Reduction of Rotational Vibration Using Coriolis Force Generated by Electromagnetic Oscillatory Actuator Moving in Radial Direction

Abstract

1 IntroductionReciprocating engines generate torque only during the explosion process and this causes a large ripple in output torque. When the torque ripple is transmitted through the engine drive system, torsional vibration is induced in a drive shaft, resulting in uncomfortable vibration and noise. This rotational vibration is a problem not only in automobiles, but also in ships, cableways, and other transportation equipment. Various vibration suppression methods have been proposed to solve this problem.Matsuhisa et al. proposed a dynamic vibration absorber (DVA) with an additional mass that oscillates in the radial direction of the cableway to suppress rotational vibration caused by wind [1]. When the natural frequency of the added mass is set to about twice the natural frequency of the gondola, the Coriolis force generated by the radial vibration of the added mass has been shown to be effective in damping the rotational vibration. However, since the additional mass is passively vibrated by the centrifugal force of the gondola, itis not possible to control the magnitude of the Coriolis damping force.This paper proposes a new hybrid DVA to reduce rotational vibration of a rotor. The additional mass operates as the mover of the electromagnetic linear oscillatory actuator (LOA). The LOA actively oscillates the mover by current control and obtains the variable damping force caused by the Coriolis effect. LOAs have been applied to various industrial equipment due to their simple structure, direct drive, and high efficiency [2].They have also been applied to an active control device for automobiles [3]. In this paper, we describe the structure and mathematical model of the proposed hybrid DVA. Moreover, we evaluate numerically the vibration suppression effect under the active operation of the LOA.2 Hybrid DVA Using Electromagnetic Oscillatory Actuator2.1 Basic Structure and Operational PrincipleFig. 1 shows the basic configuration and operational principle of the proposed hybrid DVA. 1-DOF LOA is embedded on the surface of a rotor and oscillates in the radial direction (r-direction). Excitation of the coil generates an electromagnetic force in the r-direction, which works to increase the mover stroke of the LOA. Consequently, the Coriolis force is produced in the circumferential direction (θ-direciton),which suppresses the rotational vibration in the θ-direction. Since the coil of the LOA rotates synchronously with the rotor, we need to discuss power supply. We assume a direct power supply from a battery mounted on the rotor surface. Therefore, we include the equivalent moments of inertia of the stator and the battery in the moment of inertia for the rotor.Our hybrid DVA can be also understood as a opposite phenomenon of a parametric excitation (parametric damper). Fig. 1(b) shows an example of the mover trajectories under parametric excitation and parametric damper. This figure clearly suggests that the vibration in the θ-direction increases after one oscillation period because of the positive energy provided by the parametric excitation. When the trajectory becomes reversed, the vibration decreases because of the negative energy provided by the parametric damper. The important point is that both cases require to satisfy the following relationship,ωr:ωθ = 2:1. (1)Here, ωr and ωθ are the natural frequency in the r- and θ-directions, respectively.2.2 Equations of Motion for Hybrid DVAThe Euler-Lagrange equation gives the following equations of motion in the r- and θ-directions,Jd2θ/dt2=m(l+r)d2θ/dt2+cθdθ/dt+2m(l+r)dθ/dt*dr/dt+kθθ=Tincosωt, (2)md2r/dt2+crdr/dt+krr=m(l+r)(dθ/dt)2+KtI+Kdr. (3)Here, the fourth term on the left-hand side of equation (2) is the Coriolis force, which works to decrease the vibration in the θ-direction. The first term on the right-hand side of equation (3) is the centrifugal force, which works to increase the vibration in the r-direction. The kinetic parameters of the hybrid DVA are determined as follows so that equation (1) holds,2√(kθ/(J+ml2))=√((kr-Kd)/m). (4)3 Reduction of Rotational Vibration under Active ControlIn this section, we solve the above equations of motion and verify the effectiveness of the proposed hybrid DVA through numerical simulation. Fig. 2 shows the rotational vibration of the rotor and the displacement of the mover with and without the LOA. In this simulation, The mover is oscillating passively until 4s. After that, Electromagnetic force generated by the LOA starts to oscillate the mover actively. Focusing on the envelopes of the waveform (blue solid line: without LOA, orange dashed line: with LOA), the active operation with LOA reduced the rotational vibration of the rotor by about 30%. In response to the decrease in the rotational vibration, the amplitude of the mover displacement is approximately doubled. We are able to conclude that the electromagnetic force generated by the LOA increases the displacement of the mover and the resulting damping effect by the Coriolis force also increases.In the final paper, we will describe the detailed design of the hybrid DVA and evaluate performances of the LOA through finite element analysis. Because this technique may also be applied to reduce torque ripple in AC motors, the rotational vibration by torque ripple will also be discussed. **