Vibration analysis of rotating beam structures made of functionally graded materials in a thermal environment by generalized differential quadrature method

Abstract

In recent years, there has been increasing curiosity in rotating aerospace structures such as gas turbine blades, helicopter rotary wings, wind turbine blades, tilt-rotor propellers, and flexible appendages of space vehicles due to various problems up to the chosen material type, boundary conditions, rotational speed, geometric properties, and environmental effects that have been studied for the best solution. To overcome these problems and prevent possible catastrophic failures, various analyses like vibration, buckling, static of structures, fatigue, and thermal are performed through engineering analysis software, user-written scripts based on numerical and semi-analytical methods, or specific test equipment. Then, the structure is improved by changing design parameters and operating conditions at the beginning of the design process if any unwanted cases occur. In specific design problems, user-written scripts are generally preferred as engineering analysis tools to obtain more accurate and precise results in a short time. The existence of this wide research field causes many researchers to turn their attention to the topics related to such types of rotating structures able to be modeled as beam elements. One of the extensively studied topics realized under free vibration analysis is the computation of dynamic characteristics, which is a critical design and performance evaluation criteria designating the life of a structure, operating limits, and stability. Therefore, many numerical methods have been attempted to analyze and avoid possible resonance cases by calculating the dynamic characteristics of rotating beam structures accurately. Of these numerical approaches, DQM is first introduced by Bellman et al. to solve nonlinear partial differential equations accurately by expressing them as a set of algebraic equations. This technique uses weighting coefficients to approximate the derivatives of a function at a point and employs a weighted linear sum of the function values at all discrete points. In many areas of engineering problems, it presents satisfactory results for the well-optimized spacing of grid points and well-determined weighting coefficients by using suitable functions. Being computationally less expensive, easy implementation of non-classical boundary conditions, less memory requirements, simple algorithm building, derivation of new numerical methods by combining element-based methods to solve complex geometries, etc. are some reasons why preferred by many researchers. The first objective of this study is to develop a mathematical model for a rotating double-tapered beam with a flexible root i.e. elastic restraints on the root of the beam attached to a rigid hub and present a numerical solution algorithm based on DQM to compute the dynamic characteristics of the beam. As a beam model, the Euler-Bernoulli beam theory is employed to model the system easily with a reasonable result in the first part of free vibration analysis. Using DQM, the governing differential equations of beam and boundary conditions are transformed into a set of linear algebraic equations written in the matrix form. Both of them are defined in different matrices, and boundary conditions are entered by updating corresponding rows of the system matrix created for the governing differential equations, which gives great flexibility to use various nonclassical support types defined as mixed-type partial or ordinary differential equations. Differently from previous studies based on DQM, the effect of rotary inertia, setting angle, and linear changes in taper ratios on dynamic characteristics are investigated. Also, the effect of hub radius and rotational speed are presented akin to previous research findings. To validate the solution method, the obtained results are compared with other studies in the open literature. The second objective is to investigate the elevated temperature and material effects on the mechanical behavior of the beam structures made of functionally graded materials, including shear deformations. Under a uniform temperature distribution, the variations in natural frequencies and mode shapes are investigated for temperature-dependent material properties. The findings of the second part of the solution present that the gradational composition of material, thermal loads, and shear deformation have a significant effect on the dynamic characteristics of the beam structures exposed to elevated temperatures. To overcome the softening effect of high temperatures, the composition of the ceramic-metal mixtures must be determined accurately by employing meshless numerical approaches such as DQM. To sum up, a comprehensive study about free vibration analysis of beam structures used in the aerospace industry has been presented through the thesis, providing assessments of their vibration behavior. Understanding the structural dynamics of these structures is vital in sectors like aerospace, energy, and manufacturing.