Three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam

The three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam is investigated in this paper by means of two numerical techniques. The equations of motion for the longitudinal, transverse, and rotational motions are derived using constitutive relations and via Hamilton’s principle. The Galerkin method is employed to discretize the three partial differential equations of motion, yielding a set of nonlinear ordinary differential equations with coupled terms. This set is solved using the pseudo-arclength continuation technique so as to plot frequency-response curves of the system for different cases. Bifurcation diagrams of Poincaré maps for the system near the first instability are obtained via direct time integration of the discretized equations. Time histories, phase-plane portraits, and fast Fourier transforms are presented for some system parameters.     

Identification of nonlinear anti-vibration isolator properties

Vibrations are classified among the major problems for engineering structures. Anti-vibration isolators are used to absorb vibration energy and minimise transmitted force which can cause damage. The isolator is modelled as a parallel combination of stiffness and damping elements. The main purpose of the model is to enable designers to predict the dynamic response of systems under different structural excitations and boundary conditions. A nonlinear identification method, discussed in this paper, aims to provide a tool for engineers to extract information about the nonlinear dynamic behaviour using measured data from experiments. The proposed method is demonstrated and validated with numerical simulations. Thus, this technique is applied to determine the nonlinear parameters of a commercial metal mesh isolator. Nonlinear stiffness and nonlinear damping can decrease with the increase in the amplitude of the base excitation. The softening behaviour of the mesh isolator is clearly visible.     

Output frequency properties of nonlinear systems

Nonlinear systems usually have complicated output frequencies. For the class of Volterra systems, some interesting properties of the output frequencies are studied in this paper. These properties show theoretically the periodicity of the output super-harmonic and inter-modulation frequencies and clearly demonstrate the mechanism of the interaction between different output harmonics incurred by different input nonlinearities in system output spectrum. These new results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities. Examples and discussions are given to illustrate these new results.     

Experiments on the nonlinear stages of excited and natural planar jet shear layer transition

An experimental investigation focusing on the nonlinear stages of planar jet shear layer transition is presented. Experimental results for transition under both natural and low level artificial forcing conditions are presented and compared. The local spectral dynamics of the jet shear layer is modeled as a nonlinear system based upon a frequency domain, second-order Volterra functional series representation. The local linear and nonlinear wave coupling coefficients are estimated from time-series streamwise velocity fluctuation data. From the linear coupling coefficient, the mean dispersion characteristics and spatial growth rates may be obtained. With the estimation of the nonlinear power transfer function, the total, linear and quadratic nonlinear spectral energy transfer may be locally estimated. When these measures are used in conjunction with the local quadratic bicoherency and linear-quadratic coupling bicoherency, the local system output power may be completely characterized and the effect of nonlinearity on local mean flow distortion assessed. Particular attention is focused upon quantifying the linear and nonlinear power transfer that characterizes the different stages of the jet shear layer transition for both natural and excited flows. The quadratic power transfer that occurs with deviation from the perfect resonant wavenumber matching condition is clarified as is the dynamic mechanism of subharmonic resonance. The mechanism of spectral broadening is described and contrasted for natural and artificially excited flows.     

Symmetry properties of a nonlinear acoustics model

We give a classification into conjugacy classes of subalgebras of the symmetry algebra generated by the Zabolotskaya–Khokhlov equation, and obtain all similarity reductions of this equation into (1+1)-dimensional equations. We thus show that the Lie classical reduction approach may also give rise to more general reduced equations as those expected from the direct method of Clarkson and Kruskal. By transforming the determining system for the similarity variables into the equivalent adjoint system of total differential equations, similarity reductions to odes which are independent of the three arbitrary functions defining the symmetries are also obtained. These results are again compared with those obtained by the direct method of Clarkson and Kruskal, by finding in particular equivalence transformations mapping some of the reduced equations to each other. Various families of new exact solutions are also derived.     

Comparison of waveguide properties of curved versus straight planar elastic layers

Dispersion equations are solved for the in-plane and anti-plane wave propagation in planar elastic layer with constant curvature. The classical Lamé formulation of displacements via elastic potentials is applied and appropriate simplifications are employed. The dispersion diagrams in each case are compared with their counterparts for a straight layer, e.g., the classical Rayleigh–Lamb solution. The curvature-induced symmetry-breaking effects are investigated for layers with symmetric boundary conditions. The role of curvature is also investigated in the cases, when the boundary conditions are not symmetrical. The elementary Bernoulli–Euler theory is employed to analyze the wave guide properties of a curved planar elastic beam in its in-plane deformation. The validity range of the Bernoulli–Euler theory is assessed via comparison of dispersion diagrams.     

A nonlinear contact pressure distribution model for wear calculation of planar revolute joint with clearance

As various errors result from manufacture and assembly processes or wear effect, clearance joint widely exists in mechanical system as a base component. The coupling analysis of tribology and dynamics of clearance joint is important to the reliability of mechanical system. A nonlinear contact pressure distribution mode (NLCP) is proposed to combine dynamics analysis with wear calculation together in this paper. The discrete thought of Winkler model is adopted to deal with contact problem with a high conformal rate. The contact relationship in a local microcontact area can be regarded as the contact between cylinder and plane. And the local contact pressure is acquired based on Hertz contact theory. The NLCP model has not only described the nonlinear relationship between contact pressure and penetration depth, but also avoided the complexity in contact pressure computation. The performance of NLCP model is demonstrated in comparison with asymmetric Winkler model, revealing that NLCP model has enhanced the calculation accuracy with a good efficiency. A comprehensive experimental study on the wear calculation of a slider–crank mechanism with clearance joint is presented and discussed to provide an experimental verification for NLCP model. The paper’s work has solved the contact problem with a high conformal rate and has described the nonlinear relationship between contact pressure and penetration depth. It has great value to the wear analysis of clearance joint.     

A nonlinear planar beam formulation with stretch and shear deformations under end forces and moments

A new nonlinear planar beam formulation with stretch and shear deformations is developed in this work to study equilibria of a beam under arbitrary end forces and moments. The slope angle and stretch strain of the centroid line, and shear strain of cross-sections, are chosen as dependent variables in this formulation, and end forces and moments can be either prescribed or resultant forces and moments due to constraints. Static equations of equilibria are derived from the principle of virtual work, which consist of one second-order ordinary differential equation and two algebraic equations. These equations are discretized using the finite difference method, and equilibria of the beam can be accurately calculated. For practical, geometrically nonlinear beam problems, stretch and shear strains are usually small, and a good approximate solution of the equations can be derived from the solution of the corresponding Euler–Bernoulli beam problem. The bending deformation of the beam is the only important one in a slender beam, and stretch and shear strains can be derived from it, which give a theoretical validation of the accuracy and applicability of the nonlinear Euler–Bernoulli beam formulation. Relations between end forces and moments and relative displacements of two ends of the beam can be easily calculated. This formulation is powerful in the study of buckling of beams with various boundary conditions under compression, and can be used to calculate post-buckling equilibria of beams. Higher-order buckling modes of a long slender beam that have complex configurations are also studied using this formulation.     

Effects of temperature variations on nonlinear planar free and forced oscillations at primary resonances of suspended cables

Nonlinear Dynamics - This paper is concerned with the analysis of the free and forced nonlinear vibrations of the suspended cable with thermal effects. By introducing the new thermal stressed...     

Interval analysis of nonlinear frames with uncertain connection properties

Semi-rigid connections can often be a more economical solution for a framing system than one with either fully fixed connections or fully pinned ones. In view of the fact that the properties of such ductile and partial-strength connections are not known accurately, this paper presents a method for the obtention of both upper and lower bound responses of semi-rigid frames for possible variations in their moment-rotation properties. The latter are thus assumed to be known within some key upper and lower bound values, namely a constitutive law that is still deterministic but is described in terms of a so-called “interval” model. A mathematical programming approach is used to formulate and solve the problem. In particular, for each load level, a pair of nonstandard optimization problems known as interval mathematical programs with equilibrium constraints (or interval MPECs) are solved to provide the required bounds. A number of examples are provided to highlight the important effects of considering uncertainties in semi-rigid connection properties.     

Permanence and asymptotic properties of nonlinear delay difference equations

The asymptotic behavior of a class of nonlinear delay difference equation wax studied . Some sufficient conditions are obtained for permanence and global attractivity . The results can be applied to a claxs of nonlinear delay difference equations and to the delay discrete Logistic model and some known results are included.     

Predicting the nonlinear strain properties of fibrous metal composites

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 9, pp. 45–51, September, 1989.     

Field fluctuations and macroscopic properties for nonlinear composites

A recently introduced nonlinear homogenization method [J. Mech. Phys. Solids 50 ( 2002) 737–757] is used to estimate the effective behavior and the associated strain and stress fluctuations in two-phase, power-law composites with aligned-fiber microstructures, subjected to anti-plane strain, or in-plane strain loading. Using the Hashin–Shtrikman estimates for the relevant “linear comparison composite,” results are generated for two-phase systems, including fiber-reinforced and fiber-weakened composites. These results, which are known to be exact to second-order in the heterogeneity contrast, are found to satisfy all known bounds. Explicit analytical expressions are obtained for the special case of rigid-ideally plastic composites, including results for arbitrary contrast and fiber concentration. The effective properties, as well as the phase averages and fluctuations predicted for these strongly nonlinear composites appear to be consistent with deformation mechanisms involving shear bands. More specifically, for the case where the fibers are stronger than the matrix, the predictions appear to be consistent with the shear bands tending to avoid the fibers, while the opposite would be true for the case where the fibers are weaker.     

Experimental investigation of the nonlinear resonance properties of cylindrical shells

The nonlinear resonance properties of cylindrical shells filled with a fluid are experimentally studied. It is proved that travelling waves appear under resonance conditions due to the effect of the gyroscopic moment on the shell. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 2, pp. 111–114, February, 2000.     

Continuous solutions of nonlinear functional difference equations and their properties

We establish conditions for the existence of continuous bounded solutions of systems of nonlinear functional difference equations and study their properties.     

Global solutions of systems of nonlinear difference equations and their properties

We establish sufficient conditions for the existence and uniqueness of a global solution of one class of systems of nonlinear difference equations and investigate its properties. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 291–297, July–September, 2007.     

EFFECT OF DAMAGE ON NONLINEAR DYNAMIC PROPERTIES OF VISCOELASTIC RECTANGULAR PLATES

IntroductionRecently,compositestructureshavereceivedwideapplicationsinmodernindustriesincludingastronavigation ,aviation ,petroleumandchemicalindustry ,etc.However,thiskindofmaterialsgenerallyhavethepropertiesofviscoelasticitywiththeapparentcreepphenomenonandrelaxationbehaviors.Moreover,variousdamagewillemergeinthestructuresundertheactionofloading ,temperatureandenvironment.Damagemakesthemechanicalbehaviorsdeteriorategraduallybeforethefailure .Damageeffectswillsoftenthestiffnessofthestructure…