Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells北大核心CSCD

The nonlinear vibration responses of functionally graded materials (FGMs) shells with different cone angles under external loads were studied. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the 1st-order shear deformation theory, the von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations were simplified into a 1DOF nonlinear vibration differential equation under Volmir’s assumption. Finally, the nonlinear motion equations were solved with the harmonic balance method and the Runge-Kutta method, and the amplitude frequency response characteristic curves of the FGMs conical shells were obtained. The effects of different material distribution functions and different ceramic volume fraction exponents on the amplitude frequency response curves of conical shells were discussed. The bifurcation diagrams of conical shells with different cone angles, as well as time process diagrams and phase diagrams for different excitation amplitudes, were described. The motion characteristics were characterized by Poincaré maps. The results show that, the FGMs conical shells present the nonlinear characteristics of hardening springs. The chaotic motions of the FGMs conical shells are restrained and not prone to motion instability with the increase of the cone angle. The FGMs conical shell present a process from the periodic motion to the multi-periodic motion and then to chaos with the increase of the excitation amplitude. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

The transient retardation of a rectilinear viscoplastic flow when the loading stresses are abruptly removed

The development of a viscoplastic flow in a solid layer of an elastoviscoplastic material on an inclined plane is considered when loading stresses act on its free surface. It is shown that the elastoplastic boundary starts its motion from the rigid inclined plane and, propagating through the elastic core, it can reach the free surface of the layer. An exact solution is obtained for the dynamic problem of the retardation of developed viscoplastic flow after the loading stresses are abruptly removed. The possibility of writing the equation of motion for the unloading wave in terms of the displacements is pointed out. It reduces to an inhomogeneous wave equation where the velocity of the unloading wave is found to be equal to the velocity of the equivoluminal elastic wave. Reflection of the unloading wave from a rigid boundary in the form of an inclined plane is also considered.     

带控制面机翼结构基于弧长数值连续法的颤振特征研究

以三自由度二元机翼为研究对象,将浮沉位移和俯仰位移方向的非线性刚度简化为立方非线性,对于存在间隙的控制面采用双线性刚度代替.考虑准定常气流,建立气动弹性运动方程,通过数值模拟构造峰值-峰值图,反映其在不同气流速度下的振动特征.通过弧长数值连续法构造系统的分岔图,结合Floquet算子研究其稳定性及其分岔类型,所得分岔图和数值模拟的结果相吻合.由分岔图可得系统由于控制面双线性的存在,导致机翼结构振动形态多变,存在多个分岔点和多个不稳定区间,不仅存在极限环振动和非光滑准周期振动,而且在某些不稳定区间出现混沌现象.     

Nonlinear vibration phenomenon of an aircraft rub-impact rotor system due to hovering flight

This paper focuses on the nonlinear vibration phenomenon caused by aircraft hovering flight in a rub-impact rotor system supported by two general supports with cubic stiffness. The effect of aircraft hovering flight on the rotor system is considered as a maneuver load to formulate the equations of motion, which might result in periodic response instability to the rotor system even the eccentricity is small. The dynamic responses of the system under maneuver load are presented by bifurcation diagrams and the corresponding Lyapunov exponent spectrums. Numerical analyses are carried out to detect the periodic, sub-harmonic and quasi-periodic motions of the system, which are presented by orbit diagrams, phase trajectories, Poincare maps and amplitude power spectrums. The results obtained in this paper will contribute an understanding of the nonlinear dynamic behaviors of aircraft rotor systems in maneuvering flight.     

井下钻柱纵向横向耦合振动模型建立与数值分析

针对井下钻柱运动的复杂性,基于动力学理论,建立了井下钻柱纵向和横向耦合振动的数学模型,并进行数值求解及分析.根据井下钻柱的实际工况,以整个井下钻柱为研究对象,提出了钻柱纵向和横向耦合振动的动力方程,并利用解析法和无量纲法分别求解出其动刚度和动阻尼的表达式,以及钻柱前两阶振动的固有频率.分析结果表明:当井下钻柱振动频率增大时,其动刚度呈幅值衰减的周期性变化,而其动阻尼呈幅值增强的周期性变化;井下钻柱长度和横截面面积越大,其动刚度和动阻尼的幅值越小;井下钻柱的Poisson(泊松)比对其振动的动刚度、动阻尼和前两阶固有频率没有影响;同时,井下钻柱的第二阶固有频率始终大于第一阶固有频率.该文的研究方法和模型为井下钻柱钻具分析和结果优化提供了理论参考和实际意义.     

流体诱发水平悬臂输液管的内共振和模态转换(Ⅱ)

基于得到的水平悬臂输液管非线性动力学控制方程,详细研究了由流速最小临界值诱发的3∶1内共振．通过观察内共振调谐参数、主共振调谐参数和外激励幅值的变化,发现在内共振临界流速附近,流速导致系统出现模态转换、鞍结分岔、Hopf分岔、余维2分岔和倍周期分岔等非线性动力学行为,对应的管道系统的周期运动失稳出现跳跃、颤振和更加复杂的动力学行为．通过理论结果与数值模拟比较,表明了理论分析的有效性和正确性．     

Torsional Vibration Damping Through Frictional Torsion Damper with Structural Friction and Slide Taken into Consideration

The paper is concerned with a non-linear discrete stationary mechanical system containing a frictional torsion damper. Proper effect of vibration damping in a two-degree-of freedom system can be reached by the right selection of geometrical parameters for given loads, as pre-determined by a mathematical model. Structural friction was considered, as well as small relative sliding of damper's discs cooperating with a plunger. The system vibrates under harmonic excitation. The problem was considered on the assumption of uniform unit pressure distribution between the contacting surfaces of friction discs and the plunger. When the discs are sliding, the friction coefficient varies, depending on relative angular velocity. Friction characteristics were assumed on the basis of the author's own research and experimental testing by other authors. Properties of the material were assumed to be in accordance with classical theory of elasticity. The author analysed the influence of parameters of the dynamic system upon amplitude and frequency characteristics as well as on phase and frequency characteristics. The equation of motion was solved by means of the slowly-varying-parameters method and, in order to compare the results, by means of numerical simulation. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear vibration of hybrid laminated plates resting on elastic foundations in thermal environments

This paper deals with large amplitude vibration of hybrid laminated plates containing piezoelectric layers resting on an elastic foundation in thermal environments. The motion equation of the plate that includes plate-foundation interaction is based on a higher order shear deformation plate theory and solved by a two-step perturbation technique. The thermo-piezoelectric effects are also included and the material properties of both orthotropic layers and piezoelectric layers are assumed to be temperature-dependent. The numerical illustrations concern nonlinear vibration characteristics of unsymmetric cross-ply and antisymmetric angle-ply laminated plates with fully covered or embedded piezoelectric actuators under different sets of thermal and electrical loading conditions. The results show that the foundation stiffness and stacking sequence have a significant effect on the nonlinear vibration characteristics of the hybrid laminated plate. The results also reveal that the temperature rise reduces the natural frequency, but it only has a small effect on the nonlinear to linear frequency ratios of the hybrid laminated plate. The results confirm that the effect of the applied voltage on the natural frequency and the nonlinear to linear frequency ratios of the hybrid laminated plate is marginal except the plate is sufficiently thin.     

Nonlinear analysis of a parametrically excited beam with intermediate support by using Multi-dimensional incremental harmonic balance method

In this paper, a nonlinear Euler-Bernoulli beam under a concentrated harmonic excitation with intermediate nonlinear support is investigated. Continuous expression for the kinetic energy, potential energy and dissipation function are constructed. An energy method based on the Lagrange equation combined with the Galerkin truncation is used for discretizing the governing equation. The Multi-dimensional incremental harmonic balance method (MIHBM) is derived, and the comparisons between the numerical results and the approximate analytical solutions based on the MIHBM verify the excellent accuracy of the MIHBM. The steady state dynamic of the beam is investigated by MIHBM. In order to investigate the energy transmission and understand the vibration response of the Euler-Bernoulli beam, the effects of the key parameters on the dynamic behaviors are studied and discussed, individually. The results show that the amplitude-frequency curves exhibits softening nonlinear behavior in the super-harmonic resonance region, and near resonant region the hardening nonlinear behavior is observed depending on the different parameters. Nonlinear dynamic analysis, such as bifurcation, 3-D frequency spectrum, waveform, frequency spectrum, phase diagram and Poincaré map, are also presented in order to study the influences of the key parameters on the vibration behaviors for the beam in a more accurate manner. In addition, the path to chaotic motion is observed to be through a sequence of the periodic motion and quasi-periodic motion.     

Study of non-linear dynamic behavior of open cracked functionally graded Timoshenko beam under forced excitation using harmonic balance method in conjunction with an iterative technique

The nonlinear modeling and subsequent dynamic analysis of cracked Timoshenko beams with functionally graded material (FGM) properties is investigated for the first time using harmonic balance method followed by an iterative technique. Crack is assumed to be open throughout. During modeling, nonlinear strain–displacement relation is considered. Rotational spring model is adopted in order to model the open cracks. Energy formulations are established using Timoshenko beam theory. Nonlinear governing differential equations of motion are derived using Lagrange's equation. In order to incorporate the influence of higher order harmonics, harmonic balance method is employed. This reduces the governing differential equations into nonlinear set of algebraic equations. These equations are solved using two different iterative techniques. Methodology is computationally easier and efficient as well. This is observed that although assumption of simple harmonic motion (SHM) simplifies the problem, it yields to erroneous results at higher amplitude of motion. However, accuracy of the solution is improved considerably when the contribution of higher order harmonic terms are considered in the analysis. Results are compared with the available results, which confirm the validity of the methodology. Subsequent to that a parametric study on influence of forcing term, material indices and crack parameters on large amplitude vibration of Timoshenko beams is performed for two different boundary conditions.     

Rapid gravity flow of cohesionless granular materials down inclined chutes

This paper is concerned with the analysis of the flow of cohesionless granular materials down inclined chutes. The stress tensor is assumed to be composed of two contributions, a rate-independent Coulomb part and a non-linear dynamic or viscous part. Plausible forms are chosen for these two stress contributions; the rate-dependent viscous part is based upon previous viscometric experiments of the present authors. The equations of motion are solved for the velocity and solids concentration for the case of two-dimensional stress and velocity fields. The two-dimensional analysis is then extended in a simple but approximate way to handle the effects of friction on the vertical side walls of a rectangular cross-sectioned channel. The predicted velocities and flow rates are compared with experiments of Savage for flowing polystyrene beads and reasonable agreement is found.     

Effect of fast parametric viscous damping excitation on vibration isolation in sdof systems

In this work, we investigate analytically the effect of cubic nonlinear parametric viscous damping on vibration isolation in sdof systems. Attention is focused on the case of a fast parametric damping excitation. The method of direct partition of motion is used to derive the slow dynamic and steady-state solutions of this slow dynamic are analyzed to study the influence of the fast nonlinear parametric damping on the vibration isolation. This study shows that adding periodic nonlinear damping variation to the vibration isolation device can reduce transmissibility over the whole frequency range. The results also reveals that this nonlinear parametric viscous damping enhances vibration isolation comparing to the case where the cubic nonlinear damping is time-independent.     

Nonlinear dynamics and stability analysis of piezo-visco medium nanoshell resonator with electrostatic and harmonic actuation

In this paper, nonlinear dynamics, vibration and stability analysis of piezo-visco medium nanoshell resonator (PVM-NSR) based on functionally graded (FG) cylindrical nanoshell integrated with two piezoelectric layers subjected to visco-pasternak medium, electrostatic and harmonic excitations is investigated. Nonclassical method of the electro-elastic Gurtin–Murdoch surface/interface theory with von-Karman–Donnell's shell model as well as Hamilton's principle, the assumed mode method combined with Lagrange–Euler's are considered. Complex averaging method combined with arc-length continuation is used to achieve a numerical solution for the steady state vibrations of the system. The stability analysis of the steady state response is performed. The parametric studies such as the effects of different boundary conditions, different geometric ratios, structural parameters, electrostatic and harmonic excitation on the nonlinear frequency response and stability analysis are studied. The results indicate that near the natural frequency of the nanoshell, it will lead to resonance and will have large motion amplitude and near the resonant frequency, the nanoshell shows a softening type of nonlinear behavior, and the nanoshell bandwidth increases due to nonlinear factors. In this range, nanoshell has three different ranges of motion, of which two are stable and the other unstable, and so the jump phenomenon and saddle-node bifurcation are visible in the behavior of the system. Also piezoelectric voltage influences on static deformation and resonant frequency but has no significant effect on nonlinear behavior and bandwidth and also system very sensitive to the damping coefficient and due to decrease of nano shell stiffness, natural frequency decreases. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system's instability, nonlinear behavior and bandwidth.     

Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects

Analytical solution for the steady-state response of an Euler–Bernoulli nanobeam subjected to moving concentrated load and resting on a viscoelastic foundation with surface effects consideration in a thermal environment is investigated in this article. At first, based on the Eringen's nonlocal theory, the governing equations of motion are derived using the Hamilton's principle. Then, in order to solve the equation, Galerkin method is applied to discretize the governing nonlinear partial differential equation to a nonlinear ordinary differential equation; solution is obtained employing the perturbation technique (multiple scales method). Results indicate that by increasing of various parameters such as foundation damping, linear stiffness, residual surface stress and the temperature change, the jump phenomenon is postponed and with increasing the amplitude of the moving force and the nonlocal parameter, the jump phenomenon occurs earlier and its frequency and the peak value of amplitude of vibration increases. In addition, it is seen that the non-linear stiffness and the critical velocity of the moving load are important factors in studying nanobeams subjected to moving concentrated load. Presence of the non-linear stiffness of Winkler foundation resulting nanobeam tends to instability and so, the jump phenomenon occurs. But, presence of the linear stiffness will lead to stability of the nanobeam. In the next sections of the paper, frequency responses of the nanobeam made of temperature-dependent material properties under multi-frequency excitations are investigated.     

Damping of Vibrations in a SystemWith Two Beams by Structural Friction

This paper presents the results of tests on free and forced harmonic vibrations in a system with two beams with structural friction taken into account. The beams are clamped together with uniform unitary pressure. The hysteresis loop describing the frictional-elastic properties of the system has a form of a parallelogram. The autor created a mathematical model of the vibrating system with two beams. During free vibrations of the system, its damping characteristics were tested by a digital simulation method. The vibration damping decrement as a function of amplitude displacement was determined. When vibrations were harmonically forced, the amplitude - frequency characteristics of the system were determined numerically. The system was used as a nonlinear vibration damper in a linear system with a harmonic force. The equations of motion of the nonlinear two-degree of freedom system were solved by means of a digital simulation method. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear dynamic analysis of fractional order rub-impact rotor system

Nonlinear dynamic characteristics of rub-impact rotor system with fractional order damping are investigated. The model of rub-impact comprises a radial elastic force and a tangential Coulomb friction force. The fractional order damped rotor system with rubbing malfunction is established. The four order Runge–Kutta method and ten order CFE-Euler method are introduced to simulate the fractional order rub-impact rotor system equations. The effects of the rotating speed ratio, derivative order of damping and mass eccentricity on the system dynamics are investigated using rotor trajectory diagrams, bifurcation diagrams and Poincare map. Various complicated dynamic behaviors and types of routes to chaos are found, including period doubling bifurcation, sudden transition and quasi-periodic from periodic motion to chaos. The analysis results show that the fractional order rub-impact rotor system exhibits rich dynamic behaviors, and that the significant effect of fractional order will contribute to comprehensive understanding of nonlinear dynamics of rub-impact rotor.     

具有确定运动姿势的柔性体的动力学分析研究

讨论了具有确定运动姿态的柔性多体系统的非线性动力学控制方程． 将飞行器在空间的运动看作是已知的,分析了飞行器上的挠性构件对飞行器运动和姿态的影响,利用假设模态,将挠性构件的变形,看作是空间直角坐标轴方向的线元振动所构成的,根据动力学中的Kane方法,建立了动力学方程,方程中包含表示弹性变形的结构刚度矩阵及表示变形体非线性变形几何刚度矩阵,方程推导从应力-应变关系入手,使用了有限元法．经简化,得到了带帆板结构的平面挠性体对飞行器运动影响的动力学方程,这种方程可通过计算机实现其数值解．     

一类四阶变系数振动问题的复合解

以一类斜拉索模型为背景,讨论了一类四阶变系数振动问题.用匹配法导出了方程的复合解,从而较简捷地得出了一类斜拉索的振幅的渐近表达式,并为解决相关类型的变系数问题提供了一种有效的方法.     

Transportation of dry fine powders by coordinated friction manipulation

The transportation of dry fine powders is an emerging technologic task, as in biotechnology, pharmaceutical or coatings industry particle sizes of processed powders are getting smaller and smaller. Fine powders are primarily defined by the fact that adhesive and cohesive forces outweigh the weight forces. This leads to mostly unwanted agglomeration (clumping) and adhesion to surfaces, what makes it more difficult to use conventional conveyor systems (e. g. pneumatic or vibratory conveyors) for transport. A rather new method for transporting these fine powders is based on ultrasonic vibrations, which are used to reduce friction and adhesion between powder and the substrate. One very effective set-up consists of a pipe, which vibrates harmoniously in axial direction at low frequency combined with a pulsed radial high frequency vibration. The high frequency vibration accelerates the particles perpendicular to the surface of the pipe, which in average leads to lower normal and thereby smaller friction force. With coordinated friction manipulation the powder acceleration can be varied so that the powder may be greatly accelerated and only slightly decelerated in each excitation period of the low frequency axial vibration of the pipe. The amount of powder flow is adjustable by vibration amplitudes, frequencies, and pulse rate, which makes the device versatile for comparable high volume and fine dosing using one setup. Within this contribution an experimental set-up consisting of a pipe, a solenoid actuator for axial vibration and a piezoelectric actuator for the radial high frequency vibration is described. An analytical model is shown, that simulates the powder velocity. Finally, simulation results are validated by experimental data for different driving parameters such as amplitude of low frequency vibration, pipe material and inclination angle. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

裂缝三级摩擦因数及影响因素研究(以砂岩（颗粒胶结体）为例)

讨论摩擦面的摩擦因数模型．认为砂岩的摩擦因数分为砂粒球面摩擦因数、微裂纹平面摩擦因数、凸凹构成的裂缝摩擦因数3个层次,分别代表3类不同的成因,3个层次的耦合是真实岩石摩擦因数的决定因素．岩石摩擦因数是在砂粒球面材料摩擦因数基础上,经过后两种形式的放大而形成岩石的宏观摩擦因数．裂纹表面凸起的平均角度或者分形维数是影响岩石摩擦因数分异的最大影响因素,而颗粒排布模式导致的分异相对小得多．颗粒接触的静摩擦因数大于动摩擦因数的成因与颗粒的平均接触角度有关．