On the three-dimensional vibrations of a hollow elastic torus of annular cross-section

This paper offers a three-dimensional elasticity-based variational Ritz procedure to examine the natural vibrations of an elastic hollow torus with annular cross-section. The associated energy functional minimized in the Ritz procedure is formulated using toroidal coordinates (r,q, j)({r,\theta , \varphi}) comprised of the usual polar coordinates (r, θ) originating at each circular cross-sectional center and a circumferential coordinate j{\varphi} around the torus originating at the torus center. As an enhancement to conventional use of algebraic–trigonometric polynomials trial series in related solid body vibration studies in the associated literature, the assumed torus displacement, u, v and w in the r, θ and j{\varphi } toroidal directions, respectively, are approximated in this work as a triplicate product of Chebyshev polynomials in r and the periodic trigonometric functions in the θ and j{\varphi} directions along with a set of generalized coefficients. Upon invoking the stationary condition of the Lagrangian energy functional for the elastic torus with respected to these generalized coefficients, the usual characteristic frequency equations of natural vibrations of the elastic torus are derived. Upper bound convergence of the first seven non-dimensional frequency parameters accurate to at least five significant figures is achieved by using only ten terms of the trial torus displacement functions. Non-dimensional frequencies of elastic hollow tori are examined showing the effects of varying torus radius ratio and cross-sectional radius ratio.     

Rheological–dynamical analogy: Prediction of damping parameters of hysteresis damper

This research aims to predict the damping parameters of hysteresis damper based on an analytical rheological–dynamical (RDA) visco-elasto-plastic solution of one-dimensional longitudinal continuous vibrations of a bar. A visco-elasto-plastic bar or damper is an energy dissipation device. An attempt is made to estimate quantitatively the influence of material physical parameters of materials on the damping ratio in both the linear visco-elastic analysis and the nonlinear visco-elasto-plastic analysis of damper subjected to external vibration forces. Two types of damping are considered: viscous damping in the case of linear analysis, defined as stiffness and/or mass proportional and, in the case of nonlinear analysis, hysteresis damping caused by inelastic deformations of damper. Owing to the visco-elastic nature of the materials of the damper and the frequency dependence of the viscous damping ratio ξ, it is useful to consider separately the situations arising when ξ is positive (the system is stable) and when it is negative. A negative damping ratio means that the complementary solution of the response would not die away (the system is unstable because of factor e^{ξ · ω · t}). In the case of nonlinear analysis, the force–displacement relation is nonlinear, so it is very difficult to predict the actual damping and stiffness coefficients, even if the force–displacement characteristic is simply perfect elasto-plastic. Using the RDA method, which takes into account the rate of release of visco-elasto-plastic energy of the dissipation devices; nonlinear behaviors are linearized, enabling to obtain the equivalent damping and stiffness coefficients and the effective period for the damper.     

“Non-linear normal modes” and the generalized Ritz method in the problems of vibrations of non-linear elastic continuous systems

In the study of natural vibrations of non-linear elastic systems it is shown that the mode shape of the vibration can vary with the amplitude as well as the frequency, and that the amplitude frequency relation is strongly affected by constraints imposed on the mode shape in an approximate solution. A method is developed which assumes the approximate solution in the form of a truncated series in which, instead of the set of coefficients, the set of functions of spatial variables is unknown and then determined by a procedure that can be regarded as a generalization of the Ritz method. The problem of variations of the normal mode shapes and of the associated natural frequencies with the amplitude is illustrated by two examples of beams with non-linear boundary conditions, and the amplitude-frequency relation is compared to that corresponding to the a priori assumed linear normal mode solution. Further possible consequences of the mode shape amplitude variations in forced, resonant motion of nonconservative systems are also indicated.     

Elasticity solutions for functionally graded annular plates subject to biharmonic loads

Based on England’s expansion formula for displacements, the elastic field in a transversely isotropic functionally graded annular plate subjected to biharmonic transverse forces on its top surface is investigated using the complex variables method. The material parameters are assumed to vary along the thickness direction in an arbitrary fashion. The problem is converted to determine the expressions of four analytic functions α (ζ), β (ζ), ? (ζ) and ψ (ζ) under certain boundary conditions. A series of simple and practical biharmonic loads are presented. The four analytic functions are constructed carefully in a biconnected annular region corresponding to the presented loads, which guarantee the single-valuedness of the mid-plane displacements of the plate. The unknown constants contained in the analytic functions can be determined from the boundary conditions that are similar to those in the plane elasticity as well as those in the classical plate theory. Numerical examples show that the material gradient index and boundary conditions have a significant influence on the elastic field.     

A quadrilateral element based on refined global-local higher-order theory for coupling bending and extension thermo-elastic multilayered plates

In this paper a refined higher-order global-local theory is presented to analyze the laminated plates coupled bending and extension under thermo-mechanical loading. The in-plane displacement fields are composed of a third-order polynomial of global coordinate z in the thickness direction and 1,2–3 order power series of local coordinate ζ_k in the thickness direction of each layer, which is identical to the 1,2–3 global-local higher-order theory by Li and Liu [Li, X.Y., Liu, D., 1997. Generalized laminate theories based on double superposition hypothesis. Int. J. Numer. Methods Eng. 40, 1197–1212] Moreover, a second-order polynomial of global coordinate z in the thickness direction is chosen as transverse displacement field. The transverse shear stresses can satisfy continuity at interfaces, and the number of unknowns does not depend on the layer numbers of the laminate.Based on this theory, a quadrilateral laminated plate element satisfying the requirement of C¹ continuity is presented. By solving both bending and thermal expansion problems of laminates, it can be found that the present refined theory is very accurate and obviously superior to the existing 1,2–3 global-local higher-order theory. The most attractive feature of this theory is that the transverse shear stresses can be accurately predicted from direct use of constitutive equations without any post-processing method. It is also shown that the present quadrilateral element possesses higher accuracy.     

Three-dimensional vibration analysis of prisms with isosceles triangular cross-section

This paper studies the three-dimensional (3-D) free vibration of uniform prisms with isosceles triangular cross-section, based on the exact, linear and small strain elasticity theory. The actual triangular prismatic domain is first mapped onto a basic cubic domain. Then the Ritz method is applied to derive the eigenfrequency equation from the energy functional of the prism. A set of triplicate Chebyshev polynomial series, multiplied by a boundary function chosen to, a priori, satisfy the geometric boundary conditions of the prism is developed as the admissible functions of each displacement component. The convergence and comparison study demonstrates the high accuracy and numerical robustness of the present method. The effect of length-thickness ratio and apex angle on eigenfrequencies of the prisms is studied in detail and the results are compared with those obtained from the classical one-dimensional theory and the 3-D finite element method. Sets of valuable data known for the first time are reported, which can serve as benchmark values in applying various approximate beam and rod theories.     

A unified Chebyshev–Ritz formulation for vibration analysis of composite laminated deep open shells with arbitrary boundary conditions

In this paper, a unified Chebyshev–Ritz formulation is presented to investigate the vibrations of composite laminated deep open shells with various shell curvatures and arbitrary restraints, including cylindrical, conical and spherical ones. The general first-order shear deformation shell theory is employed to include the effects of rotary inertias and shear deformation. Under the current framework, regardless of boundary conditions, each of displacements and rotations of the open shells is invariantly expressed as Chebyshev orthogonal polynomials of first kind in both directions. Then, the accurate solutions are obtained by using the Rayleigh–Ritz procedure based on the energy functional of the open shells. The convergence and accuracy of the present formulation are verified by a considerable number of convergence tests and comparisons. A variety of numerical examples are presented for the vibrations of the composite laminated deep shells with various geometric dimensions and lamination schemes. Different sets of classical constraints, elastic supports as well as their combinations are considered. These results may serve as reference data for future researches. Parametric studies are also undertaken, giving insight into the effects of elastic restraint parameters, fiber orientation, layer number, subtended angle as well as conical angle on the vibration frequencies of the composite open shells.     

A polar theory for vibrations of thin elastic shells

In relation to a polar continuum, this paper presents a 2-D shear deformable theory for the high frequency vibrations of a thin elastic shell. To begin with, the 3-D fundamental equations of the micropolar elastic continuum are expressed as the Euler–Lagrange equations of a unified variational principle. Next, the kinematic variables of the shell are represented by the power series expansions in its thickness coordinate, and then, they are used to establish the 2-D theory by means of the variational principle. The 2-D theory is derived in invariant variational and differential forms and governs all the types of vibrations of the functionally graded micropolar shell. Lastly, the uniqueness is investigated in solutions of the initial mixed boundary value problems defined by the 2-D theory, and some of special cases are indicated in the theory.     

The validity range of CPT and Mindlin plate theory in comparison with 3-D vibrational analysis of circular plates on the elastic foundation

The validity and the range of applicability of the classical plate theory (CPT) and the first-order shear deformation plate theory, also called Mindlin plate theory (MPT), in comparison with three-dimensional (3-D) p-Ritz solution are presented for freely vibrating circular plates on the elastic foundation with different boundary conditions. In order to achieve this purpose, a study of the 3-D elasticity solution is carried out to determine the free vibration frequencies of clamped, simply supported and free circular plates resting on an elastic foundation. The Pasternak model with adding a shear layer to the Winkler model is used for describing the elastic foundation. In addition to being employed the p-Ritz algorithm, the analysis is based on the linear, small strain and 3-D elasticity theory. In this analysis method, a set of orthogonal polynomial series in a cylindrical polar coordinate system is used to arrive eigenvalue equation yielding the natural frequencies for the circular plates. The accuracy of these results is verified by appropriate convergence studies and checked with the available literature and the MPT. Furthermore, the effect of the foundation stiffness parameters, thickness-radius ratio, and different boundary conditions on the ill-conditioning of the mass matrix as well as on the vibration behavior of the circular plates is investigated. Afterwards, the validity and the range of applicability of the results obtained on the basis of the CPT and MPT for a thin and moderately thick circular plate with different values of the foundation stiffness parameters are graphically presented through comparing them with those obtained by the present 3-D p-Ritz solution. Finally, the phenomenon of mode shape switching is investigated in graphical forms for a wide range of the Winkler foundation stiffness parameters.     

A coupled electromechanical analysis of a piezoelectric layer bonded to an elastic substrate: Part I,development of governing equations

This two-part contribution presents a novel and efficient method to analyze the two-dimensional (2-D) electromechanical fields of a piezoelectric layer bonded to an elastic substrate, which takes into account the fully coupled electromechanical behavior. In Part I, Hellinger–Reissner variational principle for elasticity is extended to electromechanical problems of the bimaterial, and is utilized to obtain the governing equations for the problems concerned. The 2-D electromechanical field quantities in the piezoelectric layer are expanded in the thickness-coordinate with seven one-dimensional (1-D) unknown functions. Such an expansion satisfies exactly the mechanical equilibrium equations, Gauss law, the constitutive equations, two of the three displacement–strain relations as well as one of the two electric field-electric potential relations. For the substrate the fundamental solutions of a half-plane subjected to a vertical or horizontal concentrated force on the surface are used. Two differential equations and two singular integro-differential equations of four unknown functions, the axial force, N, the moment, M, the average and the first moment of electric displacement, D₀ and D₁, as well as the associated boundary conditions have been derived rigorously from the stationary conditions of Hellinger–Reissner variational functional. In contrast to the thin film/substrate theory that ignores the interfacial normal stress the present one can predict both the interfacial shear and normal stresses, the latter one is believed to control the delamination initiation.     

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency-response curves, stability, and bifurcation points of the system.     

Axisymmetric free vibrations of infinite micropolar thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Characteristics and suppression of flow-induced vibrations of two side-by-side circular cylinders

A free-vibration experiment was conducted to examine flow-induced vibration (FIV) characteristics of two identical circular cylinders in side-by-side arrangements at spacing ratio T^⁎ (=T/D)=0.1–3.2, covering all possible flow regimes, where T is the gap spacing between the cylinders and D is the cylinder diameter. Each of the cylinders was two-dimensional, spring mounted, and allowed to vibrate independently in the cross-flow direction. Furthermore, an attempt to suppress flow-induced vibrations was undertaken by attaching flexible sheets at the rear stagnation lines of the cylinders. Based on the vibration responses of the two cylinders, four vibration patterns I, II, III and IV are identified at 0.1≤T^⁎<0.2, 0.2≤T^⁎≤0.9, 0.9<T^⁎<2.1 and 2.1≤T^⁎≤3.2, respectively. Pattern I is characterized by the two cylinders vibrating inphase^, with the maximum amplitudes occurring at the same reduced velocity Ur=10.47 almost two times that (Ur=5.25) for an isolated cylinder. Pattern II features no vibration generated for either cylinder. Pattern III exemplifies the occurrence of the maximum vibration amplitude of a cylinder at a smaller Ur and that of the other cylinder at a higher Ur compared to its counterpart in an isolated cylinder. Pattern IV represents each cylinder response resembling an isolated cylinder response; the vibrations of the two cylinders are, however, coupled inphase or antiphase. Linking maximum vibration amplitudes to fluctuating lift forces acting on fixed cylinders reveals that fluid–structure interactions between two fixed cylinders and between two elastic cylinders are not the same, even though vibration is not significant. As such, while two fixed cylinders generate narrow and wide wakes at 0.2≤T^⁎<1.7, two elastic cylinders do the same for a longer range of T^⁎ (0.2≤T^⁎<2.1). The flexible sheets effectively suppress FIV of the two cylinders in patterns III and IV, and reduce the vibration amplitude in pattern I. For the effectively controlled cases (patterns III and IV), the flexible sheet of each cylinder folds into a semicircle at the base, forming two free edges.     

Axisymmetric free vibrations of infinite thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Chaotic and asymmetrical beam response to impulsive load

Both symmetrical and asymmetrical final displacements are observed for elastic–plastic beams under symmetrical impulsive loading. A three-degree-of-freedom Shanley-type model is developed in this study, which is capable of revealing chaotic and asymmetrical responses of an elastic–plastic beam by introducing initial imperfections. To identify the asymmetrical displacement, the beam response is decomposed into three vibration modes. Corresponding modal participation factors are derived based on the displacement of the three-degree-of-freedom beam model. Phase plane trajectories, Poincaré maps and power spectral density diagrams are derived to illustrate both the symmetrical and asymmetrical chaotic vibrations. Numerical simulations using a general-purpose FE code LS-DYNA are carried out for an elastic–plastic beam subjected to impulsive load. The simulation results indicate that the elastic–plastic beam demonstrates chaotic and asymmetrical vibration when the applied impulsive load exceeds a critical value, which agrees with experimental observations.     

Vibration analysis of rotating twisted and open conical shells

Based on a non-linear strain–displacement relationship of a non-rotating twisted and open conical shell on thin shell theory, a numerical method for free vibration of a rotating twisted and open conical shell is presented by the energy method, where the effect of rotation is considered as initial deformation and initial stress resultants which are obtained by the principle of virtual work for steady deformation due to rotation, then an energy equilibrium of equation for vibration of a twisted and open conical shell with the initial conditions is also given by the principle of virtual work. In the two numerical processes, the Rayleigh–Ritz procedure is used and the two in-plane and a transverse displacement functions are assumed to be algebraic polynomials in two elements. The effects of characteristic parameters with respect to rotation and geometry such as an angular velocity and a radius of rotating disc, a setting angle, a twist angle, curvature and a tapered ratio of cross-section on vibration performance of rotating twisted and open conical shells are studied by the present method.     

A Variational Model for Plastic Slip and Its Regularization via Γ-Convergence

A variational model is presented able to interpret the onset of plastic deformations, here modeled as displacement jumps occurring along slip surfaces at constant yielding stress. The corresponding strain energy functional, leading to a free-discontinuity problem set in the space of SBV functions, is then approximated by a sequence of regularized elliptic functionals following the seminal work by Ambrosio and Tortorelli (Commun. Pure Appl. Math. 43, 999–1036, 1990) within the framework of Γ-convergence. Comparisons between the results obtainable with the free-discontinuity model and its regularized approximation, in terms of stability of the pure elastic phase, irreversibility of plastic slip and response under unloading, are presented, in general, for the 2-D case of antiplane shear and exemplified, in particular, for the 1-D case.     

Vibration of skew plates at large amplitudes including shear and rotatory inertia effects

An approach to the large amplitude free, undamped flexural vibration of elastic, isotropic skew plates is developed with the aid of Hamilton's principle taking into consideration the effects of transverse shear and rotatory inertia. On the basis of an assumed vibration mode of the product form, the relationship between the amplitude and period is studied for skew plates of various aspect ratios and skew angles clamped along the boundaries. It is found that the time differential equation, i.e. modal equation when numerically integrated provides interesting information about the effects of transverse shear and rotatory inertia on aspect ratios and skew angles of thin and moderately thick skew plates both at small and at large amplitudes.     

Buckling and its effect on the confined flow of a model capsule suspension

The rheology of confined flowing suspensions, such as blood, depends upon the dynamics of the components, which can be particularly rich when they are elastic capsules. Using boundary integral methods, we simulate a two-dimensional model channel through which flows a dense suspension of fluid-filled capsules. A parameter of principal interest is the equilibrium membrane perimeter, parameterized by ξ_o, which ranges from round capsules with ξ_o=1.0 to ξ_o=3.0 capsules with a dog-bone-like equilibrium shape. It is shown that the minimum effective viscosity occurs for ξ_o≈1.6, which forms a biconcave equilibrium shape, similar to a red blood cell. The rheological behavior changes significantly over this range; transitions are linked to specific changes in the capsule dynamics. Most noteworthy is an abrupt change in behavior for ξ_o≈2.0, which correlates with the onset of capsule buckling. The buckled capsules have a more varied orientation and make significant rotational (rotlet) contributions to the capsule–capsule interactions.     

基于无网格法的弹性地基加肋斜板频率数值解

为了给实际地基工程中的经济效益提供技术指标参考,应用无网格法计算加肋斜板在地基上的自由振动行为,应用移动最小二乘近似MLS和一阶剪切变形原理描述加肋斜板的位移场,分别建立斜板与肋条的势能泛函,使用Winkler弹性地基模型处理加肋板与地基之间的接触势能。通过斜板与肋条的位移协调关系寻找斜板和肋条节点参数转换公式,确立加肋斜板的自由振动控制方程。本文运用了无网格法的优势,即使肋条位置改变也不需重置网格。与有限元解的对比验证了本文方法的有效性和精确性。