Analysis of nonlinear vibrations of a two-degree-of-freedom mechanical system with damping modelled by a fractional derivative

Free damped vibrations of a mechanical two-degree-of-freedom system are considered under the conditions of one-to-one or two-to-one internal resonance, i.e., when natural frequencies of two modes – a mode of vertical vibrations and a mode of pendulum vibrations – are approximately equal to each other or when one natural frequency is nearly twice as large as another natural frequency. Damping features of the system are defined by the fractional derivatives with fractional parameters (the orders of the fractional derivatives) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. The model put forward allows one to obtain the damping coefficient dependent on the natural frequency of vibrations, so it has been shown that the amplitudes of vertical and pendulum vibrations attenuate by an exponential law with damping ratios which are exponential functions of the natural frequencies. Damped soliton-like solutions have been found analytically.     

路面基础激励下分数导数型包装系统 核心部件振动特性研究

研究分数导数型包装系统关键部件在路面激励下的动力学响应,建立了二自由度分数导数型包装系统动力学方程并求解,得到了核心部件位移的解析解和振幅比的表达式。并讨论了质量比、频率比、阻尼比等结构参数对隔振系统隔振效果的影响。     

High Damping Alloys and Their Application

Damping alloys show prospective applications in the elimination of unwanted vibrations and acoustic noise. The basic definitions and characterization methods of damping capacity are reviewed in this paper. Several physical mechanisms controlled by the alloy microstructure are responsible for the damping behavior in the damping alloys. Composite, dislocation, ferromagnetic and planar defect types are commonly classified for the alloys, which show the different damping behavior against temperature, frequency of vibration,amplitude of vibration and damping modes. Development of practically applicable damping alloys requires the higher mechanical properties and adequate workability, besides the high damping capacity. A new Mn-Cu damping alloy, named as M2052 alloy, is recently developed with possible industrial applications.     

Non-linear vibration of a centrally clamped rotating disc

Deflection profiles were calculated for vibration modes with up to four nodal diameters. Strain energy at large amplitudes was calculated. When vibrating with a particular number of nodal diameters, a disc of slightly imperfect axial symmetry has two normal modes with slightly different natural frequencies. When vibration amplitude becomes large, elastic coupling affects the way in which the two vibrations combine to form two traveling waves. Calculated frequency-response curves compared well with experimental curves. For low values of damping, a regular regime was found, giving a simple relation between frequency interval and the minimum vibration amplitude required for producing a relatively pure traveling wave. The predicted relation was numerically confirmed by tests on steel discs having accentuated frequency intervals. The results have significance in self-excited vibration which derives excitation from a single traveling wave.     

Linear dynamical analysis of fractionally damped beams and rods

The aim of this study is to develop a general model for beams and rods with fractional derivatives. Fractional time derivatives can represent the damping term in dynamical models of continuous systems. Linear differential operators with spatial derivatives make it possible to generalize a wide range of problems. The method of multiple scales is directly applied to equations of motion. For the approximate solution, the amplitude and phase modulation equations are obtained in terms of the operators. Stability boundaries are derived from the solvability condition. It is shown that a fractional derivative influences the stability boundaries, natural frequencies, and amplitudes of vibrations. The solution procedure may be applied to many problems with linear vibrations of continuous systems.     

Nonlinear damping and forced vibration analysis of laminated composite beams

The purpose of the present work it to study the damping and forced vibrations of three-layered, symmetric laminated composite beams. In the analytical formulation, both normal and shear deformations are considered in the core by using the higher-order zig-zag theories. The harmonic balance method is coupled with a one mode Galerkin procedure for a simply supported beam. The geometrically nonlinear coupling leads to a nonlinear frequency amplitude equation governed by several complex coefficients. In the first part of the paper, linear and nonlinear damping parameters of laminated composite beams are obtained. In the second part, nonlinear forced vibration analysis is carried out for small and large vibration amplitudes. The frequency response curves are presented and discussed for various geometric and material properties.     

Modeling of hybrid vibration control for multilayer structures using solid-shell finite elements

A self-control method of vibrations is presented in this paper. This method combines the passive damping capabilities afforded by viscoelastic materials with the active control properties associated with piezoelectric materials. Active control is introduced, using the piezoelectric properties, in order to improve the reduction in vibration amplitudes that can be obtained by viscoelastic passive damping alone. To this end, a filter has been mounted between the sensors and the actuators. The resulting nonlinear problem is discretized using the recently developed solid-shell finite element SHB20E, due to the advantages it offers in terms of accuracy and efficiency as compared to standard finite elements with the same geometry and kinematics. In order to solve the discretized problem, a resolution method using DIAMANT approach is developed. A set of selective and representative numerical tests are performed on multilayer plates to demonstrate the interest of the proposed damping model.     

Dynamic viscoelastic modeling of magnetorheological elastomers

A micromechanics-based viscoelastic constitutive model is proposed to estimate the zero-magnetic-field- and magnetic-field-dependent dynamic shear stiffness and damping behavior for magnetorheological elastomers (MREs). The effect of imperfect interfacial condition between the ferromagnetic particles and the elastomeric matrix on those properties is incorporated in the proposed model. A concept of effective volume fraction of particles is introduced to take into account the particle agglomeration in MREs. The magnetic dipole interaction is further employed to evaluate the magnetic-field-induced increase in shear stiffness of MREs. Numerical simulations are conducted and compared with experimental data to verify the proposed model.     

Effect of break of the elastic axis of a rotor with transverse crack on its vibrational characteristics

We perform a numerical analysis of the effect of break of the elastic axis on the characteristics of bending vibrations of a multispanned unbalanced rotor with transverse crack mounted on elastic damping supports. The break of the elastic axis in the cross section with crack is caused by local plastic strains. The solution of the problem is sought in the form of a sum of the contributions of the forces of gravity, residual disbalance, and the crack responsible for the break of the axis. The problem is reduced to the numerical analysis of the trajectory of the shaft with its subsequent improvement by the method of iterations. Amplitudes and phases of several first harmonics of vibrational displacements are found as functions of sizes, initial orientation of the crack, and the ratio of vibrational and static displacements. In the presence of the break of the elastic axis, the magnitudes of vibrational characteristics undergo significant changes. The amplitudes of vibrations induced by the crack are comparable with the amplitudes of vibrations caused by disbalance. It is shown that the behavior of the trajectory of motion regarded as a function of principal parameters is nonlinear. Institute for Problems in Mechanical Engineering, National Academy of Sciences of Ukraine, Kharkov, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 82–89, July–August, 1997.     

Experimental studies on high-cycle fatigue and damping properties of R2MA rotor steel in torsion

High-cycle fatigue curves in symmetrical torsion were plotted, the endurance limit–mean cycle stress relation was derived, and damping of torsional vibrations was evaluated by the results of R2MA rotor steel specimen testing.     

钢管混凝土轴压构件耗能计算及单位体积耗能公式建立

提出了计算钢管混凝土轴压构件材料阻尼的新方法,根据Lazan材料阻尼与应力关系的理论研究,计算弹性阶段钢管混凝土构件在轴向重复荷载下的滞回曲线包围面积,得到钢管混凝土材料单位体积损耗能量。结果表明,单位体积损耗能量随最大应力幅值的提高而增大,且随混凝土强度和截面含钢率增大而减小。通过SPSS统计分析软件回归建立了钢管混凝土构件在轴向重复荷载下材料单位体积耗能与最大应力幅值、混凝土强度和截面含钢率的关系式。为进一步准确分析钢管混凝土结构动力性能及进行钢管混凝土结构的抗震分析打下了基础。     

Failures of cranes due to wind induced vibrations

Self-excited vibrations with large amplitudes in natural wind may occur at slender structural elements with low damping. Because of the different designs (e. g. using solid sections today instead of cables for tension elements in the past) the susceptibility to wind induced oscillations has increased. Those wind induced vibrations of profiles with specific cross section geometry which are motion induced and therefore self-exciting are called “galloping vibrations”. Especially systems with elements that are highly tensile loaded and undamped, like hangers of bridges or tension bars of cranes, are sensitive to wind induced vibrations. Therefore more and more fatigue problems caused by galloping oscillations have occurred in the 1990s. This paper describes exemplary the collapses of two modern cranes of different design and manufacturers. During standstill periods, both cranes suffered from wind induced vibrations of the tension bars, which bear up the counterweights. The failure analysis process to identify and explain the fatigue fractures as well as the comparative experiments and simulation to verify that they were caused by wind induced galloping-vibrations is described. It is shown, which parameters led to galloping-vibrations of the tension bars and how their onset wind speed and the amplitudes can be estimated with more accuracy by a non-linear and non-stationary approach. Furthermore it is shown that such dynamic stresses caused fatigue failure of the tension bars for the counter weights and subsequently collapsing of the cranes.For loss prevention knowledge and results gained by these investigations should be put at disposal to engineers working on this field of design. In the meantime, a contribution to development of appropriate technical standards on structural steelwork was given by the research works on galloping. Although new standards were introduced, which consider wind induced vibrations, such failures still occur. (Reference to the paper “Fatigue crack in railway bridge hanger due to wind induced vibrations – failure analysis, measures and remaining service life estimation” in this same Special Issue “A tribute to A. Martens” 2014).     

INFLUENCE OF DAMPING MODELS ON DYNAMIC ANALYSES OF A BASE-ISOLATED COMPOSITE STRUCTURE UNDER EARTHQUAKES AND ENVIRONMENTAL VIBRATIONS

Structural design simultaneously governed by earthquakes and environmental vibrations has received a lot of attention in recent years.Base-isolated composite structures are typically used in the above-mentioned structural design.The corresponding analysis involves validating structural safety under earthquakes and human comfort under environmental vibrations through a time-history analysis.Thus,a reasonable damping model is essential.In this work,the representatives of viscous damping model and rate-independent damping model,namely the Rayleigh damping model and uniform damping model,were adopted to investigate the influence of damping models on the time-history analysis of such structural designs.The energy dissipation characteristics of the above-mentioned damping models were illustrated via a dynamic test of recycled aggregate concrete specimens.A case study was performed on a base-isolated steelconcrete composite structure.The dynamic responses under the excitation of earthquakes and environmental vibrations were compared using different damping models.The uniform damping model was found to be more flexible than the Rayleigh damping model in dealing with excitations with different frequency components.The uniform damping model is both theoretically advantageous and easy to use,demonstrating its potential in dynamic analysis of structures designed simultaneously governed by earthquakes and environmental vibrations.     

Évaluation expérimentale des caractéristiques modales et ajustement du modèle dynamique d’un bâtiment de portiques

This article presents the evaluation of the dynamic properties (frequencies, damping ratios, and modal shapes) of a five-story framed building by various experimental procedures (free, harmonic and ambient vibration tests). The results were very similar, even though ambient vibrations tests deal with amplitudes 20 times lower than free vibration. Special attention is given to free vibration in the frequency domain. Mathematical model adjustment is also carried out. A model including masonry, soil structure iteration, stairs, and slab influences reproduced quite accurately the experimental values of the dynamic properties of the first three modes.     

Numerical analysis of strongly nonlinear extensional vibrations in elastic rods

In the framework of transduction, nondestructive testing, and nonlinear acoustic characterization, this article presents the analysis of strongly nonlinear vibrations by means of an original numerical algorithm. In acoustic and transducer applications in extreme working conditions, such as the ones induced by the generation of high-power ultrasound, the analysis of nonlinear ultrasonic vibrations is fundamental. Also, the excitation and analysis of nonlinear vibrations is an emergent technique in nonlinear characterization for damage detection. A third-order evolution equation is derived and numerically solved for extensional waves in isotropic dissipative media. A nine-constant theory of elasticity for isotropic solids is constructed, and the nonlinearity parameters corresponding to extensional waves are proposed. The nonlinear differential equation is solved by using a new numerical algorithm working in the time domain. The finite-difference numerical method proposed is implicit and only requires the solution of a linear set of equations at each time step. The model allows the analysis of strongly nonlinear, one-dimensional vibrations and can be used for prediction as well as characterization. Vibration waveforms are calculated at different points, and results are compared for different excitation levels and boundary conditions. Amplitude distributions along the rod axis for every harmonic component also are evaluated. Special attention is given to the study of high-amplitude damping of vibrations by means of several simulations. Simulations are performed for amplitudes ranging from linear to nonlinear and weak shock.     

形状记忆合金复合材料系统非线性动力学研究

研究了形状记忆合金复合材料系统的非线性动力学特性。基于Brinson的形状记忆合金本构关系和Dejonghe的内耗模型,提出了单自由度简化系统的模型建立方法。考虑材料系统刚度和阻尼的变化,在弱非线性和较强非线性两种情况下给出近似-解析解法。算例表明:形状记忆合金可以作为良好的耗能材料,用于结构的被动控制。     

薄壁圆柱壳构件受迫振动的响应特征研究

针对固支-自由约束条件下受径向谐波激励或径向冲击激励的薄壁圆柱壳构件,开展其受迫振动下的响应特征分析。首先基于Love壳体理论建立了薄壁圆柱壳构件的动力学模型,然后,根据固支-自由约束条件特点,采用轴向梁函数和周向三角函数组合的振型函数以及振型叠加法,获得了考虑粘性阻尼的薄壁圆柱壳模态坐标振动方程,进而求解受径向谐波激励或冲击激励的振动响应。通过一个具体算例,进行了不同位置上的响应幅度与相位的变化分析,并对比了模态阻尼比和激励力幅值对响应幅值的影响。     

信道阻尼边界对井下钻杆声传输的影响

为改善低频声波沿钻杆管肇的传输性能，分析了影响声遥测的阻尼机制。考虑钻杆内外沿轴向流动的钻井液的阻尼影响，引入流体的粘性阻尼力，建立了一维纵波波动方程。基于圆柱源辐射理论，研究纵波的径向耦合损耗，并将地层等效为Kelvin粘弹性介质，应用有限单元法求解时域波动方程，讨论了钻井液粘性阻尼和粘弹性地层边界对行波传播特性的影响。理论分析表明，径向辐射和由此产生的波型耦合是声阻尼损耗的主要形式，地层粘弹性系数的变化列低频特性影响很大，但并不改变信道通阻带交替的梳状滤波器频谱。     

Magneto-Mechanical Finite Element Analysis of Single Crystalline Ni2MnGa Ferromagnetic Shape Memory Alloy

Based on an existing micromechanical constitutive model for Ni2MnGa ferromagnetic shape memory alloy single crystals, a three-dimensional quasi-static isothermal incremental constitutive model that is suitable for finite element analysis is derived by using Hamilton's variational principle. This equation sets up the coupling relation between the magnetic vector potential and the mechanical displacement. By using the incremental equation and ANSYS software, the mechanical behaviors of martensitic variant reorientation for Ni2MnGa single crystals are analyzed under magneto-mechanical coupling action. And the finite element results agree well with the experimental data. The methods used in the paper can well describe the mechanical behaviors of the material in complex fields.     

循环荷载下奥氏体型和双相型不锈钢材料本构关系研究

为研究循环荷载下不锈钢材料的本构关系,对奥氏体型S30408不锈钢和双相型S220503不锈钢材料进行了单调拉伸和大应变超低周循环加载试验。采用三种常用的单调拉伸本构模型对所得应力-应变曲线进行拟合,得到相应单调荷载下材料本构参数;采用Ramberg-Osgood本构模型对循环骨架曲线进行拟合,得到材料循环强化参数;利用Chaboche塑性本构模型,标定了两种材料的循环本构参数。结果表明:在单调拉伸荷载下,G-R-O本构模型更适用于拟合不锈钢材料的单调拉伸本构;在循环荷载下,不锈钢材料滞回曲线饱满,且随着应变增大,两种材料在加载后期均表现出了明显的循环强化现象;Ramberg-Osgood本构模型对骨架曲线拟合较好,有限元计算结果和试验滞回曲线吻合度高;表明该文标定出的强化参数、循环本构参数可用于结构体系地震响应分析之中,为准确分析不锈钢结构在地震作用下的受力性能提供参考。