Effects of anti-symmetric nonlinear viscous damping on the force transmissibility of multi-degree of freedom structures

In the present study, the concept of the output frequency response function is applied to theoretically investigate the force transmissibility of multi-degree of freedom (MDOF) structures with a nonlinear anti-symmetric viscous damping. The results reveal that an anti-symmetric nonlinear viscous damping can significantly reduce the transmissibility over all resonance regions for MDOF structures while it has almost no effect on the transmissibility over non-resonant and isolation regions. The results indicate that the vibration isolators with an anti-symmetric damping characteristic have great potential to overcome the dilemma in the design of linear viscously damped vibration isolators where an increase of the damping level reduces the force transmissibility over resonant region but increases the transmissibility over non-resonant regions.     

Non-linear output frequency response functions of MDOF systems with multiple non-linear components

In engineering practice, most mechanical and structural systems are modelled as multi-degree-of-freedom (MDOF) systems such as, e.g., the periodic structures. When some components within the systems have non-linear characteristics, the whole system will behave non-linearly. The concept of non-linear output frequency response functions (NOFRFs) was proposed by the authors recently and provides a simple way to investigate non-linear systems in the frequency domain. The present study is concerned with investigating the inherent relationships between the NOFRFs for any two masses of non-linear MDOF systems with multiple non-linear components. The results reveal very important properties of the non-linear systems. These properties clearly indicate how the system linear characteristic parameters govern the propagation of the non-linear effect induced by non-linear components in the system. One potential application of the results is to detect and locate faults in engineering structures which make the structures behave non-linearly.     

粘滞和粘弹性阻尼器减震结构的等效阻尼

利用积分型本构关系,建立了带支撑的一般粘滞和粘弹性阻尼器单自由度耗能结构的微分-积分混合地震响应方程;基于与随机平均分析完全相同的等效准则,推导了可直接应用反应谱的阻尼器的等效刚度和等效阻尼的解析公式;得到了带支撑广义Maxwell阻尼器和广义微分模型阻尼器的等效刚度和等效阻尼的一般结果。通过与一些典型问题的数值计算结果比较,表明了所提方法的有效性。     

On a modal damping identification model of building structures

The simple modal damping identification model of Hart and Vasudevan is generalized. The model works in the frequency domain and provides time-invariant modal damping ratios of building structures under seismic excitations in terms of modal participation factors and the roof-to-basement transfer function. Translational as well as torsional modes of vibration are considered. The performance of the model is assessed through a small, yet indicative, number of numerical examples involving steel plane and space frames under seismic excitations and on the basis of a number of criteria an ideal identification model should satisfy. It is concluded that the presented model, in spite of its simplicity, gives very good results for low-amplitude seismic excitations resulting in linear elastic structural behavior (with damping) even for cases of closely spaced modes, local modes, and very small or large amounts of damping. Some numerical pitfalls regarding the application of the model are mentioned and carefully treated. The limitations of the model when used in conjunction with inelastic structural behavior are determined and discussed. Experimental verification of the model is also provided.     

Transmissibility of non-linear output frequency response functions with application in detection and location of damage in MDOF structural systems

Transmissibility is a well-known linear system concept that has been widely applied in the diagnosis of damage in various engineering structural systems. However, in engineering practice, structural systems can behave non-linearly due to certain kinds of damage such as, e.g., breathing cracks. In the present study, the concept of transmissibility is extended to the non-linear case by introducing the Transmissibility of Non-linear Output Frequency Response Functions (NOFRFs). The NOFRFs are a concept recently proposed by the authors for the analysis of non-linear systems in the frequency domain. A NOFRF transmissibility-based technique is then developed for the detection and location of both linear and non-linear damage in MDOF structural systems. Numerical simulation results verify the effectiveness of the new technique. Experimental studies on a three-storey building structure demonstrate the potential to apply the developed technique to the detection and location of damage in practical MDOF engineering structures.     

The complete infinite series solution of systems governed by the wave equation with boundary damping

A common method of solving initial boundary value problems is separation of variables, denoted as modal analysis in the field of flexible structures. For systems with undamped boundary conditions the method is well-established, but for systems with boundary damping it does not provide closed form solutions. In this paper the exact modal series solution for second order systems with damped boundaries is derived with explicit expressions for the series coefficients. Knowledge of these coefficients enables practical applications of the solution, such as finite dimension approximation. The key element of the derivation is a new orthogonality condition for the damped eigenfunctions. The modal series is also transformed into a traveling wave form. The solution, which is the extension of the classical D’Alembert formula, is represented by a single equivalent propagating wave. A component of the solution, denoted by “end waves”, is identified to provide the continuity of the systems displacement response.     

Linear and non-linear analyses of convection in a micropolar fluid occupying a porous medium

Linear and weakly non-linear analyses of convection in a micropolar fluid occupying a high-porosity medium are performed. The Brinkman–Eringen momentum equation is considered. The linear and non-linear analyses are, respectively, based on the normal mode technique and truncated representation of Fourier series. The linear theory for a two-phase system reiterates that the preferred mode of convection is stationary as in the case of a single-phase system. An autonomous system of differential equations representing cellular convection arising in the study is considered to analyse the critical points. The Nusselt number is obtained as a function of micropolar and porous medium parameters.     

Random vibrations of a rotating shaft with non-linear damping

Bending vibrations of a rotating shaft due to external random excitation are considered for the case of potential instability of the shaft's linear model due to the presence of internal or “rotating” damping. A two-degree-of-freedom model is studied which accounts for non-linearity in external or “non-rotating” damping. An explicit expression is obtained for a stationary joint probability density of displacements and velocities as an exact analytical solution to the corresponding Fokker-Planck-Kolmogorov equation. The results are used to develop criterion for on-line detection of instability for the operating shaft from its measured response.     

Response of a non-linear system with strong damping to multifrequency excitations

Summary In this paper, a new asymptotic calculation method is presented for a class of nonlinear nonautonomous vibration systems with multiple external periodic interferences. Simple calculation formulae of an asymptotic solution for resonance and off-resonance vibration are derived. The paper is concerned with a vibration system representing a class of nonlinear oscillators. Consequently, the calculation of a class of nonlinear oscillators is routinized. Two different Duffing equations are verified which shows that the results are completely in accordance with the solutions of references [3, 4, 6]. The derivation of solutions of Duffing equations becomes easier and simpler. In addition, some errors in reference [6] are pointed out.     

大型非比例阻尼线性系统的地震反应复振型分析方法

对于大型非比例阻尼线性系统,当采用基于复振型的振型叠加方法进行动力反应分析时,按照通用算法(例如雅克比法)求解结构全部振型向量的计算工作量很大,甚至是不现实的.本文将经典阻尼系统中行之有效的Lanczos法和子空同迭代法加以推广和改进,给出了一组可对原来的方程进行自由度缩减的实向量基,然后与Foss变换相结合,得到一组实用的复向量基,使之适用于求解复杂非比例阻尼线性系统的任意低阶复振型和相应的复特征值,适用于任意扰力作用下的动力反应分析.理论推导和实例计算表明,本文所给出的复向量基概念清晰,计算效率高,能够适应对具有非比例阻尼特性的大型复杂结构进行动力分析的实际需要,其中包括地震作用下的时程分析和反应谱振型叠加分析.     

New estimations of the added mass and damping of two cylinders vibrating in a viscous fluid,from theoretical and numerical approaches

This paper deals with the small oscillations of two circular cylinders immersed in a viscous stagnant fluid. A new theoretical approach based on an Helmholtz expansion and a bipolar coordinate system is presented to estimate the fluid forces acting on the two bodies. We show that these forces are linear combinations of the cylinder accelerations and velocities, through viscous fluid added coefficients. To assess the validity of this theory, we consider the case of two equal size cylinders, one of them being stationary while the other one is forced sinusoidally. The self-added mass and damping coefficients are shown to decrease with both the Stokes number and the separation distance. The cross-added mass and damping coefficients tend to increase with the Stokes number and the separation distance. Compared to the inviscid results, the effect of viscosity is to add a correction term which scales as $S{k}^{-1∕2}$. When the separation distance is sufficiently large, the two cylinders behave as if they were independent and the Stokes predictions for an isolated cylinder are recovered. Compared to previous works, the present theory offers a simple and flexible alternative for an easy determination of the fluid forces and related added coefficients. To our knowledge, this is also the first time that a numerical approach based on a penalization method is presented in the context of fluid–structure interactions for relatively small Stokes numbers, and successfully compared to theoretical predictions.     

Dynamic characteristics of a cantilever beam with partial self-sensing active constrained layer damping treatment

The equations of motion governing the vibration of a cantilever beam with partially treated self-sensing active constrained layer damping treatment(SACLD) are derived by application of the extended Hamilton principle. The assumed-modes method and closed loop velocity feedback control law are used to analyze and control the flexural vibration of the beam. The influence of the bonding layer and piezoelectric layer thickness, material properties, placements of the piezoelectric patch and feedback control parameters on the actuation ability of the vibration suppression are investigated. Some design considerations for pure passive, pure active control, and self-sensing active constrained layer damping are discussed. The present work is supported by the National Natural Science Foundation of China (No. 59635140).     

The asymptotic behavior of solutions to the Kirchhoff equation with a viscous damping term

We study the asymptotic behavior of solutions to an equation describing non-linear vibration of a string with viscosity. In the case when the string is unstretched (the degenerate case), we determine the decay order of solutions by investigating the dynamics near an infinite-dimensional center manifold. Moreover, we classify the asymptotic behavior of all solutions from a dynamical systems point of view. We also deal with the case where the string is stretched (the nondegenerate case).     

The dynamic plastic behavior of a simply supported circular plate in a damping medium with finite-deflections

A theoretical analysis is presented for the dynamic plastic behavior of a simplysupported rigid,perfectly plastic circular plate in damping medium with finite-deflectionssubjected to a rectangular pressure pulse.Analytical solutions of every moving stage underboth medium and high loads are developed.     

Safety and comfort analysis of a 3-D vehicle model with optimal non-linear active seat suspension

A generalized nonlinear model is formulated for the dynamic analysis of suspension seats with passive, semi-active and active dampers. The model incorporates coulomb friction due to suspension linkages and bushings, forces arising from interactions with the elastic limit stops, a linear suspension spring and nonlinear damping force for passive, semi-active and active dampers, while the contribution due to biodynamics of the human operator is considered to be negligible. The semi-active and active dampers are characterized by force generators in accordance with the control laws based upon suspension mass velocity. Two different suspension seats are experimentally assessed in the laboratory under sinusoidal and random excitations arising from an urban bus, and the measured data is used to demonstrate the validity of the proposed generalized model. The results showed reasonably good agreement between the model results and the measured data. Optimal model parameters are selected using the sequential unconstrained minimization technique with an objective to minimize the acceleration due to vibration transmitted to the occupant mass. The comfort and safety performance characteristics of the optimal suspension seat with semi-active and active dampers are evaluated under both the sinusoidal and random excitations based on the guidelines provided by ISO-2631. From these results, it is concluded that the comfort performance of a suspension seat with semi-active and active dampers can be considerably enhanced by 20–30%.     

Geometric non-linear analysis of space frame with rotation greater than 90°, with Euler angles and quasi-fixed local axes system

The geometric non-linear analysis of space frames with definition of the local axes through rotations similar to Euler angles, presents difficulties when the rotations reach 90° in the fixed local axes. To surmount these difficulties this work presents a new technique using a system coined quasi-fixed local axes. In this technique, the direction cosines of these axes of the actual increment of loading are defined through the direction cosines of the moving local axes of the last iteration of the previous increment and of the quasi-fixed local axes of the previous increment (or fixed, in case of the first increment). The technique of definition of the quasi-fixed local axes is used with updated Lagrangian reference. The numerical examples presented herein show the great numerical stability of the technique.     

The interactions between wave-currents and offshore structures with consideration of fluid viscosity

Study of the flow field around the large scale offshore structures under the action of waves and viscous currents is of primary importance for the scouring estimation and protection in the vicinity of the structures. But very little has been known in its mechanism when the viscous effects is taken into consideration. As a part of the efforts to tackle the problem, a numerical model is presented for the simulation of the flow field around a fixed vertical truncated circular cylinder subjected to waves and viscous currents based on the depth-averaged Reynolds equations and depth-averagedk-ɛ turbulence model. Finite difference method with a suitable iteration defect correct method and an artificial open boundary condition are adopted in the numerical process. Numerical results presented relate to the interactions of a pure incident viscous current with Reynolds numberRe=10⁵, a pure incident regular sinusoidal wave, and the coexisting of viscous current and wave with a circular cylinder, respectively. Flow fields associated with the hydrodynamic coefficients of the fixed cylinder, as well as corresponding free surface profiles and wave amplitudes, are discussed. The present method is found to be relatively straightforward, computationally effective and numerically stable for treating the problem of interactions among waves, viscous currents and bodies. The project supported by the National Natural Science Foundation of China and Foundation of State Key Laboratory of Ocean Engineering at Shanghai Jiao Tong University.     

Stability and Bifurcation Analysis of a Modified Geometrically Nonlinear Orthotropic Jeffcott Model with Internal Damping

In this paper, a modified Jeffcott model is proposed and studied in order to shed light into the dynamics of a complex system, the Short Electrodynamic Tether (SET), which is similar to an unbalanced rotor. Due to the internal damping, a geometrically linear SET model appears to be unstable as predicted by the linear rotordynamics theory. Some studies in the field of rotordynamics suggest that this instability caused by internal damping do not appear if geometric nonlinearities are taken into account in the system equations of motion. Stability and bifurcation analysis have been carried out on the modified Jeffcott model, which accounts for geometric nonlinearities, orthotropy in the shaft's cross section, and a viscous damping-based internal damping model. The stability results analytically obtained have been compared with a nonlinear multibody model by means of time simulations and good agreement has been found.