Three-Axis Theorem in Moment of Inertia Computation

A very important property in the study of rigid body dynamics, moment of inertia describes the resistance of an object to any change in its angular velocity, given a certain amount of torque. Although many novel methods have been developed to simplify its calculation, this paper presents a remarkable theorem in moment of inertia that has never been widely used, the three-axis theorem. The theorem provides an alternative way for moment of inertia computation and better visualization in integrating each infinitesimal constituent mass element of a rigid body. The key idea is to focus on the distance from this infinitesimal mass to the intersection of the three axes, instead of its distance to a certain rotational axis.

Dynamic responses of an inclined FGSW beam traveled by a moving mass based on a moving mass element theory

Dynamic analysis of an inclined functionally graded sandwich (FGSW) beam traveled by a moving mass is studied. The beam is composed of a fully ceramic core and two skin layers of functionally graded material (FGM). The material properties of the FGM layers are assumed to vary in the thickness direction by a power-law function, and they are estimated by Mori-Tanaka scheme. Based on the first-order shear deformation theory, a moving mass element, taking into account the effect of inertial, Coriolis and centrifugal forces, is derived and used in combination with Newmark method to compute dynamic responses of the beam. The element using hierarchical functions to interpolate the displacements and rotation is efficient, and it is capable to give accurate dynamic responses by small number of the elements. The effects of the moving mass parameters, material distribution, layer thickness ratio and inclined angle on the dynamic behavior of the FGSW beam are examined and highlighted.

A Theory of Inertia Based on Mach’s Principle

A non-relativistic theory of inertia based on Mach’s principle is presented as has been envisaged, but not achieved, by Ernst Mach in 1872. The central feature is a space-dependent, anisotropic, symmetric inert mass tensor. The contribution of a mass element d m to the inertia of a particle m 0 experiencing an acceleration from rest is proportional to cos 2 α , where α is the angle between the line connecting m 0 and d m and the direction of the acceleration. Apsidal precession for planets circling around a central star is not a consequence of this theory, thereby avoiding the prediction of an apsidal precession with the wrong sign as is done by Mach-like theories with isotropic inert mass.

A Theory of Inertia Based on Mach's Principle

A non-relativistic theory of inertia based on Mach's principle is presented as has been envisaged but not achieved by Ernst Mach in 1872. Central feature is a space-dependent, anisotropic, symmetric inert mass tensor. The contribution of a mass element $dm$ to the inertia of a particle $m_0$ experiencing an acceleration from rest is proportional to $\cos^2\alpha$, where $\alpha$ is the angle between the line connecting $m_0$ and $dm$ and the direction of the acceleration. Apsidal precession for planets circling around a central star is not a consequence of this theory, thereby avoiding the prediction of an apsidal precession with wrong sign as is done by Mach-like theories with isotropic inert mass.