Numerical integration to obtain second moment of inertia of axisymmetric heterogeneous body
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 r–z plane as shown in Fig. 1. The mass Ma in axisymmetric problems is defined as
The second moments of inertia Ir(p) and Iz(p) about the r’- and z’-axes of a nonhomogeneous body are given by
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 aswhere r is the distance from the center of the sphere and ρ0 = 1 . The exact mass Ma can be obtained as
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)
- et al.
A boundary formulation for calculating of an arbitrary closed planar region
Eng Anal Bound Elem
(1999) Numerical integration to obtain moment of inertia of nonhomogeneous material
Eng Anal Bound Elem
(2019)- et al.
The modified dual reciprocity boundary elements method and its application for solving stochastic partial differential equations
Eng Anal Bound Elem
(2015) - et al.
Improved method generating a free-form surface using integral equation
Comput Aided Geom Des
(2000) - et al.
Moments of inertia for solids of revolution and variational methods
Eur J Phys
(2006) On the effect of functionally graded materials on resonances of rotating beams
Shock Vib
(2012)- et al.
Handbook of Monte Carlo methods
(2011) - et al.
Boundary element techniques – theory and applications in engineering
(1984)
Cited by (1)
Mass moments of functionally graded 2D domains and axisymmetric solids
2024, Applied Mathematical Modelling