Force-displacement characteristics of simply supported beam laminated with shape memory alloys

As a preliminary step in the nonlinear design of shape memory alloy(SMA) composite structures,the force-displacement characteristics of the SMA layer are studied.The bilinear hysteretic model is adopted to describe the constitutive relationship of SMA material.Under the assumption that there is no point of SMA layer finishing martensitic phase transformation during the loading and unloading process,the generalized restoring force generated by SMA layer is deduced for the case that the simply supported beam vibrates in its first mode.The generalized force is expressed as piecewise-nonlinear hysteretic function of the beam transverse displacement.Furthermore the energy dissipated by SMA layer during one period is obtained by integration,then its dependencies are discussed on the vibration amplitude and the SMA’s strain(Ms-Strain) value at the beginning of martensitic phase transformation.It is shown that SMA’s energy dissipating capacity is proportional to the stiffness difference of bilinear model and nonlinearly dependent on Ms-Strain.The increasing rate of the dissipating capacity gradually reduces with the amplitude increasing.The condition corresponding to the maximum dissipating capacity is deduced for given value of the vibration amplitude.The obtained results are helpful for designing beams laminated with shape memory alloys.     

A new beam element for analyzing geometrical and physical nonlinearity

Based on Timoshenko＇s beam theory and Vlasov＇s thin-walled member theory, a new model of spatial thin-walled beam element is developed for analyzing geometrical and physical nonlinearity, which incorporates an interior node and independent interpolations of bending angles and warp and takes diversified factors into consideration, such as traverse shear deformation, torsional shear deformation and their coupling, coupling of flexure and torsion, and the second shear stress. The geometrical nonlinear strain is formulated in updated Lagarange （UL） and the corresponding stiffness matrix is derived. The perfectly plastic model is used to account for physical nonlinearity, and the yield rule of von Mises and incremental relationship of Prandtle-Reuss are adopted. Elastoplastic stiffness matrix is obtained by numerical integration based on the finite segment method, and a finite element program is compiled. Numerical examples manifest that the proposed model is accurate and feasible in the analysis of thin-walled structures.     

Stability of nonlinear periodic vibrations of 3D beams

The geometrically nonlinear periodic vibrations of beams with rectangular cross section under harmonic forces are investigated using a p-version finite element method. The beams vibrate in space; hence they experience longitudinal, torsional, and nonplanar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The theory employed is valid for moderate rotations and displacements, and physical phenomena like internal resonances and change of the stability of the solutions can be investigated. Green’s nonlinear strain tensor and Hooke’s law are considered and isotropic and elastic beams are investigated. The equation of motion is derived by the principle of virtual work. The differential equations of motion are converted into a nonlinear algebraic form employing the harmonic balance method, and then solved by the arc-length continuation method. The variation of the amplitude of vibration in space with the excitation frequency of vibration is determined and presented in the form of response curves. The stability of the solution is investigated by Floquet’s theory.     

Bending of orthotropic plates resting on Pasternak’s foundations using mixed shear deformation theory

The mixed first-order shear deformation plate theory(MFPT) is employed to study the bending response of simply-supported orthotropic plates.The present plate is subjected to a mechanical load and resting on Pasternak’s model or Winkler’s model of elastic foundation or without any elastic foundation.Several examples are presented to verify the accuracy of the present theory.Numerical results for deflection and stresses are presented.The proposed MFPT is shown simplely to implement and capable of giving satisfactory results for shear deformable plates under static loads and resting on two-parameter elastic foundation.The results presented here show that the characteristics of deflection and stresses are significantly influenced by the elastic foundation stiffness,plate aspect ratio and side-to-thickness ratio.     

Vibration of two beams connected by nonlinear isolators: analytical and experimental study

To study the coupling vibration of nonlinear isolators and flexible bodies, test rigs of two flexible beams connected by wire mesh isolators are constructed and investigated both experimentally and analytically. A five-parameter polynomial model of wire mesh isolators is derived by identifying parameters in the frequency domain with the sine-sweep test. For obtaining the parameters that are valid in a wide range of frequency, a numerically assisted identification method is developed. With this model, the vibration of two flexible beams connected by wire mesh isolators is studied. The frequency response is obtained analytically by employing the Green’s function method and harmonic balance method. Sine-sweep test results with three test rigs show good coherence with the corresponding numerical results. With obtained experimental results and numerical results, effect of connection parameters is studied in detail. It is found that traditional design rules for isolators are no longer effective and the coupling vibration must be investigated in the design phase. Another phenomenon is that the damping has a function of weakening the effect of nonlinear stiffness. Nonlinear stiffness and nonlinear damping can decrease the transmissibility along with the increase of the excitation level.     

Second-order approximation of primary resonance of a disk-type piezoelectric stator for traveling wave vibration

Considering the quadratic nonlinear constitutive relations of piezoelectric materials, a traveling wave dynamic model for a lead zirconate titanate stator of a traveling wave ultrasonic motor is established using Hamilton’s principle and the Rayleigh–Ritz method. Applying the method of multiple scales, the second-order approximation of the primary resonance for traveling wave vibration of the stator is investigated. The second harmonic component is found in the primary response of the stator, which arises from the quadratic stiffness in the condition of weak excitation. In the region of the resonance, the two coupled modals are split and the lower-order peak bends to the left, hence a jump and delay exist in the response. In this way numerical results are given to verify the feasibility of the analytical approach. The results provide a theoretical foundation for further nonlinear dynamic analysis and design of the traveling wave ultrasonic motor.     

Static analysis of ultra-thin beams based on a semi-continuum model

A linear semi-continuum model with discrete atomic layers in the thickness direction was developed to investigate the bending behaviors of ultra-thin beams with nanoscale thickness.The theoretical results show that the deflection of an ultra-thin beam may be enhanced or reduced due to different relaxation coefficients.If the relaxation coefficient is greater/less than one,the deflection of micro/nano-scale structures is enhanced/reduced in comparison with macro-scale structures.So,two opposite types of size-dependent behaviors are observed and they are mainly caused by the relaxation coefficients.Comparisons with the classical continuum model,exact nonlocal stress model and finite element model (FEM) verify the validity of the present semi-continuum model.In particular,an explanation is proposed in the debate whether the bending stiffness of a micro/nano-scale beam should be greater or weaker as compared with the macro-scale structures.The characteristics of bending stiffness are proved to be associated with the relaxation coefficients.     

Free vibration of axially loaded thin-walled composite Timoshenko beams

Based on shear-deformable beam theory, free vibration of thin-walled composite Timoshenko beams with arbitrary layups under a constant axial force is presented. This model accounts for all the structural coupling coming from material anisotropy. Governing equations for flexural-torsional-shearing coupled vibrations are derived from Hamilton’s principle. The resulting coupling is referred to as sixfold coupled vibrations. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for thin-walled composite beams to investigate the effects of shear deformation, axial force, fiber angle, modulus ratio on the natural frequencies, corresponding vibration mode shapes and load–frequency interaction curves.     

Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.     

Coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments

Delaminations in structures may significantly reduce the stiffness and strength of the structures and may affect their vibration characteristics. As structural components, beams have been used for various purposes, in many of which beams are often subjected to axial loads and static end moments. In the present study, an analytical solution is developed to study the coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments. Euler–Bernoulli beam theory and the "free mode" assumption in delamination vibration are adopted. This is the first study of the influences of static end moments upon the effects of delaminations on natural frequencies, critical buckling loads and critical moments for lateral instability. The results show that the effects of delamination on reducing natural frequencies, critical buckling load and critical moment for lateral instability are aggravated by the presence of static end moment. In turn, the effects of static end moments on vibration and instability characteristics are affected by the presence of delamination. The analytical results of this study can serve as a benchmark for finite element method and other numerical solutions.     

Observation of the shape of a water drop on an oscillating Teflon plate

This is an experimental investigation of the dynamics of water drops under large-amplitude forcing within a frequency range of 10–700 Hz. Water drops on an oscillating horizontal plate displayed various behaviors, such as bouncing, dividing, ejecting, axisymmetric polygonal vibration, and polygonal but not axisymmetric vibration. These last two were alike in that the capillary waves of a water drop gave rise to various polygons. However, the planar directions of the polygons of the non-axisymmetric vibrations seemed stable, and their frequencies were the same as the induced frequencies of the oscillating plate. These two features make non-axisymmetric polygonal vibration different from the axisymmetric polygonal vibration reported to date, and the phenomenon has not yet been successfully analyzed. Trials of simulating some shapes of non-axisymmetric polygonal vibrations according to Rayleigh’s theory are described in this paper.     

飞机槽型大开口结构刚度加强研究

为了飞机典型的方舱型机身大开口结构能够满足刚度设计要求,设计了一种对槽型大开口结构增加4个边梁进行刚度加强的设计方案,并构建了工程分析模型,与无开口槽型结构的弯曲刚度及扭转刚度进行了对比研究,得到了两种构型的刚度比,进一步得到了满足一定刚度指标下的边梁面积计算公式,提出了方舱型机身大开口结构刚度设计流程,可以用于方案阶段的飞机大开口结构加强设计。     

Dissipation-induced instabilities and symmetry

The paradox of destabilization of a conservative or non-conservative system by small dissipation,or Ziegler’s paradox(1952),has stimulated a growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations.Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary,associated with Whitney’s umbrella.The first explanation of Ziegler’s paradox was given(much earlier)by Oene Bottema in 1956.The aspects of the mechanics and geometry of dissipation-induced instabilities with an application to rotor dynamics are discussed.     

A nonlinear microbeam model based on strain gradient elasticity theory with surface energy

A microscale nonlinear Bernoulli–Euler beam model on the basis of strain gradient elasticity with surface energy is presented. The von Karman strain tensor is used to capture the effect of geometric nonlinearity. Governing equations of motion and boundary conditions are obtained using Hamilton’s principle. In particular, the developed beam model is applicable for the nonlinear vibration analysis of microbeams. By employing a global Galerkin procedure, the ordinary differential equation corresponding to the first mode of nonlinear vibration for a simply supported microbeam is obtained. Numerical investigations show that in a microbeam having a thickness comparable with its material length scale parameter, the strain gradient effect on increasing the beam natural frequency is higher than that of the geometric nonlinearity. By increasing the beam thickness, the strain gradient effect decreases or even diminishes. In this case, geometric nonlinearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for beams with some specific thickness-to-length parameter ratios, both geometric nonlinearity and size effect have significant role on increasing the frequency of nonlinear vibration.     

Vibration analysis of bi-layered FGM cylindrical shells

In the present study, a vibration frequency analysis of a bi-layered cylindrical shell composed of two independent functionally graded layers is presented. The thickness of the shell layers is assumed to be equal and constant. Material properties of the constituents of bi-layered functionally graded cylindrical shell are assumed to vary smoothly and continuously through the thickness of the layers of the shell and are controlled by volume fraction power law distribution. The expressions for strain–displacement and curvature–displacement relationships are utilized from Love’s first approximation linear thin shell theory. The versatile Rayleigh–Ritz approach is employed to formulate the frequency equations in the form of eigenvalue problem. Influence of material distribution in the two functionally graded layers of the cylindrical shells is investigated on shell natural frequencies for various shell parameters with simply supported end conditions. To check the validity, accuracy and efficiency of the present methodology, results obtained are compared with those available in the literature.     

Natural vibration of a sandwich beam on an elastic foundation

The natural vibration of an elastic sandwich beam on an elastic foundation is studied. Bernoulli’s hypotheses are used to describe the kinematics of the face layers. The core layer is assumed to be stiff and compressible. The foundation reaction is described by Winkler’s model. The system of equilibrium equations is derived, and its exact solution for displacements is found. Numerical results are presented for a sandwich beam on an elastic foundation of low, medium, or high stiffness __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 57–63, May 2006.     

纳米圆轴在弹性介质中的扭转振动分析

基于非局部应变梯度理论,考虑周围弹性介质的影响,研究纳米圆轴的扭转自由振动。首先通过Hamilton原理推导纳米圆轴扭转振动的控制方程及边界条件,接着采用微分求积法得到控制方程及三类边界条件的离散形式,最后由数值计算结果分析扭转振动特性。讨论了两个小尺度参数和弹性介质刚度的变化对振动频率的影响,并通过小尺度参数比对振动频率的影响分析两个尺度参数的耦合作用。研究结果表明,扭转自由振动频率随非局部参数增加而减小,随应变梯度尺度参数、弹性介质刚度增加而增大;当非局部参数大于应变梯度尺度参数时,小尺度效应体现为非局部效应,相反则体现为应变梯度效应。     

Effect of elastic foundation on the vibration of orthotropic elliptic plates with varying thickness

Two-dimensional boundary characteristic orthonormal polynomials are used in the Rayleigh–Ritz method to study the free vibration of rectangular orthotropic elliptic plates resting on a Winkler elastic foundation. Two types of varying thickness are considered: (1) linearly varying with respect to the concentric ellipses, (2) linearly varying along both the principal axes simultaneously. Numerical results for frequencies are presented in tables. Nodal lines are shown in figures. Effects of stiffness of elastic foundation, aspect ratios of plates, degree of orthotropy, and types of thickness variation on vibration frequencies of elliptical plates are also discussed.     

Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions

Axially moving beams are often discussed with several classic boundary conditions, such as simply-supported ends, fixed ends, and free ends. Here, axially moving beams with generalized boundary conditions are discussed for the first time. The beam is supported by torsional springs and vertical springs at both ends. By modifying the stiffness of the springs, generalized boundaries can replace those classical boundaries.Dynamic stiffness matrices are, respectively, established for axially moving Timoshenko beams and Euler-Bernoulli(EB) beams with generalized boundaries. In order to verify the applicability of the EB model, the natural frequencies of the axially moving Timoshenko beam and EB beam are compared. Furthermore, the effects of constrained spring stiffness on the vibration frequencies of the axially moving beam are studied. Interestingly, it can be found that the critical speed of the axially moving beam does not change with the vertical spring stiffness. In addition, both the moving speed and elastic boundaries make the Timoshenko beam theory more needed. The validity of the dynamic stiffness method is demonstrated by using numerical simulation.     

A discrete model of a rope with bending stiffness or viscous damping

A discrete model of a rope is developed and used to simulate the plane motion of the rope fixed at one end.Actually,two systems are presented,whose members are rigid but non-ideal joints involve elasticity or dissipation.The dissipation is reflected simply by viscous damping model, whereas the bending stiffness conception is based on the classical curvature-bending moment relationship for beams and simple geometrical formulas.Equations of motion are derived and their complexity is discussed from the computational point of view.Since modified extended backward differentiation formulas(MEBDF)of Cash are implemented to solve the resulting initial value problems,the technique scheme is outlined.Numerical experiments are performed and influences of the elasticity and damping on behaviour of the model are analyzed.Basic energy principles are used to verify the obtained results.