Finite Element Equations Of Motion Research Articles

A method for dynamic analysis and design of joined conical-cylindrical shells based on the model condensation

In the vibration analysis of joined conical-cylindrical shells, mode methods are widely used. However, from the practical results, boundary conditions affect significantly on the vibration behaviors of these structures. Therefore, the assumed modes may not obtain accurate results, which is terrible for structural design or vibration control. Finite element method (FEM) can be used to solve such problems, whereas the time and hardware costs are unacceptable, especially for the structural design and vibration control. The Craig-Bampton (C-B) method takes into account the advantages of both FEM and mode theory, and is very suitable for the dynamic analysis. This paper studies the advantages and feasibility of the C-B method in vibration analysis of joined conical-cylindrical shells. Using the eight-node super-parametric shell element and Hamilton's principle, the finite element equation of motion for the joined conical-cylindrical shell is established. The reduced FE model of the structure is obtained by using the C-B method. A process for efficient structural dynamic design is proposed. The correctness of the model reduction based on C-B method is verified through modal measurement experiments. In addition, some examples are given to illustrate the advantages of the C-B method in the optimization design and dynamic analysis of joined conical-cylindrical shell structures. Finally, a solution idea of coordinate transformation matrix is proposed when there are much more interfaces between substructures.

Active vibration suppression of wind turbine blades integrated with piezoelectric sensors

Abstract As the wind turbine size gets larger, the optimal design of blades, which is a major source of energy for the wind turbines and also the cause of loads, is becoming more important than anything else. Therefore, reducing the load on the blade should be the top priority in designing a blade. In this article, we studied the vibration control of the stiffened wind blades subjected to a wind load with piezoelectric sensors and actuators to mitigate fluctuations in loading and adding damping to the blade. The model is a laminated composite blade with a shear web and the PZT piezomaterial layers embedded on the top and bottom surfaces act as a sensor and actuator, respectively. A uniformly distributed external wind load is assumed over the entire plate surface for simplicity. The first-order shear deformation (FSDT) theory is adopted, and Hamilton’s principle is used to derive the finite element equation of motion. The modal superposition technique and the Newmark- β \beta method are used in numerical analysis to calculate the dynamic response. Using the constant gain negative velocity feedback control algorithm, vibration characteristics and transient responses are compared. Furthermore, vibration control at various locations of the shear webs subjected to an external load is discussed in detail. Through various calculation results performed in this study, this article proposes a method of designing a blade that can reduce the load by actively responding to the external load acting on the wind turbine blade.

Vibration response of Timoshenko perforated microbeams under accelerating load and thermal environment

This article presents a dynamic finite elements procedure capable of analyzing the dynamic behavior of perforated Timoshenko microbeams in thermal environment and subjected to moving mass for the first time. An analytical geometrical model is presented to incorporate the perforation effect into the governing equations. The microstructure size dependent effect is incorporated based on the modified couple stress theory. Based on the presented geometrical model, the dynamic equations of motion including the interaction terms of the mass inertia are derived from the exact differential of the displacement functions of the beam relative to the mass contact point. To investigate the temperature effect on the dynamic behavior, thermal loading with different temperature rise profiles are considered. Thermal stresses, due to thermal loads are converted to mechanical stresses and then the thermal rigidity matrix is combined with the stiffness matrix of the beam. The dynamic finite element equations of motion are derived including the thermal effect. The system equations are solved numerically by using implicit time integration method due to its stability. The proposed numerical procedure is checked by comparing the obtained results with the available solutions and good agreement is observed. Effects of perforation parameters, thermal loading profile, moving mass characteristics, as well as the moving speed on the dynamic behavior of perforated microbeams are studied. The obtained results are helpful in the design and manufacturing of perforated microbeams structural systems.

Nonlinear dynamics and active control of smart beams using shear/extensional mode piezoelectric actuators

In this work, the nonlinear dynamics and active control of smart beams are investigated using the shear mode and extensional mode piezoelectric actuators. The numerical analysis is performed considering two smart beams for separate use of the two kinds of piezoelectric actuators. These smart beams are considered to operate under the transverse or axial compressive harmonic load while the actuators are activated by supplying the external electric field according to the velocity feedback control law. The dynamics of the smart beams is analysed numerically by deriving the geometrically nonlinear electro-elastic incremental equations of motion in the finite element (FE) framework. The FE equations of motion are solved using Bathe time integration method for evaluation of the transient responses while the frequency responses are evaluated by implementing the harmonic balance method in conjunction with a numerical continuation technique. The results for the transverse harmonic load reveal that both the actuators are capable of attenuating the large amplitude vibration of the smart beams; however, the application of the shear/extensional mode actuator is mainly limited to a certain extent of the load-amplitude. Under the axial compressive harmonic load, the dynamics of the smart beams in the pre-buckled state appears through the parametric instabilities. However, once the buckling occurs, the smart beams undergo complex nonlinear dynamics due to the appearance of the snap-through periodic motion and the chaotic motion through the symmetry breaking, period-doubling/demultiplying and saddle-node bifurcations. The presented results mainly reveal the suitability of the shear mode and extensional mode piezoelectric actuators in active control of such dynamic instabilities and the associated complex nonlinear dynamics of the smart beams.

Flutter of variable stiffness composite laminates in supersonic flow with temperature effects

The nonlinear thermal flutter behavior of variable stiffness composite laminates (VSCL) with curvilinear fibers in high supersonic flow is investigated. The first order shear deformation theory (FSDT) combining von Karman large-deflection strain-displacement relations, quasi-steady first-order piston theory aerodynamics and quasi-steady thermal stress theory are used to formulate the nonlinear panel flutter finite element equations of motion. The fiber orientation within a layer is assumed to vary linearly from [Formula: see text] at the center to [Formula: see text] at the vertical edges of the rectangular lamina. The flutter characteristics of variable stiffness composite laminates with different temperature distributions are then studied. The results show that the critical dynamic pressure decreases as [Formula: see text] or [Formula: see text] increases, whereas the limit cycle amplitude increases as [Formula: see text] or [Formula: see text] increases for the same dynamic pressure. The critical dynamic pressure and limit cycle amplitude both increase when the temperature gradient along panel thickness increases. Simple harmonic motions, unharmonic but periodic motions, and chaotic motions can be observed on VSCL under different temperatures. It also turns out that temperature distribution has similar influence on both the critical dynamic pressure and limit cycle amplitude of VSCL.

Finite element based stress and vibration analysis of axially functionally graded rotating beams

This study presents a comprehensive numerical dynamic finite element analysis to investigate the dynamic behavior and induced stresses of axially functionally graded rotating beam, for the first time. The material properties of the rotating beam are assumed to continuously vary nonlinearly along the longitudinal direction according to the power law. Based on Timoshenko beam theory (TBT), the Hamiltonian principle is applied to derive governing equations of motion. The dynamic finite element equation of motion for axially functionally straight rotating cantilever beam is derived. Both stress and vibration responses are detected and analyzed. The proposed computational procedure is verified by comparing the obtained results with the corresponding results in the literature and good agreement is observed. Effects of the material gradation index and the rotating speed on the dynamic behavior of functionally graded rotating cantilever are investigated and analyzed. The obtained results show the significant effect of the material gradation index and the rotating speed on the dynamic behavior of axially functionally graded beams. The proposed model can be used effectively in design of wind turbine, rotation shafts and turbomachinery systems.

Nonlinear dynamics of viscoelastic flexible structural systems by finite element method

This article presents a nonlinear displacement based finite elements model to study and analyze the nonlinear dynamic response of flexible double wishbone structural vehicle suspension system considering damping effect which was not previously discussed elsewhere. Due to large deflection and moderate rotation encountered during passing over road bumps, the kinematic nonlinearity is included through von Karman strain component. Elastic undamped as well as viscous and viscoelastic damping mechanism are considered and compared. Considering the viscoelastic damping mechanism, the viscoelastic damping mechanism is modeled based on the integral constitutive form, which is recast into an incremental form suitable for finite element implementation. Additionally, the revolute joint element is adopted to incorporate the joint flexibility in the double wishbone system. The plane frame element is adopted to model the suspension links by using Timoshenko beam theory. The developed nonlinear finite element equations of motion are solved through the incremental iterative Newmark technique. The developed procedure is verified by comparing the obtained results with analytical solution and excellent agreement is observed. The applicability of the developed procedure is demonstrated by conducting parametric studies to show the effects of the road irregularities profiles, the vehicle speed, and the material damping coefficients on the nonlinear vibrations response of the double wishbone suspension systems. The obtained results are supportive in the design and manufacturing processes of these structural systems.

Orientation-dependent mechanical properties of planar microtubule-based bio-nanometamaterials

Microtubules are protein-based filaments that carry out essential biological roles of eukaryotic cells. They play, in vivo, critical roles in a wide variety of physical situations such as maintaining cell shape, supporting cellular motions, and playing significant roles in both intracellular transport and cellular division. To perform some of these functions, microtubules may need to form different topologies in various orientations. Therefore, the need to introduce different structures of microtubules arises. Orientation-dependent architected structures are a type of periodic architected materials that can exhibit various mechanical properties in different orientations. In the present study, several periodic architectures of various topologies are investigated numerically, and their mechanical properties (elastic modulus, Poisson’s ratio, and natural frequencies) are analyzed in different spatial orientations. For this purpose, the representative volume elements are constructed by considering unit cells of different models aligned in different spatial angles, and their elastic moduli and Poisson’s ratios are obtained by the finite element method. Then the finite element equation of motion for the given structures are solved to find their six natural frequencies. Besides, Bloch’s theorem is implemented to study wave propagation and compare the bandgap widths of the structures in different orientations. It is found that while the elastic properties of some architected structures may increase as the orientation angle increases from zero, for some others, a reduction in the elastic moduli can be observed as the angle rises. Further, it is seen that some structures may exhibit positive (and high) values of Poisson’s ratio in some angles, while for a different orientation, these values turn out to be negative. Also, the effects of orientation angle on the natural frequencies and stop-band performance are observed to be significant in some topologies and negligible in some others. A better understanding of the behavior of architected structures in different orientations may provide new ways to design and manufacture various structures with enhanced and tunable mechanical properties.

An Inverse Method to Predict NEMS Beam Properties From Natural Frequencies

Abstract This paper presents a method to simultaneously predict the elastic modulus, axial load, and boundary conditions of a nanoelectromechanical system (NEMS) beam from a minimum of two measured natural frequencies. The proposed method addresses the challenges of the inverse problem at the nano scale, which include high natural frequencies, small geometric beam dimensions, and measurements limited to natural frequencies. The method utilizes a finite element model of an Euler–Bernoulli beam under axial loading to predict the response of the beam with axial loading and flexible boundary conditions. By expressing the finite element model in terms of dimensionless beam parameters, the proposed method may be applied to nano scale beams while maintaining numerical stability of the finite element equation of motion. With the stabilized finite element model, the NEMS beam properties are predicted by iterating through values of dimensionless beam parameters until the normalized error between predicted and measured natural frequencies is minimized. A key feature of the proposed method is the simultaneous prediction of the elastic modulus during the iterative search, resulting in a reduction of the search space and significant computational savings. Additionally, the proposed method readily accommodates an arbitrary number of measured natural frequencies without the reformulation of procedures and analyses. Numerical examples are presented to illustrate the proposed method’s ability to predict the elastic modulus, axial load, and boundary conditions. The proposed method is applied to experimental measurements of a NEMS beam, where the normalized error between predicted and measured natural frequencies is reduced below 10−3.

Dynamic Finite Element Analysis of Flexible Double Wishbone Suspension Systems with Different Damping Mechanisms

Suspension systems in running vehicles keep the occupants comfortable and isolated from road noise, disturbances, and vibrations and consequently prevent the vehicle from damage and wearing. To attain comfortable and vibration isolation conditions, both material flexibility and damping should be considered in the considered suspension model. This paper presents an incremental finite element model to study and analyze the dynamic behavior of double wishbone suspension systems considering both material flexibility and damping effects. The flexibility of the suspension links are modeled with plane frame element based on Timoshenko beam hypothesis (TBH). On the other hand, the flexibility of joints connecting the suspension links together and with the vehicle chassis is modeled with the revolute joint element. To incorporate the damping effect, viscoelastic, viscous and proportional damping are considered. An incremental viscoelastic constitutive relations, suitable for finite element implementation, are developed. The developed finite element equations of motion are solved using the Newmark technique. The developed procedure is verified by comparing the obtained results with that obtained by the developed analytical solution and an excellent agreement is found. The applicability and effectiveness of the developed procedure are demonstrated by conducting parametric studies to show the effects of the road irregularities profiles, the vehicle speed, and the material damping on the transverse deflection and the resultant stresses of suspension system. Results obtained are supportive in the mechanical design, manufacturing processes of such type of structural systems.

Finite Element Analysis of Double Wishbone Vehicle Suspension System

Suspension systems play an important role in vehicles. These systems provide passenger comfort and vibration isolation from road bumps. Thus an efficient finite element model is required to study and analyze the dynamic response of these suspension systems. This paper presents a finite element model to analyze the dynamic response of double wishbone vehicle suspension system taking into consideration both links and joints flexibilities. Links are modeled using plane frame element based on Timoshenko beam theory (TBT) kinematic relations. On the other hand, the revolute joint element, developed in ANSYS, is adopted to model joints flexibility.Both internal viscoelastic and external viscous as well as the modal proportional damping models are adopted to simulate the damping effect. The resulting dynamic finite element equations of motion are solved using Newmark numerical technique. The proposed numerical methodology is checked by comparing the obtained results with the developed analytical solution and good agreement is noticed. The applicability of the proposed procedure is demonstrated by operating parametric studies to illustrate the effects of the road irregularity configurations, the vehicle travelling velocity, as well as the material damping on the dynamic response characteristics of the double wishbone suspension system. The obtained results are helpful for the mechanical design of these structural systems.

A robust algorithm for behavior and effectiveness investigations of super-elastic SMA wires embedded in composite plates under impulse loading

The spatial and time variations of the martensite volume fraction due to the localized and time variations of the stress distribution along shape memory alloy wires embedded in composite plates have not been considered in the majority of the available researches. In the present research, an accurate algorithm is proposed to determine the localized time variations of the martensite volume fraction and carefully assess the sudden changes in the magnitudes of the induced stresses and strains, and consequently, the effectiveness of the SMA wires, under the impulse loadings. The governing finite element equations of motion of the hybrid composite plate are derived based on Hamilton’s principle. The resulting non-linear and piecewise-defined time-dependent finite element formulation is solved through a FORTRAN code, employing Newmark’s numerical procedure to evaluate the dynamic responses of the plate. Results reveal that unlike almost all of the available reports, the shape memory alloy wires embedded in the composite host plate cannot always strengthen the plates. Indeed, the behavior of the SMA wires is governed by two opposite phenomena: the significant degradation in the elastic modulus due to the austenite to martensite phase-transformation and the hysteretic behavior due to super-elasticity that consumes the strain energy in the phase-transformation process and leads to smaller stresses and lateral deflections. Therefore, when elasticity modulus of the host plate is higher than that of the SMA, the super-elasticity advantage cannot compensate the disadvantage of the degradation in the elastic modulus and leads to a plate with inferior performances.

Nonlinear Response of Laminated Panels under Blast Load

Nonlinear response of laminated composite plates and cylindrical panels subjected to a parabolically distributed blast load is studied here using a shear deformable finite element method. The nonlinear finite element equations of motion are solved in the time domain by an energy and momentum conserving time integration technique. The state-of-stress inside the laminated panels is calculated at each time-step to predict the occurrence of damage using Tsai-Wu quadratic failure criterion. The effects of span-to-thickness ratio, radius-to-span ratio and peak pressure on the nonlinear dynamic response of laminated composite plates and curved panels are investigated.

Vibration of Train-Rail-Bridge Interaction Considering Rail Irregularity with Arbitrary Wavelength

A generation method for the rail random irregularity with arbitrary wavelength interval (WI) is developed, and its accuracy and efficiency are demonstrated. Then a moving wheel-rail-bridge interaction element is derived to establish the finite element equations of motion for the train-rail-bridge interaction system, and the flow chart of assembly and calculation for the system equations is given. According to the sub-interval principle, the influences of the irregularities with the large WI and the sub-intervals on the dynamic responses are analyzed by a numerical example, and the sensitive WI of each response is discussed. The results indicate that the bridge acceleration and the contact force are both more sensitive to the irregularity with WI (1~5 m). The irregularity with WI (0.1~1 m) has less influence on the car body acceleration but mainly contributes to the rail acceleration. However, all the irregularities with wavelengths in interval (1~150 m) can have significant influences on the car body acceleration. Meanwhile, the transient jump of wheel can be simulated and should be taken into account for the derailment risk assessment.

Piezo-viscoelastically damped nonlinear frequency response of functionally graded plates with a heated plate-surface

A geometrically nonlinear frequency-domain analysis of functionally graded plates integrated with an active constrained layer damping (ACLD) arrangement is performed by developing an incremental nonlinear closed-loop dynamic finite element model of the overall plate. The active constraining layer is made of piezoelectric fiber reinforced composite (PFRC) and a heated substrate-plate surface is considered. The analysis is mainly for investigating the effect of temperature on the nonlinear vibration characteristics of the overall plate in the frequency domain and also, on the corresponding control authority of the PFRC constraining layer. A negative velocity feedback control strategy is utilized to achieve active damping. The temperature dependent material properties of the substrate plate are graded in the thickness direction according to a power law, and expressed in terms of the power law exponent and the constituent material (metal and ceramic) properties. Using the Golla-Hughes-McTavish method for modeling the viscoelastic material, the incremental nonlinear finite element equations of motion are derived in the frequency domain assuming periodic motion of the overall plate. An arc-length extrapolation solution technique is used in combination with a new strategy for determination of incremental arc-length. The numerical illustrations show a potential use of PFRC actuator in the ACLD arrangement and suggest an optimal thickness of viscoelastic layer for more effective use of PFRC. The analysis reveals the significant effects of initial thermal bending of the overall smart plate on its nonlinear dynamic behavior in the frequency domain. The effects of temperature, metal-volume fraction in substrate, fiber volume fraction in PFRC and the fiber orientation angle in the PFRC on the control authority of the ACLD layer are presented. For the use of the ACLD layer in the form of a patch, a new numerical strategy for determining its optimal location and optimal size for effective control is presented.

Dynamic Analysis of Train-Rail-Bridge Interaction considering Concrete Creep of a Multi-Span Simply Supported Bridge

A moving wheel-rail interaction element incorporating rail irregularity is derived, and the finite element equations of motion for the train-rail-bridge interaction system are established. Then, the long-term creep deformation increment of an actual pre-stressed concrete girder is predicted based on the modified creep-prediction model, and the real rail creep irregularity of a multi-span simply supported bridge is analyzed in consideration of the rail smoothing effect. Finally, the influence of the rail smoothing effect on the dynamic response is discussed by a numerical example. Also, the influences of the creep irregularities at different periods are investigated, and the effects of a control method of delaying the track-laying time for about half year for mitigating the creep irregularity are studied. The results indicate that the periodic creep irregularity can excite significantly both the natural vibration and the forced vibration for the vehicle, and leads to the sharp impact at the beam end, thereby increasing dramatically the car body acceleration and the contact force. The continuous rail structure can smooth obviously the turning angle of the creep deformation and the impact at the beam end, and should be taken into account for a more accurate analysis. Moreover, the control method has distinct control effects, especially for the car body acceleration as well as the bridge acceleration.

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.

Vibration Characteristics Analysis of Orthotropic Rectangular Sandwich Plate with Magnetorheological Elastomer

Vibration characteristics of orthotropic rectangular sandwich plate with magnetorheological (MR) elastomer core and constraining layer is presented in this study. In order to improve the vibrational behaviors of the system, the orthotropic rectangular plate is covered a MR fluid core and a constraining layer. Finite element method is adopted to derive the finite element equations of motion for the orthotropic sandwich system. The effects of the natural frequencies and loss factors of the orthotropic sandwich plate are discussed. The rheological property of the MR material can be changed and controlled by applying various magnetic fields. The modal dampings and the natural frequencies for the orthotropic sandwich plate are calculated for various magnetic fields. Besides, the effects of the strength ratio of the base plate on the natural frequencies and modal loss factors are also discussed.

On the vibration control capability of shape memory alloy composite beams

In this paper, the vibration control capability of shape memory alloy (SMA) composite beams subjected to impulsive loads is examined. In order to simulate the SMA response, a one-dimensional constitutive model is introduced which is able to reproduce pseudo-elasticity, martensite transformation/orientation and in particular ferro-elasticity effects. A numerical algorithm is presented to solve non-linear SMA constitutive model by means of an elastic-predictor inelastic-corrector return map procedure. The equivalent single layer theory of Rayleigh–Euler–Bernoulli is used to describe displacement field of SMA laminated composite beams. Geometrical non-linearity is also considered in the von Karman sense. Considering rotary inertia effects, finite element equations of motion are developed using the Hamilton principle. Newmark and Newton–Raphson methods are utilized to obtain an incremental solution of the problem. Extensive numerical results are presented to provide an insight into the influence of pre-strain, temperature, location and thickness of SMA layers on the vibration control of SMA composite beams subjected to various blast pulses. Considering ferro-elasticity effect, results reveal the fact that SMA layers with high pre-strain have a passive vibration control capability in low temperatures and yield a better efficiency in comparison with pseudo-elastic SMA layers.

Active control of geometrically non-linear transient response of sandwich beams with a flexible core using piezoelectric patches

Active control of geometrically non-linear dynamic response of sandwich beams impacted by blast pulses with integrated piezoelectric sensor/actuator patches is addressed. The analysis is based on a higher-order structural model using the first-order shear deformation theory for the face sheets and piezoelectric patches, and the extended high-order sandwich theory for the flexible core. Geometrical non-linearity is considered in the von Karman sense. Hamilton’s principle is employed to introduce a novel smart sandwich beam element which has eighteen physical nodes with mechanical and electric potential degrees of freedom plus two electric nodes for the electric potentials of the electroded surfaces. The presented smart sandwich beam element is used to develop finite element equations of motion. Constant negative velocity feedback algorithm is adapted in the active control. The solution of non-linear equations is sought by Newmark and the modified Newton–Raphson methods for dynamic analysis. Furthermore, the model behavior in free vibrations is assessed by using a direct iteration technique suitably modified for non-linear eigenvalue problems based on the QZ algorithm. A detailed analysis of the influence of geometrical non-linearity, core flexibility and piezoelectric patch arrangement on the dynamic response, frequency and loss factor of active sandwich beams is carried out.