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Research Papers

# Switch-Mode Continuously Variable Transmission: Modeling and Optimization

[+] Author and Article Information
James D. Van de Ven

Department of Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609vandeven@wpi.edu

Michael A. Demetriou

Department of Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609mdemetri@wpi.edu

The correct notation for the output $y(t)$ should be $y(t;ts)$ since the output of the switched dynamical system depends on both the time $t$ in $[0,Δt]$ and on the switch time instance $ts$. However, we hope that this seemingly minor abuse of mathematical notation will not cause any confusion.

J. Dyn. Sys., Meas., Control 133(3), 031008 (Mar 24, 2011) (7 pages) doi:10.1115/1.4003373 History: Received December 15, 2009; Revised August 10, 2010; Published March 24, 2011; Online March 24, 2011

## Abstract

Hybrid vehicles are an important step toward reducing global petroleum consumption and greenhouse gas emissions. Flywheel energy storage in a hybrid vehicle combines high energy density and high power density, yet requires a highly efficient continuously variable transmission with a wide operating range. This paper presents a novel solution to coupling a high-speed flywheel to the drive train of a vehicle, the switch-mode continuously variable transmission (CVT). The switch-mode CVT, the mechanical analog of a boost converter from power electronics, utilizes a rapidly switching clutch to transmit energy from a flywheel to a spring, which applies a torque to the drive train. By varying the duty ratio of the clutch, the average output torque is controlled. This paper examines the feasibility of this concept by formulating a mathematical model of the switch-mode CVT, which is then placed in state-space form. The state-space formulation is leveraged to analyze the system stability and perform simple optimization of the switch time and damping rate of the spring over the first switching period. The results of this work are that a stable equilibrium does exist when the speed of the output shaft is zero, but the system will not reach and stay at a desired torque if this condition is not met, but requires continuous switching between the two states. An optimal switching time and damping ratio were found for the given parameters, where the lowest error occurred with low values of damping ratio. This work builds a foundation for future work in increasing the complexity of the model and the optimization method.

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## Figures

Figure 1

A boost power converter increases the voltage from the input source to the output by controlling the duty cycle of the switch

Figure 2

The switch-mode CVT controls the torque transmitted to the output shaft by specifying the duty cycle of the clutch

Figure 3

The switch-mode CVT with the additional components required for regenerative operation. In regenerative operation, the brake is pulsed to create deflection in the spring and the over-running clutch transmits torque to the flywheel.

Figure 4

Simplified free-body-diagrams of the components of the switch-mode CVT

Figure 5

Evolution of output y(t) over 10 cycles

Figure 6

Evolution of the output y(t) for different switch times. Switch times are expressed as a percentage of the first duty cycle, in the range 0.2–0.7%.

Figure 7

Effect of damping ratio on optimal switch time and optimal error norm. Switch time is expressed as a percentage of the first duty cycle.

## Errata

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