Element Equations Of Motion Research Articles

A Simple-FSDT-Based Isogeometric Method for Piezoelectric Functionally Graded Plates

An efficient isogeometric analysis method (IGA) based on a simple first-order shear deformation theory is presented to study free vibration, static bending response, dynamic response, and active control of functionally graded plates (FGPs) integrated with piezoelectric layers. Based on the neutral surface, isogeometric finite element motion equations of piezoelectric functionally graded plates (PFGPs) are derived using the linear piezoelectric constitutive equation and Hamilton’s principle. The convergence and accuracy of the method for PFGPs with various mechanical and electrical boundary conditions have been investigated via free vibration analysis. In the dynamic analysis, both time-varying mechanical and electrical loads are involved. A closed-loop control method, including displacement feedback control and velocity feedback control, is applied to the static bending control and the dynamic vibration control analysis. The numerical results obtained are accurate and reliable through comparisons with various numerical and analytical examples.

Kane’s Method-Based Simulation and Modeling Robots with Elastic Elements, Using Finite Element Method

The Lagrange’s equation remains the most used method by researchers to determine the finite element motion equations in the case of elasto-dynamic analysis of a multibody system (MBS). However, applying this method requires the calculation of the kinetic energy of an element and then a series of differentiations that involve a great computational effort. The last decade has shown an increased interest of researchers in the study of multibody systems (MBS) using alternative analytical methods, aiming to simplify the description of the model and the solution of the systems of obtained equations. The method of Kane’s equations is one possibility to do this and, in the paper, we applied this method in the study of a MBS applying finite element analysis (FEA). The number of operations involved is lower than in the case of Lagrange’s equations and Kane’s equations are little used previously in conjunction with the finite element method (FEM). Results are obtained regardless of the type of finite element used. The shape functions will determine the final form of the matrix coefficients in the equations. The results are applied in the case of a planar mechanism with two degrees of freedom.

Auto-parametric resonance of framed structures under periodic excitations

A framed structure may be composed of two sub-structures, which are linked by a hinged joint. One sub-structure is the primary system and the other is the secondary system. The primary system, which is subjected to the periodic external load, can give rise to an auto-parametric resonance of the second system. Considering the geometric-stiffness effect produced by the axially internal force, the element equation of motion is derived by the extended Hamilton's principle. The element equations are then assembled into the global non-homogeneous Mathieu-Hill equations. The Newmark's method is introduced to solve the time-history responses of the non-homogeneous Mathieu-Hill equations. The energy-growth exponent/coefficient (EGE/EGC) and a finite-time Lyapunov exponent (FLE) are proposed for determining the auto-parametric instability boundaries of the structural system. The auto-parametric instabilities are numerically analyzed for the two frames. The influence of relative stiffness between the primary and secondary systems on the auto-parametric instability boundaries is investigated. A phenomenon of the "auto-parametric internal resonance" (the auto-parametric resonance of the second system induced by a normal resonance of the primary system) is predicted through the two numerical examples. The risk of auto-parametric internal resonance is emphasized. An auto-parametric resonance experiment of a <TEX>${\Gamma}$</TEX>-shaped frame is conducted for verifying the theoretical predictions and present calculation method.

The Hori–Deprit method for averaged motion equations of the planetary problem in elements of the second Poincaré system

We consider an algorithm to construct averaged motion equations for four-planetary systems by means of the Hori–Deprit method. We obtain the generating function of the transformation, change-variable functions and right-hand sides of the equations of motion in elements of the second Poincare system. Analytical computations are implemented by means of the Piranha echeloned Poisson processor. The obtained equations are to be used to investigate the orbital evolution of giant planets of the Solar system and various extrasolar planetary systems.

Effects of shear deformation and rotary inertia on the natural frequencies of axially loaded beams

The purpose of this study is to investigate how the axial load in beams influences the relationships between the natural frequencies and the effects of shear deformation and rotary inertia. Four beam theories are considered in this study. Finite element equations of motion for the beams under a tensile load are formulated to allow the application of various axial loads as well as to impose any type of boundary conditions. The results demonstrate that the stiffening effect by a tensile load may not reduce the frequency error of the Euler beam theory, unlike the results reported in other studies.

Nonlinear Aeroelastic Panel Flutter Based on Proper Orthogonal Decomposition

It is commonly accepted that 36 in vacuo natural modes (NMS) are needed for converged, limit-cycle oscillations (LCOs) of isotropic or laminated anisotropic rectangular panels in supersonic air flow. It’s computationally costly for nonlinear aeroelastic panel response using such a large number of modes, and it also causes complexity and difficultly in designing controllers for panel flutter suppression. Based on Hamilton principle, the aeroelastic finite element motion equations of the 3-D panel are established by using the von Karman large deflection theory, first-order piston theory aerodynamics, the proper orthogonal decomposition (POD) method are adopted to construct a reduced order model of the panel, then the reduced panel flutter equations are solved in time domain using a numerical integration method. Comparing with the LCOs calculated by using 36NMS, the results obtained by using the reduced order model based on POD method (POD/ROM) show a good agreement.

Effect of Boundary Conditions on Transient Response of Sandwich Plates with Electrorheological Fluid Core

Electrorheological (ER) fluids are a kind of smart material whose rheological properties can be controlled by an external electric field. In the present paper, the transient vibration of a rectangular three layer sandwich plate with electrorheological fluid core is analyzed based on the classical plate theory. The Bingham plastic model is used to consider the post-yield behavior of ER fluid. The structure is modeled using a finite element method. Hamilton’s principle is employed to derive the finite element equations of motion. The constant average acceleration scheme is used to integrate the equations of motion. The effects of change in electric field and core thickness on the structure settling time and its natural frequencies are studied for various boundary conditions. The results show that the thickness of the core layer and the electric field strength has significant effects on damping behavior of the sandwich plate. When the applied electric field increases a linear decay in transient response of the structure is observed. It is also found that the electric field changes have no influence on the system natural frequencies.

Investigation of direct integration for nonlinear finite element equations of motion by the modified central difference method

The investigation of stability and precision of the previously proposed implicit scheme of direct integration for the finite element equations of motion is presented. The scheme parameter values that ensure its unconditional stability in the nonlinear problems are determined and comparison with existing procedures is given.

Thermal Post-Buckling and Vibration Analysis of Composite Conical Shell Structures Using Layerwise Theory

The thermal post-buckling and vibration characteristics of composite conical shells are investigated using a finite element method. Based on the layerwise theory and the von Karman displacement strain relationships, the nonlinear finite element equations of motion are derived for the thermoelastic response of the composite conical shell structure. The cylindrical arc-length method is used to account for the snapping phenomenon. The influence of the structural parameters, such as the semi-cone angle, thickness ratio, and shallowness angle (curvature), on the structural stability of the composite conical shell subjected to the thermal load is also observed.

Vibration Control of Smart Composite Laminated Spherical Shell using Neural Network

This study presents a neural network approach for the identification and control of a smart composite laminated spherical shell. The spherical shell is in the form of a layered composite shell having a sensor and an actuator layer. The vibratory response of the shell is modeled using FEM. A degenerate shell element is implemented to model composite laminated spherical shell. Modeling is based on the first-order shear deformation theory and linear piezoelectricity theory. The mode superposition method has been used to transform the coupled finite element equations of motion in the physical coordinates into a set of reduced uncoupled equations in the modal coordinates. The reduced uncoupled equations are transformed into discrete state space form. An identifier neural network has been trained using the results of the FEM program to predict the future response of the structure from the immediate history of the system's response. Then a controller neural network has been trained with the aid of the identifier neural network so that the overall behavior of the controlled system can be described by a prescribed reference model. Numerical results have been presented for the vibratory response of the laminated composite spherical shell. The controlled response of the shell is found to exactly follow the reference model.

무 베어링 로터 시스템의 정지 및 전진 비행시 공력탄성학적 해석

In this study, the aeroelastic response and stability of bearingless rotors are investigated using a large deflection beam theory. The outboard main blade, flexbeam, and torque tube are all assumed to be an elastic beam undergoing arbitrary large displacements and rotations. The finite element equations of motion obtained from Hamilton`s principle. Two-dimensional quasi-steady strip theory is used to evaluate aerodynamic forces. In hover, the modal approach method based on coupled rotating natural modes is used for the stability analysis. In forward flight, the nonlinear periodic blade steady response is obtained by integrating the full finite element equation in time through a coupled trim procedure with a vehicle trim. The results of the full finite element analysis using the large deflection beam theory are compared with those of a previously published modal analysis using the moderate deflection-type beam theory.

Stabilization of Thermoelastic Unstable Postbuckling in Cylindrical Piezolaminated Panels

In this study, the stabilization of thermoelastic unstable postbuckling in cylindrical piezolaminated panels was investigated by applying piezoelectric actuation. In order to derive the finite element equations of motion of the active piezolaminated cylindrical panels, the total Lagrangian formulation was derived using the Hamilton’s virtual work principles for largely deformed structures with small strain. For the purpose of finding the load-displacement tracking, a cylindrical arc-length method was applied to the iterative Newton-Raphson scheme. Present results show that the eccentric piezoelectric patch can passively remove the unstable postbuckling and the snap-through without piezoelectric actuation. Also, the thermoelastic snap-through can be successfully suppressed by using a piezoelectric actuation.

Quantum hydrodynamic analysis of decoherence: quantum trajectories and stress tensor

Quantum trajectories, obtained by integrating equations of motion for elements of the probability fluid, are used to analyze decoherence in a model two-mode system. Analysis of trajectories, flux maps, and the stress tensor for two composite systems, in one of which the system is uncoupled from the environment, leads to a hydrodynamic interpretation of the decoherence process.

Passive suppression of nonlinear panel flutter using piezoelectric materials with resonant circuit

In this study, a passive suppression scheme for nonlinear flutter problem of composite panel, which is believed to be more reliable than the active control methods in practical operations, is proposed. This scheme utilizes a piezoelectric inductor-resistor series shunt circuit. The finite element equations of motion for an electromechanically coupled system is derived by applying the Hamilton’s principle. The aerodynamic theory adopted for the present study is based on the quasi-steady piston theory, and von-Karrnan nonlinear strain-displacement relation is also applied. The passive suppression results for nonlinear panel flutter are obtained in the time domain using the Newmark-β method. To achieve the best damping effect, optimal share and location of the piezoceramic (PZT) patches are determined by using genetic algorithms. The effects of passive suppression are investigated by employing in turn one shunt circuit and two independent shunt circuits. Feasibility studies show that two independent inductor-resistor shunt circuits suppresses flutter more effectively than a single shunt circuit. The results clearly demonstrate that the passive damping scheme that uses piezoelectric shunt circuit can effectively attenuate the flutter.

Nonlinear finite element analysis of composite shell under impact

Large deflection dynamic responses of laminated composite cylindrical shells under impact are analyzed by the geometrically nonlinear finite element method based on a generalized Sander’s shell theory with the first order transverse shear deformation and the von-Karman large deflection assumption. A modified indentation law with inelastic indentation is employed for the contact force. The nonlinear finite element equations of motion of shell and an impactor along with the contact laws are solved numerically using Newmark’s time marching integration scheme in conjunction with Akay type successive iteration in each step. The ply failure region of the laminated shell is estimated using the Tsai-Wu quadratic interaction criteria. Numerical results, including the contact force histories, deflections and strains are presented and compared with the ones by linear analysis. The effect of the radius of curvature on the composite shell behaviors is investigated and discussed.

A Note on Computational Rotor Dynamics

Concise equations for improvements in computational efficiency on dynamics of rotor systems are presented. Two coordinate ordering methods are introduced in the element equations of motion. One is in the real domain and the other is in the complex domain. The two coordinate ordering algorithms lead to compact element matrices. A station numbering technique is also proposed for the system equations during the assembly process. The proposed numbering technique can minimize the matrix bandwidth, the memory storage and can increase the computational efficiency. Numerical examples are presented to demonstrate the benefit of the proposed algorithms.

Dynamic Modeling and Neural Control of Composite Shells Using Piezoelectric Devices

A modal dynamic model was developed for the active vibration control of laminated doubly curved shells with piezoelectric sensors and actuators. The dynamic effects of the mass and stiffness of the piezoelectric patches were considered in the model. Finite element equations of motion were developed based on shear deformation theory and implemented for an isoparametric shell element. The mode superposition method was used to transform the coupled finite element equations into a set of uncoupled equations in the modal coordinates. A robust controller was developed using Linear Quadratic Gaussian with Loop Transfer Recovery (LQG/LTR) design methodology to calculate the gain and actuator voltage requirements. A Neural Network controller was then designed and trained offline to emulate the performance of the LQG/LTR controller. Numerical results have been presented for a flat plate and a spherical shell showing the variation in initial conditions and structural parameters. The neural network controller was shown to effectively emulate the LQG/LTR controller.

Static and dynamic analysis of composite box beams using large deflection theory

The static and dynamic behavior of composite box beams is investigated using a large deflection beam theory. The finite element equations of motion for beams undergoing arbitrary large displacements and rotations, but small strains, are obtained from Hamilton's principle. The importance of non-classical structural phenomena is systematically investigated for composite box beams. The sectional elastic constants including warping deformations have been determined from the refined cross-sectional finite element method. The effects of fiber orientations and stacking sequences on the static deformation and vibration characteristics have been investigated. Numerical results are compared with the previously published experimental and theoretical results. The present results are proved to be very accurate.

Dynamic Response Analysis in End Milling Using Pretwisted Beam Finite Element

This paper develops an analytical model to estimate the dynamic responses in end milling, i.e., dynamic milling cutter deflections and cutting forces, by using the finite element method along with an adequate end milling cutting force model. The whole cutting system includes spindle, bearings and cutter. The spindle is structurally modeled with the Timoshenko-beam element, the milling cutter with the pretwisted Timoshenko-beam element due to its special geometry, and the bearings with lumped springs and dampers. Because the damping matrix in the resulting finite element equation of motion for the whole cutting system is not of proportional damping due to the presence of bearing damping, we use state-vector approach and convolution integral to find the solution of equations of motion. To assure the accuracy of dynamic response predication, the associated cutting force model should be sufficiently precise. Since the dynamic cutting force is proportional to the chip thickness, a quite accurate algorithm for the calculation of chip thickness variation due to tool geometry, runout and spindle-tool vibration is developed. A number of dynamic cutting forces and tool deflections obtained from the present model for various cutting conditions are compared with the experimental and analytical results available in the literature, and good agreement is demonstrated for these comparisons. Therefore the present model is useful for the prediction of end milling instability. Also, the tool deflections obtained by using the pretwisted beam element are found smaller than those by straight beam elements without pretwist angle. Hence, neglecting the pretwist angle in the structural model of milling cutter may overestimate the tool deflections.

Effect of the coupling between stretching and bending in the large displacement analysis of plates

AbstractThe effect of the rotary inertia on the non‐linear dynamics of plates that undergo a large reference displacement is examined in this paper. An assumed displacement field that accounts for the coupling between the stretching and bending of the plate as the result of considering the effect of the rotary inertia is used to identify the configuration of the plate. Furthermore, the coupling between the stretching and bending of the plate as the result of finite rotation is also considered in this investigation. Based on the assumed displacement field that accounts for the effect of the rotary inertia, a non‐linear finite element formulation is developed for the large displacement analysis of plates. The element equations of motion are expressed in terms of a set of element invariants that depend on the assumed displacement field as well as the rotary inertia. The use of the formulation presented in this paper is demonstrated using numerical examples.