Free vibration behavior of rotating bidirectional functionally-graded micro-disk for flexural and torsional modes in thermal environment

Highlights

• A MCST-based mathematical model to study free vibration of a rotating BFGM annular micro-disk.

• Both the asymmetric and axisymmetric flexural modes as well as the torsional mode are studied.

• This model involving asymmetric flexural and torsional modes are presented for the first time.

• Effects of thermal loading and of different material and geometric parameters are studied.

• Size-effect is significant on the flexural modes but insignificant on the torsional mode.

Abstract

A mathematical model for investigating the asymmetric as well as the axisymmetric free vibration behavior of a rotating annular micro-disk is presented for the first time. The disk is assumed to be functionally-graded (FG) along the radial and thickness directions, and is considered to be operating in high-temperature environment. An energy based approach involving minimum potential energy principle and Hamilton's principle is used to derive the governing equations of motion considering Kirchhoff plate theory. The size-effect is addressed employing modified couple stress theory. A novel tangent stiffness based formulation is employed to derive the governing equations of vibratory motion in the neighborhood of the centrifugally and thermally deformed disk configuration. The governing equations are solved following Ritz method. The model captures both the axisymmetric and asymmetric flexural vibration modes, as well as the torsional mode. The model is successfully validated with the available results for some reduced problems. Numerical results are presented in tabular and graphical form for various material and geometric parameters, and some illustrative mode shape plots are presented showing the mode-switching phenomenon. The work presents a generalized model which can be reduced to theoretically model a wide variety of practical problems.

Introduction

Due to the emerging needs of miniaturization, the use of rotating micro-size disks is increasing rapidly in a wide range of application fields such as mechanical, aerospace, chemical, bio-medical, manufacturing etc. The present and potential use of rotating micro-disks involve micro-gas-turbines, micro-steam-turbines, micro-propulsion systems, micro-pumps, micro-machining devices, and various other small-scale rotating machineries and miniaturized robots [1], [2], [3], [4], [5]–6]. In many of these applications, the rotating micro-disks need to operate in high-temperature environment. Recent literatures reveal that metal-ceramic functionally-graded (FG) materials (FGMs) are preferred over conventional composites for high-temperature applications due to its continuous gradation of effective material properties. In view of this background, the present work studies the free vibration behavior of a rotating micro-disk, made of metal-ceramic bidirectional FGM (BFGM) and operating in high-temperature environment. The term ‘BFGM’ implies that the disk material is graded along both the radial and thickness directions. The through-the-thickness gradation is assumed to be symmetric where the material composition is graded from pure metal at the geometric mid-plane to pure ceramic at the outer surfaces of the disk. This type of gradation helps protect the disk from thermal effect as well as avoids bending of the disk during rotation and temperature rise [7], [8], [9], [10]–10]. On the other hand, it is experimentally established that the classical theories are size-independent and are incapable of simulating the size-dependent behavior of micro- and nano-size components. In this regard, modified couple stress theory (MCST) [11] is one of the well-established theories that is capable of studying the size-dependent behavior of micro-size components and thus is used for the present work. In the above-mentioned applications, a rotating micro-size disk, called the rotor, forms an important component of the machines. For mechanical design of such machines, the designer has to know or apprehend the free vibration behavior of the in-built rotating micro-disk. In that effort, special attention is paid to avoid coincidence of the free vibration frequencies with the speed of the micro-disk in order to avoid critical condition. This is important because such coincidence may lead to catastrophic failure. Again, experimental investigation of the mechanical behavior of micro-size components needs state-of-the-art and costly instrumentation and facilities, and may not always be readily feasible. That is why the researchers worldwide have resorted to theoretical studies involving micro-size components. Hence, the authors are motivated to conduct the present work.

The theoretical formulation of a rotating disk is based on circular plate theory. Research articles dealing with the mechanical behavior of circular micro-size plates involving MCST have been published in the last few years and some of them [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]–23] are mentioned here. As far as the studies on static behavior of circular micro-plates are concerned: Ref. [12] dealt with homogeneous annular plates; Refs. [13–15] dealt with FGM solid plates and Refs. [15,16] dealt with FGM annular plates. On the other hand, dynamic behaviors of circular micro-plates have been investigated in Refs. [17,18] for homogeneous solid plates and in Ref. [16] for FGM annular plates. As FGMs are suitable to be used for load-bearing components subjected to thermal loading, the mechanical behavior of circular FGM micro-plates under thermal loading were also studied, and are found in Refs. [19–23] for solid plates and in Ref. [20] for annular plates. It must be noted that the problems dealt in Refs. [12–23] are based on axisymmetric formulation.

It is known that the natural frequencies of a rotating disk for its various modes of vibration are important design parameters. As rotating homogeneous annular macro-size disks, mounted on shafts, are very commonly found in various rotating machineries, investigation of its free vibration behavior based on classical theory (size-independent) have been an interesting research area since long, and some of these investigations are found in Refs. [24], [25], [26]–27]. Since the development of FGMs, research articles on mechanical behavior of rotating FGM annular macro-size disks based on classical theory have been on the rise. It is found that most of these articles dealt with the stress and deformation behavior [28], [29], [30], [31]–32] and study on free vibration is scarce [33]. Kermani et al. [33] investigated the free vibration behavior of a rotating annular disk with exponential functional gradation in the radial direction. They studied both the axisymmetric and asymmetric flexural modes of vibration for clamped inner edge with three different outer radius support configurations namely clamped, simply supported and free. It is to be mentioned that a flexural mode without any nodal diameter is called an axisymmetric mode and that containing one or more nodal diameters is called an asymmetric mode.

The literatures involving rotating micro- and nano-disks, and rotating micro- and nano-circular plates are briefly described here. Pei [34] investigated the free vibration behavior of a homogeneous micro-size annular disk under thermal loading, considering classical theory only and presented results for both axisymmetric and asymmetric modes. Based on MCST, Mahinzare et al. [35,36] and Shojaeefard et al. [37] investigated the axisymmetric free vibration behavior of rotating BFGM solid circular plates supported at the outer boundary. In Ref. [35] a uniform thickness micro-size plate in high-temperature environment supported by Winkler-Pasternak foundation was considered, in Ref. [36] a uniform thickness nano-size plate with piezoelectric properties was considered, and in Ref. [37] a micro-size plate of non-uniform thickness was considered. In these works, the functional gradation is considered along the radial and thickness directions, and the results are presented for the first two axisymmetric vibration modes for clamped and hinged boundary conditions. Some recently published research works [38], [39], [40]–41] dealing with rotating disks involving other size-dependent theories are also mentioned here. Based on strain gradient theory (SGT), Danesh and Asghari [38] carried out axisymmetric deformation and stress analysis of a homogeneous annular micro-disk. The stress analysis of rotating FGM annular nano-disks were reported by Hosseini et al. [39] for non-uniform thickness disks, by Shishesaz et al. [40] for uniform thickness disks under thermoelastic loading and by Hosseini et al. [41] for non-uniform thickness disks under thermoelastic loading. All these works considered gradation along the radial direction and are based on axisymmetric formulation.

Literature review reveals that the research problem involving mechanical behavior of rotating BFGM annular micro-disk based on MCST with inner edge clamped and outer edge free is not yet investigated. Ref. [34] studied similar type of problem for homogeneous disk in the absence of high-temperature environment but did not use any size-dependent theory. Refs. [35–37] considered BFGM based on MCST but for a rotating solid circular micro-plate supported at the outer edge which is not a typical clamped-free rotating disk-type problem. In addition, only the axisymmetric modes are considered in Refs. [35–37] whereas in the present work both the axisymmetric and asymmetric modes as well as the torsional mode are analysed. Moreover, in the present work, the symmetric through-the-thickness material gradation does not cause bending of the disk either due to thermal loading or due to the centrifugal loading imparted through rotational speed, which is unlike the case considered in Refs. [35–37]. To the best knowledge of the authors, the non-axisymmetric mathematical model for a rotating micro-disk based on MCST is presented for the first time through this work. The present model is quite general in nature and is applicable for a wide range of problems such as size-independent rotating disk problem, axisymmetric rotating micro-disk problem, axisymmetric solid micro-plate problem, radial-FG rotating micro-disk/plate problem, thickness-FG rotating micro-disk/plate problem etc.

The numerical results are presented in non-dimensional speed-frequency plane to show the effects of disk-size in micron level, radial gradation index, thickness gradation index, outer-inner radius ratio, high operating temperature and material composition. Some illustrative mode shape plots are presented to demonstrate various mode shapes and the mode-switching phenomenon. The presented results are new of its kind. The results indicate the following significant findings: (i) The increase in speed leads to increase in the flexural vibration frequencies due to centrifugal stiffening effect, but does not influence the torsional vibration frequency. (ii) The reduction in disk thickness toward the order of the material length scale parameter imparts significant stiffening effect on the flexural vibration frequencies but shows almost negligible effect on the torsional mode. (iii) The radial and thickness gradation indices significantly affect the flexural and torsional vibration frequencies, and this effect is found to be more prominent for the higher flexural modes. (iv) The radius ratio exhibits very strong influence on the flexural modes and the torsional mode. (v) The high operating temperature affects the higher flexural modes only, and the thermal effect is strongly influenced by the size effect.