Approximate solutions of axially moving viscoelastic beams subject to multi-frequency excitations

Approximate solutions of axially moving viscoelastic beams subject to multi-frequency excitations are presented. A non-linear partial-differential equation governing the transverse vibration of the beams is derived from Newton's second law, the Kelvin constitutive relationship, and the Lagrangian strain. Based on 1-term Galerkin's truncation, the governing equation is reduced to an ordinary differential equation. Three cases, including superharmonic resonance case, subharmonic resonance, and combination resonance are studied. The approximate solutions of the transverse vibration of the beams are obtained. Numerical results show that the approximate solutions are in good agreement with numerical results.     

轴向运动正交各向异性板的超谐波共振及稳定性

研究了轴向运动正交各向异性条形薄板在线载荷作用下的超谐波共振问题。通过哈密顿原理导出了几何非线性下正交各向异性条形板的非线性振动方程。运用伽辽金积分法,推得了关于时间变量的量纲归一化非线性振动微分方程组。应用多尺度法求解三阶超谐波共振问题,得到了稳态运动下一阶、二阶、三阶共振形式的共振幅值响应方程。利用Liapunov方法推得不同共振形式稳态解的稳定性判据,并据此分析不同参数对系统稳定性的影响。绘制了振幅特性变化曲线图和与之对应的激发共振多解临界点曲线图,分析系统参数对共振的影响,并预测系统进入非线性共振区域的临界条件。得出激励在特定位置区间时可激发系统的超谐波共振,随着激励幅值的增加,上稳定解支减小,下稳定解支增加,且一阶模态振幅大于二阶、三阶振幅。     

A time domain collocation method for obtaining the third superharmonic solutions to the Duffing oscillator

In this study, a simple time domain collocation method (TDC) is applied to investigate the third superharmonic solutions of the Duffing oscillator. Upon using the proposed scheme, the multivaluedness, jump phenomenon, and transitional region of the third superharmonic response are explored. The amplitude frequency response curves for various values of damping, nonlinearity, and external force are obtained and compared. In addition, instead of collocating at N points so that the resulting nonlinear algebraic system is well determined, we extend the time domain collocation method to a new version by collocating at M>N points. The resulting over determined system is solved by the least square method. The extended time domain collocation method can significantly relieve the nonphysical solution phenomenon, which may be severe in the time domain collocation method, and its equivalent high dimensional harmonic balance method. Finally, numerical examples confirm the simplicity, efficiency, and accuracy of the proposed scheme.     

双稳态电磁式振动能量捕获器超谐波响应研究

超谐波响应是非线性振动系统在较大激励下表现的特性,在某种条件下双稳态振动能量捕获系统的超谐波响应可使系统产生优越的输出功率。本文将质量-非线性弹簧-阻尼系统与双稳态振动能量捕获器相结合,提出了附加非线性振子的双稳态电磁式振动能量捕获器,建立系统的力学模型及控制方程。采用两项式谐波平衡法,获得了双稳态系统在简谐激励下产生大幅运动的基谐波和超谐波响应的解析解,借助数值仿真分析了质量比和调频比对双稳态振动能量捕获器产生大幅运动的影响规律,获得了双稳态系统的结构参数的最佳配置范围,且当外部激励频率处于低频段时,系统发电主要表现为超谐波发电,随着激励频率的增大,振动发电系统主要呈现基谐波发电。上述研究,为双稳态能量捕获装置的理论研究提供了参考。     

静载荷对夹层圆板超谐波共振的影响

研究静载荷作用下夹层圆板的超谐波共振问题.基于Hoff型夹层板理论,给出了静载荷作用下夹层圆板的非线性动力学方程.应用Galerkin法推导了静载荷作用下夹层圆板的轴对称非线性振动方程.运用多尺度法分别对系统的三次超谐波问题和二次超谐波问题进行了求解,并依据Lyapunov稳定性理论得到了系统稳态运动的稳定性判据.通过...     

磁场中轴向运动条形板强非线性超谐共振及混沌运动

针对磁场环境中周期外载作用下轴向运动导电条形板的非线性振动及混沌运动问题进行研究。应用改进多尺度法对横向磁场中条形板的强非线性振动问题进行求解,得到超谐波共振下系统的分岔响应方程。根据奇异性理论对非线性动力学系统的普适开折进行分析,求得含两个开折参数的转迁集及对应区域的拓扑结构分岔图。通过数值算例,分别得到以磁感应强度、轴向拉力、激励力幅值和激励频率为分岔控制参数的分岔图和最大李雅普诺夫指数图,以及反映不同运动行为区域的动力学响应图形,讨论分岔参数对系统呈现的倍周期和混沌运动的影响。结果表明,可通过相应参数的改变实现对系统复杂动力学行为的控制。     

Response and stability analysis of piecewise-linear oscillators under multi-forcing frequencies

Steady-state solutions of a piecewise-linear oscillator under multi-forcing frequencies are obtained using the fixed point algorithm (FPA). Stability analysis is also performed using the same technique. For the periodic solutions of a piecewise-linear oscillator with single forcing frequency, the harmonic balance method (HBM) is also used along with the FPA. Although both FPA and HBM generate accurate solutions, it is observed that the HBM failed to converge to solutions in the superharmonic range of the forcing frequency.The fixed point algorithm was also applied to the oscillator under multifrequency excitation. The algorithm proved to be very effective in obtaining torus solutions and in locating corresponding bifurcation thresholds. A piecewise-linear oscillator model of an offshore articulated loading platform (ALP) subjected to two incommensurate wave frequencies is found to exhibit chaotic behavior. A second order Poincaré mapping technique reveals the hidden fractal-like nature of the resulting chaotic response. A parametric study is performed for the response of the ALP.     

Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber

The present investigation deals with the dynamics of a two-degrees-of-freedom system which consists of a main linear oscillator and a strongly non-linear absorber with small mass. The non-linear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model. The periodic solutions of this system are studied and their calculation is performed through an averaging procedure. The study of non-linear modes and their stability shows, under specific conditions, the existence of localization which is responsible for a passive irreversible energy transfer from the linear oscillator to the non-linear one. The dissipative effect of the non-linearity appears to play an important role in the energy transfer phenomenon and some design criteria can be drawn regarding this parameter among others to optimize this energy transfer. The free transient response is investigated and it is shown that the energy transfer appears when the energy input is sufficient in accordance with the predictions from the non-linear modes. Finally, the steady-state forced response of the system is investigated. When the input of energy is sufficient, the resonant response (close to non-linear modes) experiences localization of the vibrations in the non-linear absorber and jump phenomena.     

Local bifurcation theory of nonlinear systems with parametric excitation

This paper summarizes the authors' research on local bifurcation theory of nonlinear systems with parametric excitation since 1986. The paper is divided into three parts. The first one is the local bifurcation problem of nonlinear systems with parametric excitation in cases of fundamental harmonic, subharmonic and superharmonic resonance. The second one is the experiment investigation of local bifurcation solutions in nonlinear systems with parametric excitation. The third one is the universal unfolding study of periodic bifurcation solutions in the nonlinear Hill system, where the influence of every physical parameter on the periodic bifurcation solution is discussed in detail and all the results may be applied to engineering.     

Periodic oscillations of a class of non-autonomous non-linear elastic continua

A modified perturbation method for obtaining periodic solutions to a class of non-autonomous non-linear partial differential equations is developed. The classic small divisor is discussed in detail and a general method for its elimination is presented. New terminology is introduced for the purpose of discussing forcing functions that produce in a system a response that is of the same form as a non-linear periodic mode for the same system. Specific examples are examined to verify the results of this work.     

Large amplitude non-linear oscillations of a general conservative system

This paper presents new, approximate analytical solutions to large-amplitude oscillations of a general, inclusive of odd and non-odd non-linearity, conservative single-degree-of-freedom system. Based on the original general non-linear oscillating system, two new systems with odd non-linearity are to be addressed. Building on the approximate analytical solutions of odd non-linear systems developed by the authors earlier, we construct the new approximate analytical solutions to the original general non-linear system by combinatory piecing of the approximate solutions corresponding to, respectively, the two new systems introduced. These approximate solutions are valid for small as well as large amplitudes of oscillation for which the perturbation method either provides inaccurate solutions or is inapplicable. Two examples with excellent approximate analytical solutions are presented to illustrate the great accuracy and simplicity of the new formulation.     

Nonclassical Responses of Oscillators with Hysteresis

The responses and codimension-one bifurcations in Masing-type andBouc–Wen hysteretic oscillators are investigated. The pertinent statespace is formulated for each system and the periodic orbits are soughtas the fixed points of an appropriate Poincaré map. The implementedpath-following scheme is a pseudo-arclength algorithm based on arclengthparameterization. The eigenvalues of the Jacobian of the map, calculatedvia a finite-difference scheme, are used to ascertain the stability andbifurcations of the periodic steady-state solutions. Frequency-responsecurves for various excitation levels are constructed consideringrepresentative hysteresis loop shapes generated with the two models inthe primary and superharmonic frequency ranges. In addition to knownbehaviors, a rich class of solutions and bifurcations, mostly unexpectedfor hysteretic oscillators – including jump phenomena,symmetry-breaking, complete period-doubling cascades, fold, andsecondary Hopf – is found. Complex (mode-locked) periodic andnonperiodic responses are also investigated thereby allowing to draw amore comprehensive picture of the dynamical behavior exhibited by thesesystems.     

Non-linear dynamics of asymmetric structures under 2:2:1 resonance

The non-linear dynamic behavior of a novel model of a single-story asymmetric structure under earthquake and harmonic excitations and near two-to-one internal resonance is investigated. The non-linearities of the proposed model, ignored in conventional linear models, are caused by non-linear inertial coupling between translational and torsional degrees of freedom defined in the directions of a non-inertial rotational system of reference, attached to the center of mass of the floor. The multiple scales method is used to achieve approximately linear solutions for the originally non-linear equations near a two-to-one ratio of external and internal resonant conditions. The suitability of the proposed model is justified by the similarity between the simulated response of the non-linear model and the experimental results. The numerical results of time history and frequency domain analyses illustrate the difference between the non-linear and linear models. Energy transfer from a lower natural frequency excited mode to a higher one due to non-linear interaction in the novel model is shown. The effects of amplitude, frequency detuning parameters, uncoupled lateral and torsional frequencies, and damping ratio on the responses are inspected and some non-linear phenomena such as hysteresis, jumping, hardening, and softening are observed.     

Determination of the non-linear parameter (mobility factor) of the Giesekus constitutive model using LAOS procedure

The paper introduces a novel procedure to determine the non-linear parameter of the Giesekus model, in relation to the characterization of the non-linear oscillatory shear regime of viscoelastic polymer solutions based on polyacrylamide. Instead of using the shear-thinning viscosity as the representative non-linear effect, the third harmonic in the Fourier spectrum of the shear stress response signal is considered for computing the mobility factor. The fluid is subjected to large amplitude oscillatory shear (LAOS) and its response is recorded. Deviations of this signal from the sinusoidal form are specific to each material and gives both qualitative and quantitative measures of the non-linearity. By fitting the material response with the corresponding numerical solutions of the n-modes Giesekus constitutive relation, one can extract the values of the non-linear α_i-parameters that describe the fluid rheology. It is demonstrated that this procedure, which can be successfully applied to semi-concentrated polymer solutions, provides better results than the classical viscosity-fit method.     

Saddle-Node Bifurcations of Cycles in a Relief Valve

We extend the asymptotic perturbation (AP) method to the studyof a linear partial differential equation with nonlinear boundaryconditions. A relief valve under the combined effects of static anddynamic loadings is considered with the following resonances between thenth linear mode and the external periodic excitation: primaryresonance, subharmonic resonance of order one-half or one-third,superharmonic resonance of order-two or order-three and combinationresonance. The AP method uses two different procedures for thesolutions: introducing an asymptotic temporal rescaling and balancing ofthe harmonic terms with a simple iteration. We obtain amplitude andphase modulation equations and determine external force-response andfrequency-response curves. Stability of steady-state motions is alsoinvestigated. Saddle-node bifurcations of cycles are observed and underappropriate conditions the performance of the relief valve may beunsatisfactory due to the presence of jumps and hysteresis effects inthe system response. Global analysis is used in order to exclude theexistence of modulated motion. The validity of the method is highlightedby comparing approximate solutions with results of the numericalintegration.     

Forced oscillations of a piecewise-linear non-linear dynamic system with several degrees of freedom

Exact periodic solutions are derived for a dynamic system with several degrees of freedom consisting of a series of ‘Reid springs’ with piecewise-linear, non-linear characteristis; however, the solutions are restricted to a class of harmonic excitation in the ‘modal form’ described subsequently in the paper. Conditions are derived for the asymptotic stability of the periodic solution and an example has been worked out in detail on the response of a dynamic system with two degrees of freedom.     

Nonlinear Dynamics of a Beam on Elastic Foundation

The nonlinear dynamics of a simply supported beam resting on a nonlinear spring bed with cubic stiffness is analyzed. The continuous differential operator describing the mathematical model of the system is discretized through the classical Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms. This model can be regarded as a simple system describing the oscillations of flexural structures vibrating on nonlinear supports and then it can be considered as a simple investigation for the analysis of more complex systems of the same type. Indeed, the possibility of the model to exhibit actually interesting nonlinear phenomena (primary, superharmonic, subharmonic and internal resonances) has been shown in a range of feasibility of the physical parameters. The singular perturbation approach is used to study both the free and the forced oscillations; specifically two parameter families of stationary solutions are obtained for the forced oscillations.     

Dynamic response of composite sandwich plates subjected to time-dependent pressure pulses

The dynamic response of orthotropic sandwich composite plates impacted by time-dependent external blast pulses is studied by use of numerical techniques. The theory is based on classical sandwich plate theory including the large deformation effects, such as geometric non-linearities, in-plane stiffness and inertias, and shear deformation. The equations of motion for the plate are derived by the use of the virtual work principle. Approximate solutions are assumed for the space domain and substituted into the equations of motion. Then the Galerkin Method is used to obtain the non-linear differential equations in the time domain. The finite difference method is applied to solve the system of coupled non-linear equations. The results of theoretical analyses are obtained and compared with ANSYS results. Effects of the face sheet number, as well as those related to the ply-thickness, core thickness, geometrical non-linearities, and of the aspect ratio are investigated. Detailed analyses of the influence of different type of pressure pulses on dynamic response are carried out.     

Multiple solutions generated by statistical linearization and their physical significance

Three cases are examined where the statistical linearization (SL) procedure can yield multiple solutions for the first and second moments of the response. The first is an oscillator with a hardening non-linear stiffness excited by a narrow-band random excitation, the second is an oscillator with two potential wells excited by wide-band random excitation, and the third is an oscillator where the non-linear features present in the first two problems are combined. The results of an SL analysis are quantitatively compared with the behaviour of digitally simulated sample functions of the displacement response. In all cases a definite correspondence is found between the occurrence of multiple solutions generated by the SL method and the appearance of noticeable jumps in sample functions of the response. In some cases a quantitative agreement exists between the first and second moment values of the multiple solutions and the magnitude of “local” moments of the response.