The author has investigated an approximate method for numeric simulation of acoustic fields of the two- and three-dimensional domains with arbitrary shapes. In the method, the collocation method is applied to analytic solutions represented in the polar coordinates. Results obtained by this method are very accurate in spite of simplicity of the method. The theoretical validity of the method, however, has not been explained sufficiently so far. In this paper, it is shown that the natural modes of some domains can be resolved into plane waves propagating in all directions. Some properties of the natural modes can be explained by the propagating direction spectra. And then a consideration is made about the case in which the collocation method brings ill results.

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This paper describes the active control systems for reducing coor impulse noise. The control system is comprised of actively controlled modular thin plates, which wholly cover the surface of concrete slab structure in the ceiling, and in each of which the mechanical vibration is controlled independently with bimorph type piezoelectric actuators. An active control system of modular plate s presented, where ave sensors and actuators are applied to control low frequency modes of bending vibration in the plate. The control system is now under development, main subjects of which are; 1) designing of vibration characteristics of modular thin plate, 2) the active control system appropriate to the characteristics and cost reduction. Subject 1) is mainly discussed below.

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This report describes the active control systems for reducing floor impulse noise. The control system is comprised of actively controlled modular thin plates, which wholly cover the surface of concrete slab structure in the ceiling, and in each of which the mechanical vibration is controlled independently with bimorph type piezoelectric actuators. In this report, first, modeling of the active control system and control system design are presented. Second, the control system is evaluated via numerical simulations. Finally, the effectiveness of the control system is examines through control experiment.

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The nearfield acousticholography(NAH)method is frequentlyresearchednowadays.AndIamproposingtheconvertedNAHmethod using extra k-domain spectrums. In this report, the theoretical argumentisdescribed. Fromtheresult of mathematicalargument, it is proved that theproposing method may be a substitute of conventional method, and thatthereconstructed value by proposingmethod is able torecognizedastheroughapproximationofphysical values.

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High-speed diesel engines for motor vehicles and construction machineries are required sound reduction and good sound quality along with high performance and clean exhaust gas emission. In order to produce the good engine sound, it is important that engineers estimate the engine sound and evaluate it in the early design phase. This paper presents a theoretical approach to estimate the time history of engine sound and an evaluation approach of engine sound quality using wavelet transform.

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This paper describes the generation of the arbitrary phase difference between self-excited oscillators in mutual entrainment. We used the self-excited oscillators, of which amplitude was controlled by variable velocity feedback gain. The self-excited oscillators which vibrate at the respective natural frequency are entrained in the same frequency by synchronous control force which is the sum of the signals proportional to the velocity and the displacement of the adjoining self-excitation oscillators. In this mutual entrainment phenomenon, arbitrary phase difference is generable by giving suitably the gain about velocity, and the gain about displacement.

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This paper describes the frequency entrainment that simulates synchronous flashing of fireflies. We used self-excited vibrator, which consists of signal amplification element, saturation element and phase delay element. In the forced entrainment and the mutual entrainment, when the difference of frequency between vibrators is small, they synchronize by small influence, when the difference of frequency between vibrators is large, they synchronize by large influence. As we considered the saturation element as variable gain, we could obtain the synchronization area from simulation by using describing function.

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This paper describes the generation of phase difference in elliptical vibration machine by mutual entrainment and control of transportation velocity. Elliptical vibration machine which has a horizontal vibratory system and a vertical vibratory system has local feedback loops for generating self-excited vibration. The driving by self-excited vibration, follows resonance point, and reduces energy consumption. The synchronization control using a Mutual entrainment generates phase difference between horizontal vibration and vertical vibration by cross feedback of velocity and displacement as synchronization control force. Transportation velocity also including the direction of positive/negative can be controllable by using this phase difference.

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The rubber surface of a size press roll of a paper making machine is deformed into a certain convex polygon at a high nipforce and at an operating speed. This problem is similar to other pattern formation phenomena, for instance, polygonal deformations of the smoother roll in paper making machine, regenerative chatters in cutting, and the polygonal deformation of the work roll in a hot leveler. In the size press part, the surface of the paper is coated with starch between the rubber rolls. As the operating speed increases, the pattern formation phenomena are likely to occur. This paper describes the countermeasures to avoid the pattern formation by changing the diameter ratio of the top to the bottom roll. It is found from the numerical analysis that the optimum ratio exists to decrease the instability effectively.

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While the contact rotating systems are operating, the periodic polygonal deformation patterns are often formed on the peripheral surfaces of rolls due to viscoelastic deformation, plastic deformation, cutting, grinding, wear and so on. Such phenomena are called pattern formation phenomena. The phenomena are caused by the unstable vibration generated in the time delay system. In the previous reports, the authors suggested a simplified method to analyze the pattern formation phenomena and formulated a simplified criterion for the stability analysis. In this report, concepts of optimum design method are applied to design system parameters for generalized model efficiency. The validity of the present method is confirmed from the numerical computational results.

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It is generally accepted that self-excited vibration in cylindrical grinding caused by the regenerative effect. In this paper, we discuss effect of the work speed variation on the pattern formation with analytically and numerically. The results are follows: A work speed variation affect number of polygon formed on work piece, but it doesn't always depend on its stability. It is clarified a relationship of critical work speed to phase difference from a viewpoint of energies put into the system.

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In this study, liquefaction is simulated by using Cellular Automata (CA) method. At first, the local interaction rules among sand, water and empty cells are constructed. The parameters including in the local rules are determined from the simulation of dynamic behavior of sand cells only and the drop behavior of one sand cell in the water. By using the local rules, the behavior of sand cells can be simulated and the liquefaction also can be simulated.

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In the present study, we simulated the dynamic behavior of granular media excavated by a cutting blade using Cellular Automata (CA). The space is divided into triangular grids, where particles are distributed two dimensionally on cellular space. The local interaction rules are defined based on two assumptions that are related to the determination of moving vector of particle cells due to the force action of blade, and also the free drop motion of particles. The present CA model provides basic idea for understanding dynamic behavior of granular media, and also the model can easily be extended to more practical scenario.

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A Neural Network is a nonlinear mapping system derived in reference to the brain. It is conposed of neurons and their connections, and the structure of network has close relation to the mapping capability. However, the structure has been decided by experience or by "trial and error". In this study, a decision technique for network structure was proposed. The present technique referenced just local information of the whole network and it worked in the local part of the network. Two-Spiral Problem was used to verify the learning capability of the self-organized network presented in this paper.

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The elastic parameters of skis are essential for simulation on skiing. In the present study, an inverse analysis method toidentify equivalent elastic parameters of skis is proposed. First,by applying the experimental modal analysis technique to aski with free boundary conditions, natural frequencies and mode shapes are obtained. Secondly,from the obtained naturalfrequencies and mode shapes, the equivalent elastic parameters of the ski are identified numerically. Finally, to justify theapplication of this approach,the bending stiffness and twisting stiffness in longitudinal direction calculated by the obtainedequivalent elastic parameters of the ski are compared with those obtained by the other experimental method.

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The purpose of the present investigation is to design a piezoelectric speaker which is composed with the flat plate and the piezoelectric vibrator connected by rod. The rubber-weight is stuck on piezoelectric vibrator in order to improve the acoustic characteristics of piezoelectric speaker. In the experiment, in addition to the measurement of sound pressure level, the displacement of flat plate has been measured. The experimental results show that the radiation sound could be improved by connecting rod which supported in the center of vibrator. And, by sticking rubber-weight on the circumferential parts of piezoelectric vibrator, the acoustic characteristics of piezoelectric speaker could be planarized.

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In this paper, the SMA actuator, which has stockinet' structure has been fabricated, and its structure, principle and dynamics characteristics have been argued according to the bionics-theory. Moreover mechanical characteristics of the actuator, which is knitted with SMA wire of 0.15mm diameter was tested and analyzed. The experimental result shows that the actuator has enough output force and amount of contraction, when it is used to artificial muscles. And using mathematic model, the relations among temperature, amount of contraction and output force were carried.

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This paper deals with impact vibration in a mass-spring system for two-degree-of-freedom excited by periodic force with arbitrary functions. The analytical model is steady impact vibration for two-degree-of- freedom having two masses in which one mass is subjected to an exciting vibration. Then the restoring force, which has characteristics of an unsymmetric piecewise-linear system, collides elastically to anather mass when amptitude of the mass increases farther than clearance. In order to analyze resulting vibration for the main resonance, the Fourier series method is applied and is analyzed for this system. Next, following the theoretical analysis, numerical calculations are performed, and the resonance curves are made by using resulting vibration. Effects of amplitude ratio of excitation, stiffness of clamped spring and mass ratio on the resonance curves are shown by numerical results. For verification of the analytical results, analog experiments are performed by using simulator, and numerical results are compared with analog results on the resonance curves. The analytical results are in a fairy good agreement with the simulated ones.

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A new method is proposed for the fault detection and health monitoring of smart structures. The Eigensystem Realization Algorithm (ERA), a time-domain system identification algorithm, is applied to estimate the system matrix and to detect a system failure by monitoring the variation the eigenvalues of the estimated system matrix. Numerical simulations are carried out for a flexible cantilever beam equipped with piezoelectric actuators and sensors. The results of the numerical simulations show that the present method is feasible to detect the small variation in the stiffness parameter of the flexible beam by using the estimated eigenvalues.

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The objective was to apply a structural health monitoring concept to detect loosening of a bolt without human involvement. This paper proposes a new method of a bolt loosening detection by adopting a smart washer which uses a piezoelectric material with sub-space state spacce identification algorithm (4SID). The proposed washer is a cantilevered plate type patched piezoelectric material. A self-sensing and actuation function featuring this material is applied to the washer. An explanation that natural frequency decreases with decrease of the bolt tightening axial tension by using the system identification, the concept of the proposed method and experiments verifying its effectiveness are described. The natural frequency of the smart washer was identified with sub-space state space identification method. There was good correlation between the change of the natural frequency of the smart washer system and the decrement of bolt tightening axial tension.

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This paper describes a study of active noise isolation using an advanced composite board with embedded piezoelectric elements, which are used as both sensors and actuators. The identification system, which estimates the radiated sound from the voltage signals of the piezoelectric elements in the smart board, was constructed based on Rayleigh's integral formula so that no external microphone is needed for noise control. Since the control system was implemented to minimize the estimated value of noise by suppressing the vibration of the smart board, it can reduce the sound transmission through the board. The technique of self-sensing actuator was also used to reduce the number of piezoelectric need for the control. Experiments were performed and the results showed that the transmitted noise level was successfully reduced at several resonant peaks.

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This study aims to fabricate [Pb(Nb_<2/3>Ni_<1/3>)O_3]_<0.55>-[Pb(Zr_<0.3>Ti_<0.7>)O_3]_<0.45>(PNN-PZT) piezoelectric ceramic fibers which has Pt core by extrusion. Since the Pt core can be used as an electrode, piezoelectric ceramic fibers with core can be used as actuators or sensors with the other electrode coated on the outer surface of fiber. In this paper, the process to fabricate PNN-PZT piezoelectric ceramic fibers with Pt core is explained. The fine PNN-PZT fibers with Pt core are fabricated by extrusion of the mixture of ceramic powder and an organic binder with some addictives. Green fibers with a diameter of about 300μm were successfully extruded from the mixture of PNN-PZT powder and each admixture. The extruded fibers sintered at 1200℃ have the microstructure with 2-6μm grains but no cracks. The result of displacement measurement at the center of its cross-section shows that a PNN-PZT fiber with Pt core sintered at 1200℃ in this study has desired piezoelectric properties.

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This paper aims at proposing a piezoelectric generator for self-powered system by vibration energy regenaration. In order to demonstrate the effectiveness of the proposed design method, numerical simulation and laboratory experiment will be done using a cantilever beam model equipped with 4 surface bonded PZT tiles pairs.

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This paper describes a method of vibration control for a flexible cantilever beam with piezoelectric actuators. A Sliding mode control is applied to the vibration control of the smart structure in the presence of model parameter errors. In this paper, it is shown by numerical simulations and experiments that the sliding mode control has high performance and robustness against system parameter errors.

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This paper presents characteristics of morphological change in organisms and their applications to artificial deployable membrane structures. We observe deployment procedures of some membrane structures in organisms including deployment of flowers, eclosions of insects, and flaps of bats' wings. As well as flaps of bats' wings, we also observe furling behaviors. It is important to furl up membrane structures for future space structures. Because it is said to that membrane structures are suitable for next large space structures like solar power satellites, space antennas, solar sails, and so on.

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A design approach of statically indeterminate geometry adaptive truss with wire member actuators is discussed from the viewpoint of its dynamic property. Generally, the dynamic property of a mechanism is evaluated based on its natural frequency calculated as the eigenvalue in terms of its inertia and stiffness matrices. It is difficult, however, to adopt this kind of approach in our case. The reason is that the stiffness matrix drastically changes during the dynamic vibration because of the characteristics of wire members that they do not have stiffness against any compressive force. In this study, we develop a criterion for dynamic property evaluation based on dynamic simulation of the adaptive truss. Example designs of adaptive truss with wire member actuatros based on a kind of combinatorial design variables are also demonstrated.

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H_2 controller is designed with the reduced order model of smart structure which is obtained by finite element and modal analyses. The control force is applied by the piezoelectric film actuator and the feedback signals are detected by the piezoelectric sensor and the accelerometer in the system. The vibration response is suppressed by modal control in which the modal coordinate displacement is taken as the controlled variable. The placement of the piezoelectric actuator and the H_2 control system are simultaneously optimized by using GA to achieve an enhanced vibration control performance. It is verified by some applications that the enhanced performance for the vibration suppression can be achieved by the present method.

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In hard disk drives, vibration suppression is very important to boost the performance of information-processing equipments. It has beenexpected that technology of smart structure will contribute to the development of small and light-weight mechatronics devices with therequired performance. The smart structure is composed of the piezoelectric sensor and actuator in order to reduce the structural vibration. Theplacement of the piezoelectric actuator and H2 control system are simultaneously optimized based on genetic algorithm to improve the effecton the vibration suppression. It has been verified by some applications with a magnetic disk drive and an L-shaped beam that an enhancedperformance for the vibration suppuression can be achieved by the proposed optimal design method.

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Research and development of the variable damper using magnetorheological fluid (MR damper) is carried out in the fields with various, brakes, clutch, and vibration response control system etc. The rheological properties of the fluids can be changed dramatically by applying a magnetic field, and this behavior can be applied to control dynamic properties of some building structures. In this paper, the fundamental characteristic was grasped by the dynamic loading tests. And, the shaking table test by one-degree-of-freedom system was carried out, and performance and controllability of the MR damper were verified.

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Magnetorheological (MR) fluids can vary the rheological characteristics according to applied magnetic field strength and has been applied to various kinds of devices such as dampers, clutches, etc. In this paper, typical characteristics of MR fluids were measured under two types of modes, shear deformation and pressure deformation of MR fluids, which typically appear in MR devices. Shear stress in shear mode and pressure difference in pressure mode were measured on changing flow rate and magnetic flux density in MR fluids, types of MR fluids, etc. Using the experimental result, generated force by MR devices was modeled to each mode.

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Passive piezoelectric shunt damping is a vibration suppression method using a piezoelectric element which is electrically shunted by a passive electrical circuit. This paper shows a method of optimizing the values of the elements of the shunt circuits based on the robust performance index against the model uncertainties. For the optimization, the worst-case peak gain of the transfer function is selected as a cost function. From the results of the optimization for the LR shunt circuit, we investigate some required properties for the robustness of shunt circuits. As an example of the robust shunt circuits, the optimization for the LR-LRC shunt circuit is also presented. The experimental results demonstrate the efficiency of the proposed method.

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This paper considers decoupling control of a model helicopter with 3 degrees of freedom. Since its position control system is not decouplable via static-state feedback, we try to apply dynamic-state feedback. In most practical cases, however, the model parameter cannot be measured accurately due to a difficulty of system parameter identification. Here, the online parameter estimation technique is introduced in the derived control loops. In this paper, a decoupling control method with the online parameter estimation technique is proposed under the presence of the parameter uncertainties and applied to an experimental system. The experimental results show the effectiveness of the proposed algorithm.

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In this paper, we propose the model based control system design for autonomous flight and guidance control of a small unmanned helicopter. A small unmanned helicopter has been studied with fuzzy and neural network theory, but it is very difficult for unmodel based control to stabilize it. For this reason, we design a mathmetical model(Mettler's model) and a model based controller for a small unmanned helicopter system. In order to realize a full automatic controlled small unmanned helicopter, we have designed the attitude controller and trajectory controller with kalman filter based LQI for a small unmanned helicopter. The characteristic of the attitude closed loop dynamics is taken into consideration to design a trajectory controller . By way of the simulations and experiments, it has been clarified that the proposed scheme for attitude control and guidence control is very useful.

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The double inverted pendulum has been known as a good example of nonlinear system. In the double inverted pendulum there are four equilibrium points. In this paper, we present a control method for transition between equilibrium points of the double inverted pendulum. The swing-up controller is designed based on energy. The stabilizing controllers are designed on the basis of the linear models at each unstable equilibrium point. Radial Basis Function Neural Network (RBFNN) learned by genetic algorithm is used to integrate the swing-up controller and the stabilizing controllers. The effectiveness of the proposed method is shown by simulation and experimental results.

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This paper describes the nonlinear vibration and the control of a vibratory machine which has a heavy work. Vibration of a vibratory machine and the movement of work are coupled through normal force and friction force. If a vibratory machine receives a large friction force, vibratory wave will be formed into triangular wave.Then, the amplitude cannot be control because of difficulty of the amplitude estimation. The triangular wave changes into a sine wave when the vibratory machine is driven at high frequency. Therefore, stable self-excited vibration is practiced using the control method which vibrates the vibratory machine at high frequency.

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This paper describes the measuring method of the viscoelasticity on the human body. In order to realize this method, we used the self-exited vibration generated by the positive feedback of velocity. This method could measure time-varying dynamic stiffness on body surface. Then we showed that the elastic material constant and the viscous material constant on the human body, which did not depend on measurement conditions, could be quantified from a spring constant and a viscous damping coefficient using Hertz's formula and the correspondence principle between the theory of elasticity and viscoelasticity.

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A suspension of a vehicle is the support system between a vehicle body and wheels.The task of a suspension is to support the vehicle body and develop ride comfort.But if the attenuation force becomes large, the passenger will feel hard at high frequencies If the attenuation force becomes small, the passenger will feel soft at low frequencies.If the coefficient of spring becomes low the vehicle's naturel frequency of vibration becomes low,thus the heave,rolling and pitching will become large. In this study, a new method has been proposed.A non-linear seven degrees of freedom vehicle model is used in order to design and check the performance of sliding mode controlled active suspensions.From this basement the active suspension control input can be obtained,but in the semi-active suspention systerm the control input force can not be used directly.There are many restrictions on dampers.Putting restrictions on the active control input force one can realize the semi-active control.Experiment results are presented to confirm effectiveness of this new algorithm.

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In this paper we present the grasping control system based on the position control. The slide mode control, which excels in robustness, is used for the position controller. Due to grasping is made by the position control, it is not necessary that the change of the control rule accompanying approach/grasp phase. Therefore the control system is stable. Grasping objects may slip from fingertips, when the mass of grasping objects and the attitude of the manipulators, which attached hands on, changes. To regard the coefficient as a threshold, the grasping force is made to increase by making the compliance of joints increase. The proposed control system is actually applied to a multifingerd hand, and the usefulness was verified.

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This paper describes H_∞ horizontal velocity/position control and flight experiments for the hobby-class small-scale unmanned helicopter. Simple system identification method was applied to acquire single-input/single-output blackbox mathematical models. Cross-validation results have shown that those models were reflected in the actual dynamics very well. Attitude control has been constructed by proportional-integral feedback loop with derivative feed-forward compensator for improvement of reference following performance. H_∞ control theory was applied for horizontal velocity control synthesis. Four closed-loop transfer functions were shaped according to the design specifications given in the frequency domain. Also, horizontal position control has been designed as proportional gain feedback. In the flight experiment, good reference following performance has been shown through 15 [m] square point-to-point traveling guidance control, and this also showed high accuracy of the models. It is considered that successful horizontal guidance flight using H_∞ control theory for small-scale unmanned helicopter is the first result in the world.

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An experimental identification technique by using transfer functions is developed. The technique requires neither application of the excitation on nor measurement at the boundary. The technique can treat the nonlinear boundary conditions. The first step in the proposed technique is to measure transfer functions of the unset structure. The second step is to replace the boundaries by springs and dampers as well as equivalent masses and moments of inertia. The third step is to apply the periodic excitation and measure steady state responses except boundaries. The last step is to determine the characteristics of the boundary elements by applying the principle of harmonic balance. Numerical simulation is conducted, and applicability of the technique is confirmed.

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A simplified analytical method is applied to vibration analysis of a commuter railway vehicle. Vibrationmeasurement test results for a real commuter vehicle show that the vibration characteristic of the vehicle is verycomplex; therefore, usual beam model cannot be applied. In the proposed method, the carbody is modeled as abox-type structure; this enables to express three-dimensional elastic vibration of carbody. A numerical model forthe commuter vehicle is established by using the method. Numerical results are compared with the experimentalresults. Good agreement is observed for natural vibration mode properties and frequency response functions.Some vibration suppression measures are examined by using the numerical model, and it is found that to increasebending rigidity of the floor is more effective than increasing vertical bending rigidity of the side lengthwisemember.

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Modeling for an action simulation is performed focusing on the suspension system of a car using the modeling technique called the functional model that had been developed by one of the authors. Simulation analysis of the suspension system of a car was performed in the three dimensional field. It was shown that the method based on the modeling concept of functional model can express the general dynamic characteristic of the automobile suspension.

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This paper presents improvement of isolation performance of the single-axle truck for railway vehicle. It has several advantages over conventional two axle truck in decreasing weight of truck and curve performance. However, it was said that the isolation performance in vertical excitation, such as rail joints had been problem to be solved. The authors propose the modification of vibration system in the single axle truck to improve the reduction of vibration in vertical direction. The two axle truck has three degrees of freedom with pitching motion and it is effective to reduce the vibration of the vertical direction. Therefore, in the single axle truck the pitching motion of the truck frame is added by modification of the setting of secondary suspension. The theoretical examination, numerical simulation were made to confirm the effectiveness of the proposed system. Finally experimental setup was made and the improved performance was proved by experiments.

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This paper presents a design procedure of H∞ controller for bending-torsional coupled vibration of flexible robot arm. In the case that the arm does not hold the center of the payload, it processes bending-torsional coupled vibration. In this research, H∞ robust control design with structured uncertainties is utilized to describe the uncertainty in coupling of bending-torsional vibration. By taking account of uncertainties in coupling, the arm is controlled with robustness against it. In this research, an experimental system is set up and its physical model is made by utilizing the modeling method presented by Seto. According to this result, presented modeling and controller design method is certificated.

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This paper deals with motion control of 1-link flexible arm, using computed torque method and H_∞ control. We considered a method to apply computed torque method for a 1-link flexible arm, and we proposed a desired trajectory considering the first natural mode of the link. Also, we considered a compensator using H_∞ control. According to the experimental study, it is confirmed that the flexible arm follows the desired trajectory smoothly, and spill over phenomenon does not occur.

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In this study, optimal control of a golf swing robot that is used to evaluate the performance of golf clubs is described. A mathematical model is derived by Hamilton's principle. A linear quadratic regulator (LQR) that considers the vibration of the club shaft is used to stop the robot during the follow-through. Since the robot moves fast and has strong non-linearity, an ordinary state observer for a linear system cannot accurately estimate the states of the system. A state observer that considers disturbances accurately estimates the state variables. Results of numerical simulation are compared with those of experiment.

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The functional model is a modeling method which defined more clearly than before the function of each part unit and the flow of the energy transmitted in them. In this research, it tries to express movement of two link arm in the two dimensional plane by the functional model. Furthermore, the validity of the control system design in the functional model is examined by building into the model of two link arms.

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In this paper, we propose a control method for changing the nonlinear characteristics of a parametrically excited beam by linear feedback based on the bifurcation control. The control is achieved by using a piezoelectric actuator. Appropriate linear feedback gain can change the nonlinear characteristics so that the jumping phenomena in the sweep of the excitation amplitude are avoided. It is experimentally confirmed that discontinuous bifurcation is transformed into continuous one, and the jumping phenomena can be reduced for the sweep of the excitation amplitude. Key Words: Parametric response, bifurcation control, linear feedback, piezoelectric

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This paper discusses the control problems of deployment and retrieval of a tethered satellite system. First, a linear tangent model of the tethered satellite system which is independent of its operating point is derived by using a pseudolinearization technique. Then, a nonlinear control law is designed based on the linear tangent model and a nonlinear change of state coordinates. The present feedback control law is simulated numerically, and the results show excellent controlled response of the tethered subsatellite system.

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This paper describes vibration analysis using finite element method for an elastic block supported by a nonlinear concentrated spring and connected to another block by a linear concentrated spring. The restoring force of the spring has cubic nonlinearity. Finite element for the nonlinear spring is expressed and is connected to the elastic blocks modeled by linear solid finite elements. Further, the discretized equations in physical coordinate are transformed into the nonlinear ordinary coupled equations using normal coordinate corresponding to linear natural modes. This transformation yields computation efficiency. Influences of Young's modulus on nonlinear frequency responses for the elastic block are clarifed

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