On the Axial-Bending-Torsional Vibration Coupling Effect Occurring in a Cantilever Beam with an Asymmetrically Attached Spatial Rigid Body

Abstract

Purpose The analysis pertains to the coupled axial-bending-torsional vibration of an axially functionally graded cantilever beam of a non-uniform cross-section to which a rigid spatial body is fixed at its free end. The centre of mass of the rigid body is positioned eccentrically in the space. Methods Both the Euler-Bernoulli and Timoshenko beam theories are employed in this investigation. For cross-sections that are not circular, the Saint-Venant theory of torsional vibrations is applied. By employing the extended Hamilton’s principle,the governing differential equations alongside the respective boundary conditions are derived. The task of determining the associated frequency equations and mode shapes is transformed into solving a corresponding two-point boundary value problem (TPBVP), which consists of a system of first-order ordinary differential equations subject to the relevant boundary conditions. The TPBVP is resolved utilising the symbolic-numeric method of initial parameters. Results and Conclusions The impact of the eccentricity parameters of the mass centre and the inertia products of the spatial rigid body on the kind of vibration coupling is scrutinised. All possible forms of vibration coupling effects are recognised. For the coupling of all vibration types (axial vibration, bending vibration, and torsional vibration), it is essential that all three type of the eccentricities of the rigid body mass center are concurrently non-zero, irrespective of the values associated with the inertia products of the rigid body. The proposed numerical approach for determining frequencies and the corresponding mode shapes is applicable to non-uniform axially functionally graded beams of arbitrary cross-sectional shape. The orthogonality conditions for the coupled axial-bending-torsional vibration modes of the scrutinised cantilever beam are established within the frameworks of both the Euler-Bernoulli and Timoshenko beam theories. A numerical example is provided in which a high degree of agreement is shown between the results obtained using the presented method and the corresponding results obtained using the finite element method