Nonlinear vibrations in beams and frames: The effect of the deformed equilibrium state

In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies. The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude.     

Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.     

Non-linear interactions in imperfect beams at veering

Non-linear interactions in a hinged-hinged uniform moderately curved beam with a torsional spring at one end are investigated. The two-mode interaction is a one-to-one autoparametric resonance activated in the vicinity of veering of the frequencies of the lowest two modes and results from the non-linear stretching of the beam centerline. The excitation is a base acceleration that is involved in a primary resonance with either the first mode only or with both modes. The ensuing non-linear responses and their stability are studied by computing force- and frequency-response curves via bifurcation analysis tools. Both the sensitivity of the internal resonance detuning—the gap between the veering frequencies—and the linear modal structure are investigated by varying the rise of the beam half-sinusoidal rest configuration and the torsional spring constant. The internal and external resonance detunings are varied accordingly to construct the non-linear system response curves. The beam mixed-mode response is shown to undergo several bifurcations, including Hopf and homoclinic bifurcations, along with the phenomenon of frequency island generation and mode localization.     

Principal parametric and three-to-one internal resonances of flexible beams undergoing a large linear motion

A set of nonlinear differential equations is established by using Kane‘s method for the planar oscillation of flexible beams undergoing a large linear motion. In the case of a simply supported slender beam under certain average acceleration of base, the second natural frequency of the beam may approximate the tripled first one so that the condition of 3 : 1 internal resonance of the beam holds true. The method of multiple scales is used to solve directly the nonlinear differential equations and to derive a set of nonlinear modulation equations for the principal parametric resonance of the first mode combined with 3 : 1 internal resonance between the first two modes. Then, the modulation equations are numerically solved to obtain the steady-state response and the stability condition of the beam. The abundant nonlinear dynamic behaviors, such as various types of local bifurcations and chaos that do not appear for linear models, can be observed in the case studies. For a Hopf bifurcation,the 4-dimensional modulation equations are reduced onto the central manifold and the type of Hopf bifurcation is determined. As usual, a limit cycle may undergo a series of period-doubling bifurcations and become a chaotic oscillation at last.     

Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances

Nonlinear planar oscillations of suspended cables subjected to external excitations with three-to-one internal resonances are investigated. At first, the Galerkin method is used to discretize the governing nonlinear integral–partial-differential equation. Then, the method of multiple scales is applied to obtain the modulation equations in the case of primary resonance. The equilibrium solutions, the periodic solutions and chaotic solutions of the modulation equations are also investigated. The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency/force–response curves. The supercritical Hopf bifurcations are found in these curves. Choosing these bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. Numerical simulations are used to illustrate the cascades of period-doubling bifurcations leading to chaos. At last, the nonlinear responses of the two-degree-of-freedom model are investigated.     

悬索在考虑1:3内共振情况下的动力学行为

研究了悬索在受到外激励作用下考虑1∶3内共振情况下的两模态非线性响应.对于一定范围内悬索的弹性-几何参数而言,悬索的第三阶面内对称模态的固有频率接近于第一阶面内对称模态固有频率的三倍,从而导致1∶3内共振的存在.首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动得到主共振情况下的平均方程.接下来对平均方程的稳态解、周期解以及混沌解进行了研究.最后利用Runge-Kutta法研究了悬索两自由度离散模型的非线性响应.     

Stress relaxation,creep, and internal stresses in high density polyethylene filled with calcium carbonate

The effect of filling high density polyethylene (HDPE) with calcium carbonate (up to 50% by weight) on the stress relaxation and the creep in uniaxial extension at room temperature was investigated. The addition of CaCO₃ was found to have a strong influence on the flow behaviour of HDPE. In particular, it was observed that the internal stress level, calculated from relaxation data, increased markedly with the filler content. The reduction in creep rate of the filled samples suggested that the CaCO₃-particles induce a change in the structure of the HDPE-interphase close to the filler surface. This was supported by dynamic mechanical measurements performed at low temperatures on swollen HDPE-CaCO₃ samples.     

On the spectrum of natural frequencies of circular cylindrical shells completely filled with a fluid

The natural frequencies of cylindrical shells filled with a fluid and having the ends either simply supported or clamped are determined. Conditions are studied under which the natural frequencies of the shell are close or multiple __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 42–49, May 2006.     

Some studies on rotors with polynomial type non-linear external and internal damping

Non-linear internal damping in rotating cylindrical shafts leads to isotropic non-linear circulatory and dissipative force fields. Orbital instability in such class of systems arises when the regenerative work due to circulatory forces exceeds the dissipative work done over an orbit. In this paper, stability condition for rotor shafts with polynomial type non-linear internal damping is derived and also it is shown that distortion of the critical orbit, which results in larger change in orbital path as compared to the change in enclosed area, may lead to stable orbit at normally unstable operating conditions. This principle is applied to stabilize a simple rotor with discontinuous stiffness characteristics, which come into effect on exceeding some threshold displacement of the rotor itself.     

斜交连续梁桥数值分析研究

在分析了矩阵位移法和传递矩阵法的理论基础上，将二者组合而成矩阵混合法。矩阵混合法的实质就是将结构物的一端未知向量通过矩阵的简单乘法传递到另一端，再将已知条件代入求解的过程。在矩阵传递的过程中引入位移法中的前进代入的概念，即将未知向量不断地以新的未知向量替换，可以防止误差的传递和积累。将矩阵混合法运用于斜交连续梁桥内力计算，完成了矩阵混合法在斜交连续梁桥内力计算中的理论推导和程序的设计，给出了实际算例并与有限元法进行了比较。