Finite Element Equations Of Motion Research Articles

Aeroelastic stability of smart sandwich plates with electrorheological fluid core and orthotropic faces

The flutter of orthotropic sandwich plates with an electrorheological fluid layer subjected to supersonic airflow is discussed in this article. The sandwich plate consists of an electrorheological fluid layer, a base and a constraining orthotropic layer. The first piston theory is used to model the aerodynamic forces. Hamilton’s principle is employed to derive the finite element equations of motion. Taking the aerodynamic damping into account, an iterative complex eigenvalue solution is conducted to determine the flutter boundaries. The effects of electric field strength, electrorheological layer thickness, electrorheological fluid type, constraining layer thickness, and fiber angle of orthotropic faces on the critical aerodynamic pressure are investigated. Both simply supported and clamped boundary conditions are considered. The results show that the electrorheological core layer is capable of shifting the flutter instability of the system. It is also found that the electrorheological fluid type and the thickness ratios have significant effects on the flutter bounds.

The effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk

It is well known that either the asymmetric disk or transverse crack brings parametric inertia (or stiffness) excitation to the rotor-bearing system. When both of them appear in a rotor system, the parametric instability behaviors have not gained sufficient attentions. Thus, the effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk is studied. First, the finite element equations of motion are established for the asymmetric rotor system. Both the open and breathing transverse cracks are taken into account in the model. Then, the discrete state transition matrix (DSTM) method is introduced for numerically acquiring the instability regions. Based upon these, some computations for a practical asymmetric rotor system with open or breathing transverse crack are conducted, respectively. Variations of the primary and combination instability regions induced by the asymmetric disk with the crack depth are observed, and the effect of the orientation angle between the crack and asymmetric disk on various instability regions are discussed in detail. It is shown that for the asymmetric angle around 0, the existence of transverse (either open or breathing) crack has attenuation effect upon the instability regions. Under certain crack depth, the instability regions could be vanished by the transverse crack. When the asymmetric angle is around π/2, increasing the crack depth would enhance the instability regions.

Parametric instability of a rotor-bearing system with two breathing transverse cracks

Abstract When the rotor rotates at a constant speed, the transverse crack opens and closes alternatively, due to gravity, and thus a “breathing effect” occurs. This variance in shaft stiffness is time-periodic, and hence a parametrically excited system is expected. The parametric excitation from the time-varying stiffness causes instability and severe vibration under certain operating conditions. Current research mostly focused on the rotor with single transverse crack. There are few studies on the multi-cracked rotor system. In fact, the interaction between the multiple parametric excitations with various phasing and amplitude, which are induced by the multiple breathing transverse cracks, would make the instability behavior of the system differ distinctly from that of the single cracked rotor system. Moreover, how the instability regions change with various crack breathing mechanisms should also be investigated. Thus, the parametric instability of a rotor-bearing system with two breathing transverse cracks is studied in the paper. First, the finite element equations of motion are established for the cracked rotor system. Two types of crack breathing mechanisms, of which one is more accurate (new) and the other is empirical (old), are adopted in the finite element formulation. Then, a generalized Bolotin's method is introduced for determining the boundaries of the primary and secondary instability regions. Based upon these, instability analysis for a practical used rotor-bearing system with single and two cracks are conducted, respectively. The instability regions induced by the single transverse crack with new and old breathing mechanisms are compared with each other. For the two-cracked rotor system, the variations of the unstable boundaries with crack depths, orientation angles and positions are observed and discussed in detail. It is shown from the results that the dynamic instability of the two-cracked rotor-bearing system indeed have some unique features that differ from that of the single cracked rotor system.

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.

자이로스코픽 효과와 유체 동압 베어링에 의한 비대칭성을 고려한 회전 유연 디스크-스핀들 시스템의 유한요소 강제 진동 해석

This paper presents an efficient method for determining the forced response of a spinning flexible disk-spindle system supported by fluid dynamic bearings(FDBs) in a computer hard disk drive(HDD). The spinning flexible disk-spindle system is represented by the asymmetric finite element equations of motion originating from the asymmetric dynamic coefficients of the FDBs and the gyroscopic moment of a spinning disk-spindle system. The proposed method utilizes only the right eigenvectors of the eigenvalue problem to transform the large asymmetric finite element equations of motion into a small number of coupled equations, guaranteeing the accuracy of their numerical integration. The results are then back-substituted into the equations of motion to determine the forced response. The effectiveness of the proposed method was verified by comparing it with the responses from the classical methods of mode superposition with the general eigenvalue problems, and mode superposition with modal approximation. The proposed method was shown to be effective in determining the forced response represented by the asymmetric finite element equations of motion of a spinning flexible disk-spindle system supported by FDBs.

New breathing functions for the transverse breathing crack of the cracked rotor system: Approach for critical and subcritical harmonic analysis

The actual breathing mechanism of the transverse breathing crack in the cracked rotor system that appears due to the shaft weight is addressed here. As a result, the correct time-varying area moments of inertia for the cracked element cross-section during shaft rotation are also determined. Hence, two new breathing functions are identified to represent the actual breathing effect on the cracked element stiffness matrix. The new breathing functions are used in formulating the time-varying finite element stiffness matrix of the cracked element. The finite element equations of motion are then formulated for the cracked rotor system and solved via harmonic balance method for response, whirl orbits and the shift in the critical and subcritical speeds. The analytical results of this approach are compared with some previously published results obtained using approximate formulas for the breathing mechanism. The comparison shows that the previously used breathing function is a weak model for the breathing mechanism in the cracked rotor even for small crack depths. The new breathing functions give more accurate results for the dynamic behavior of the cracked rotor system for a wide range of the crack depths. The current approach is found to be efficient for crack detection since the critical and subcritical shaft speeds, the unique vibration signature in the neighborhood of the subcritical speeds and the sensitivity to the unbalance force direction all together can be utilized to detect the breathing crack before further damage occurs.

Dynamic response of structure coupled with liquid sloshing in a laminated composite cylindrical tank with baffle

Baffles are used effectively to reduce the sloshing response of liquid in the liquid storage containers. In this paper, the dynamic interaction that exists between the liquid and elastic tank–baffle system has been considered to evaluate the coupled response of liquid and tank–baffle system using the finite element method. The baffle and tank wall are made up of lightweight laminated composite materials. The liquid is assumed to be incompressible and inviscid, and the flow is irrotational. The finite element equations of motion of liquid and structure domains are numerically integrated by Newmark’s integration scheme. The interaction effect between the two fields is studied by transferring the structural normal acceleration to the liquid domain and liquid pressure to the structure domain. Effects of different parameters, such as being composite baffles, lamination scheme on the slosh frequencies and coupled vibration frequencies in the liquid filled composite tanks have been studied. The effects of composite baffle parameters, such as being its dimensions and position in the tank wall, on the sloshing response of liquid and structural response of liquid filled composite tank are studied under translational base excitation.

粘弾性体の分数階微分有限変形理論と非線形有限要素法 : 減衰マトリックスと運動方程式の定式化(機械力学,計測,自動制御)

Fractional derivative finite deformation theory and nonlinear finite element method were developed for evaluating dynamic characteristics of viscoelastic bodies used in shock and vibration reduction. Fractional differential orders from the static and dynamic testing of viscoelastic material with small strain were obtained. Then a fractional derivative constitutive model for finite deformation theory was constructed and nonlinear finite element equations of motion, involving a newly developed viscoelastic damping matrix as well as standard mass and stiffness matrices, were formulated. Based on this work, a nonlinear finite element computer program for the evaluation of vibration characteristics of viscoelastic products was developed. Simple numerical examples were computed and compared with the experimental results. Both results are in good agreement.

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.

Dynamic simulation of viscoelastic soft tissues in harmonic motion imaging application

A finite element model was built to simulate the dynamic behavior of soft tissues subjected to sinusoidal excitation during harmonic motion imaging. In this study, soft tissues and tissue-like phantoms were modeled as isotropic, viscoelastic, and nearly incompressible media. A 3D incompressible mixed u– p element of eight nodes, S1P0, was developed to accurately calculate the stiffness matrix for soft tissues. The finite element equations of motion were solved using the Newmark method. The Voigt description for tissue viscosity was applied to estimate the relative viscous coefficient from the phase shift between the response and excitation in a harmonic case. After validating our model via ANSYS simulation and experiments, a MATLAB finite element program was then employed to explore the effect of excitation location, viscosity, and multiple frequencies on the dynamic displacement at the frequency of interest.

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.

Aeroelastic Analysis of Bearingless Rotors Using Large Deflection Beam Theory

The aeroelastic response and stability of bearingless rotors are investigated using large deflection beam theory. The bearingless rotor configuration consists of a single flexbeam with a wraparound-type torque tube and pitch links located at the leading edge and trailing edge of the torque tube. The outboard main blade, flexbeam, and torque tube are all assumed to be an elastic beam undergoing arbitrary large displacements and rotations, which are discretized into beam finite elements. The finite element equations of motion for beams are obtained from Hamilton's principle. Two-dimensional quasi-steady strip theory is used to evaluate aerodynamic forces in both hover and forward flight. For the analysis of hover flight, the nonlinear equations of motion are solved for an equilibrium position through an iterative procedure. The modal approach method based on coupled rotating natural modes is used for the stability analysis. For the analysis of 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. After the coupled trim response is computed, the aeroelastic response is calculated through a time-marching solution procedure under small perturbations assumption. A stability analysis is then performed using a moving block analysis. The results of the full finite element analysis using the large deflection beam theory are quite different from those of a previously published modal analysis using the moderate deflection-type beam theory.

Finite element dynamic analysis of orthotropic sandwich plates with an electrorheological fluid core layer

The finite element dynamic analysis of an orthotropic rectangular sandwich plate with orthotropic base plate and constraining layer and an electrorheological (ER) fluid core is analyzed in the present paper. The orthotropic rectangular plate is covered an ER fluid core and a constraining layer to improve the vibrational behavior of the system. The finite element method and Hamilton’s principle are employed to derive the finite element equations of motion for the orthotropic sandwich plate. The effects of the natural frequencies and loss factors on the dynamic behavior of the orthotropic sandwich plate are studied. The rheological property of an ER material, such as viscosity, plasticity, and elasticity may be changed when applying an electric field. The ER fluid core is found to have a significant effect on the vibrational behaviors of the orthotropic sandwich plate. The modal dampings and the natural frequencies for the orthotropic sandwich plate are calculated for various electric fields. When an electric field is applied, the damping of the system is more effective. The effects of the thickness ratio of the base plate, ER fluid core, and the constraining layer on the natural frequencies and modal loss factors are also discussed.

Vibration and stability of rotating polar orthotropic sandwich annular plates with a viscoelastic core layer

A sandwich annular plate is built with two face layers of polar orthotropic material and a viscoelastic core layer. The non-axisymmetric vibration and stability problem of the rotating sandwich plate are analyzed by using the finite element method. By employing the discrete layer annular element and Hamilton’s principle, the finite element equations of motion that consider the effects of transverse shear and rotary inertia are derived. The viscoelastic material in the core layer is assumed to be incompressible, and its extensional and shear moduli are described by complex quantities. Complex-eigenvalued problems are then solved. The effects of various properties, such as orthotropic properties of face layers, and the viscoelastic core layer stiffness and thickness are discussed. Numerical results show that the larger modulus ratio and higher rotational speed tend to increase the nature frequency of the sandwich annular plate system. In addition, the critical speeds decrease while increasing the thickness of the viscoelastic core layer.

Axisymmetric dynamic instability of rotating polar orthotropic sandwich annular plates with a constrained damping layer

A sandwich annular plate built with two constrained layers of polar orthotropic material and a viscoelastic core layer is subjected to a periodic uniform radial stress while rotating. The axisymmetric dynamic instability of such a rotating sandwich plate is analyzed using finite elements. By employing a discrete layer axisymmetric annular element and Hamilton’s principle, the finite element equations of motion that facilitate considerations of transverse shear effect shear effect are derived. The viscoelastic material in the core layer is assumed to be incompressible, and its extensional and shear moduli are described by complex quantities. The regions of dynamic instability are determined by Bolotin’s method. Numerical results show that the constrained viscoelastic core layer tends to stabilize the sandwich annular plate system. In addition, the widths of instability regions decrease as the rotation speed increases.

Axisymmetric parametric resonance of polar orthotropic sandwich annular plates

Parametric resonance of polar orthotropic sandwich annular plates with a viscoelastic core layer subjected to a periodic uniform radial stress is studied by the finite element method. The axisymmetric discrete layer annular element and Hamilton's principle are employed to derive the finite element equations of motion for a sandwich plate including the transverse shear effect. The viscoelastic material in the core layer is assumed to be incompressible, and the extensional and shear moduli are described by complex quantities. The regions of dynamic instability are determined by Bolotin's method. The effects of various parameters on the dynamic instability regions are investigated.

A Non-Incremental Nonlinear Finite Element Solution for Cable Problems

In this investigation, a nonlinear finite element method for the large deformation and rotation of cable problems is presented. This method is based on finite element absolute nodal coordinate formulation that guarantees the continuity of all displacement gradients and leads to a constant mass matrix. The classical cable theory is first reviewed and the assumptions used in this linear theory are defined in order to demonstrate the basic differences between the linear theory and the nonlinear finite element formulation proposed in this paper for cable applications. The elastic cable forces in the absolute nodal coordinate formulation are obtained using a general continuum mechanics approach that accounts for the effect of all geometric nonlinearities. It is shown in this investigation that the use of the general continuum mechanics approach leads to a simpler and more efficient formulation as compared to the use of the assumptions of the linear theory that employs a local finite element coordinate system. The results obtained using the absolute nodal coordinate formulation show a good agreement with the results obtained using the classical cable theory when linear cable problems are considered. In particular it is shown that the use of perturbation methods to linearize the finite element equations of motion leads to modal characteristics results that are in a good agreement with the linear theory. The results of this investigation obtained using explicit numerical integration also show the potential of the proposed finite element formulation in the nonlinear analysis of cables that experience large rotations and deformations. The generalization of the procedure presented in this paper to three-dimensional cable problems is demonstrated and the computer implementation in multibody algorithms is discussed.

Finite element dynamic analysis of rotating orthotropic sandwich annular plates

The finite element dynamic analysis of rotating orthotropic sandwich annular plates with a viscoelastic central layer and two polar orthotropic laminated face layers are analysed in the present paper. The axisymmetric discrete layer annular element and Hamilton’s principle are employed to derive the finite element equations of motion for a rotating sandwich plate including transverse shear effects. The viscoelastic material in the central layer is assumed to be incompressible, and the extensional and shear moduli are described by complex quantities. Complex-eigenvalued problems are then solved, and the modal frequencies and loss factors of the sandwich plate are extracted. The results of the symmetric and non-symmetric sandwich annular plates are both presented. The effects of outer radius to thickness ratio, orthotropic properties of face layers, the thickness of the viscoelastic central layer, and thickness of face layers on the natural frequencies and modal loss factors of sandwich annular plates are discussed.