Designing Strength-Proportional Hydraulic Resistance for an Elbow Flexion-Extension Exercise Machine

[+] Author and Article Information
Brian A. Garner

Department of Mechanical Engineering, Baylor University, Waco, TX 76798-7356

J. Med. Devices 1(1), 3-13 (Jul 19, 2006) (11 pages) doi:10.1115/1.2355684 History: Received February 01, 2006; Revised July 19, 2006

The mechanical linkage of a hydraulic-resistance, elbow flexion and extension exercise machine was redesigned to provide a resistance response that varies in proportion to female joint strength over the range of motion. The aim was to integrate into a simple, passive exercise machine the respective benefits of hydraulic resistance and isokinetic exercise. Hydraulic resistance facilitates bidirectional, concentric-concentric exercise that naturally scales to accommodate users of varying strength. Strength-proportional resistance emulates the response of isokinetic exercise to work muscles at their maximum capacity throughout the range of motion. Methods: Independent mathematical models of elbow joint strength and machine resistance were derived as a function of machine arm flexion angle and angular velocity. The strength model was based on experimental data measured from female subjects during maximum-effort trials on the exercise machine. The resistance model was based on the force-velocity response of the hydraulic cylinder and the linkage’s mechanical advantage as determined from its geometric parameters. The intersection of these two models was assumed to represent conditions of equilibrium between strength and resistance, and was used to predict exercise operating speed for any angle of elbow flexion. Numerical optimization methods were applied to compute optimal parameters for two-bar and four-bar linkage configurations with the objective of achieving predicted operating speed patterns that were constant (isokinetic) over the range of motion. Results: A four-bar linkage configuration was found to be more effective at providing the desired resistance response than a two-bar configuration. Experimental trials using the optimized linkages confirmed model predictions and demonstrated that user operating speeds over the range of motion were closer to the desired isokinetic patterns than with the original linkage. The design method was effective at predicting user operating speeds and targeting desired shapes and magnitudes for the operating speed patterns. The method should be applicable to the design of other exercise machines that seek the advantages of a strength-proportional response derived from hydraulic resistance.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

Adjustable prototype configured like the original elbow flexion and extension exercise machine. The user sits with the upper arms resting against the elbow pads, and operates the machine arms in cyclic, reciprocal elbow flexion and extension motion as fast as possible against the hydraulic cylinder resistance.

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Figure 2

Female maximum voluntary elbow flexion and extension moments at isokinetic speeds of 108deg∕s and 180deg∕s over a range of elbow joint angles. Data collected on the prototype exercise machine of this study (thick lines) compares well with data (thin lines) reported by Knapik (14).

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Figure 3

Measured force-velocity response of the hydraulic cylinder that provides resistance for the elbow flexion and extension exercise machine. Resistance naturally increases with piston speed, and was found to be slightly higher when the cylinder was loaded in tension (corresponding to elbow flexion) than in compression (corresponding to elbow extension).

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Figure 4

Mechanical advantage (leverage) transmitting hydraulic cylinder force into resistance moment at the machine arm (inset). The higher mechanical advantage in the middle of the range of motion provides increased resistance where the elbow joint is strongest.

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Figure 5

Three-dimensional surface models representing female elbow flexion strength and machine arm flexion resistance, respectively, as a function of machine flexion angle and angular velocity. Exercise is assumed to occur over the range of motion at speeds where strength is equalized by resistance (line of intersection between surfaces, and its projection).

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Figure 6

Predicted (thick lines) and observed (thin lines) user operating speeds for maximal-effort elbow flexion and extension exercise over the range of motion. The predicted values compare well with the observed data from two representative subjects (filled and unfilled triangles, respectively).

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Figure 7

Linkage configurations of (a) the original two-bar design and (b) the modified four-bar design. In both cases the cylinder piston rod is displaced by user actuation of the padded machine arm.

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Figure 8

Mechanical advantage as a function of machine arm flexion angle for the original two-bar linkage (thick solid line) and the optimized four-bar linkages (thin and dashed lines). Four-bar linkage designs were optimized to effect fast, moderate, and slow target operating speeds, and to prioritize only the extension stroke (ext, thin lines), only the flexion stroke (flx, thin lines), or both strokes equally (dashed lines). See text and Table 1 for details.

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Figure 9

Target (dashed lines), predicted (thick lines), and observed (thin lines) exercise operating speeds for the three four-bar linkage solutions optimized to prioritize both the flexion stroke (flx) and extension stroke (ext) equally. The observed operating speed data represents the average over all subjects.

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Figure 10

The average (lines) and ranges of observed user operating speeds over all subjects for maximal-effort elbow flexion and extension exercise using the original two-bar linkage (dashed line and hatched region) and the moderate-speed four-bar linkage (solid line and shaded region). With the four-bar linkage the range of operating speeds is narrower across subjects, and is more constant over the range of motion.

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Figure 11

Geometric parameters and other variables for describing the two-bar and four-bar linkages, and for deriving values of the mechanical advantage of each, respectively (see Appendix). Geometric parameters varied in order to optimize (see text) the respective mechanical advantages include: d (the length of link OD), α, Hx, and Hy in both linkages, and Rx, Ry, and c and r (the lengths of links DC and CR, respectively) in the four-bar linkage. Note that in Fig. 7 the optimized four-bar linkage is shown with parameter α set to zero.



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