Nonlinear free vibrations of beams in space due to internal resonance

The geometrically nonlinear free vibrations of beams with rectangular cross section are investigated using a p-version finite element method. The beams may vibrate in space, hence they may experience longitudinal, torsional and non-planar 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 geometrical nonlinearity is taken into account by considering Green’s nonlinear strain tensor. Isotropic and elastic beams are investigated and generalised Hooke’s law is used. The equation of motion is derived by the principle of virtual work. Mostly clamped–clamped beams are investigated, although other boundary conditions are considered for validation purposes. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. One constant term, odd and even harmonics are assumed in the Fourier series and convergence with the number of harmonics is analysed. The variation of the amplitude of vibration with the frequency of vibration is determined and presented in the form of backbone curves. Coupling between modes is investigated, internal resonances are found and the ensuing multimodal oscillations are described. Some of the couplings discovered lead from planar oscillations to oscillations in the three dimensional space.     

Difference equation analysis of non-linear subharmonic and superharmonic oscillations

This paper deals with sub- and superharmonic oscillations in strongly excited non-linear systems described by difference equations. An adaptation of the discrete multiple time perturbational technique [1] is used. A Duffing type of equation is considered and the possible sub- and superharmonic oscillations are investigated. The results deduced from the proposed method are compared with those obtained by using the well known method of harmonic balance.     

Non-linear structural vibrations under combined parametric and external excitations

A set of second order equations with weak quadratic and cubic non-linearities is considered. Simultaneous parametric and external (forcing) excitations are included. The frequency of the parametric excitation is near a natural frequency of the system, and three cases are analyzed: (i) the external excitation is absent; (ii) the external excitation is present but is not involved in a resonance; and (iii) the external frequency is the same as the parametric frequency. Results are obtained by the method of multiple scales. Frequency-response curves are presented for various combinations of excitation amplitudes, damping coefficients, and phase shift between the excitations. It is found that stable multi-modal responses may exist in the first-order asymptotic solution, even though only one mode is involved in the resonance and no internal resonance condition is present.     

参量激励表面次谐波内共振现象的实验研究

研究了参量激励非线性表面波的内共振现象。在实验中,除了观测到已有的Faraday1/2分频次谐波外,还观测到1/4,1/6,1/8,1/10,1/12等大振幅次谐波。实验研究表明,分频次谐波的产生是由于本征频率接近于对应的低阶表面波本征模式所共振激发的,而低阶模式的激发是由于模式之间的强烈非线性耦合所导致的,即为内共振现象。同时,强烈的非线性耦合还引发了混合模式表面波的模式竞争现象等复杂非线性动力学行为,且这种模式竞争具有明显的无规性和混沌特征。级联内共振激发的多模式共存以及相互之间的竞争,是导致水槽中产生混沌的一种重要途径。     

On the non-linear vibrations of rectangular plates

A solution, based on a one-term mode shape, for the large amplitude vibrations of a rectangular orthotropic plate, simply supported on all edges or clamped on all edges for movable and immovable in-plane conditions, is found by using an averaging technique that helps to satisfy the in-plane boundary conditions. This averaging technique for satisfying the immovable in-plane conditions can be used to resolve many anisotropic and skew plate problems where otherwise, when a stress function is used, the integration of the u and v equations becomes difficult, if not impossible. The results obtained herein are compared with those available in the literature for the isotropic case and excellent agreement is found. Results available for the one-term mode shape solutions of these problems are compared and the non-linear effect is presented as functions of aspect ratio and of the orthotropic elastic constants of the plate. The results are further compared with those based on the Berger method and the detailed comparative studies show that the use of the Berger approximation for large deflection static and dynamic problems and its extension to anisotropic plates, skew plates, etc., can lead to quite inaccurate results.     

The influence of porosity on vibrations of elastic solids

Wave scattering due to inhomogeneities in a solid leads to damping of wave motion in a random mixture of elastic materials. In particular, the phenomenon is observed in a material containing random porosity. A general dissipative model is presented to evaluate the influence of small randomly distributed pores on the dynamic response of elastic structures. The numerical results are given for transverse wave propagation and for vibrations of a beam. It is shown that analysis of vibrations of elastic solids containing random porosity can be carried out by the methods of viscoelasticity.     

The magneto-elastic subharmonic resonance of current-conducting thin plate in magnetic filed

Based on Maxwell equations and corresponding electromagnetic constitutive relations, the electrodynamic equations and electromagnetic force expressions of a current-conducting thin plate in electromagnetic field are deduced. Nonlinear magneto-elastic vibration equations of the thin plate are given. In addition, nonlinear subharmonic resonances of the thin plate with two opposite sides simply supported which is under the mechanic live loads and in constant transverse magnetic field are studied. The corresponding vibration differential equation of Duffing type is deduced by the Galerkin method. The method of multiple scales is used to solve the equation, and the frequency-response equation of the system in steady motion under subharmonic responses is obtained, and the stability of solution is analyzed. According to the Liapunov stability theory, the critical conditions of stability are obtained. By the numerical calculation, the curves of resonance amplitude changing with the detuning parameter, the excitation amplitude and the magnetic intensity and corresponding state planes are obtained. The existing regions of nontrivial solutions and the changing law of stable and unstable solutions are analyzed. The time history response plots, the phase charts and the Poincare mapping charts are plotted. And the effect of the magnetic intensity on the system is discussed, and some complex dynamic performances as period-doubling motion and quasi-period motion are analyzed.     

Strain hardening under structural superplasticity conditions

A model is proposed for describing the hardening of fine-grained materials deformed under structural superplasticity conditions. Under these conditions, the strain dependence of the flow stress is shown to be caused by the internal stress fields induced by the defects introduced into grain boundaries during intragranular slip. Expressions describing the dependences of the flow stress on the rate and temperature of superplastic deformation and the structural parameters of the material are obtained.     

Effects of edge restraints on the non-linear flexural vibrations of an imperfect cross-ply laminated plate resting on an elastic foundation

A solution is presented, for the non-linear Marguerre dynamic equilibrium and compatability equations, for the large amplitude free flexural vibrations of an imperfect, cross-ply, laminated plate, having elastically restrained edges and resting on an elastic foundation. The analysis is used to study the effects of edge restraints and elastic foundation constants on the frequency ratios of isotropic and CFRP plates. Numerical results are presented graphically.     

Experimental study of the ultrasonic transducer non-linear vibrations

Conclusion The method introduced proved to be useful in obtaining information on transducer behaviour at high power levels. With respect to ultrasonic generator design several conclusions may be drawn from the transducer characteristics. First, it was found that the current and the pick-up voltage characteristics are equivalent with respect to the information content on the transducer vibrations. Therefore, the feedback signal may be derived both from the currentI and from the voltageU _p .It was also found that the resonance frequency of the transducerf _r varies strongly both with the power level and with the mechanical load. However, it is the resonance phase angle _r that is almost independent of these external changes.Lastly, because of the jumps, the transducer may be tuned to the resonance only by lowering the frequencyf starting from an arbitrary frequency higher thanf ₁. In designing a feedback system this requirement should be taken into account.The author wishes to thank M. Los and L. Kozák from the Ultrasonic Laboratory of the Czech Technical University in Prague for kindly lending him the ultrasonic transducer for this study.     

Forced non-linear vibrations of a damped sandwich beam

Forced, damped, non-linear, low-frequency flexural motions of a clamped-clamped sandwich beam with thin face sheets and a soft viscoelastic core are examined experimentally and theoretically. The theory employed neglects the extensional rigidity of the core and treats the face sheets as membranes. The non-linearity stems from axial stretching of the face sheets. Damping is taken into account by modeling the core as a Kelvin solid, with the material parameters used being obtained experimentally as functions of frequency and temperature. Theoretical frequency-amplitude relations are obtained using Galerkin's procedure and the method of harmonic balance. Results on fundamental natural frequencies, mode shapes, and stability are also presented. In the experiment, mechanical contact with the specimen was avoided by employing electromagnetic forcing and using a proximeter to measure displacements. Also, special attention was given to the interface bonds and to the reproduction, as close as possible, of clamped-clamped conditions. Agreement between the theoretical and experimental results is, in general, quite good.