Vibration analysis of an axially functionally graded material non-prismatic beam under axial thermal variation in humid environment

The vibration analysis of an axially functionally graded material non-prismatic Timoshenko beam under axial thermal variation in humid environment is carried out using the harmonic differential quadrature method. In this modeling, the length and width of the beam remains constant whereas thickness of the beam is linearly varied along the axis of the beam. The material properties are temperature dependent and are assumed to be varied continuously along the axial direction according to power law distribution. Three types of temperature variations are considered in this study, that is, uniform temperature rise, linear temperature rise, and non-linear temperature rise. The temperature of the beam remains constant under uniform temperature rise condition and it is varied linearly and nonlinearly along the length of beam for rest of the conditions. The beam is subjected to uniform moisture concentration to impose humidity. Hamiltonian’s approach is used to derive the governing equations of motion. The resultant governing equations are then solved using the harmonic differential quadrature method to obtain the natural frequencies of the axially functionally graded material non-prismatic beam. The results obtained using the harmonic differential quadrature method are compared with results obtained for special cases. The effects of thermal variation, humidity, non-homogeneity parameter, and end conditions on natural frequencies of the non-prismatic beam are reported.

An approach in deformation and stress analysis of elasto-plastic sandwich cylindrical shell panels based on harmonic differential quadrature method

Using harmonic differential quadrature method, an approach to analyze sandwich cylindrical shell panels with any sort of boundary conditions under a generally distributed static loading, undergoing elasto-plastic deformation is proposed. The faces of the sandwich shell panel are made of some isotropic materials with linear work hardening behavior while the core is assumed to be an isotropic material experiencing only elastic behavior. The faces are modeled as thin cylindrical shells obeying the Kirchhoff–Love assumptions. For the core material, it is assumed to be thick and the in-plane stresses are negligible. Upon application of an inner and outer general lateral loading, the governing equations are derived using the principle of virtual displacements. Using an iterative approach, named elasto-plastic harmonic differential quadrature method (EP-HDQM), the equations are solved. The obtained results are compared with the results from finite element software Ansys for different sandwich shell panel configurations. Then, the effects of changing different parameters on the stress and displacement components of sandwich cylindrical shell panels in different elasto-plastic conditions are investigated.

Harmonic Differential Quadrature Method for Surface Location Error Prediction and Machining Parameter Optimization in Milling

This paper presents a semi-analytical numerical method for surface location error (SLE) prediction in milling processes, governed by a time-periodic delay-differential equation (DDE) in state-space form. The time period is discretized as a set of sampling grid points. By using the harmonic differential quadrature method (DQM), the first-order derivative in the DDE is approximated by the linear sums of the state values at all the sampling grid points. On this basis, the DDE is discretized as a set of algebraic equations. A dynamic map can then be constructed to simultaneously determine the stability and the steady-state SLE of the milling process. To obtain optimal machining parameters, an optimization model based on the milling dynamics is formulated and an interior point penalty function method is employed to solve the problem. Experimentally validated examples are utilized to verify the accuracy and efficiency of the proposed approach.