Kenichiro Matsuzaki

Effect of Stiffness Ratio of Piecewise-Linear Spring on the Occurrence of Subharmonic Nonlinear Vibration in Automatic Transmission Powertrain

In the torque converter, a damper with a piecewise-linear spring is used to reduce the forced vibration, and the subharmonic vibration occurs when the spring restoring torque characteristics approach the switching point. This research analyzed the effect of stiffness ratio between the neighboring piecewise-linear springs on the occurrence of the subharmonic nonlinear vibration in automatic transmission powertrain. The powertrain is modeled with multi degree-of-freedom nonlinear system as an actual vehicle. The result shows higher value of the stiffness ratio between the neighboring springs creates larger value of the subharmonic vibration.

Optimal Design of Spring Characteristics of Damper for Subharmonic Vibration in Automatic Transmission Powertrain

In the torque converter, the damper of the lock-up clutch is used to effectively absorb the torsional vibration. The damper is designed using a piecewise-linear spring with three stiffness stages. However, a nonlinear vibration, referred to as a subharmonic vibration of order 1/2, occurred around the switching point in the piecewise-linear restoring torque characteristics because of the nonlinearity. In the present study, we analyze vibration reduction for subharmonic vibration. The model used herein includes the torque converter, the gear train, and the differential gear. The damper is modeled by a nonlinear rotational spring of the piecewise-linear spring. We focus on the optimum design of the spring characteristics of the damper in order to suppress the subharmonic vibration. A piecewise-linear spring with five stiffness stages is proposed, and the effect of the distance between switching points on the subharmonic vibration is investigated. The results of our analysis indicate that the subharmonic vibration can be suppressed by designing a damper with five stiffness stages to have a small spring constant ratio between the neighboring springs. The distances between switching points must be designed to be large enough that the amplitude of the main frequency component of the systems does not reach the neighboring switching point.

Fundamental study of subharmonic vibration of order 1/2 in automatic transmissions for cars

A torque converter is an element that transfers torque from the engine to the gear train in the automatic transmission of an automobile. The damper spring of the lock-up clutch in the torque converter is used to effectively absorb the torsional vibration caused by engine combustion. A damper with low stiffness reduces fluctuations in rotational speed but is difficult to use because of space limitations. In order to address this problem, the damper is designed using a piecewise-linear spring with three stiffness stages. However, the damper causes a nonlinear vibration referred to as a subharmonic vibration of order 1/2. In the subharmonic vibration, the frequency is half that of the vibrations from the engine. In order to clarify the mechanism of the subharmonic vibration, in the present study, experiments are conducted using the fundamental experimental apparatus of a single-degree-of-freedom system with two stiffness stages. In the experiments, countermeasures to reduce the subharmonic vibration by varying the conditions of the experiments are also performed. The results of the experiments are evaluated through numerical analysis using the shooting method. The experimental and analytical results were found to be in close agreement.

237 Analysis of Pulse Wave in Blood Vessel by Concentrated Mass Model

A waveform of a pulse wave in a blood vessel often changes because of nonlinear effect. To analyze this nonlinear phenomenon, the finite difference method has been used. However, the treatment of the method is cumbersome. In order to overcome this problem, we propose a concentrated mass model to analyze the nonlinear pulse wave problems. This model consists of masses, connecting nonlinear springs, connecting dampers, base support dampers, and base support springs. The characteristic of connecting nonlinear spring is derived from the relationship between pressure and diameter of a blood vessel, and the base support damper and the base support spring are derived from the shear stress from a wall of a blood vessel. The pulse waves in the blood vessel of the dog measured by Laszt are analyzed numerically by using the proposed model in order to confirm the validity of the model. Numerical computational results agree very well with the experimental results. Especially, “steepening phenomenon” generated by the nonlinear effect of fluid is numerically reproduced. Therefore, it is concluded that the proposed model is valid for the numerical analysis of nonlinear pulse wave problem.

A high-performance method of vibration analysis for large-scale nonlinear systems (application to flexural vibration of straight-line beam structure with nonlinear supports)

A rational method of dimensional reduction is developed in order to analyze accurately a nonlinear vibration generated in a large-scale structure with locally strong nonlinearity. In the proposed method, the state variables of linear nodes are transformed into the modal coordinates by using the real constrained modes that is obtained by fixing the nonlinear nodes, and a small number of modal coordinates that have a significant effect on the computational accuracy of the solution are selected and utilized in the analysis by combining them with the state variables of nonlinear nodes that are expressed in the physical coordinates. The remaining modes that have little effect on the computational accuracy are appropriately approximated and are eliminated from the system. From the reduced model constructed by these procedures, the steady state periodic solution and the stability, the transient solution and the quasi-periodic solution can be computed with a very high degree of computational accuracy and at a high computational speed. The effectiveness of the proposed method is verified by the computational results obtained for a straight-line beam structure with nonlinear supports.

A Layout Technique for Multiple Floor Layout Problem Considering the Area of an Elevator

This paper presents a layout technique for the multiple floor layout problems. Compared with the carriage in horizontal way, the carriage in vertical way needs more cost dramatically and gets complicated with the assignment for each carriage, because of using vertical transportation equipment such as elevator. Then we need multiple floor layout algorithms that consider proper location and number of vertical transportation equipment. We considered about the area of elevator inside the facility and several kinds of vertical transportation equipment, draw up the approximate optimum layout by Space Filling Curve, examine the possibility about the layout in practice consideration about timing of production, and suggest more realistic multiple floor layout algorithm.

Some Problems in the Precise Numeration of Fourier Expansions of Jacobian Elliptic Functions.

Jacobian elliptic functions are useful in solving certain types of nonlinear problems, particularly in obtaining exact solutions of homogeneous Duffing equations. In addition to the theoretical formulation, the task of accurate numeration requires much effort. Finding the effective relationship inherent in the theory of elliptic functions may be very useful in developing algorithms/programs. This paper introduces some new formulations which assist the precise Fourier expansion of elliptic functions. The authors propose a set of optimal development neighborhoods of the respective values of modulus parameter m = 0, 1/2 and 1, which successfully renders uniform accuracy over the whole range of m. Also discussed are an exact solution of the snap-through spring Duffing free oscillator, an anomalous error propagation in the numerical transmission between words in the computer and the explicit polynomial solution of the so-called q-factor as a function of m.

Exact Solutions and the Numeration Program for Free Oscillation in a Nonlinear System with a Quadratic Spring Function and a Single Degree of Freedom.

A classical nonlinear oscillator possesses a spring function expressed in terms of the 2nd-or the 3rd-order polynomial, viz., the so-called Helmholtz or Duffing systems. The latter, a symmetrical system, has been studied extensively since Duffing proposed the mathematical models ; however, the former, an asymmetrical system, is less well known than the latter. The exact solution of their free oscillation seems to be at the same stage of development, in that an expression in terms of the Jacobian elliptic functions, i. e., sn, cn or dn for the Duffing system, exists already, whereas sn2 or cn2 have only recently been found for the Helmholtz system. This paper summarizes existing numeration algorithms and proposes a program for exact solutions of the free oscillation in the Helmholtz system. This solution successufully verifies the accuracy of a numerical or an approximate scheme applicable to solve a nonlinear problem which may essentially have no exact solution.