to a rigid substrate, undergoing coupled flexural and torsional vibrations with a finite mass attached in the middle. Two of the beams carry piezoelectric patch actuators on top, while the other two possess piezoelectric patch sensors. Using extended Hamilton's principle, the resulting eight coupled partial differential equations of motion with their corresponding boundary conditions are derived. In spite of the need for a high computational power, the system is analysed in the frequency domain using an exact method and the closed-form characteristic equations for two cases of fixed and rotating base support are obtained. Furthermore, a detailed parameter sensitivity analysis is carried out to determine the effects of different parameters on the complex natural frequencies of the system. Results presented are valuable in the design of this type of gyroscope as the exact resonant conditions and the sensitivity of the system parameters play important roles in the dynamic performance of gyroscopes.

Due to the wide range of the applications of the vibrating mass gyroscopes; they are being used in many navigational applications, namely, aerospace, marine and automobile industries. Hence, the detailed study of such systems has always been of great interest to engineers and researchers. In most of these types of gyroscopes, the bending and torsional vibrations are coupled. The theory of coupled flexural–torsional vibrations for thin-walled beams was first developed by Timoshenko and Young [1]. The free flexural/torsional vibration of an Euler–Bernoulli beam with a rigid tip mass was studied by Oguamana [2]. He presented explicit expressions for the frequency equation, mode shapes and their orthogonality relationship and investigated the effects of different parameters on the fundamental frequencies of the system. Salarieh and Gorashi [3] continued his work, but used the Timoshenko beam theory. They studied the effects of the shear deformation and the rotary inertia on the free vibration response of a Timoshenko beam with a rigid tip mass. Gokdag and Kopmaz [4] extended the work of Oguamana [2] by studying the coupled flexural/torsional vibrations of a beam with either the tip or in the span mass attachments. In a series of studies, Jalili and his research team [5], [6] and [7] worked on the vibrating gyroscopic systems experiencing coupled flexural/torsional vibrations. Their first work was to develop a thorough modeling framework for vibrating gyroscopes subjected to general support motion by considering both the flexural and torsional vibrations [5]. In a subsequent work, they include a novel piezoelectric actuation for the vibrating beam gyroscope, which was modeled as an Euler-Bernoulli beam with a tip load subjected to the base rotation. They investigated the effect of the cross-axis in single beam vibratory gyroscopes [6], and also the influence of the substrate motion on the performance of the ring microgyroscopes [7]. Although vibrating beam gyroscopes are becoming the most widely used gyroscopes in many applications [6], but they possess a very important drawback, which produces the cross-coupling error in the measurements [6] and [8]. The vibrating beam gyroscope is typically used to measure the rotational rate around one of the axes. In practice, however, there are always some secondary rotations present in the system. These secondary base rotations can produce significant errors in the measurement of the gyroscope output (cross-axis error). The gyroscopic output increases significantly even for a small secondary rotation. This increased output could be interpreted as the gyroscope output due to the primary base rotation and can hence, develop errors in the measurement [6]. In spite of single beam gyroscopes, the rocking-mass gyroscope does not have those drawbacks, and can accurately measure the rate of rotation. Due to the complexities involved in the modelling and performance analysis of this kind of gyroscope, only few studies have been carried out in this area. The fabrication and design of a rocking-mass gyroscope was studied by Tang and Gutierrez [9], but the operating principle of the device was not fully discussed. Royle and Fox [10] presented an analysis of the mechanics of an oscillatory rate gyroscope that is actuated and sensed using thin piezoelectric actuators and sensors. A modeling framework for these systems, which forms the basis of this paper, is an extension to the work reported by Bhadbhade [11]. The present research undertakes the vibration analysis of a rocking-mass gyroscope, which comprises of a rotating rigid substrate and an assembly of four cantilever beams with a rigid mass attached to them in the middle, as shown in Fig. 1. The objective of the research is to develop a detailed mathematical modeling of the system. The governing equations of motion, using the extended Hamilton's principle, are derived. Since the closed-form solutions can serve as the benchmarks for validating the results obtained from either the numerical calculations or experimental results, the closed-form equations are developed for the frequency characteristic equations of the system for either a fixed supporting base or a rotating one. These exact equations are very important and useful, since their solutions would not only provide exact information about the system fundamental resonant frequencies and their corresponding mode shapes, but also they serve as the bases for the time-domain analyses. Full-size image (82 K) Fig. 1. Schematic of a rocking-mass gyroscope: (a) regular view; (b) zoomed view. Figure options In most of the works done so far, the time-domain analysis has been performed using the assumed mode method (AMM). In fact, instead of determining the exact complex natural frequencies and performing mode superposition method (MSM), in these works the mode shapes of the system have been assumed to have certain forms (AMM). In contrast, this paper offers a closed-form frequency equation and consequently exact fundamental frequencies which can serve as the basis of an accurate time domain analysis (MSM). Moreover, a thorough parametric sensitivity analysis is carried out to determine the effects of different parameters on the complex natural frequencies of the system.

The traditional single beam gyroscopes often encounter with cross-axis error in their measurements as the secondary rotation is always available in reality. The rocking-mass gyroscope, consists of four elastic beams with the piezoelectric sensors and actuators and a mass attached to them in the middle, provides a better performance. However, due to the complexities of the analysis of such a system, this area had remained intact. Along this line of reasoning, the vibration of a rocking-mass gyroscope was studied in this paper. The eight coupled partial differential equations as well as the twenty four boundary conditions were derived for the system using the extended Hamilton's principle. The closed-form characteristic equation of the system was obtained for the two different cases of the fixed supporting base and the rotating one. The parameter sensitivity analysis was also performed on the system and the effects of the primary and the secondary base rotations as well as the value and the length of the rocking-mass on the complex natural frequencies of the system were determined. Results obtained demonstrate that the increase in the primary base rotation would increase the real and the imaginary parts of the frequencies; however, the secondary base rotation has almost no effect on them. In addition, an increase in rocking-mass will decrease the frequencies, while also increasing the rocking-mass length causes a higher values of the real and the imaginary parts of the complex natural frequencies.