This paper is to study how the vibration modes of a cyclic symmetric rotor evolve when it is assembled to a flexible housing via multiple bearing supports. Prior to assembly, the vibration modes of the rotor are classified as “balanced modes” and “unbalanced modes.” Balanced modes are those modes whose natural frequencies and mode shapes remain unchanged after the rotor is assembled to the housing via bearings. Otherwise, the vibration modes are classified as unbalanced modes. By applying fundamental theorems of continuum mechanics, we conclude that balanced modes will present vanishing inertia forces and moments as they vibrate. Since each vibration mode of a cyclic symmetric rotor can be characterized in terms of a phase index (Chang and Wickert, “Response of Modulated Doublet Modes to Travelling Wave Excitation,” J. Sound Vib., 242, pp. 69–83; Chang and Wickert, 2002, “Measurement and Analysis of Modulated Doublet Mode Response in Mock Bladed Disks,” J. Sound Vib., 250, pp. 379–400; Kim and Shen, 2009, “Ground-Based Vibration Response of a Spinning Cyclic Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects,” ASME J. Vibr. Acoust. (in press)), the criterion of vanishing inertia forces and moments implies that the phase index by itself can uniquely determine whether or not a vibration mode is a balanced mode as follows. Let N be the order of cyclic symmetry of the rotor and n be the phase index of a vibration mode. Vanishing inertia forces and moments indicate that a vibration mode will be a balanced mode if n≠1,N−1,N. When n=N, the vibration mode will be balanced if its leading Fourier coefficient vanishes. To validate the mathematical predictions, modal testing was conducted on a disk with four pairs of brackets mounted on an air-bearing spindle and a fluid-dynamic bearing spindle at various spin speeds. Measured Campbell diagrams agree well with the theoretical predictions.
Abstract-In many physical human-robot interaction scenarios, such as haptic virtual environments for training and rehabilitation, it is desirable to carefully control the apparent inertia of a robot. Inertia compensation can be used to mitigate forces felt by the user during free-space motion, and rendering of additional inertia is desired for particular rehabilitation and training procedures. Many factors influence the stability and accuracy of rendering for haptic display of a pure mass, including device mechanical properties, sample rate, control structure, and human behavior. Inspired by the "Z-Width" approach to haptic device stability and performance analysis, we introduce "M-width", which we define as the dynamic range of virtual masses renderable in a stable manner. We identify the important parameters for system stability, find stability boundaries, and describe the expected accuracy of the haptic rendering for a canonical haptic system. These results serve as a design tool for creating stable and accurate haptic virtual environments, establish limits of performance, and lay the groundwork for new controllers to improve mass rendering.
Impedance-type kinesthetic haptic displays aim to render arbitrary desired dynamics to a human operator using force feedback. To render realistic virtual environments, the difference between desired and rendered dynamics must be small. In this paper, we analyze the closed-loop dynamics of haptic displays for three common virtual environments: a spring, a damper, and a spring-damper, including the effects of time delay and low-pass filtering. Using a linear model, we identify important parameters for the rendered dynamics in terms of effective impedances, a conceptual tool that decomposes the displays closed-loop impedance into components with physical analogs. Our results establish bandwidth limits for rendering effective stiffness and damping. The effective stiffness bandwidth is limited by the virtual stiffness and device mass, and the effective damping bandwidth is limited by the cut-off frequency of the low-pass filter which filters the device velocity estimate. We show that a general system impedance can be characterized by a mass, damper, and spring optimally by the solution to a convex optimization problem, and we present a quantitative metric, the Average Distortion Error (ADE), to describe the fidelity of this model. Time delay has no significant effect on characterized stiffness, and reduces characterized damping by the product of virtual stiffness and total time delay. Reducing the low-pass filter cut-off frequency reduces the characterized damping. Experimental data gathered with a Phantom Premium 1.5 validates the theoretical analysis. We also conducted human user experiments to investigate the effects of time delay and low-pass filtering on perceived stiffness and damping. Similar to the characterized dynamics results, we observed no significant effect of time delay on perceived stiffness, and increasing time delay resulted in reduced perceived damping. Lower filter cut-off frequencies resulted in lower perceived damping. This work informs haptic display design by presenting how closed-loop behavior changes with key parameters.
The stable, quantization-error noise-free, rendering of high-stiffness dynamics can be challenging using impedance-type haptic displays. In this paper we examine a canonical, one degree of freedom, haptic display rendering a virtual spring and damper, including the effects of the device and human dynamics, sampling, position quantization, time delay, and the low-pass filter operating on the device velocity estimate. We construct various stability and quantization-error regions as a function of the system parameters and show the necessary trade-offs that occur between them. Although we apply the quantization-error analysis to virtual spring and damper rendering, it applies to a general virtual environment. We present sufficiency for quantization-error passivity, necessity for no malicious-touch limit cycles, and necessity for no uncoupled-touch limit cycles. Using these results, aided by the presented supplementary code, we find control parameters to render the largest, renderable, virtual stiffness for a given haptic display. The analytical results are experimentally verified using a Phantom Premium 1.5 haptic device.
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