Journal of System Design and Dynamics
Online ISSN : 1881-3046
ISSN-L : 1881-3046
Volume 1 , Issue 3
Special Issue on The Third Asian Conference on Multibody Dynamics 2006
Showing 1-25 articles out of 25 articles from the selected issue
Special Issue on The Third Asian Conference on Multibody Dynamics 2006
Papers (Special Issue)
  • Etsujiro IMANISHI, Takao NANJO, Eiko HIROOKA, Naoki SUGANO
    2007 Volume 1 Issue 3 Pages 387-397
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    Recently, HILS (Hardware in the Loop Simulation) has been investigated in the field of the multibody dynamics (MBD). The fast calculation is necessary for the HILS system in order to require the real time simulation. This paper presents a fast simulation technique using the domain decomposition method. The domain decomposition method is widely used in the dynamic simulation for the mechanical system involving the hydraulic control system. This method is, however, not absolutely stable as the numerical integration. Fujikawa proposed a numerical stable solution scheme by introducing the iteration calculation. This paper applies the method to actual simulations of flexible multibody system in which the flexible linkage system and the hydraulic drive system are coupled with each other, and examines the speedup by parallel computing with the common memory in the calculation time. It is shown that using the present method in a multi-degrees-of freedom model can shorten the computing time. The present method is effective for the speedup in the calculation time by applying the dynamic simulation of the actual digging works on the hydraulic excavator.
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  • Jin-Rae CHO, Joo-Hyoung CHOI, Hong-Woo LEE, Jong-Shik WOO, Wan-Suk YOO
    2007 Volume 1 Issue 3 Pages 398-409
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    In this paper, we present a wet-road braking distance estimate for the vehicles equipped with ABS (Anti-lock Brake System). In order to effectively compute the interval-wise braking times and the resulting total braking distance, we divide the entire speed interval at braking into finite number of uniform sub-intervals and apply the energy conservation law to individual sub-intervals. The proposed method is based on a numerical-analytical approach such that the frictional energy loss of the patterned tire is computed by 3-D hydroplaning analysis while the other at the disc pad is analytically derived. The hydroplaning simulation is performed by generally coupling an Eulerian finite volume method and an explicit Lagrangian finite element method. The operation of ABS is numerically implemented by controlling the tire angular velocity such that the preset tire slip ratio on the wet road is maintained. Numerical results are presented to illustrative and verify the the proposed numerical estimate.
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  • Kai SEDLACZEK, Peter EBERHARD
    2007 Volume 1 Issue 3 Pages 410-421
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    The problem of optimizing nonlinear multibody systems is in general nonlinear and nonconvex. This is especially true for the dimensional synthesis process of rigid body mechanisms, where often only local solutions might be found with gradient-based optimization methods. An attractive alternative for solving such multimodal optimization problems is the Particle Swarm Optimization (PSO) algorithm. This stochastic solution technique allows a derivative-free search for a global solution without the need for any initial design. In this work, we present an extension to the basic PSO algorithm in order to solve the problem of dimensional synthesis with nonlinear equality and inequality constraints. It utilizes the Augmented Lagrange Multiplier Method in combination with an advanced non-stationary penalty function approach that does not rely on excessively large penalty factors for sufficiently accurate solutions. Although the PSO method is even able to solve nonsmooth and discrete problems, this augmented algorithm can additionally calculate accurate Lagrange multiplier estimates for differentiable formulations, which are helpful in the analysis process of the optimization results. We demonstrate this method and show its very promising applicability to the constrained dimensional synthesis process of rigid body mechanisms.
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  • Javier CUADRADO, Urbano LUGRÍS, Daniel DOPICO, Manuel GONZ&Aacu ...
    2007 Volume 1 Issue 3 Pages 422-433
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    The simulation of flexible multibody systems is a very demanding task that needs improvements in efficiency in order to achieve real-time performance. One of the possible improvements may be the use of topological formulations, which have provided good results in the simulation of large rigid multibody systems. In this work, a topological formulation for rigid bodies is extended to the flexible case, and tests are carried out in order to compare its performance with that of a global formulation. Two systems are simulated, a double four-bar mechanism and a vehicle suspension. As it happened in the rigid case for these two examples, the topological formulation shows lower performance than its global counterpart for such small systems, but the difference decreases as more bodies are modeled as flexible. Since in the rigid case the topological formulation became faster for large systems, further tests must be performed in order to check whether this advantage is kept or even increased in the flexible case.
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  • Suguru ARIMOTO, Morio YOSHIDA, Ji-Hun BAE
    2007 Volume 1 Issue 3 Pages 434-446
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    This paper derives a mathematical model that expresses motion of a pair of multi-joint robot fingers with hemi-spherical rigid ends grasping and manipulating a 3-D rigid object with parallel flat surfaces. Rolling contacts arising between finger-ends and object surfaces are taken into consideration and modeled as Pfaffian constraints from which constraint forces emerge tangentially to the object surfaces. Another noteworthy difference of modeling of motion of a 3-D object from that of a 2-D object is that the instantaneous axis of rotation of the object is fixed in the 2-D case but that is time-varying in the 3-D case. A further difficulty that has prevented us to model 3-D physical interactions between a pair of fingers and a rigid object lies in the problem of treating spinning motion that may arise around the opposing axis from a contact point between one finger-end with one side of the object to another contact point. This paper shows that, once such spinning motion stops as the object mass center approaches just beneath the opposition axis, then this cease of spinning evokes a further nonholonomic constraint. Hence, the multi-body dynamics of the overall fingers-object system is subject to non-holonomic constraints concerning a 3-D orthogonal matrix expressing three mutually orthogonal unit vectors fixed at the object together with an extra non-holonomic constraint that the instantaneous axis of rotation of the object is always orthogonal to the opposing axis. It is shown that Lagrange's equation of motion of the overall system can be derived without violating the causality that governs the non-holonomic constraints. This immediately suggests possible construction of a numerical simulator of multi-body dynamics that can express motion of the fingers and object physically interactive to each other. By referring to the fact that human grasp an object in the form of precision prehension dynamically and stably by using opposable force between the thumb and another finger (index or middle finger), a simple control signal constructed from finger-thumb opposition and an object-mass estimator is proposed and shown to realize stable grasping in a dynamic sense without using object information or external sensing (this is called “blind grasping”).
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  • Frederico Grases RAUTER, João POMBO, Jorge AMBRÓSIO, J&e ...
    2007 Volume 1 Issue 3 Pages 447-457
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    In the great majority of railway networks the electrical power is provided to the locomotives by the pantograph-catenary system. The single most important feature of this system consists in the quality of the contact between the contact wire(s) of the catenary and the contact strips of the pantograph. The work presented here proposes a new methodology to study the dynamic behavior of the pantograph and of the interaction phenomena in the pantograph-catenary system. The catenary is described by a detailed finite element model while the pantograph is described by a detailed multibody model. The dynamics analysis of each one of these models uses different time integration algorithms: the finite element model of the catenary uses a constant time step Newmark type of integration algorithm while the multibody model uses a variable order and variable time step Gear integration algorithm. The gluing element between the two models is the contact model, it is through the representation of the contact and of the integration schemes applied to the referred models that the needed co-simulation is carried on. The work presented here proposes an integrated methodology to represent the contact between the finite element and multibody models based on a continuous contact force model that takes into account the co-simulation requirements of the integration algorithms used for each subsystem model. The discussion of the benefits and drawbacks of the proposed methodologies and of their accuracy and suitability is supported by the application to the real operation scenario considered and the comparison of the obtained results with experimental test data.
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  • Christian STUDER, Christoph GLOCKER
    2007 Volume 1 Issue 3 Pages 458-467
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    In this paper we show how inclusions originating from set-valued force laws can be written as non-linear equations for finite-dimensional second order dynamical systems. We express the set-valued force laws as subdifferentials of the indicator function to a given convex set C, apply the augmented Lagrangian and arrive at a proximal point problem which is solved by Jacobi or Gauss Seidel like iterative schemes. This proceed provides a simple unified access to frictional contact problems in dynamics and can be used either in event-driven or time-stepping simulation approaches.
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  • Dan BRABIE
    2007 Volume 1 Issue 3 Pages 468-480
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    The current paper establishes the necessary prerequisites for studying post-derailment dynamic behavior of high-speed rail vehicles by means of multi-body system (MBS) software. A finite-element (FE) model of one rail vehicle wheel impacting a limited concrete sleeper volume is built in LS-DYNA. A novel simulation scheme is employed for obtaining the necessary wheel-sleeper impact data, transferred to the MBS code as pre-defined look-up tables of the wheel's impulse variation during impact. The FE model is tentatively validated successfully by comparing the indentation marks with one photograph from an authentic derailment for a continuous impact sequence over three subsequent sleepers. A post-derailment module is developed and implemented in the MBS simulation tool GENSYS, which detects the wheel contact with sleepers and applies valid longitudinal, lateral and vertical force resultants based on the existing impact conditions. The accuracy of the MBS code in terms of the wheels three-dimensional trajectory over 24 consecutive sleepers is successfully compared with its FE counterpart for an arbitrary impact scenario. An axle mounted brake disc is tested as an alternative substitute guidance mechanism after flange climbing derailments at 100 and 200 km/h on the Swedish high-speed tilting train X 2000. Certain combinations of brake disc geometrical parameters manage to stop the lateral deviation of the wheelsets in circular curve sections at high lateral track plane acceleration.
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  • Hiroyuki SUGIYAMA, Ahmed A. SHABANA
    2007 Volume 1 Issue 3 Pages 481-490
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    In this investigation, the formulation of trajectory coordinate constraints in terms of the absolute coordinates is developed for the use in the computer simulations of railroad vehicle system applications. Examples of trajectory coordinate constraints are the specified forward velocity of a vehicle or a wheelset along a curved track or specified yaw angle of a vehicle or a wheelset with respect to a body trajectory coordinate system. The relationship between the trajectory coordinates and the absolute coordinates is defined and then used to write the trajectory constraints in terms of the absolute coordinates at the position, velocity, and acceleration levels. The motion of the trajectory coordinate system can be uniquely defined in the track body coordinate system using the arc-length coordinate defined along the space curve. It is demonstrated that the use of the trajectory coordinates leads to simple linear constraint equations, while the trajectory constraint equations become highly nonlinear functions of the absolute coordinates. This is mainly due to the nature of the nonlinear relationship between the absolute Cartesian and trajectory coordinates. A numerical example is presented in order to demonstrate the use of the proposed formulations in the analysis of multibody railroad vehicle systems.
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  • Keisuke KAWAGUTI, Yoshiaki TERUMICHI, Shoichiro TAKEHARA, Stefan KACZM ...
    2007 Volume 1 Issue 3 Pages 491-500
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    In this study, the modeling and formulation for tether motion with time-varying length, large rotation, large displacement and large deformation are proposed. A tether or cable is an important element in lift systems, construction machines for transportation and often is used with a time-varying length. In some cases, these systems are large and the tether has a long length, large deformation and large displacement. The dynamic behavior of a tether in extension and retraction using the proposed method is discussed in this paper. In the passage through resonance, significant tether motions with large rotation and large deformation result. In the analysis of this phenomenon, the transient fluctuations of the motion amplitudes are examined and compared with the corresponding steady state motions. The accuracy and the cost of the calculations are also verified by comparison with the experimental results.
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  • Tetsuya MATSUMOTO, Nobuyuki KOBAYASHI, Yajun ZHANG, Friedrich PFEIFFER
    2007 Volume 1 Issue 3 Pages 501-512
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    This paper deals with the nonlinear seismic response characteristics of tall and flexible structures without anchorage system. It is very important for designing the tall and flexible structure to estimate the slipping displacement and the rocking characteristics of the tall structure more precisely due to the earthquake excitation. Reaction and friction forces caused in the supports of the structure are not uniform due to the overturning moment by the horizontal seismic force. Then contact problem should be taken into account. From this point of view, we developed a model for simulating the nonlinear seismic response of the tall and flexible structure including multi contact problem as the unilateral contact problem. The linear complimentarity between contact forces and the relative displacement of supports against the ground was focused and formulated. Numerical simulations on the seismic response of a container crane were carried out as the example of a tall and flexible structure. As the result, it was demonstrated that the seismic response of the container crane is affected by the slip and rocking coupled motion, and the transition between the static friction and kinetic friction.
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  • Martin FÖRG, Thomas ENGELHARDT, Heinz ULBRICH
    2007 Volume 1 Issue 3 Pages 513-523
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    In this work different contact models within valve train simulations are investigated. Two basic approaches are opposed, namely rigid and flexible models. The rigid model is described by a unilateral constraint, for the group of flexible models a linear spring-damper-element and different variants of HERTZIAN contacts are used. Frictional effects are considered by COULOMB's law in the original and a regularized formulation.
    The models are discussed in terms of their influence on the dynamics of the valve train system, especially when they are used in diverse configurations.
    For the numerical analysis the system size has been varied and extended to the entire valve train with all cylinders and valve mechanisms.
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  • Yasuhiro KAWAMOTO, Yoshihiro SUDA, Hirofumi INOUE, Takuhiro KONDO
    2007 Volume 1 Issue 3 Pages 524-535
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    In this paper, the modeling of the electromagnetic damper (EMD) for automobile suspension is presented and the validation of the model is demonstrated by comparing the numerical results with the experimental results obtained using shaker tests. EMD is used as an active suspension and controlled to have output force calculated from velocities of sprung and unsprung masses. The formulation of EMD system for active suspensions is first developed, and the validation of the EMD model is demonstrated by experiments of the EMD for automobile suspensions. The validity of the formulation of the EMD developed in this investigation is shown for the frequency responses as well as energy balance for its active use.
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  • Chih-Keng CHEN, Thanh-Son DAO
    2007 Volume 1 Issue 3 Pages 536-547
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    Due to its non-holonomic constraints and a highly unstable nature, the autonomous bicycle is difficult to be controlled for tracking a target path while retaining its balance. As a result of the non-holonomic constraint conditions, the instantaneous velocity of the vehicle is limited to certain directions. Constraints of this kind occur under the no-slip condition. In this study, the problem of optimization of fuzzy logic controllers (FLCs) for path-tracking of an autonomous robotic bicycle using genetic algorithm (GA) is focused. In order to implement path-tracking algorithm, strategies for balancing and tracking a given roll-angle are also addressed. The proposed strategy optimizes FLCs by keeping the rule-table fixed and tuning their membership functions by introducing the scaling factors (SFs) and deforming coefficients (DCs). The numerical simualtions prove the effectiveness of the proposed structure of the genetic fuzzy controller for the developed bicycle system.
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  • Kyosuke ONO, Mie KATSUMATA
    2007 Volume 1 Issue 3 Pages 548-557
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    Walking with prosthesis has not been well analyzed mathematically and it seems that the design of powered prosthesis has been done empirically so far. This paper presents a dynamic simulation of a normal human walking and walking with an active prosthesis. We also studied the two controlling methods of a powered thigh prosthesis based on multi-body simulation of human walking. First we measured the normal human walking gait, then, we showed that a 3-DOF human walking model can walk on level ground by applying tracking control to the measured walking gait within a certain range of tuned walking period. Next, we applied the tracking control and self-excited control to the powered thigh prosthesis and compared the robustness and efficiency of the two control methods by numerical simulation. As a result, we found that the self-excited control can significantly decrease the hip joint torque and specific cost to 1/3 compared with the tracking control. Moreover, the self-excited control is superior to the tracking control because tuning for the walking period is not needed for the active prosthetic leg.
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  • Kangsoo KIM, Hiroaki GOMI
    2007 Volume 1 Issue 3 Pages 558-569
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    In this paper, modeling of human speech articulator dynamics and its motor control are presented. The model dynamics of human articulators consists of lip soft tissue around a mouth, combined with surrounding muscles and jaw bone structures. The lip soft tissue is modeled as a discrete model approximation composed of networked lumped nodal masses interconnected with adjacent ones via viscoelastic elements. Stiffness of each element is adjusted to ensure the compatibility in static deformation between the discrete model and its soft tissue prototype considering each compartment size. Muscle motor commands to drive the articulator model are estimated using a control strategy of mimicking human speech motion. An inverse dynamics algorithm based on Gradient Descent Search (GDS) selectively adjusts the muscle motor command in order to produce the reference speech motion. Simple articulatory motions of the model are demonstrated by the activation of muscle motor commands, estimated from the measured human articulatory motions.
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  • Masahiro WATANABE, Yutaka KOYAMA
    2007 Volume 1 Issue 3 Pages 570-582
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    This paper deals with numerical simulations of an active feedback control of leakage-flow-induced sheet flutter in a narrow passage. In this paper, a new non-contact control device and active feedback control technique by using moving baffle-plates to suppress the sheet flutter are proposed and its numerical model is developed by using finite-segment method based on multi-body dynamics formulation. The control performance of the proposed control device and active feedback control method is evaluated by the numerical simulations. In the modeling of the system, the flexible sheet consists of some rigid bodies connected with rotational springs and dampers based on multi-body dynamics formulation. The nonlinear fluid forces acting on the rigid bodies are derived from the basic equations of leakage fluid flow in a narrow passage. The nonlinear equation of motion of the flexible sheet coupled with the fluid flow controlled by the moving baffle-plate actuators is calculated numerically. By the numerical simulations, it is shown that sheet flutter is effectively suppressed by the proposed non-contact control device and active feedback control. Moreover, the control performance is evaluated with changing controller gain and phase shift angle of the control signal.
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  • Min-Soo KIM, HuiJe CHO, Soon-Geul LEE, Jinhwan CHOI, DaeSung BAE
    2007 Volume 1 Issue 3 Pages 583-592
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    This study presents the DFSS optimization for the dynamic responses of a paper feeding mechanism. The flexible paper is idealized as a series of rigid bars connected by revolute joints and rotational spring dampers. In this mechanism, a paper is fed by a contact and friction mechanism on rollers or guides. The design objective is to minimize the slip amounts between paper and mechanisms and satisfy the 6-sigma constraint for the nip forces of rollers. In order to avoid the difficulty of design sensitivity analysis and overcome the numerical noise, a meta-model based optimization is employed. In this approach, first, the space filling methods and the classical DOE methods are used to generate sampling points. Second, the meta-models are constructed from the Kriging, RBF and RSM methods. Finally, a well-developed numerical optimizer sequentially solves the approximate optimization problem. In the numerical test, the DFSS for the paper feeding mechanism problem, having 8-random design variables, is solved in only 23 analyses.
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Papers
  • (Vibration Control of Lateral and Rolling Motions)
    Ken WATANABE, Hiroshi YOSHIOKA, Erimitsu SUZUKI, Takayuki TOHTAKE, Mas ...
    2007 Volume 1 Issue 3 Pages 593-604
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    The superconducting magnetically levitated transport (Maglev) system is conceptualized as a next-generation high-speed transportation system. For practical use, it is important to achieve adequate ride comfort particularly in high-speed running. Maglev vehicles are composed of lightweight car bodies and relatively heavy bogies which are mounted with devices such as superconducting magnets (SCMs) and an on-board refrigerating system. In this magnetically levitated system, the passive electromagnetic damping in the primary suspension between the SCMs and ground coils is very small. Therefore, it is effective to add active electromagnetic damping to this primary suspension, and to adjust the secondary suspension between the car body and bogie. This paper examines vibration control systems of the Maglev vehicle using actuators for the secondary suspension. Moreover, the estimated electromagnetic damping, which interacts between the SCMs and the guideway, is also considered in the model to improve the ride comfort.
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  • Kazuhiko HIRAMOTO, Hironori INAYOSHI
    2007 Volume 1 Issue 3 Pages 605-616
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    A unified methodology of structural and/or control design for mechanical systems based on a polynomial matrix approach is proposed. A closed-loop D-stability specification is considered in this paper. By employing the polynomial matrix approach structural design parameters of mechanical systems, e.g., the mass, the damping and the stiffness and a feedback controller that is allowed to have a specified structure can be obtained simultaneously by solving LMIs. A simple iterative algorithm to obtain a better D-stable closed-loop system and a better central polynomial matrix is proposed. Design examples are provided to show the effectiveness of the present idea.
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  • Ji Woo YOO
    2007 Volume 1 Issue 3 Pages 617-628
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    This paper deals with a built-up structure consisting of a stiff beam and a flexible plate, where the plate is regarded as an energy absorber while the excited beam is regarded as being an energy path. For such a configuration, Grice showed the so-called blocking effect in that the isotropic plate behaves as an energy absorber in narrow frequency bands. This study finds that such energy dissipation can occur in relatively broad frequency bands, if different stiffnesses are used in the rectangular plate consisting of a series of strips. It was experimentally verified by Heckl that the energies, in terms of one-third octave band averages, transferred to the plate (or dissipated in the plate) increase with increased plate damping. This paper, however, shows that the energy absorption suddenly reduces at certain narrow frequency bands where the plate damping effect upon the coupled beam is maximized. Also, in order to minimize energy transfer through the beam in terms of one-third octave band averages, it is advantageous to increase the plate damping closer to the excitation point. All these results are based on the wave method.
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  • Naoyuki TAKESUE, Akihito SANO, Hiromi MOCHIYAMA, Ryo KIKUUWE, Taiki IS ...
    2007 Volume 1 Issue 3 Pages 629-637
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    In this paper, we propose a sensor that detects both pressure and shear information by a helical spring and strain gages. The stresses working on the wire of spring are analyzed. The guideline to attach strain gages on the spring is discussed and the calibration procedure is shown. Finally, experiments are carried out and the effectiveness of the sensor is shown.
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  • Takeo FUJII, Noritoshi NAKAGAWA
    2007 Volume 1 Issue 3 Pages 638-647
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    In the operation of a centrifugal fan installed on a steel plate floor reinforced with steel frames, applied to the heated air recirculation for the paint drier plant, abnormal vibration of the fan was observed in the transient condition of the plant operation. In view of the complexity of the vibration on the fan at the site, an experimental study on the vibration characteristics was conducted by using an identical fan and a simulated steel plate floor. As a result, there were three characteristic vibration phenomena during heating-up or cooling-down operation. The instantaneous increase of vibration occurred when the bearing case was new and there was insufficient lubrication even on the concrete slab. The temporary increase of vibration in vertical direction was caused by raising or sinking of the floor and the fan base due to thermal expansion. And, the fluctuation of vibration in axial direction was caused by friction, which yielded when the bearing moved inside the bearing case due to thermal expansion or contraction of the shaft. It seems that this fluctuation was due to self-excited vibration by friction.
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  • Kai FENG, Shigehiko KANEKO
    2007 Volume 1 Issue 3 Pages 648-659
    Published: 2007
    Released: August 31, 2007
    JOURNALS FREE ACCESS
    Foil bearings are supposed to be one of the best candidates of supporting component for turbo-machineries because of their design simplicity, reduced weight and size, high speed and temperature capability, and easy maintenance. Among various types of foil bearings, multi wound foil bearing (MWFB), which had been designed and fabricated in our lab, is easy to analyze static characteristics even though load capability of which is small compared with other types of foil bearings.
    In this study, a theoretical model of MWFB taking account of the effect of the foil deformation is developed to predict its static performance. Reynolds equation is solved using Finite Difference Method (FDM) to yield air pressure distribution, while the elastic deformation equation is solved by Finite Element Method (FEM) to predict the deformation of the foil. Then, the above two equations are coupled by several iterations until the convergence criterion is reached. Based on such calculations, static characteristics of MWFB such as load capacity, torque are presented.
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