Nonlinear transient response analysis for double walls with a porous material supported by nonlinear springs using FEM and MSKE method

In this paper, we newly propose a fast computation method for the nonlinear transient responses including coupling between nonlinear springs and sound proof structures having porous materials using FEM. In this method, we extend our numerical method named as Modal Strain and Kinetic Method (i.e. MSKE method proposed previously by Yamaguchi who is one of the authors) from linear damping analysis to nonlinear dynamic analysis. We assume that the restoring force of the spring has cubic nonlinearity and linear hysteresis damping. To calculate damping properties for soundproof structures including elastic body, viscoelastic body and porous body, displacement vectors as common unknown variable are solved under coupled condition. The damped sound fields in the porous materials are defined by complex effective density and complex bulk modulus. The discrete equations in physical coordinate for this system are transformed into nonlinear ordinary coupled differential equations using normal coordinates corresponding to linear natural modes. Further, using MSKE method, modal damping can be derived approximately under coupled conditions between hysteresis damping of viscoelastic materials, damping of the springs and damping due to flow resistance in porous materials. The modal damping is used for the nonlinear differential equation to compute nonlinear transient responses.Moreover, using the proposed method, we demonstrate new vibration phenomena including nonlinear coupling between nonlinear springs and soundproof structures by use of a simplified model. As a typical numerical example of the soundproof structure, we adopt double walls with a porous material. The double walls are supported by nonlinear concentrated springs. We clarify influences of amplitude of the impact force on nonlinear transient responses. We focused on the vibration modes, which magnify the amplitudes of the double walls. In these modes, the internal air of the porous material played a role of a pneumatic spring. Under a very large impact force as a severe condition, there exist the complicated nonlinear couplings between these modes and the super harmonic components of the rigid modes of the whole structure with large deformations in the nonlinear springs.     

FEA for vibrated structures with non-linear concentrated spring having hysteresis

This paper describes vibration analysis using finite element method for structures connected with non-linear concentrated spring. The restoring force of the spring is assumed to be expressed as power series of displacement. The restoring force also has linear hysteresis damping. Thus, complex stiffness is introduced for the linear component of the restoring force. Finite element for the spring is expressed and is connected to viscoelastic structures modelled by linear solid finite elements. Further, the discretised equations in physical coordinate are transformed into the non-linear ordinary coupled equations using normal coordinate corresponding to linear natural modes. Note that modal damping is also transformed in this procedure. The transformed equations were integrated numerically in fairly small degree of freedom. This transformation yields computation efficiency. Effectiveness of this analysis was shown for a basic block-spring model. And then, this numerical method was applied for a sophisticated measurement system named as the Levitation Mass Method proposed by Fujii for evaluating viscoelastic parameters.     

IDENTIFICATION OF NON-PROPORTIONAL MODAL DAMPING MATRIX AND REAL NORMAL MODES

Equations of motion for non-proportionally damped structures cannot be decoupled by using the real normal modes. For such structures, the complex normal modes are in common use for this purpose, but for the validation of finite element mass and stiffness matrices where physical damping matrices are not available, the related experimental real normal modes must be known. In previous publications, an identification theory using the real normal modes and the non-diagonal modal damping matrix for the non-proportionally damped system and some applications with the computer code ISSPA were presented. However, the theory cannot assure the symmetry of the identified modal damping matrix, which must be theoretically symmetric. In this paper, a method for identifying the symmetric non-proportional modal damping matrix using undamped modal parameters obtained from ISSPA is presented and the validity of the method is demon-strated through both numerical and experimental examples.     

Comparative study of damped FE model updating methods

Most of finite element (FE) model updating techniques do not employ damping matrices and hence, cannot be used for accurate prediction of complex frequency response functions (FRFs) and complex mode shapes. In this paper, a detailed comparison of two approaches of obtaining damped FE model updating methods are evaluated with the objective that the FRFs obtained from damped updated FE models is able to predict the measured FRFs accurately. In the first method, damped updating FE model is obtained by complex parameter-based updating procedure, which is a single-step procedure. In the second method, damped updated model is obtained by the FE model updating with damping identification, which is a two-step procedure. In the first step, mass and stiffness matrices are updated and in the second step, damping matrix is identified using updated mass and stiffness matrices, which are obtained in the previous step. The effectiveness of both methods is evaluated by numerical examples as well as by actual experimental data. Firstly, a study is performed using a numerical simulation based on fixed–fixed beam structure with non-proportional viscous damping model. The numerical study is followed by a case involving actual measured data for the case of F-shaped test structure. The updated results have shown that the complex parameter-based FE model updating procedure gives better matching of complex FRFs with the experimental data.     

Determination of the dynamic coefficients of the coupled journal and thrust bearings by the perturbation method

This paper proposes a method to calculate the stiffness and the damping coefficients of the coupled journal and thrust bearings. Reynolds equations and their perturbation equations of journal and thrust bearings are transformed to the finite element equations by considering the continuity of pressure and flow at the interface between the journal and the thrust bearings. It also includes the Reynolds boundary condition in the numerical analysis to simulate the cavitation phenomenon. The stiffness and damping coefficients of the proposed mathematical method are compared with those of the numerical differentiation of the bearing force with respect to finite displacements and finite velocities of bearing center. It shows that the proposed method can calculate the dynamic coefficients of a coupled journal and thrust bearing more numerically stable and computationally efficient than the differentiation method. It also investigates the coupling effect of the coupled journal and thrust bearing and it shows that the proposed method makes it possible to calculate the cross-coupled dynamic coefficients in the radial–axial direction of the coupled journal and thrust bearing.     

Mechanical joint parameter estimation using frequency response functions and component mode synthesis

This paper treats the problem of finite element model updating of structures consisting of substructures connected through mechanical joints whose stiffness and damping properties are unknown. The model is updated by estimating the mechanical joint parameters via the curve fit of measured frequency response functions using a non-linear least-squares method. A damped component mode synthesis formulation is used to calculate the theoretical frequency response functions of the assembled structure given the joint stiffness and damping parameter values. It is shown that a good approximation of the sensitivity matrix may be obtained by finite differences at a smaller computational cost than the closed form expression. Identifiability criteria and estimation errors are addressed. A numerical example consisting of two free-free beams connected through axial springs and dampers illustrates the proposed method.     

Finite element model updating of vibrating structures under free-free boundary conditions for modal damping prediction

A method to predict resonance frequencies and modal loss factors of bare and damped samples, using constrained layer damping treatment, under free-free boundary conditions is proposed. In a first phase, measurements of the frequency response functions of these two specimens are performed. In a second phase, a finite element model of the undamped sample is developed. The novelty lies in the consistent modelling of the suspension with spring-damper elements defined with stiffness and damping coefficients with fixed values over the whole considered frequency range. By updating these, the agreement between experiments and simulation is further improved. In a third phase, a finite element model of the damped sample, with constrained layer damping material, is realized. A good agreement with experimental results is obtained thanks to an optimization algorithm used to determine the material parameters of the viscoelastic layer at various frequency. A comparison with experimental results, from a Dynamic Mechanical Analysis, confirms the consistency of the results from the optimization process.     

阻尼材料中气泡的存在对结构阻尼性能的影响

为了研究粘弹性阻尼结构的阻尼层中的气泡对结构阻尼性能的影响,制备了两种自由阻尼层结构,其中一个阻尼层中含有较多的气泡,而另一个则基本没有气泡。对两种结构进行试验模态分析,研究比较了两种结构的阻尼比,同时还比较了两结构对应点的加速度导纳曲线和相同激励下的线性谱曲线。研究结果表明,阻尼材料中气泡的存在将会降低结构的阻尼性能。     

Modeling and analysis of rotating plates by using self-sensing active constrained layer damping

This paper proposes a new finite element model for active constrained layer damped (CLD) rotating plate with self-sensing technique. Constrained layer damping can effectively reduce the vibration in rotating structures. Unfortunately, most existing research models the rotating structures as beams that are not the case many times. It is meaningful to model the rotating part as plates because of improvements on both the accuracy and the versatility. At the same time, existing research shows that the active constrained layer damping provides a more effective vibration control approach than the passive constrained layer damping. Thus, in this work, a single layer finite element is adopted to model a three-layer active constrained layer damped rotating plate. Unlike previous ones, this finite element model treats all three layers as having the both shear and extension strains, so all types of damping are taken into account. Also, the constraining layer is made of piezoelectric material to work as both the self-sensing sensor and actuator. Then, a proportional control strategy is implemented to effectively control the displacement of the tip end of the rotating plate. Additionally, a parametric study is conducted to explore the impact of some design parameters on structure??s modal characteristics.     

A continuous force model for the impact analysis of flexible multibody systems

A new continuous impact model applicable to multibody systems consisting of interconnected rigid and flexible bodies is presented. The continuous impulsive force that acts during the short-lived interval of impact is written in terms of the relative displacement and velocity of the impacting bodies. In the method developed in this paper, the material compliance and damping coefficients are determined from energy balance relations. In order to account for the kinematic constraints between the two impacting bodies and other bodies in the system, an effective mass compensation is proposed. Flexible bodies in the system are discretized using the finite element methods. The generalized impulsive forces associated with the system generalized coordinates are then derived using the virtual work and are written in terms of a coupled set of reference and modal elastic coordinates. Numerical examples are presented in order to demonstrate the feasibility of the mathematical model developed in this paper.     

Frequency-dependent damping model for the hydroacoustic finite element analysis of fluid-filled pipes with diameter changes

The integration of a model for longitudinal hydroacoustic fluid damping in thin hydraulic pipes in 3D finite element models is presented in this paper. In order to perform quantitative prediction of the vibroacoustic behavior and resulting noise levels of such fluid-structure coupled system due to hydraulic excitation, an accurate frequency-dependent fluid damping model including friction effects near the pipe wall is required. This step is achieved by matching complex wave numbers from analytical derivation into a parameterized damped wave equation and consecutive translation into finite element modeling.Since the friction effect close to the pipe wall changes locally with the inner pipe radius, the fluid damping model is applied segment-wise in order to model the influence of cross-sectional discontinuity, such as orifices, on the oscillating pressure pulsations. A component synthesis approach, which uses pipe segments as substructures, allows a simple model generation and fast computation times.The numerical harmonic results are compared to experimental frequency response functions, which are performed on a hydraulic test bench driven by a dynamic pressure source in the kHz-range.     

The response of viscoelastic-frictionless bodies under normal impact

This paper presents a computational finite element model capable of simulation of the impact response of viscoelastic-frictionless bodies. The variational method is developed to drive the equations of motion. The Lagrange multiplier method is exploited to impose the nonholmonic contact constraints and convert the problem form constraint problem to unconstraint one. The constitutive equations are expressed in integral form. and involving relaxation moduli are adopted to describe the nonaging simple material. Generalized Standard Linear Solid (Wiechert) model is adopted to simulate the internal damping (viscoelastic) behavior of the structure. Wiechert model is exploited in order to drive a recursive relationship and thereby eliminate storage problem. Newmark constant average acceleration is proposed to integrate the governing equation numerically and transfer the problem into a static one for each time increment.     

Identifying the frequency dependent material property of a hydraulic engine mount through an iterative procedure using 3D finite element modeling

Achieving very restricted noise, vibration and harshness targets in modern vehicles, makes using the hydraulic engine mount crucial. Hydraulic engine mounts have both solid and fluid media in their structures that make their dynamic behavior complex to figure out. We present a three-dimensional model of HEM with using finite element method that encompasses elastomeric material’s nonlinearity and fluid-structure-interaction. Dynamic equivalent modulus of elasticity for elastomeric material is identified through iterative model updating procedure. To do model updating, the results (here, namely, natural frequencies and frequency response function graphs) are compared with real hydraulic engine mount behavior that derived from modal tests. The results showed that the dynamic characteristic of elastomeric material is frequency dependent and can be divided into two distinct regions: below 30 Hz (low frequency) and above 30 Hz (high frequency) with different trends.     

Characterisation and modelling of viscoelastically damped sandwich structures

A passive damping technique with viscoelastic materials is analysed in the present work. Concretely, the constrained layer damping (CLD) configuration, also known as viscoelastically damped sandwich structure, is considered. For an efficient dynamical analysis of viscoelastically damped structures, an accurate material characterisation and suitable mathematical models are necessary for reproducing the real material behaviour. With this aim, a CLD structure has been characterised in the frequency domain by means of experimental tests, considering both homogenised and multilayered approaches. The material-parameter extraction procedure is presented, and mathematical models are reviewed. For the homogenised characterisation, a mathematical model based on the Ross et al. model has been proposed, whereas for the multilayered characterisation, the constraining layers have been considered as linear elastic, and the viscoelastic core has been represented by a four-parameter fractional derivative model. Both material models have been considered for defining both homogenised and layered multimaterial numerical models by the finite element method. The numerical models have been validated by correlating the numerical and the experimentally measured dynamic responses. Both numerical models have reproduced the experimental dynamic response accurately for the bandwidth considered herein.     

计算机磁头/磁盘超薄气膜润滑稳定性

以任意拉森数的超薄气体润滑方程为基础,给出磁头刚体小扰动对空气轴承滑块 (ABS)气膜压强摄动方程。采用算子分裂法求解气膜压强和非结构三角网格的有限元法解压强摄动方程,得到气膜的刚度系数和阻尼系数矩阵。模态分析得到磁头气固耦合系统的固有频率,衰减率和振型。以Ω型磁头为例,分析了在不同气膜厚度和磁盘转速下的磁头稳定性。研究结果表明,磁头稳定性对气膜厚度很敏感,在小气膜厚度运行时,系统固有频率高,稳定性好;磁头升沉和纵倾运动的动力耦合,使磁头系统固有频率和衰减率降低,对稳定性不利;高转速的磁盘对磁头稳定性不利,但影响不大。     

Model updating of damped structures using FRF data

Due to the important contribution of damping on structural vibration, model updating of damped structures becomes significant and remains an issue in most model updating methods developed to date. In this paper, the frequency response function(FRF) method, which is one of the most frequently referenced model updating methods, has been further developed to identify damping matrices of structural systems, as well as mass and stiffness matrices. In order to overcome the problem of complexity of measured FRF and modal data, complex updating formulations using FRF data to identify damping coefficients have been established for the cases of proportional damping and general non-proportional damping. To demonstrate the effectiveness of the proposed complex FRF updating method, numerical simulations based on the GARTEUR structure with structural damping have been presented. The updated results have shown that the complex FRF updating method can be used to derive accurate updated mass and stiffness modelling errors and system damping matrices.     

弹性机构振动被动控制中模态损耗因子的确定

针对组成弹性机构的特点,给出了一种描述组成弹性机构的２节点８自由度约束阻尼平面刚架单元的形函数,建立了含大阻尼材料时弹性机构的动力学方程,基于此研究了弹性连杆机构模态损耗因子的预测问题。文中方法概念清晰、计算简单、预测可靠。可用于解决阻尼处理中的机构结构参数的优化设计问题。     

多孔介质空气轴承静态特性仿真分析

多孔介质空气轴承作为气浮导轨的关键零部件,其静态特性对导轨性能至关重要。采用有限体积法进行了多孔介质空气轴承静态特性仿真,设计了多孔介质平板试验。采用可压缩Forchheimer方程获得多孔介质的渗透率系数;采用X-ray断层扫描方法分离了多孔介质内部孔隙结构,并将孔隙结构细分获得孔隙度。分析了多孔介质CT三维扫描图像的最小重复单元。基于CFD方法对多孔介质空气轴承静态特性进行了数值模拟,对网格数量无关性进行了验证。结果表明:通过合理设计多孔介质的形状和大小以及气膜的厚度,空气轴承能够获得较好承载力和静刚度。     

Characterization of coupled-domain multi-layer microplates in pull-in phenomenon, vibrations and dynamics

This paper presents a model to analyze pull-in phenomenon, vibrational behavior and dynamics of multi-layer microplates using coupled finite element and finite difference methods (FDM). First-order shear deformation theory (FSDT) is used to model dynamical system using finite element method, while FDM is applied to solve nonlinear Reynolds equation of squeeze film damping. Using this model, pull-in analysis of single- and multi-layer microplates are studied. Vibrational behavior of single- and multi-layer microplates are analyzed to compute resonance frequencies and mode shapes of the system. Also, an algorithm is presented to study dynamics of microplates under the actuation of nonlinear electrostatic force and squeeze film damping. Results for simplified single-layer microplates are validated and in good agreement with the published literature. This investigation can be implemented in the design of multi-layer microplates.     

3-RRS柔性并联机器人的振动特性分析

基于有限元法、Lagrange方程和运动协调条件,建立了3-RRS柔性并联机器人的弹性动力学模型。分析了含有Rayleigh阻尼的3-RRS柔性并联机器人的振动特性。通过算例,揭示了系统阻尼固有频率与模态衰减系数的变化规律。研究3-RRS柔性并联机器人系统的振动特性可为此类机器人的机构优化设计、控制和工程应用提供指导。