Experimental Investigation and Semi-Active Control Design of A Magnetorheological Engine Mount

In this paper; the dynamic characteristics of a semi-active magnetorheological fluid (MRF) engine mount are studied. To do so, the performance of the MRF engine mount is experimentally examined in higher frequencies (50~170 Hz) and the various amplitudes (0.01 ~ 0.2 mm). In such an examination, an MRF engine mount along with its magnetically biased is fabricated and successfully measured. In addition, the natural frequencies of the system are obtained by standard hammer modal test. For modelling the behavior of the system, a mass-spring-damper model with tuned PID coefficients based on Pessen integral of absolute error method is used. The parameters of such a model including mass, damping ratio, and stiffness are identified with the help of experimental modal tests and the recursive least square method (RLS). It is shown that using PID controller leads to reducing the vibration transmissibility in the resonance frequency (=93.45 Hz) with respect to the typical passive engine mount by a factor of 58%. The average of the vibration transmissibility decreasing is also 43% within frequency bandwidth (50~170 Hz).     

Hand-arm vibration part III: A distributed parameter dynamic model of the human hand-arm system

An anatomically analogous distributed parameter dynamic model of the human arm is proposed and quantitatively validated. Distributed mass and stiffness parameters have been obtained by representing each long bone of the arm as a flexural beam. A distributed damping parameter was introduced by allowing the beam stiffness to be a complex quantity. Hand properties were modelled as a lumped parameter damped spring-mass system. Mechanical driving point impedance techniques were used to verify the model. A dual beam model of the forearm was first proposed, and its frequency response was compared with impedance data collected on the forearm. After having established the validity of the forearm model, it was then extended to include the upper arm. The frequency response of the whole-arm model was then compared with impedance data on the whole arm collected by a previous investigator. It is concluded that the beam model of the human arm adequately represented its dynamic behavior as measured by mechanical driving point impedance techniques. The amount of information concerning the dynamic behavior of the arm yielded by the distributed parameter model is found to be vastly greater than that yielded by lumped parameter models.     

NON-LINEAR MODELLING OF HYDRAULIC MOUNTS: THEORY AND EXPERIMENT

This paper focuses on the development of a complete non-linear model of a hydraulic engine mount and the evaluation of the model using a unique experimental apparatus. The model is capable of capturing both the low- and high-frequency behavior of hydraulic mounts. The results presented here provide a significant improvement over existing models by considering all non-linear aspects of a hydraulic engine mount. Enhancements to already published non-linear models include a continuous function that follows a simplistic yet effective approach to capture the switching effect and leakage through the decoupler, and upper chamber bulge damping. It is shown that the model developed here provides the appropriate system response over the full range of loading conditions (frequency and amplitude) encountered in practice. In order to obtain the parameter values for the non-linear model, a unique test apparatus is introduced. Using the experimental set-up, it is possible to verify the model of individual components of the mount, and later on test the behavior of the whole assembly. These data also establish the relative importance of several damping, inertia and stiffness terms. In addition, the measured responses of the mounts to loading at various frequencies and amplitudes are compared to the predictions of the mathematical model. The comparisons generally show a very good agreement (better than 10%), which corroborate the non-linear model of the mount. It is felt that this work will help engineers in reducing mount design time, by providing insight into the effects of various parameters within the mount.     

Identification of the dynamic properties of joints using frequency–response functions

The dynamic properties of joints are extremely difficult to model accurately using a purely analytical approach. However, these properties can be extracted from experimental data. In this paper we present a method for establishing a theoretical model of a joint from the substructures and assembly frequency–response function (FRF) data. The identification process considers not only translational, but also rotational degrees of freedom (RDOFs). The validity of the proposed method is demonstrated numerically and experimentally. A combined numerical–experimental approach was used to identify the mass, stiffness and damping effects of a real bolted joint. Using the least-squares method, data from the wide frequency range were used. A substructure synthesis method with the joint effects included was used to check the extracted values.     

Passive-active vibration isolation systems to produce zero or infinite dynamic modulus: theoretical and conceptual design strategies

The application of mechanical springs connected in parallel and/or in series with active springs can produce dynamical systems characterised by infinite or zero value stiffness. This mathematical model is extended to more general cases by examining the dynamic modulus associated with damping, stiffness and mass effects. This produces a theoretical basis on which to design an isolation system with infinite or zero dynamic modulus, such that stiffness and damping may have infinite or zero values. Several theoretical designs using a mixture of passive and active systems connected in parallel and/or in series are proposed to overcome limitations of feedback gain experienced in practice to achieve an infinite or zero dynamic modulus. It is shown that such systems can be developed to reduce the weight supported by active actuators as demonstrated, for example, by examining suspension systems of very low natural frequency or with a very large supporting stiffness or with a viscous damper or a self-excited vibration oscillator. A more general system is created by combining these individual systems allowing adjustment of the supporting stiffness and damping using both displacement and velocity feedback controls. Frequency response curves show the effects of active feedback control on the dynamical behaviour of these systems. The theoretical design strategies presented can be applied to design feasible hybrid vibration control systems displaying increased control performance.     

On estimating system damping from frequency response bandwidths

As damping determines the maximum vibratory response of a system at resonance, reliable estimates of damping are critical both to the design and qualification of systems to be subjected to a vibratory environment and for the evaluation of the effectiveness of modifications or additions provided to increase damping. The sharpness of the frequency response at resonance is often used for this purpose, quantified by the width of the frequency range (bandwidth) for which the response is above some fraction of the maximum response. Several aspects of the use of bandwidth methods in interpreting test results are considered. It is shown that the use of an excessive rate of change of test frequency in a sine sweep leads to overestimates of system damping. A criterion is offered for the identification of the maximum sweep rate for which an observed frequency response function provides a true indication of system damping, rather than an erroneous value dominated by the sweep rate. Applicability of the criterion is demonstrated through the use of results from actual tests. Excessive sweep rates are shown to inflate estimates of system loss factors above the true values in proportion to the square root of the sweep rate. It is also demonstrated with a specific form for an amplitude-dependent stiffness that the resulting nonlinearity can lead to erroneous observations of bandwidth frequencies, as well as the need for further reductions in the sweep rate.     

Role of spectrally varying mount properties in influencing coupling between powertrain motions under torque excitation

The influence of spectrally varying mount properties (including stiffness and damping) on the dynamics of powertrain motions is analytically examined. To overcome the deficiency of the direct inversion method (limited to only the frequency domain analysis), two methods are developed that describe the mount elements via a transfer function (in Laplace domain) or analogous mechanical model. New analytical formulations are verified by comparing the frequency responses with numerical results obtained by the direct inversion method (based on Voigt type mount model). Eigensolutions and transient responses of a spectrally varying mounting system are also predicted from new models. Based on complex eigenstructure, new coupling indices, including modal kinetic energy fractions, are defined for each method. Complex eigenvalue problem formulation with spectrally varying properties provides a closer match with measured natural frequencies than the real eigensolution with frequency-independent mounts. Given spectral variance in the mount properties, a simple roll mode decoupling scheme is suggested for the powertrain isolation system. Finally, an axiom for torque roll axis decoupling is provided by employing direct and adjoint eigenvalue problems.     

Experiments and numerical results on non-linear vibrations of an impacting Hertzian contact. Part 1: harmonic excitation

The purpose of this paper is to investigate experimental and numerical dynamic responses of a preloaded vibro-impacting Hertzian contact under sinusoidal excitation. Dynamic response under random excitation is analyzed in the second part of this paper. A test rig is built corresponding to a double sphere-plane contact preloaded by the weight of a moving cylinder. Typical response curves are obtained for several input levels. Time traces and spectral contents are explored. Both amplitude and phase of harmonics of the dynamic response are investigated.Linearized resonance frequency and damping ratio are identified from the almost linear behaviour under very small input amplitude. Increasing the external input amplitude, the softening behaviour induced by Hertzian non-linear stiffness is clearly demonstrated. The resonance peak is confined to a narrow frequency range. Jump discontinuities are identified for both amplitude and phase responses. The forced response spectrum exhibits several harmonics because of a non-linear Hertzian restoring force. Numerical simulations show a very good agreement with experimental results.For higher input amplitudes, the system exhibits vibro-impacts. Loss of contact non-linearity clearly dominates the dynamic behaviour of the vibro-impacting contact and leads to a wide frequency range softening resonance. The spectral content of the response is dominated by both the first and the second harmonics. Evolution of the experimental downward jump frequency vs. input amplitude allows the identification of the non-linear damping law during intermittent contact. Simulations of the vibro-impacting Hertzian contact are performed using a shooting method and show a very good agreement with experimental results.     

Fatigue Life Prediction of a Rubber Mount Based on the Continuum Damage Mechanics

Abstract

Failure analysis and fatigue life prediction are important steps in the design procedure of industrial products to assure the safety and reliability of their components. A new methodology to predict the fatigue life of a rubber mount based on the continuum damage mechanics is proposed in this study. The hyperelastic constitutive model of the natural rubber material in the mount was fitted using the three parameter Mooney-Rivlin model. A damage variable was introduced and the evolution function of cumulative damage in the rubber material was derived. The parameters in the damage function were acquired based on uniaxial tensile tests and fatigue life tests of the natural rubber specimens. Then the finite element analysis (FEA) models of the rubber mount for loads in the X and Y directions were established and the strain contours and the maximum principal strains of the rubber mount under various loads were calculated. The maximum principal strain was used as the fatigue parameter, which was substituted into the natural rubber’s fatigue life damage function to predict the fatigue life of the rubber mount. Finally, the fatigue lives of the rubber mount under various loads were measured on a fatigue test rig to validate the accuracy of the fatigue life prediction method. The test results indicated that the fatigue lives predicted agreed fairly well with the test results and the fatigue prediction method should be applicable to both rubber and other types of components.     

Investigation of multi-layer sandwich beams through single degree-of-freedom transformation

The dynamic behaviors of multi-layer sandwich beams are investigated through single degree-of-freedom (SDOF) transformation. The frequency response of the multi-layer sandwich beam is obtained using finite element code COMSOL and is transformed to a SDOF system with the same frequency response. Hence, the mass, spring constant and damping coefficient of the sandwich beams with different lengths and number of visco-elastic layers can be investigated. Further, viscous damping and structural damping models are individually employed to simulate the damping effect of the sandwich beam. The frequency responses from both models are compared with that from COMSOL and experiment. The resonant peak and resonant frequency of the SDOF system using structural damping model is more consistent with that from COMSOL. The experimental result demonstrates that the response of the sandwich beam can be predicted through COMSOL and SDOF transformation.     

Vibration characteristics of a cantilever plate with attached spring-mass system

Coupled free vibration analysis has been performed on a cantilever thin plate carrying a spring-mass system attached on an arbitrary point by using Rayleigh-Ritz method. Influence of an attached ‘spring-mass’ system, i.e., attached position, relative values of mass and spring constant, on the coupled vibration characteristics of the system has been clarified comparing with those of uncoupled ones. Optimal attached position to maximize coupled plate natural frequency is also investigated and shown in contour diagrams. The influence of an attached mass has also been investigated, as the limiting case whereby the spring stiffness of the ‘spring-mass’ system approaches infinity.     

Experimental study of hydraulic engine mounts using multiple inertia tracks and orifices: Narrow and broad band tuning concepts

Hydraulic engine mount tuning concepts with one inertia track and one decoupler are well understood. However, the dynamic response with multiple tracks or orifices is not. To overcome this void in the literature, dynamic tuning concepts of hydraulic engine mounts, with emphasis on multiple (n-)inertia tracks/orifices, are experimentally examined. A new prototype mount concept is designed, built, and experimentally evaluated in a controlled manner. Refined linear time-invariant models of fixed decoupler-type designs are developed to critically assess the dynamic stiffness measurements and to explore a family of alternate designs. Three narrowband devices are investigated for accurately predicting the frequencies corresponding to peak loss angles for the first time, in addition to examining and validating an n = 3 track mount. Two broadband devices are also successfully evaluated by tuning damping introduced by orifice-type tracks. A special broad-tuned design utilizing a controlled ‘leakage’ path flow area is then suggested, and the role of fluid resistance in achieving the desired performance is clarified. Finally, a production mount with unknown configuration is diagnosed using the proposed models with n tracks.     

Transverse hysteretic damping characteristics of a serpentine belt: Modeling and experimental investigation

In this paper, the transverse dynamic hysteretic damping characteristics (HDC) of a serpentine belt are investigated. The variable stiffness and variable damping model (VSDM) constituted of a variable-stiffness spring and a variable-damping damper is developed to estimate the HDC of the belt. A test rig is designed to test the force–displacement hysteresis damping curve and resonance frequencies of serpentine belts with different lengths under diverse loading conditions. The force–displacement hysteresis damping curve getting from the experiment is then used to determine the transverse stiffness and damping coefficients needed for the VSDM. The experiment particularly shows that the orientation of the hysteresis curve swings left and right around each natural frequency as it is a symmetrical point. This interesting phenomenon is explicated in detail with the loss angle which is calculated by two methods. Moreover, two sub-analytical models included in the VSDM are proposed to model the dependence of transverse dynamic stiffness and damping coefficient of a belt on belt length, pretension and excitation frequency. A comparison of the hysteresis curves obtained from the VSDM and experiment indicates that they are in good agreement.     

A decrement method for quantifying nonlinear and linear damping parameters

A method is presented which can estimate the linear and nonlinear damping parameters in a lightly damped system. Only a single response measurement from a free decay test is required as input. This ensures that the magnitude of the damping parameters is not compromised by phase distortion between measurements. The method uses the instantaneous energy to describe the long-term evolution of the system. Practically this is achieved by using only the peak amplitudes in each period. In this way the stiffness is effectively ignored, and only the damping forces are considered. For this reason, the method is not unlike the familiar decrement method, which can be used to estimate the apparent linear damping. The method is developed in the context of a weakly nonlinear, lightly damped system, with both linear and cubic damping. Simulated response data is used to demonstrate the accuracy of the technique. The nonlinear damping parameter is extracted from the response data to within 5% of the exact value, even though the nonlinear term contributes less than 1% to the total force in the system.     

Integrated design of piezoelectric damping system for flexible structure

This paper aims at developing an integrated design method of the active/passive hybrid type of piezoelectric damping system for reducing the dynamic response of the flexible structures due to external dynamic loads. The design method is based on the numerical optimization technique whose objective function is a control effort of the active damping. A vibration suppression performance, which is evaluated by the maximum value of the gain of the frequency response function of the structure, is constrained. In order to demonstrate the structural damping capability of the hybrid type of piezoelectric damping system designed by proposed method, numerical simulation and laboratory experiment will be done using a three-story flexible structure model equipped with 12 surface bonded PZT tiles pairs. Both numerical and experimental results indicate that the optimally designed hybrid piezoelectric damping system can be successfully achieving excellent performance as compared to a conventional purely active piezoelectric damping system.     

On the response of a harmonically excited two degree-of-freedom system consisting of a linear and a nonlinear quasi-zero stiffness oscillator

There are many systems which consist of a nonlinear oscillator attached to a linear system, examples of which are nonlinear vibration absorbers, or nonlinear systems under test using shakers excited harmonically with a constant force. This paper presents a study of the dynamic behaviour of a specific two degree-of-freedom system representing such a system, in which the nonlinear system does not affect the vibration of the forced linear system. The nonlinearity of the attachment is derived from a geometric configuration consisting of a mass suspended on two springs which are adjusted to achieve a quasi-zero stiffness characteristic with pure cubic nonlinearity. The response of the system at the frequency of excitation is found analytically by applying the method of averaging. The effects of the system parameters on the frequency-amplitude response of the relative motion are examined. It is found that closed detached resonance curves lying outside or inside the continuous path of the main resonance curve can appear as a part of the overall amplitude-frequency response. Two typical situations for the creation of the detached resonance curve inside the main resonance curve, which are dependent on the damping in the nonlinear oscillator, are discussed.     

Modelling the spindle–holder taper joint in machine tools: A tapered zero-thickness finite element method

This study presents a tapered zero-thickness finite element model together with its parameter identification method for modelling the spindle–holder taper joint in machine tools. In the presented model, the spindle and the holder are modelled as solid elements and the taper joint is modelled as a tapered zero-thickness finite element with stiffness and damping but without mass or thickness. The proposed model considers not only the coupling of adjacent degrees of freedom but also the radial, tangential and axial effects of the spindle–holder taper joint. Based on the inverse relationship between the dynamic matrix and frequency response function matrix of a multi-degree-of-freedom system, this study proposes a combined analytical–experimental method to identify the stiffness matrix and damping coefficient of the proposed tapered zero-thickness finite element. The method extracts those parameters from FRFs of an entire specimen that contains only the spindle–holder taper joint. The simulated FRF obtained from the proposed model matches the experimental FRF quite well, which indicates that the presented method provides high accuracy and is easy to implement in modelling the spindle–holder taper joint.     

一类非线性相对转动系统的组合谐波分岔行为研究

针对一类具有非线性刚度、非线性阻尼的非线性相对转动系统, 应用耗散系统的拉格朗日原理建立在组合谐波激励作用下非线性相对转动系统的动力学方程. 构造李雅普诺夫函数, 分析相对转动系统的稳定性, 研究自治系统的分岔特性. 应用多尺度法求解相对转动系统的非自治系统在组合激励作用下的分岔响应方程. 最后采用数值仿真方法, 通过分岔图、时域波形、相平面图、Poincaré截面图等研究外扰激励、系统阻尼、 非线性刚度对相对转动系统经历倍周期分岔进入混沌运动的影响. 关键词： 相对转动 组合激励 分岔 混沌     

On the design of a high-static-low-dynamic stiffness isolator using linear mechanical springs and magnets

The frequency range over which a linear passive vibration isolator is effective is often limited by the mount stiffness required to support a static load. This can be improved upon by incorporating a negative stiffness element in the mount such that the dynamic stiffness is much less than the static stiffness. In this case, it can be referred to as a high-static-low-dynamic stiffness (HSLDS) mount. This paper is concerned with a theoretical and experimental study of one such mount. It comprises two vertical mechanical springs between which an isolated mass is mounted. At the outer edge of each spring, there is a permanent magnet. In the experimental work reported here, the isolated mass is also a magnet arranged so that it is attracted by the other magnets. Thus, the combination of magnets acts as a negative stiffness counteracting the positive stiffness provided by the mechanical springs. Although the HSLDS suspension system will inevitably be nonlinear, it is shown that for small oscillations the mount considered here is linear. The measured transmissibility is compared with a comparable linear mass-spring-damper system to show the advantages offered by the HSLDS mount.