Small Deflection Assumption Research Articles

Large Deflection Effects on the Energy Release Rate and Mode Partitioning of the Single Cantilever Beam Sandwich Debond Configuration

Abstract This paper investigates the effects of large deflections on the energy release rate and mode partitioning of face/core debonds for the single cantilever beam sandwich composite testing configuration, which is loaded with an applied shear force and/or a bending moment. Studies in this topic have been done by employing geometrically linear theories (either Euler–Bernoulli or Timoshenko beam theory). This assumes that the deflection at the tip of the loaded debonded part is small, which is not always the case. To address this effect, we employ the elastica theory, which is a nonlinear theory, for the debonded part. An elastic foundation analysis and the linear Euler Bernoulli theory are employed for the “joined” section where a series of springs exists between the face and the substrate (core and bottom face). A closed form expression for the energy release rate is derived by the use of J-integral. Another closed-form expression for the energy release rate is derived from the energy released by a differential spring as the debond propagates. Furthermore, a mode partitioning angle is defined based on the displacement field solution. Results for a range of core materials are in very good agreement with the corresponding ones from a finite element analysis. The results show that large deflection effects reduce the energy release rate but do not have a noteworthy effect on the mode partitioning. A small deflection assumption can significantly overestimate the energy release rate for relatively large applied loads and/or relatively long debonds.

Dynamic Characteristics of Microring Driven by the Symmetrically Distributed Electrostatic Force

This paper aims at investigating the dynamic characteristics of a microring driven by dual arch electrodes because they are basic elements of microelectrostatic motors. The dual arch electrodes surround the periphery of the microring and are arranged symmetrically to the center of the ring. The electrodes are fixed while the microring is flexible. The electrostatic force will deform the microring, while the deflection of the microring changes the gap between the microring and the electrodes, thereby changing the electrostatic force. Therefore, this is an electromechanical coupling effect. The nonlinear partial-differential equation that governs the motion of the microring is derived based on thin shell theory. Then, based on the assumption of small deflection, the nonlinear governing equation is linearized by truncating the higher-order terms of the Taylor series expansion of the nonlinear electrostatic force. After that, the linearized governing equation is discretized into a set of ordinary differential equations using Galerkin method in which the mode shape functions of the ring are adopted. The influences of the structural damping of the microring and the span of the arch electrodes on the forced response and dynamical stabilities of the microring are investigated. The results show that the damping ratio has a great influence on the system instability during high-frequency excitation. The unstable region of the system can increase with the increase of the electrode span; the response amplitude can also be increased within a certain range.

Design, analysis and validation of the bridge-type displacement amplification mechanism with circular-axis leaf-type flexure hinges for micro-grasping system

For stable and flexible manipulation, realizing parallel grasping and large output stroke simultaneously has become a key issue in the researches on micro-grasping system. The typical solution is to use compliant orthogonal displacement amplification mechanism (such as bridge-type mechanism), in which the flexure hinges are generally axial load-dominated, whereas the deflection of traditional straight-axis flexure hinges is not related to the axial load under small deflection assumption. In this study, a bridge-type mechanism with circular-axis leaf-type (CALT) flexure hinges is designed, analyzed and validated. Through modeling the generalized displacements of CALT flexure hinge, the positive direction of its axis in the bridge-type mechanism is designed. The precise models of load relations for the CALT flexure hinges in the bridge-type DAM are established, and the multi-objective parametric optimization is further given. Small deflection-based static FEA results verify the generalized displacements’ models of CALT flexure hinge as well as the parametric optimization result. Compared with the corresponding bridge-type mechanism with the largest output displacement in the traditional researches, the output displacement of optimal result is improved effectively. For the design validation, the optimal design result is further applied to construct a piezoelectric-driven microgripper, which grasps parallelly and enlarges the output stroke simultaneously.

Free vibration behavior of viscoelastic annular plates using first order shear deformation theory

In this paper, an analytical procedure based on the perturbation technique is presented to study the free vibrations of annular viscoelastic plates by considering the first order shear deformation theory as the displacement field. The viscoelastic properties obey the standard linear solid model. The equations of motion are extracted for small deflection assumption using the Hamilton

Effect of linear and non-linear blade modelling techniques on simulated fatigue and extreme loads using Bladed

There is a trend in the wind industry towards ever larger and more flexible turbine blades. Blade tip deflections in modern blades now commonly exceed 10% of blade length. Historically, the dynamic response of wind turbine blades has been analysed using linear models of blade deflection which include the assumption of small deflections. For modern flexible blades, this assumption is becoming less valid. In order to continue to simulate dynamic turbine performance accurately, routine use of non-linear models of blade deflection may be required. This can be achieved by representing the blade as a connected series of individual flexible linear bodies - referred to in this paper as the multi-part approach. In this paper, Bladed is used to compare load predictions using single-part and multi-part blade models for several turbines. The study examines the impact on fatigue and extreme loads and blade deflection through reduced sets of load calculations based on IEC 61400-1 ed. 3. Damage equivalent load changes of up to 16% and extreme load changes of up to 29% are observed at some turbine load locations. It is found that there is no general pattern in the loading differences observed between single-part and multi-part blade models. Rather, changes in fatigue and extreme loads with a multi-part blade model depend on the characteristics of the individual turbine and blade. Key underlying causes of damage equivalent load change are identified as differences in edgewise- torsional coupling between the multi-part and single-part models, and increased edgewise rotor mode damping in the multi-part model. Similarly, a causal link is identified between torsional blade dynamics and changes in ultimate load results.

Comparison of linear and non-linear blade model predictions in Bladed to measurement data from GE 6MW wind turbine

The length and flexibility of wind turbine blades are increasing over time. Typically, the dynamic response of the blades is analysed using linear models of blade deflection, enhanced by various ad-hoc non-linear correction models. For blades undergoing large deflections, the small deflection assumption inherent to linear models becomes less valid. It has previously been demonstrated that linear and nonlinear blade models can show significantly different blade response, particularly for blade torsional deflection, leading to load prediction differences. There is a need to evaluate how load predictions from these two approaches compare to measurement data from the field.In this paper, time domain simulations in turbulent wind are carried out using the aero-elastic code Bladed with linear and non-linear blade deflection models. The turbine blade load and deflection simulation results are compared to measurement data from an onshore prototype of the GE 6MW Haliade turbine, which features 73.5m long LM blades. Both linear and non-linear blade models show a good match to measurement turbine load and blade deflections. Only the blade loads differ significantly between the two models, with other turbine loads not strongly affected. The non-linear blade model gives a better match to the measured blade root flapwise damage equivalent load, suggesting that the flapwise dynamic behaviour is better captured by the non-linear blade model. Conversely, the linear blade model shows a better match to measurements in some areas such as blade edgewise damage equivalent load.

An analytical procedure for transient response determination of annular FSDT and CPT nanoplates via nonlocal elasticity theory

In this paper, an analytical procedure is introduced for transient response determination of annular nanoplates under impulsive transverse load and radial stress. The equations of motion are extracted using the Hamilton’s principle by small deflection assumption and considering the Eringen nonlocal elasticity theory in conjunction with the first order shear deformation theory as the displacement field. These equations which are a system of partial differential equations with variable coefficients are solved using the perturbation technique and the eigenfunction expansion method. The results are compared with those obtained by the classical plate theory and finite elements method and the effects of nonlocal parameter, radial stress and geometries on the response are studied.

A free vibration analysis of toroidal composite shells in free space

The main aim of this paper is to present a dynamic analysis of toroidal shell structures in free space. The simplified thin shell theory, based on the typical small deflection assumption, but enriched by the transverse shear terms, was used. The constituting material is not isotropic but a multilayer composite angle-ply laminate, with non-uniform thickness.A numerical procedure based on the classical Rayleigh–Ritz method, was utilized. The dynamic independent variables were expressed in terms of a double Fourier series expansion as function of the azimuth and internal meridian circular angle. This allowed to determine easily the strain and kinetic energy expressions, and consequently the stiffness and mass matrices.Then, the numerical problem was simplified to the classical generalized eigenvalue problem relation by differentiating the whole energetic functional, as in the Rayleigh–Ritz method. The solutions were determined by using an appropriate algorithm. The high frequency vibration modes were considered and their peculiarities with respect to the low frequency modes were pointed out. The dependence of the obtained results on the laminate winding angle was also considered.Comparisons with the other authors results were necessary to validate the utilized numerical approach.

Unbound motion of massive particles in the Schwarzschild metric: Analytical description in case of strong deflection

Deflection angles of massive test particles moving along an unbound trajectory in the Schwarzschild metric are considered for the case of large deflection. We analytically consider the strong deflection limit, which is opposite to the commonly applied small deflection approximation and corresponds to the situation when a massive particle moves from infinity, makes several revolutions around a central object and goes to infinity. For this purpose we rewrite an integral expression for the deflection angle as an explicit function of the parameters determining the trajectory and expand it. Remarkably, in the limiting case of strong deflection, we succeed in deriving for the first time the analytical formulas for deflection angles as explicit functions of parameters at infinity. In particular, we show that in this case the deflection angle can be calculated as an explicit function of the impact parameter and velocity at infinity beyond the usual assumption of small deflection.

Vibration of Visco-Elastic Parallelogram Plate with Thickness Variation Linearly in one Direction and Parabolic in another Direction

The main objective of the present investigation is to study the vibration of visco-elastic parallelogram plate whose thickness varies bi-directionally. It is assumed that the plate is clamped on all the four edges and that the thickness varies linearly in one direction and parabolically in another direction. Using the separation of variables method and Rayleigh-Ritz technique with a two-term deflection function, the governing differential equation has been solved for vibration of visco-elastic parallelogram plate. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The assumption of small deflection is made. Visco-elastic of the plate is taken of the “Kelvin Type”. Time period and deflection function at different point for the first two modes of vibration are calculated for various values of taper constant, aspect ratio and skew angle and results are presented in tabular form. Alloy “Duralumin” is considered for all the material constants used in numerical calculations.

Research on Semi-Rigid Connections Calculation of Laminated Wood Frame Beam-Column

Generally laminated wood frame beam-column uses the additional steel bolts to connect, its performance is a semi-rigid characteristics under the loads. In the small deflection assumption conditions, this paper is based on the theory of semi-rigid connection, derived the semi-rigid rod end stiffness matrix and moment calculation formula of laminated wood structural frame beam-column element under the arbitrary loads. By SAP2000 the type of flexible coefficient is adopted to establish finite element numerical analysis based on 1# wood frame structure-building of science and technology R&D center in Hebei province. The result shows that each component is basically the same with the original design internal forces while the flexible coefficient is range of 0.41 to 0.82, the framework of semi-rigid connections in the actual calculation should be considered. The internal force calculation method is proposed, which provide a basis for wood design calculation that takes into account the semi-rigid connections.

Simplified buckling and ultimate strength analysis of composite plates in compression

The work presented here concerns the ultimate strength predictions of simply supported, square plates of laminated composite material subjected to uniaxial in-plane compressive load. Plates having a range of thicknesses and initial geometric imperfections have been investigated. Several models are established, each based on first order shear deformation theory and assumption of small deflections. The approaches give reasonable but somewhat conservative estimates for the thicker plates considered, while for the thinner plates, neglect of the post-buckling behaviour makes the results very conservative. It will be necessary to use a large deflection plate theory for some of the models to realise their full potential.

On the Dynamics of the Micro-Ring Driven by Traveling Bias Voltages

There is no literature mentioned the modeling of the microstructures subjected to traveling electrostatic forces. This paper is the first time to present an analytical approach to investigate the dynamics of a micro-ring structure driven by the traveling bias voltage. The traveling electrostatic forces may come from the sequentially-actuated actuating electrodes arranged around the flexible ring. A linearized distributed model considering the electromechanical coupling effect is derived based on the small deflection assumption. According to the analytical results, the stiffness of the micro-ring will be softened periodically with the traveling speed of the driving voltage and the variation increases with the increasing of the voltage.

Vibration of Visco-Elastic Parallelogram Plate with Parabolic Thickness Variation

The main objective of the present investigation is to study the vibration of visco-elastic parallelogram plate whose thickness varies parabolically. It is assumed that the plate is clamped on all the four edges and that the thickness varies parabolically in one direction i.e. along length of the plate. Rayleigh-Ritz technique has been used to determine the frequency equation. A two terms deflection function has been used as a solution. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The assumption of small deflection and linear visco-elastic properties of “Kelvin” type are taken. We have calculated time period and deflection at various points for different values of skew angles, aspect ratio and taper constant, for the first two modes of vibration. Results are supported by tables. Alloy “Duralumin” is considered for all the material constants used in numerical calculations

Thermal Effect on Vibration of Clamped Visco-Elastic Rectangular Plate with Parabolic Thickness Variation in Both Directions

The analysis presented here is to study the effect of thermal gradient on the vibration of visco-elastic rectangular plate (having clamped boundary condition on all the four edges) of variable thickness whose thickness varies parabolically in both directions. The effect of linear temperature variation has been considered. A frequency equation of plate has been obtained by Rayleigh-Ritz technique with two terms of deflection function. The assumption of small deflection and linear visco-elastic properties of ‘Kelvin’ type are taken. Deflection and time period corresponding to the first two modes of vibrations for clamped plate have been computed for various combinations of aspect ratio, thermal constants, and taper constants. Numerical computations have been performed for an alloy ‘Duralium’ and the results obtained are depicted graphically.

Thermal effect on vibration of clamped visco-elastic rectangular plate with parabolic thickness variation in both directions

The analysis presented here is to study the effect of thermal gradient on the vibration of visco-elastic rectangular plate (having clamped boundary condition on all the four edges) of variable thickness whose thickness varies parabolically in both directions. The effect of linear temperature variation has been considered. A frequency equation of plate has been obtained by Rayleigh-Ritz technique with two terms of deflection function. The assumption of small deflection and linear visco-elastic properties of ‘Kelvin’ type are taken. Deflection and time period corresponding to the first two modes of vibrations for clamped plate have been computed for various combinations of aspect ratio, thermal constants, and taper constants. Numerical computations have been performed for an alloy ‘Duralium’ and the results obtained are depicted graphically.

Topology optimization for synthesis of contact-aided compliant mechanisms using regularized contact modeling

A topology optimization technique for systematically designing contact-aided compliant mechanisms (CCM) is presented in this paper. A CCM is a single piece elastic body that uses intermittent contacts in addition to elastic deformation to transmit force and motion. Contact interactions give rise to interesting nonlinear and nonsmooth behaviors even under the small deflection assumption made in this work. The difficulties associated with the nondifferentiability inherent in the CCM systematic synthesis problem are circumvented by using a regularized contact model. This model uses a smooth approximation of the unilateral displacement constraints that are used to model contact interactions. The use of a regularized contact model in the underlying state problem makes it possible to use efficient smooth optimization algorithms for the systematic synthesis of CCMs. The formulation of the design problem for CCMs, sensitivity analysis, and solution methodology are presented. The paper includes CCM designs that exhibit nonsmooth motion and force transmission characteristics, which are not possible or practical with compliant mechanisms that do not use intermittent contact interactions.

CONTROL OF SLEW MANEUVER OF A FLEXIBLE BEAM MOUNTED NON-RADIALLY ON A RIGID HUB: A GEOMETRICALLY EXACT MODELLING APPROACH

The positioning control using a joint torque is studied for a hub–beam system with a tip payload. In order to make the axial vibration of the beam accessible to the joint torque, the flexible beam mounted non-radially on the rigid hub is considered. Neither model truncation nor small deflection assumption is employed throughout the process of dynamic modelling and controller design. A joint PD controller with additional feedback of joint acceleration and root strain of the beam is derived using a Lyapunov-type method. Global asymptotic stability of the desired equilibrium position is proved rigorously. The geometrically exact formulation presented in this work can be viewed as a generalization of the related work based on the small deflection assumption investigated in the existing literature.

An analysis of thermal stress and distortion in bead-on-plate welding using laminated isotropic plate theory

In this study, thin-plate bending theory using the assumption of small deflection has been adopted for a two-dimensional analysis of the thermo-elasto-plastic behavior of plates in welding to overcome some difficulties arising from a fully three-dimensional approach. For simulating the temperature gradient through the plate thickness, the concept of a layered plate has been proposed, and the thermo-elasto-plastic behavior of the plate during GMA bead-on-plate welding analyzed by using the finite-element method. It was revealed that the suggested method can predict the trend of welding distortions quite satisfactorily.

Closed-form dynamic model of planar multilink lightweight robots

Closed-form equations of motion are presented for planar lightweight robot arms with multiple flexible links. The kinematic model is based on standard frame transformation matrices describing both rigid rotation and flexible displacement, under small deflection assumption. The Lagrangian approach is used to derive the dynamic model of the structure. Links are modeled as Euler-Bernoulli beams with proper clamped-mass boundary conditions. The assumed modes method is adopted in order to obtain a finite-dimensional model. Explicit equations of motion are detailed for two-link case assuming two modes of vibration for each link. The associated eigenvalue problem is discussed in relation with the problem of time-varying mass boundary conditions for the first link. The model is cast in a compact form that is linear with respect to a suitable set of constant parameters. Extensive simulation results that validate the theoretical derivation are included. >