Introduction
The Smith machine is a simple weight-training apparatus. In its most basic form, it consists of a barbell constrained by a set of rails to a straight-line translatory motion perpendicular to the axis of the barbell (^{18} ). A typical generic example of a modern design is illustrated in Figure 1 . Because of its versatility, the Smith machine is one of the most common pieces of exercise equipment found in commercial and noncommercial gymnasiums alike.

Figure 1: The appearance of a typical modern Smith machine. The viewing direction corresponding to the schematic diagram in

Figure 2 is indicated.

Some of the reasons why in certain instances it may be preferred to perform an exercise on a Smith machine rather than using an unconstrained, free barbell include the following: (a) exercises performed on the Smith machine generally require less developed coordination skills compared with the same exercises performed using a free barbell, making them attractive to a wide range of recreational, noncompetitive trainees; (b) by substantially decreasing balancing demands, the Smith machine has the advantage of allowing strength athletes to safely employ supramaximal loads in a partial range of motion when specialized, targeted training is desired (^{1,15} ); (c) motion constrained to a single degree of freedom may offer a more controlled, safer environment for resisted plyometric exercise (^{5} ).

Outstanding Questions and Empirical Challenges
The relationship between Smith machine–based and free barbell exercises is a source of dispute among trainees, coaches, and strength training researchers. A frequent objection to the use of the Smith machine is that by constraining the motion of the barbell to a straight-line movement, the risk of injury is increased. However, scientific evidence in support of this is lacking, mostly comprising case studies after accidents associated with improper lifting technique in biomechanically complex exercises such as the squat (^{9} ). Meanwhile, the Smith machine remains a commonly employed apparatus in strength training research (^{6,13} ). Comparative studies examining its effectiveness and muscle recruitment relative to free weight–based exercises have also produced conflicting, inconclusive results (^{8,19} ).

Retaining the central idea of a motion-constrained barbell, various modifications to the basic Smith machine design have been developed with the aim of overcoming its real or perceived limitations. Such machines are increasingly often found in commercial gymnasiums. In this article, we are specifically interested in comparing the effects of Smith machines with common alterations to the form of effective resistance offered by the constrained barbell. These are briefly summarized next, followed by a statement of aims of this study and a detailed mechanical analysis in the section that follows.

Counterweighted Mechanism
The simplest variation on the original Smith machine design involves a counterweight . It comprises a cord attached to the barbell, which passes over a pulley placed above the maximal allowable barbell elevation and a counterweight suspended at the other end. In the schematic drawing of the machine in Figure 2 , the counterweight is represented by the mass m .

Figure 2: A schematic diagram of the key components of the most general Smith machine resistance mechanism considered in this work. Forces acting on the system when the direction of the velocity of both the adjustable load and the counterweight is positive (and thus, respectively, in the upward and downward directions, because of differently oriented axes measuring the 2 displacements). g = acceleration due to gravity; m = mass of the adjustable load; m _{0} = mass of the counterweight; x _{1} and x _{2} = displacements of the adjustable load and the counterweight; x _{1} = velocity of the adjustable load; T = cord tension (T ≥ 0), c = viscous resistance; F _{f1} and F _{f1} = friction forces.

Viscous Resistance
Many commercial Smith machines available on the market now contain a viscous resistance component. This means that the effective resistance experienced by the trainee exerting a force against the barbell is approximately linearly increased with the barbell velocity (this notion is formalized in the “Methods” section). In Figure 2 , the viscous resistance component corresponds to the force

(relevant mathematical notation is explained in detail in the next section) and is conceptually represented as fluid drag.

Statement of Aims
To date, there have been no attempts to comparatively analyze the training effects of the aforementioned resistance components found in Smith machines. Such an analysis is the goal of this article. Our aim is to systematically investigate the differences in the imposed adaptational stimulus and furthermore the dependence of this stimulus on the context provided by the athlete's training background. This difficult task is made tangible by our adoption of a mathematical model of resistance training performance and induced adaptation and is thus computer simulated rather than being an empirical approach to the problem. The methodology and outcomes of our study are detailed in the section that follows.

Methods
General Design: Counterbalanced and Constrained Load with Viscous Resistance
The most general design of the Smith machine, as far as the aspects of interest in this article are concerned, is shown schematically in Figure 2 . It consists of a weighted barbell represented by the total mass m (which is adjustable) connected by a cord to the counterweight m _{0} (which is not adjustable). If the elevation of the barbell and the counterweight are, respectively, x _{1} and x _{2} (without loss of generality, we shall consider that x _{1} = 0 corresponds to the bottommost position of the barbell in a particular lift and x _{2} = 0 to the topmost position of the counterweight), further viscous resistance proportional in magnitude to that of the velocity of the barbell

(using the standard notion whereby

and

) but opposite in direction, is experienced. In addition, friction forces F _{f1} and F _{f2} are acting on, respectively, the barbell and the counterweight—in Figure 2 their directions are shown for an upward motion of the barbell and a downward motion of the counterweight.

The motion of the system can then be described by the following system of differential equations:

and

Observe the need to explicitly distinguish between 2 cases, x _{1} = x _{2} and x _{1} > x _{2} , depending on whether the upward displacement of the barbell is equal to or greater than the downward displacement of the counterweight (note that it cannot be smaller because it is assumed that the cord is inextensible). Furthermore, noting that in modern Smith machines the 2 friction forces F _{f1} and F _{f2} are generally negligible in magnitude in comparison to the dominant forces acting on the 2 masses, we make a further assumption that F _{f1} = F _{f2} = 0. Then, discretizing time to 0, Δt , 2Δt ,… and using a superscript n to refer to a particular time step n Δt , equations 1 and 2 can be approximated by the following finite differences:

Displacement:

Velocity:

Acceleration:

Equations 3–6 describe the motion of the loaded barbell in a Smith machine with the most general resistance characteristics. The basic design, comprising a constrained load is obtained by setting m _{0} = 0 and c = 0, whereas in the simple counterweighted setup, m _{0} > 0 and c = 0.

The Driving Force—Athlete's Performance Model
The motion of the system governed by equations 3–6 is driven by the force exerted against the barbell by the lifter. At time n Δt , this is represented by F^{n} . We follow the model introduced in (^{2} ) whereby F^{n} is represented as a function of (a) the displacement (elevation relative to the initial position) of the barbell x^{n} _{1} , (b) its velocity

, and (c) the accumulated fatigue. Specifically, Arandjelović shows that F^{n} can be modeled as a product of 2 terms:

The first term,

is what Arandjelović terms the capability profile . It is a compact representation of the athlete's ability to exert effective force (force transferred to the load) in a particular exercise, as a function of the displacement of the barbell and its velocity. Conceptually, it can be thought of as a generalization of the combined force-length (^{12,22} ) and force-velocity (^{11,17,21} ) characteristics of an isolated striated muscle, extended to an arbitrary exercise. Thus, it implicitly captures biomechanical factors transferring the force produced in individual muscles to their contribution to the effective force experienced by the load. The second term in equation 7 is a simple exponential decay modeling fatigue-induced reduction in force output as a function of time under tension (^{4} ). The quantity τ is the fatigue time constant, which is dependent on biomechanical, genetic, and nutritional factors, and the athlete's training history, among others.

Experimental Approach to the Problem
Following the mathematical model described in (^{2} ), we conducted a series of experiments simulating exercise characteristics using different forms of the athlete's capability profile and different design parameters of the Smith machine. In this article, we report a selection of some of the most illustrative findings.

Capability Profiles
We report the results obtained using 2 capability profiles. These were defined by the following separable functions of elevation and velocity:

and

where H (…) is a linear gating function, which leaves a positive argument unchanged and replaces a negative one with a zero:

Its purpose is to model the maximal velocity of the barbell, which is analogous to the maximal contraction rate for an isolated muscle (^{11} ), adjusted for biomechanical factors.

These capability profiles, that is to say hypothetical athletes, are chosen as representative of 2 performance extremes, with the weakest points at the beginning and terminal portions of the lift (for model-based analysis of weakest points, see [^{3} ]; recent empirical studies include [^{16, 10} ]). The first athlete experiences a sticking point at the beginning of the lift, that is, at the bottom of the range of motion. In contrast, the second athlete experiences a sticking point at the end of the lift, that is, at the top of the range of motion. Yet, both athletes have identical one repetition maximum (1RM) loads. Throughout the article the same notation will be adopted to refer to the maximal load which can be used for a set of multiple repetitions. Thus, n RM signifies the greatest load that can be used for n consecutive repetitions. The 2 profiles and the corresponding 1RM efforts are shown in Figure 3 . An increase of 1% in the load used for the shown performance results in failed lifts for both athletes.

Figure 3: The 2 different capability profiles used in the evaluation reported in this work. They are shown as surface plots on the left, where the height of the surface at (

x_{1} ,x_{1} ) is proportional to the value

_{1} (

x_{1} ,x_{1} ) (

i = 1,2). The corresponding variations of the barbell elevation (solid lines) and velocity (dashed lines) throughout a repetition using the same 1 repetition maximum (1RM) load of 125 kg are shown on the right, exhibiting mutually significantly differing characteristics. A) Capability profile 1 and B) capability profile 2.

Loading Parameters
A comparison between 3 linear Smith machine types was made: (a) basic linearly constrained load (no counterweight or viscous resistance), (b) counterweighted (no viscous resistance), and (c) with viscous resistance (no counterweight). For designs b and c, the following values of, respectively, the mass of the counterweight and the viscosity constant were used:

With the aforementioned parameters fixed, 4 loading protocols—in which the mass of the adjustable and constrained weight was varied—were examined using each of the 2 capability profiles. The parameters of each protocol are summarized in Table 1 .

Table 1: Four sets of loading parameters for each of the 3 resistance mechanisms used to simulate exercise and predict the adaptation thus induced.

Measured Outcomes
The effects of a particular Smith machine design were inferred using a computer simulation governed by finite differences approximating the differential equations of motion—see equations 1 and 2 and equations 3–6. Starting from the initial condition x _{1} = 0 and

, corresponding to a stationary barbell at its bottommost position, and discretizing time to intervals of length Δt = 0.01 seconds, at each time step n the value of the capability profile

is employed in equation 6, predicting the elementary change in the state of the system. Iteration is performed until failure is reached. Failure in a set is declared when the estimate of the upward velocity of the barbell becomes negative, that is, when

. Thus, each simulated repetition corresponds to a trajectory of the state vector

through the elevation-velocity plane, each starting at

and all but the last ending at

for some terminal velocity v _{T} (in general different for each repetition).

The differences between the 3 Smith machine designs were assessed qualitatively and quantitatively using the following performance variables inferred from the simulations:

(1) Force-velocity characteristics [x_{1} (t ) x_{1} (t ) ]^{T} as a function of time obtained directly from the simulated exercise:
(2) Force variation over time F (t ) inferred from the values of the capability profile crossed by the elevation-velocity trajectories, modulated by the accumulated fatigue:
(3) Average force exerted by the trainee estimated from F (t ):
where T is the total duration of the corresponding set and

the floor function (sometimes referred to as the greatest integer function ). Formally, if

and

are, respectively, the sets of real and integer numbers, then

(4) Total external work performed estimated from force variation F (t ):
(5) Rate of 1RM increase estimated using the model of strength adaptation described in (^{2} ) using maximal repetition sets at the 85% intensity loading for all 3 machine designs.
Results
Force Variation
The variation in the effective force exerted against the barbell using capability profiles JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{1} and JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{2} is shown, respectively, in Figures 4 and 5 . Although the global force characteristics corresponding to the 3 loading mechanisms exhibited similar functional features, notable differences include (a) a time lag, (b) lower mean velocity (or, equivalently, longer lasting repetitions), and (c) an increase in the mean force as loading intensity is decreased, of the counterweighted and viscosity -containing designs in comparison with the linearly constrained barbell.

Figure 4: The variation in the effective force exerted by the athlete—using the capability profile

JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{1} —against the barbell during a maximal repetition set (set taken to failure) at different loading intensities (

Table 1 ). A) 100% 1RM, B) 85% 1RM, C) 75% 1RM, and D) 55% 1RM. 1RM = 1 repetition maximum.

Figure 5: The variation in the effective force exerted by the athlete—using the capability profile

JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{2} —against the barbell during a maximal repetition set (set taken to failure) at different loading intensities (

Table 1 ). A) 100% 1RM, B) 85% 1RM, C) 75% 1RM, and D) 55% 1RM. 1RM = 1 repetition maximum.

Workload and Average Force
The average force exerted over the duration of an entire set taken to failure and the total external work thus performed across sets of different intensities are shown in Figures 6 and 7 . Bright, narrow bars in the foreground, and the right-hand side vertical axis, correspond to the average force estimates, whereas the dark, wide bars in the background and the left-hand side vertical axis correspond to the external work estimates.

Figure 6: The average force exerted by the athlete (bright narrow bars and the right-hand ordinate) and the total external work (dark wide bars and the left-hand ordinate) in maximal repetition sets at different loading intensities, using the capability profile JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{1} . A) 100% 1RM, B) 85% 1RM, C) 75% 1RM, and D) 55% 1RM. 1RM = 1 repetition maximum.

Figure 7: The average force exerted by the athlete (bright narrow bars and the right-hand ordinate) and the total external work (dark wide bars and the left-hand ordinate) in maximal repetition sets at different loading intensities, using the capability profile JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{2} . A) 100% 1RM, B) 85% 1RM, C) 75% 1RM, and D) 55% 1RM. 1RM = 1 repetition maximum.

Across all intensities and for both capability profiles, the average forces that the athlete could exert against the barbell were closely matched for the 3 Smith machine designs. Deviations from this general uniformity could be observed in favor of the 2 altered designs, with a more significant trend at lower intensities (75 and 55%). The difference in the total workload that could be performed at a particular intensity using different loading mechanisms exhibited greater variability. At medium-level intensities (85 and 75%) for both capability profiles the greatest external work was achievable using a counterweighted mechanism. At the lowest-level intensity of 55%, and again for both capability profiles, a significant transition in favor of part-viscous resistance was observed. Lastly, at high intensities, the observed trend was found to be greatly dependent on the capability profile. This is discussed in detail in the section that follows.

Elevation-Velocity Characteristics
The trajectories corresponding to the repetitions performed at different intensities are shown in Figures 8 and 9 . Each solid line is the trajectory of the state vector

in the elevation-velocity plane traced by a single repetition performed using the basic constrained load Smith machine design. Dashed lines are the trajectories corresponding to the 2 alterations considered, that with a counterweight shown on the left and that with a viscous resistance component on the right. Generally, highest velocities are achieved using the basic, constrained load Smith machine design, followed by the counterweighted setup. This trend was particularly consistent at the lower levels of loading intensity. As with the previous characteristic measured (the total external workload performed), as the loading intensity was increased, the relationship between results achieved using different loading mechanisms was found to be more influenced by the underlying capability profile of the athlete. For the first capability profile, shown in Figure 3A , the elevation-velocity trajectories at 100% intensity exhibited similar characteristics for all 3 loading designs considered. In contrast, for the second capability profile, shown in Figure 3B , a consistently higher velocity was maintained when viscous resistance was applied.

Figure 8: Comparison of elevation-velocity characteristics of a basic, constrained load Smith machine with a counterweighted Smith machine (left), and a basic constrained load Smith machine with a viscous Smith machines, corresponding to the experiments using the capability profile JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{1} See equation 8. A) 100% 1RM, B) 85% 1RM, C) 75% 1RM, and D) 55% 1RM. 1RM = 1 repetition maximum.

Figure 9: Comparison of elevation-velocity characteristics of a basic, constrained load Smith machine with a counterweighted Smith machine (left), and a basic, constrained load Smith machine with a viscous Smith machines, corresponding to the experiments using the capability profile JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{2} —see equation 9. A) 100% 1RM, B) 85% 1RM, C) 75% 1RM, and D) 55% 1RM. 1RM = 1 repetition maximum.

Rate of Increase in Maximal Strength
The rate of improvement in the 1RM effort using the basic constrained load resistance, after a single bout of exercise at 85% intensity, is summarized in Table 2 . Each row shows the rate of 1RM increase relative to that induced using the basic, unaltered machine setup. The result is revealing in that vastly different results are obtained for different initial capability profiles. For the first profile, both types of altered resistance are shown to be preferable in terms of maximal strength improvement, with a substantial advantage of 30.8% offered by the counterweighted setup and a minor advantage of 5.3% with that containing viscous resistance. The opposite is seen for the second capability profile, the basic constrained load achieving the greatest rate of improvement of the 3 and the counterweighted resistance the least.

Table 2: The rate of improvement in the 1 repetition maximum effort using constrained load resistance.*

Discussion
The observations summarized in the previous section show that the experiments conducted produced a wide spectrum of qualitatively different results, depending on the loading scenario in question. A particular resistance design found to be the superior in some instances was inferior in others; yet, in some cases, no difference between the 3 designs was observed. This highlights the power of the adopted computational model-based methodology—the range of experimental parameters that could be automatically evaluated in a matter of seconds would have been practically prohibitively challenging to consider using a purely empirical approach.

The variation of the effective force exerted by the lifter exhibits similar features across the 3 machine designs for both capability profiles. This is not surprising given that the resistance is in all 3 cases dominated by the component corresponding to the adjustable load. The key difference—also universally observed—is in the phase of force variation, both modified designs exhibiting a time delay in comparison to the basic setup comprising only a constrained load. The reason for this lag when a counterweight is used stems from the decreased tension T acting on the barbell—and thus greater resistance experienced by the lifter—as the acceleration of the barbell is increased. Intuitively, the counterweight can be considered as providing a “reactive resistance” component, decreasing the effective load at the loci of strength deficit. A similar effect can be achieved with the inclusion of a viscous resistance component, however, with a more pronounced delay in the change of effective force. This is a consequence of the nature of the origin of the 2 reactive components—unlike in the counterweighted design, in the case of viscous resistance it is not dependent directly on the change in the acceleration of the barbell

(or, equivalently, the net force acting on it) but rather its integral, that is, the velocity of the barbell

. When the athlete's force development characteristics (represented by the corresponding capability profile) are such so as to result in rapid, high-frequency force changes—as seen at the beginning of a maximal intensity repetition in Figure 5A —the result of the first-order delay introduced by viscous resistance is a lift with resistance and velocity characteristics very much unlike that experienced when unaltered constrained load is used, see Figure 9A —right. Thus, limited strength carryover between the 2 can be expected. Indeed, this is directly corroborated by the results summarized in Table 2 , which show that for the second capability profile, the addition of viscous resistance worsens the rate of improvement in the maximal performance with an unaltered constrained load.

In contrast, consider the case when maximal-effort lifts exhibit lower frequency, more gradual variations in the effective force, such as those effected by the capability profile JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{1} and shown in Figure 4A . Qualitatively, the same changes in the resistance and velocity characteristics between different machine designs can be observed. However, they are generally lower in magnitude. The reactive resistance component provided by the counterweighted and viscous mechanisms then presents a similar lifting environment to a simple constrained load, with a slightly greater challenge presented at those stages of the lift when the athlete is able to exert a significantly greater force than that because of the constrained load. Direct results in Table 2 quantitatively support this, showing that both types of altered resistance should be preferred in this case, the counterweighted setup increasing the rate of 1RM increase by 30.8% in comparison to the basic constrained load design.

Up to now, we concentrated the discussion on the results at high-intensity lifting conditions (100 and 85%), which are of most interest to strength athletes, such as power lifters. In contrast, trainees whose focus is on maximizing muscle hypertrophy tend to orient their training around improving exercise performance at medium-level intensities (75–85%, which in this population typically corresponds to 12RM–4RM effort [^{7,14} ]) and a compromise between high-force training and the duration of its application (^{20} ). Unlike in the previously considered performance at high-intensity loads, it is interesting to note that the resulting relative trends between different resistance mechanisms observed at medium intensities are less dependent on the athlete. Indeed, for both capability profiles JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{1} and JOURNAL/jscr/04.03/00124278-201202000-00006/OV0405/v/2021-02-09T093832Z/r/image-png _{2} , the counterweighted Smith machine resulted in the greatest amount of performed external work, and the highest average force, see Figures 6B, C and 7B, C.

Lastly, in low-intensity lifting conditions (55%) for both capability profiles, the loading design containing viscous resistance was consistently found to result in both the highest average effective force and the greatest amount of total external work performed in a maximal repetition set. This can be attributed to the reactive force offered by viscous resistance, resulting in lower velocity lifting conditions, which lend themselves to higher force production, see Figures 8D and 9D . The measured performance characteristics achieved when the counterweighted design was employed were also found to be superior to the basic constrained load setup. The likely reason why this design was outperformed by the viscous resistance containing mechanism lies in the limit of the reactive increase in resistance that this setup can offer. Specifically, from the initial maximal cord tension T = m _{0} g , the greatest increase in the resistance experienced by the lifter that can be achieved is when T = 0. In contrast, the reactive viscous resistance can in principle rise arbitrarily with velocity of the barbell.

Practical Applications
The study presented in this article focused on the application of a computational model of weight-training performance and effected adaptation on an analysis of context-dependent utility of different resistance mechanism designs found in modern linear Smith machines. All 3 designs considered—a basic constrained barbell, a counterweighted barbell and a barbell with a viscous resistance component—were found to offer benefit in different circumstances. In summary, the key findings are as follows:

Of primary interest to weight-training practitioners (athletes and coaches): (a) At high intensities (85–100%), particular attention should be paid to the maximal lift characteristics—specifically to the variations in the athlete's force output and the velocity of the load during maximal-effort sets—so that the optimal resistance mechanism design can be prescribed. (b) At medium-level intensities (75–85%)—which may be preferred by athletes interested in hypertrophy-specific training—less athlete-specific attention appears to be required, with the counterweighted Smith machine design offering the best choice in terms of high-force development and total external work that can be performed at a particular loading intensity. (c) At low intensities (55–75%)—of interest to strength-endurance athletes—the optimal Smith machine choice of the 3 linear designs considered in this article is the design containing a viscous resistance component.

For equipment manufacturers: Our results indicate that each resistance component in a linear Smith machine—constrained load, counterweight and viscous—offers advantages for specific athletes and training aims, motivating the development of linear Smith machines, which allow independent adjustment of the corresponding parameters.

For the weight-training research community: The nature of the problems addressed in this study and the value of the insight offered by the adopted computational model provide further evidence to support continued efforts in the development of practical models of performance and adaptation, founded on the corpus of empirical data collected over the years.

Statement of Conflicts
The authors have no affiliation with or financial involvement in any organization with a direct financial interest in the subject matter discussed in the article.

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