Numerical integration to obtain second moment of inertia of axisymmetric heterogeneous body

https://doi.org/10.1016/j.enganabound.2021.09.016Get rights and content

Abstract

The second moment of inertia of a continuous axisymmetric object with an arbitrary shape made of a nonhomogeneous material is usually calculated by dividing it into small domains. However, it is a burdensome process to specify the density of the small domains. In this paper, a technique of easily calculating the second moment of inertia of an axisymmetric nonhomogeneous material using boundary integral equations is proposed. The calculations of the mass, primary moment, and center of mass of an arbitrarily shaped object made of a nonhomogeneous material are also shown. A formulization of the boundary element method is utilized, and a technique for the direct numerical integration of the axisymmetric domain using an axisymmetric interpolation method without the need to carry out domain division is proposed. Heterogeneous materials include laminated material composites, in which the density distribution is discontinuous. Axisymmetric interpolation using harmonic and biharmonic functions has a weak point that can be easily overcome using a scale method.

Introduction

Research on rotating machine elements made of functionally graded materials has been reported [1,2]. The moment of inertia of an object with an arbitrary shape made of a nonhomogeneous material is usually calculated by dividing it into small domains. Apart from the Monte Carlo method [3], there is no effective numerical integration method for an object with an arbitrary shape. We can effectively use an automatic element dividing method in the finite element method for the decomposition into standard elemental regions (triangle or quadrangle). However, it is a burdensome process to specify the density of the small domains of a heterogeneous material. When the Monte Carlo method is used for an arbitrary shape, the computation time increases.

A boundary formulation for calculating moments of an arbitrary closed region was presented by Yeih et al. [4]. The author has already presented a boundary integration formulation for the three-dimensional moment of inertia of a nonhomogeneous material without internal cells [5]. In this paper, a technique of easily calculating the moment of inertia of an axisymmetric heterogeneous material using boundary integral equations is proposed. The calculations of the mass, primary moment, and center of mass of an arbitrary object made of a heterogeneous material are also shown. A formulization of the boundary element method is utilized [6], and a technique for the direct numerical integration of an axisymmetric domain without the need to carry out domain division is proposed. The numerical integration is performed on a polyharmonic function using a previously reported interpolation method [7,8]. Moreover, this numerical integration has also been generalized to a meshless boundary element method.In those previous papers, constant elements were used; therefore, accurate values could not be obtained. An arbitrary shape is approximately formed by many straight line segments. In the numerical integral calculation, the numerical integration of an arbitrary shape is possible, and axisymmetric integration is approximately converted into one-dimensional integration using Green's theorem; this is possible even in the presence of a singularity. In the numerical integration, quadratic elements and internal points are used. This technique utilizes the same concept as that in the triple-reciprocity boundary element method [9], [10], [11]. The dual reciprocity boundary element method is widely used [12,13]; however, it is difficult to choose the most suitable approximate function. On the other hand, the proposed method is based on the conventional boundary element method without using other field techniques. Heterogeneous materials include laminated material composites, in which the density distribution is discontinuous. In these cases, it is easy to obtain the moment of inertia. Accurate interpolation can be obtained by using quadratic elements.

Section snippets

Moment of inertia

The density distribution of a nonhomogeneous body in a domain Ω is denoted as w1(q), where q is a point inside the body. Q is a point on the body. The mass Ma in axisymmetric problems about the z-axis is obtained by integration with respect to an arbitrarily given region S in the rz plane as shown in Fig. 1. The mass Ma in axisymmetric problems is defined asMa=Ωw1(q)dΩ=2πSw1(q)rdS.

The second moments of inertia Ir(p) and Iz(p) about the r’- and z’-axes of a nonhomogeneous body are given byIr(p

Numerical examples

The quadratic boundary elements are used in numerical examples [15,17]. To confirm the accuracy of the numerical integral, the moment of inertia of the heterogeneous spherical object Ω of radius R = 50 mm shown in Fig. 4 is obtained. The density distribution ρ(r) in a heterogeneous sphere of radius R is given asρ(r)=ρ0[2(rR)2],where r is the distance from the center of the sphere and ρ0 = 1 . The exact mass Ma can be obtained asMa=ρ00R[2(rR)2]4πr2dr=2815ρ0πR3=0.7330382858×106.

The Ma value

Conclusions

A simple technique for calculating the second moment of an area of an axisymmetric object made of a heterogeneous material by the boundary integral approach was proposed. Since it is not necessary to divide the domain, it is easy to perform the integration for an object with arbitrary shape and density distribution. Heterogeneous materials include laminated materials and particle-reinforced composites, in which the density distribution is discontinuous. In these cases, it is easy to obtain the

Declaration of Competing Interest

The authors declare no conflict of interest.

References (17)

There are more references available in the full text version of this article.

Cited by (1)

View full text