where m is the mass of the system (shank, foot, and lever arm), g is the acceleration due to gravity, r_T is the distance from the axis of rotation to the system center of mass (i.e., the total system radius of rotation), and is Ө the angle of displacement from vertical (where the net torque equals zero).

The total system radius of rotation (r_T) was determined by constructing a geometrical model according to the weighted position of each component (shank, foot, and lever arm) in the plane of motion. The centers of mass for the lever arm were considered to lie along the same line within the plane of motion since the axes of each are nearly parallel when the leg is secured in the distal pad and strap.

The determination of the kinetic energy associated with the limb-lever system depends on the total moment of inertia for the limb-lever system (the sum of that for the shank, foot, and lever arm) and the square of the preset isokinetic velocity: KE = ½Iω². The values of I for the limb segments were estimated; whereas, those for the lever arm were determined experimentally. As the moment of inertia of a rotating body depends on the distribution of mass and its distance from the axis of rotation, the determination of the moment of inertia (I) for complex objects involves the summation of the product of unequal portions of the total mass in relation to their square distance from the axis of rotation (Sternheim and Kane, 1991). As each increment of mass (m) is a certain distance (r) from the axis, then I was found by the integral: I = ∫r² dm.

An alternate method was used for determining the moment of inertia for a rigid body by measuring its moment and period of oscillation with the relationship: I = T²M / 4π² (Tipler, 1991) where T is the period of oscillation and M is the moment. For each unique lever arm length used in this study, the period of oscillation was determined with the photogate timer and the moment was determined on the Biodex dynamometer in "Isometric" mode with the lever arm in a horizontal position.

By substituting the variables for kinetic energy and potential energy into the expression for W_total, the total work for each experimental condition was calculated from the following:

where m, r_T, and I_limb (the estimated limb moment of inertia) were dependent on the participant's characteristics; Ө was dependent on the mode and positioning of the participant; T and M were dependent on lever arm length; and ω was preset at the dynamometer. Depending on whether the condition was concentric or eccentric, the W_kin (middle term) was either added to or subtracted from the W_grav (first term), respectively.

Because isokinetic parameters are often reported in relation to a correction factor for the influence of gravity, standard work (W_std) is defined as the sum of the W_grav and that recorded by the dynamometer (W_dyn). W_total was used in the definition of two other parameters employed in this study: kinetic error (K_err) and work measurement type (WMT). Kinetic error was defined as the ratio W_kin/W_total and represented the proportion of total work either over- or underestimated when W_kin is not considered. W_total in combination with W_std comprised the WMT group and were compared for the purpose of identifying differences between these two measurement methods.

Work measurement type (see Table 1) expressed as mean work output measurements (W_total, W_std, W_dyn) across speeds for males and females are shown in Figures 1 and 2, respectively. Although the statistical analyses of work type included only W_total and W_std, W_dyn appears in the figure as referent to the amount of uncorrected dynamometer work. Four-way (2 WMTs x 2 modes x 2 genders x 4 speeds) analysis of variance (ANOVA) was performed to determine interactions that involved modes and genders. For concentric tests, both main effects and their interaction were significant (p < 0.05) in both males and females. Analysis of the eccentric work data revealed a significant main effect (p < 0.05) for WMT only and significant interaction (p < 0.05) for both genders. Comparisons of work measurements types were conducted for both genders to reveal if the method of reporting work output introduced significant differences between measures of work at individual speeds. Three-way ANOVA (2 WMT x 2 genders x 4 speeds) showed significant differences (p < 0.005) for all paired work measurement types across all speeds and for both modes and genders except at the lowest speeds (30 deg/sec eccentric and 90 deg/sec concentric) between genders (p > 0.05). The change in work output, which was introduced by adding W_kin to W_std, increased with greater concentric angular velocities. For eccentric data, an opposite trend was noted where differences were negative and decreased with increasing velocities.

Kinetic error (K_errr) was defined as the ratio W_kin/W_total. Descriptive statistics for this variable, expressed as a percent, are given in Table 2. Two-way ANOVA (gender x speed) for concentric data and eccentric data demonstrated that all main effects and interactions were significant (p < 0.05). For concentric tests, Duncan Multiple Range Tests showed significant differences between genders (p < 0.05) only at the two fastest velocities (210 and 270 deg/s). For eccentric speeds, a between-gender difference was observed only at the fastest velocity (150 deg/s). In addition, post-hoc comparisons showed that K_err significantly increased (p < 0.05) with increasing velocity for both genders.

The magnitude of the differences across speeds is notable. As shown in Table 1, the differences due to W_kin at the slowest speeds were marginal. However, at the fastest concentric speed (270 deg/s), the percent difference was, on average, 6.2% for males and 9.1% for females. This trend is consistent with the presence of significant interactions (WMT by speed) within both modes and genders in that the differences in work measurements were not equivalent across speeds. The most direct explanation for this relationship, where the proportion of W_kin compared to W_total increases with increasing speed, relates to (a) the fact that W_kin is proportional to the square of the angular velocity and (b) the tendency for W_dyn to be inversely proportional to the concentric velocity.

Direct comparison between concentric and eccentric modes in the present study was not appropriate for four main reasons: (a) the range of speeds tested was not the same, (b) the ranges of motion were defined differently, (c) eccentric actions required an activation force prior to movement, and (d) the role of W_kin changes across modes in determining W_total. The maximal eccentric speed available on the Biodex used in this study was -150 deg/s. Since it was an intention to explore concentric speeds beyond this eccentric speed limit, speeds between modes could not be paired across the velocity spectrum examined. Different ranges of motion were chosen due to the distinct operation of the Biodex between modes. In addition, the activation force requirement during eccentric motions may have introduced dissimilar testing conditions between modes. Measures of average torque have been reported to increase with increasing activation forces along with a tendency for larger effects with eccentric actions and at higher velocities (Kramer et al., 1991). In the present study, the combined effect of ROM and activation force may have provided conditions of differential opportunity to do work between modes. Therefore, the analysis of modes remained distinct.

Due to the inability to collect eccentric data on all the male participants in the present study, the related statistical analysis may have been less robust than for the concentric data. This shortcoming was not related to any limitations of the participants involved. On the contrary, 3 males had sufficient strength to surpass the eccentric torque limit of the Biodex dynamometer. If another population with similar mean anthropometric dimensions but lower mean isokinetic leg extension strength (non-strength trained) had been selected for this study, the magnitude of the difference between total and standard isokinetic work would have been greater.

The standard rule for gravity correction during eccentric isokinetics is the same for concentric: gravity effect torque is subtracted where gravity assists and added where gravity opposes the motion (Kramer et al., 1991; Westing and Seger, 1989). Therefore, for concentric and eccentric isokinetic knee extension, the excess torque (and work) provided by the acceleration due to gravity is added to the output measured by the dynamometer. However, since some of the potential energy of the limb-lever is expended as KE in response to the force of gravity in eccentric knee extension (as opposed to being generated by the subject during concentric knee extension), attributing all of the work represented by the PE of the system to account for gravity correction is not accurate. Because the amount of work required for this controlled eccentric descent is the potential energy less the kinetic energy (PE - KE), faster motions would require less work to control the system (see Equation 2). When the final KE of the system equals the PE at the start, no muscular work would be needed to intervene and control the descent; in this case, W_total = W_dyn. The final angular velocity at which this occurs, therefore, depends on the moment of the limb-lever system and its potential energy at the start of motion.

The isolation of kinetic energy (W_kin) as a distinct component of total work (W_total) during isokinetics is unique. The present investigation introduced two factors not previously addressed in the assessment of isokinetic work: (a) the role of kinetic energy of the system during eccentric isokinetic bouts, and (b) analysis of the kinetic effects on work measurements within speeds and between genders. The present study contrasted the resulting W_std and W_total (WMTs) during concentric and eccentric tests. Post-hoc tests were conducted to reveal whether the inclusion of KE in the calculation of W_total significantly changed the mean work measurement for individual isokinetic speeds.

In order to relate the error due to ignoring kinetic work, the kinetic error K_err was calculated in the present study as W_kin/W_total (see Table 2). This error factor was significant across speeds and between genders for both concentric and eccentric modes (p < 0.05). A parallel finding was reported by Chen et al., 1994 in their analysis of concentric acceleration work. Moreover, the results of the present study indicated that 25 of 28 pair-wise contrasts of errors between concentric speeds were significant (p < 0.05); for eccentric data, 13 of 15 contrasts were significant (p < 0.05). That comparisons of K_err between genders were significant at higher, but not lower, speeds was consistent with the interaction between speed and gender revealed in the ANOVAs. Such comparisons were not reported by Chen et al., 1994.

Although the definition of W_kin in the present study differed from the acceleration work in the study by Chen et al., 1994, both encompassed measures of work accomplished outside isokinetic conditions. As presented by Chen et al., 1994, it is evident that the work done during acceleration of the limb-lever system up to a preset isokinetic velocity must also include the portion of work done with respect to gravity during that period. In the present study, the latter work was not isolated for the purposes of defining acceleration work; rather, this portion of work was represented in the potential energy term in the calculation for W_total which included work against gravity throughout the entire ROM.

The present study accounted for the change in the moment of inertia with increments in lever arm length. The overall range of the moment of inertia for the Biodex lever arm input device used in this study was 0.178 to 0.285 kg·m² (lever arm length positions 11.5 to 15 inches). Chen et al., 1994 reported a CybexII lever arm moment of inertia of 0.162 kg·m² which estimated using a model of regular shapes such as rectangular rods and cylindrical rods. However, no detail was given to indicate whether changes in lever arm length per participant were considered. Because the dimensions of these attachments vary across manufacturer, the differences in the ranges of possible moments of inertia may affect the amount of work necessary to accelerate them.

A final consideration in the calculation of W_total is the method used for estimating the limb segment parameters in determining W_kin and the potential energy. The present study employed regression equations based on cadaveric data proposed by Yeadon and Morlock, 1989 which should be valid for different populations as long as segmental mass distributions are similar. By comparison, Chen et al., 1994 utilized what was referred to as transverse centriodal moment of inertia values. The segmental centers of mass were also derived from different sources. Whereas the present study used the data provided by Plagenhoef et al., 1983, the comparison study referred to data from another source (Miller and Nelson, 1976). The lack of published evidence regarding error from excluding kinetic energy in the calculation of eccentric total work leaves knowledge about this component of muscle action incomplete.

The calculation of total work during the entire ROM makes the present study unique in light of recent investigations into rate of velocity development, load range, and expression of power. Load range as been shown to decrease as concentric knee extension velocity increases by as little as 3.9% at 60 degrees/sec and as much as 78% at 450 degrees/sec in males (Brown et al., 1995). In the same study females exhibited a decrease in load range from 95.9% at 60 deg/sec to 0% at 450 deg/sec. Taylor et al., 1991 found a decrease in load range from 64.6 degrees at 240 deg/sec to 27 at 400 deg/sec concentrically. As well, Wilk et al., 1994 found a decrease in torque range from 87% at 180 deg/sec to 19% at 450 deg/sec concentrially. Although concentric work was not a focus of either of these investigations, the degree to which load range has been shown to decrease, and oppositely the degree to which ROM during RVD increases at velocities used in the present study and beyond, exemplify the potential degree to which kinetic error would increase under these same conditions. The elucidation of such measurement errors during the acceleration phases associated with isokinetic dynamometry is further evidence that traditional measurements of isometric work are incomplete.

• Total isokinetic work is underestimated by standard gravity corrected techniques.
• Standard gravity corrected work measurements overestimate isometric eccentric total work.
• The overestimation of isometric eccentric total work increases with greater angular velocity.