A Low-Cost Apparatus for Measuring Muscle Force–Velocity Relationships

Summary

An accessible, low-cost apparatus for measuring human muscle force–velocity–power relationships was developed using inexpensive materials, a 3D-printed gear, a smartphone accelerometer, and Python freeware. Validation against a gold-standard ergometer demonstrated physiologically relevant measurements while providing equitable, hands-on learning opportunities in an undergraduate classroom setting.

Abstract

The relationship between muscle force, contractile velocity, and power represents a nearly century-old paradigm in physiology and a cornerstone of our understanding of musculoskeletal function. However, this paradigm is rarely studied directly in physiology and biomechanics classrooms because of the prohibitive cost of the required equipment. This study sought to design and validate a low-cost apparatus using a 3D-printed pulley system incorporating a ratchet-and-pawl gear that can be easily constructed and configured to collect force–velocity–power data from human finger muscles via a smartphone accelerometer using the free Phyphox application. A custom Python script converts raw accelerometer measurements into velocity and power outputs, which are then compared on a subject-by-subject basis with measurements obtained using a gold-standard dual-mode lever system. Pairwise comparisons of five summary variables (peak force, Fmax; peak speed, Vmax; peak power, Pmax; optimal speed, Vopt; and force–velocity curvature, a/Fmax) revealed no statistically significant differences between force-velocity-power data collected using the two apparatuses (paired t-tests, p > 0.05). Therefore, without a significant reduction in data quality, the inexpensive apparatus offers an accessible, safe, and cost-effective means of teaching and studying biomechanical and physiological principles. This apparatus has the potential to democratize muscle physiology and biomechanics education, particularly in underfunded and resource-limited educational settings worldwide.

Introduction

Striated muscle is the motor that generates skeletal movement in humans and other animals. Understanding its unique hierarchical composition is complicated by cross-scale factors influencing the generation of force and production of work^1,2. Perhaps because of these complexities, muscle function can be an underprioritized component in anatomy and physiology curricula, because educators lack the tools needed to properly showcase the fundamental and most fascinating aspects of muscle function^3,4,5,6.

Muscle function has historically been described through several enduring paradigms, none more widely recognized than the force–velocity trade-off first characterized nearly a century ago by Archibald Vivian Hill^1,2,3,4,5,6. This fundamental physiological relationship describes the trade-off between muscle contraction velocity and force, whereby isometric or very slow shortening contractions are generally the strongest, producing peak muscle force (F_max), whereas the fastest isotonic contractions approaching peak muscle velocity (V_max) are typically the weakest¹. When Hill described the force–velocity–power (FVP) relationship in 1938, he introduced a function that still bears his name. The Hill equation (Equation 1) relates the peak force generated during an isometric contraction (with fixed muscle ends) to the peak velocity achieved in the absence of an external load.

Equation 1: (v + b) (F + a) = b (F_max + a)

where v is shortening velocity, F is muscle force, F_max is maximum isometric force, and a and b are constants that describe the force–velocity relationship and the energetic properties of muscle shortening¹.

Embedded within this fundamental relationship are several key concepts of muscle function represented by five quantitative parameters that define the FVP relationship. The product of instantaneous muscle force and velocity is power, which typically reaches a maximum value (P_max) at approximately one-third of V_max⁷. The velocity at which  P_max occurs is referred to as the optimal velocity (V_opt), and its location within the range of shortening velocities (from V₀ to V_max) is an important characteristic of the force–velocity and power–velocity relationships. For example, a greater V_opt results in peak power being delivered at a higher shortening velocity. Within the force–velocity relationship itself, peak force (F_max) and peak velocity (V_max) are connected by intermediate force–velocity relationships in a curvilinear manner (Figure 5A). Representative force–velocity and power–velocity relationships obtained using the present methodology are shown in Figure 1. The slope of this curve is referred to as the force–velocity curvature, which describes the rate of force decline with increasing velocity. Curvature is defined as a/F_max⁸, where a is the thermodynamic constants in the Hill equation that quantifies frictional heat generation by an unloaded shortening muscle¹. Curvature influences the position of V_opt on the force–velocity curve and determines how much force a muscle can generate at a given shortening velocity. Curvature has also been described using the muscular power ratio (Equation 2)⁹. 

Equation 2:  

where P_max is the maximum muscle power output, and V_max and F_max are the maximum shortening velocity and maximum isometric force, respectively.

The intuitive nature of the force-velocity and power-velocity relationships provides effective entry points when teaching muscle function in the life sciences classroom. The importance of providing students with hands-on opportunities to build a more complete understanding of the FVP relationships is underscored by the fact that hands-on learning has been shown to improve knowledge-retention¹⁰. Moreover, experimental learning is known to strengthen the foundation for assessing muscle function, both in a fundamental scientific context and as preparation for occupations in healthcare, including injury prevention, rehabilitation, and physical training.

Traditional instruments for testing muscle physiological properties, such as dual-mode servomotor lever systems, are highly precise and accurate but incur prohibitive costs and therefore pose substantial barriers to implementation in most classroom environments⁷. To address these limitations and expand opportunities for hands-on learning, a low-cost alternative was developed. The apparatus is built around a 3D-printed pulley incorporating a ratchet-and-pawl gear and uses off-the-shelf slotted aluminum railings, stiff wire, and a caddy-mounted smartphone running a free data acquisition application (Figure 2). A smartphone was selected instead of dedicated acceleration-measurement devices because smartphones are widely available and are typically equipped with accurate and robust accelerometers¹¹. The utility of smartphones as scientific measurement devices has been demonstrated previously, including the incorporation of a commercially available smartphone accelerometer into the inertial measurement unit of the Ingenuity Mars Helicopter¹².

While there are a few existing low-cost tools for teaching muscle physiology, they target different concepts: Judge et al.⁶ developed affordable SpikerBox-based activities for recording muscle electrical activity, and Medeiros et al.¹⁰ created a low-cost 2-D sarcomere model to demonstrate titin-related mechanics. However, no existing low-cost tool allows students to directly measure the FVP relationships of a contracting human muscle and generate quantitative, research-comparable data. The present apparatus is intended to fill this gap by delivering gold-standard-validated FVP measurements for under $300.

The present study reports on the construction of a low-cost FVP apparatus with a cross-validation against the industry gold standard. The validation analyses rely on pairwise comparisons of the five recognized FVP summary variables, measured in the same subjects on both apparatuses. The analyses test the hypothesis that both apparatuses generate equivalent force-velocity and power-velocity curves. With successful cross-validation, the low-cost apparatus can democratize classroom studies of muscle physiology and musculoskeletal biomechanics by lowering the financial barrier to apparatus acquisition, whilst maintaining experimental accuracy and thus scientific rigor. It is anticipated that the low-cost apparatus can inspire and enable students and educators in physiology and biomechanics classrooms worldwide who already study and teach muscle anatomy to practically explore one of the most foundational relationships of muscle. By including a hands-on activity for generating physiological measurements, classrooms can achieve multimodal understanding and improved knowledge-retention.

Protocol

All experiments forming the basis of this publication were approved by the Institutional Review Board at the University of Massachusetts Lowell (Protocol No. 21-163-KON-XPD).

NOTE: The apparatus uses a smartphone suspended with external loads from a 3D-printed pulley system. Acceleration data are collected at 100 Hz using a smartphone-based data acquisition application. Force is calculated by multiplying the combined mass of the smartphone and external load by acceleration. Velocity is obtained by integrating acceleration, and power is calculated as the product of force and velocity (Supplemental Coding File 1).

1. Apparatus assembly

1. Attach two short aluminum beams to the underside of one long aluminum beam using aluminum frame fasteners. Position the short beams approximately 8 cm from each end of the long beam to form a stable, noncompliant base.
   NOTE: This assembly is referred to hereafter as the frame.
2. Build the pulley mount
   1. Position a large cellular polyvinyl chloride block with the 7 in side oriented vertically and the 6.75 in side oriented horizontally.
   2. Measure 2.75 in from the bottom-left corner along the left edge and mark the location.
   3. Measure 2.75 in from the top-left corner along the top edge and mark the location.
   4. Draw a straight line connecting the two marked locations.
   5. Cut along the marked line and remove the resulting triangular section.
   6. Drill a 7/16 in hole 1.25 in from the top-right corner.
   7. Repeat steps 1.2.1–1.2.6 for the second large block.
   8. Position the small cellular polyvinyl chloride block behind the frame.
   9. Secure both large blocks to the small block such that they are 8 in apart. Orient the diagonal cuts above the small block and position the drilled holes beyond the edge of the small block.
3. Clamp the pulley mount securely to the corner of a table or counter.
   NOTE: Position the mount so that the pulley extends beyond the table edge. This prevents the suspended weights and smartphone from contacting the table legs during contractions.
4. Build and attach the armrest
   1. Cut the 14 in cellular polyvinyl chloride armrest block into three pieces consisting of two 1.5 in sections and one 3.5 in section.
   2. Cut a 2 in x 1/8 in notch into the top surface of each smaller section, positioning the notch 1 in from the front edge.
   3. Secure the smaller sections to the underside of the larger section at each end. Align the notches with one another.
   4. Cover the upper surface of the larger section with hook-and-loop fastener tape.
   5. Thread the forearm strap through the two notches.
   6. Attach the armrest to the frame using bolts and drop-in T-nuts. Position the armrest 14 in from the pulley mount with the forearm strap facing away from the pulley.
5. Insert ball bearings into the grooves of the inner gear and outer gear components (Supplemental File 1).
6. Thread 40 cm of timing belt through the lateral slot of the inner gear component. Create a loop with a 2 cm overlap and secure the overlap within the slot.
7. Insert the metal axle through the left pulley mount, a stopper, the inner gear component, the force-carry component, the outer gear component, a second stopper, and the right pulley mount.
   NOTE: Orient the lateral slot of the inner gear toward the frame. Position both stoppers 1 mm from either side of the pulley assembly.
8. Install two additional stoppers on the axle, positioning each stopper 1 mm from the pulley mounts to prevent lateral movement.
9. Loop 40 cm of multistrand steel wire through the finger brace and secure it with a crimp. Loop the opposite end through Carabiner A and secure it with a second crimp.
   NOTE: Carabiners A and B are identical and are labeled only for identification during assembly.
10. Connect the inner gear component to Carabiner A using the attached timing belt. Secure the overlap with Key A.
    NOTE: Keys A, B, and C are identical. Additional keys may be printed because they are designed as sacrificial components that fail before other parts of the apparatus are damaged.
11. Loop 10 cm of multistrand steel wire around Carabiner A and secure it with a crimp. Loop the opposite end around an aluminum frame fastener, crimp the wire, and secure the fastener to the frame.
    NOTE: This assembly is referred to hereafter as the safety bolt.
12. Create a loop using 40 cm of timing belt with 2 cm overlap and insert the loop into Key B. Loop the opposite end around Carabiner B and secure the overlap using Key C.
13. Insert Key B into the largest tier of the outer gear component.
14. Attach the smartphone armband holder to Carabiner B using either the armband loop or the headphone-cable openings.
15. Suspend the desired starting weight from Carabiner B using a looped and crimped steel wire.

2. Data acquisition setup

1. Install the data acquisition application on a smartphone equipped with an accelerometer.
2. Open the Acceleration (without g) module.
3. Place the smartphone in the armband holder attached to Carabiner B.
4. Collect data according to section 3.
5. Export the recorded data as an .xls file. Assign a unique filename corresponding to the participant.
6. Transfer the .xls file to a computer with internet access.
7. Process the data using Supplemental Coding File 1
   NOTE: The script computes velocity using forward Euler integration of the measured vertical acceleration (Equation 3). Position is computed using the constant-acceleration kinematic equation (Equation 4). The calculations use raw acceleration values without smoothing or filtering.
   Equation 3: vi = v_i-1 + a_i ∆t
   Equation 4:  
   1. Install the Supplemental Coding File 1.
   2. Install Python on the computer used for data processing.
   3. Open a terminal window and execute: pip install streamlit pandas matplotlib scipy openpyxl xlrd.
   4. Navigate to the folder containing Supplemental Coding File 1 and execute: streamlit run muscle_velocity_gui.py.
      NOTE: When using Windows PowerShell, execute: python -m streamlit run muscle_velocity_gui.py. If prompted for an email address, leave the field blank and press Enter.
   5. Click on Upload and select the exported .xls file.
   6. Click on Process Data to load and analyze the dataset.
   7. Review the acceleration, velocity, and position plots generated for the first trial.
   8. Select the detected peaks corresponding to valid muscle contractions using the checkboxes adjacent to each peak.
      NOTE: Valid peaks typically occur before the acceleration trace becomes negative and are associated with positive velocity and position values. The automated peak-detection algorithm may incorrectly identify peaks and therefore requires manual verification.
   9. Adjust peak detection parameters
      NOTE: If the software does not correctly identify peaks, expand Advanced Tweakables > Peak Detection Parameters and adjust the detection settings.
      1. Adjust Zero Threshold to define the acceleration range considered stationary.
         NOTE: Increase this value when acceleration traces contain excessive noise. The threshold is displayed as red reference lines on the acceleration plot. Default value = 0.30. Suggested range = 0.10–5.00.
      2. Adjust Min Velocity for Peak to define the minimum accepted velocity peak.
         NOTE: Decrease this value if valid peaks are excluded. The threshold is displayed as the lower green reference line on the velocity plot. Default value = 0.30. Suggested range = 0.10–10.00.
      3. Adjust Zero Scope to define the minimum number of consecutive near-zero acceleration samples required to identify a stationary period.
         NOTE: Decrease this value if the velocity and position traces fail to reset. Default value = 10. Suggested range = 3–25.
      4. Adjust Max Velocity for Peak to define the maximum accepted velocity peak.
         NOTE: Increase this value if valid peaks exceed the threshold. The threshold is displayed as the upper green reference line on the velocity plot. Default value = 10.00. Suggested range = 10.00–30.00.
      5. Adjust Min Peak Distance to define the minimum number of samples between detected peaks.
         NOTE: Default value = 2. Suggested range = 1–10. Modification is generally unnecessary.
      6. Adjust Min Peak Prominence to define the prominence required for peak detection.
         NOTE: Lower values increase sensitivity and may identify additional peaks. Default value = 0.10. Suggested range = 0.01–0.90. Modification is generally unnecessary.
   10. After identifying all valid peaks for the current trial, click on Next to advance to the next trial.
   11. Repeat steps 2.7.7–2.7.10 until all trials have been processed.
       NOTE: Do not modify the peak-detection parameters after processing the first trial. If parameter changes become necessary, repeat the analysis for all previously processed trials using the updated settings.
   12. Copy the selected velocity and acceleration data into a spreadsheet for subsequent force–velocity–power analyses.

3. Experimental protocol

1. Secure the participant's forearm to the armrest using the forearm strap. Tighten the strap sufficiently to prevent the forearm from moving forward.
2. Place the finger brace on the participant's thumb and tighten the brace securely.
3. Remove slack from the pulley system by rotating the pulley through the ratchet mechanism until the timing belt is taut.
   NOTE: The ratchet mechanism produces an audible click as it advances between positions.
4. Slide the safety bolt toward the pulley until the wire connecting the safety bolt and Carabiner A is taut.
5. Select two weights of approximately equal mass, each weighing approximately 1.25 kg.
6. Fixed testing sequence
   1. Test the largest tier, the middle tier, and the smallest tier using one weight and the smartphone.
   2. Test the smallest tier, middle tier, and largest tier using two weights and the smartphone.
   3. Test the largest tier and smallest tier using only the smartphone.
      NOTE: Use the same sequence for all participants to minimize fatigue-related bias.
7. Tap Start Recording in the data acquisition application.
8. Instruct the participant to contract the thumb as rapidly and forcefully as possible.
   NOTE: Participant motivation can substantially affect performance. Provide consistent verbal encouragement throughout testing to promote maximal effort.
9. Pause data recording after each trial. Adjust the pulley tier and loading condition according to the sequence established in step 3.6.
10. Resume data recording and repeat steps 3.8–3.9 until all eight pulley-tier and loading combinations have been completed.
11. Monitor the participant throughout testing and repeat trials affected by excessive arm movement or incomplete effort.
12. Review the acceleration trace after each trial.
    NOTE: Repeat trials that exhibit abnormally low initial acceleration peaks, excessive oscillations, or continued smartphone movement between contractions, as these conditions may compromise data quality.
13. Record notes for each trial, including repeated or excluded trials, to facilitate subsequent data analysis.

4. Gold-standard ergometer usage

1. Set up the gold-standard ergometer
   1. Power on the computer and launch the data acquisition software. Open the dual-mode lever system procedure file.
   2. Set the force offset to 10 and the length offset to 9 on the lever control unit.
   3. Power on the dual-mode lever system and the data acquisition unit.
   4. Verify proper motor operation by gently rotating the lever and confirming the presence of resistance.
   5. Loop 40 cm of multistrand steel wire through the second hole from the bottom of the lever and secure the wire using a crimp.
   6. Loop the opposite end of the wire through the finger brace and secure the wire using a crimp.
   7. Attach the dual-mode lever system to the frame 3 in in front of the armrest.
2. Secure the participant's forearm to the armrest and position the thumb within the finger brace.
3. Slide the participant's arm forward until the thumb reaches maximal flexion and the cable is under light tension.
4. Tighten the forearm strap sufficiently to prevent movement of the arm in the forward or lateral directions.
5. Configure data acquisition settings
   1. Open Scan Control Device Dev6.
   2. Set Number of Samples to 60,000.
   3. Set Averaging Samples to 1.
   4. Set Sample Period to 0.0001.
   5. Set Convert Period to 0.
   6. Set Post Trigger Samples to 0.
   7. Set Reference Trigger Source to /Dev6/PFI1.
   8. Set 2x Type to Digital.
   9. Set 2x Lever1 to 0.
   10. Set 2x Lever2 to 0.
   11. Set Mode to One Shot.
   12. Set Return Immediately to Off.
   13. Set Start Trigger to Off.
   14. Set Trigger Source to /Dev6/PFI0.
   15. Set Sample Clock to Off.
   16. Set Clock Source to /Dev6/PFI7.
   17. Set Convert Clock to Off.
   18. Set Convert Clock Source to /Dev6/PFI2.
   19. Select Channel 1 + Channel 2 under Select Channels to Scan.
6. Click on Start in Scan Control Device and instruct the participant to contract the thumb as rapidly and forcefully as possible.
   NOTE: Participant motivation can substantially affect performance. Provide consistent verbal encouragement throughout testing to promote maximal effort.
7. Click on GetWaves and save the resulting data using the participant identifier and the current force offset value.
   NOTE: For the first contraction, the force offset is 10.
8. Reduce the force offset by increments of 1.0.
   NOTE: Example progression: 10.0 > 9.0 > 8.0.
9. Repeat steps 4.6–4.8 until the motor provides no noticeable resistance to thumb movement.
10. Extract peak force and peak velocity values from each trial using the analysis procedure described in Supplemental Coding File 2.

Results

Force–velocity and power–velocity data were collected from a sample of undergraduate students (N = 15) using both the gold-standard ergometer and the low-cost apparatus to evaluate the functional outputs of each system under an identical experimental protocol. Individual subject data are shown in Figure 1, and summary data for each apparatus are shown in Figure 3.

The low-cost apparatus was evaluated as a potential classroom substitute for the gold-standard ergometer using pairwise comparisons of summary variables (Table 1), two one-sided tests (TOST) of equivalence (Table 2), and statistical parametric mapping (SPM) analyses of the force–velocity and power–velocity relationships generated by both apparatuses (Figure 4). Paired t-tests were used to assess statistical differences between apparatuses, whereas TOST analyses were used to determine whether measurements from the two systems were statistically equivalent within a predefined equivalence margin. It was hypothesized that the five force–velocity–power summary variables would not differ significantly between apparatuses (p > 0.05), thereby supporting the use of the low-cost apparatus as a teaching and data-collection tool. All five summary variables are illustrated as box-and-whisker plots in Figure 5. 

Peak force (F_max) of the human flexor pollicis brevis averaged 102.99 ± 29.08 N for the gold-standard ergometer and 104.05 ± 36.25 N for the low-cost apparatus, with no significant difference detected between apparatuses (p = 0.93). Peak velocity (V_max) averaged 1,152.50 ± 451.04 mm/s for the gold-standard ergometer and 980.58 ± 524.84 mm/s for the low-cost apparatus, with no significant difference detected (p = 0.36). Peak power (P_max), calculated as force × velocity, averaged 10.81 ± 6.71 W for the gold-standard ergometer and 12.67 ± 9.45 W for the low-cost apparatus, with no significant difference detected (p = 0.55). Optimal velocity (V_opt), measured at P_max, averaged 357.06 ± 102.67 mm/s for the gold-standard ergometer and 327.39 ± 151.56 mm/s for the low-cost apparatus, with no significant difference detected (p = 0.55). When expressed relative to Vmax, peak power occurred at 0.33 ± 0.07 V/V_max. Finally, force–velocity curvature, calculated as a/F_max from the Hill equation¹, averaged 0.89 ± 1.25 for the gold-standard ergometer and 0.69 ± 0.70 for the low-cost apparatus, with no significant difference detected between apparatuses (p = 0.29).

Comparisons of the five force–velocity–power summary variables revealed no statistically significant differences between apparatuses despite appreciable within-subject and between-subject variation in curve shape (Figure 1). To determine whether the two systems were statistically equivalent rather than merely not significantly different, TOST equivalence analyses were performed using a ±20% equivalence margin. Among the five variables examined, only F_max met both TOST criteria, indicating statistical equivalence between the low-cost and gold-standard apparatuses for measuring peak muscle force. Although the remaining variables did not meet the predefined equivalence criteria, the absence of statistically significant differences suggests that the low-cost apparatus reproduces the general characteristics of the force–velocity–power relationship.

Statistical parametric mapping analyses revealed no regions of significant difference between apparatuses across either the force–velocity or power–velocity relationships (Figure 4). These findings indicate that the overall shapes of the force–velocity and power–velocity curves generated by the two systems were statistically indistinguishable.

Figure 1: Force-velocity-power plots separated by study participant. Low-cost (LC) apparatus data are blue. Gold-standard (GS) ergometer data are yellow. Despite appreciable variation in force-velocity curve shapes across subjects, most subject data fell within the range spanned by both apparatuses. Grand mean fit (thick) lines for the LC apparatus generally approximate those for the GS ergometer, with deviations remaining within one standard deviation of each other, and demonstrating reasonable consistency in force-velocity plots across apparatuses. Some individuals generated poorer data than others, most clearly revealed by within-subject differences in F_max, V_max, and FV curvature (a/F_max). Please click here to view a larger version of this figure.

Figure 2: Low-cost apparatus and custom 3D-printed components used for force–velocity–power measurements. (A) Top-view of a subject positioned in the low-cost apparatus with the finger brace, safety bolt, Carabiner A, pulley, smartphone holder, and suspended weight labeled. (B) Rear-view of the apparatus showing the pulley, Carabiner B, smartphone holder, and suspended weight. (C) Side-view rendering of the complete apparatus mounted to the frame. (D) Close-up of the tension belt and key component. (E) Exploded rendering of the custom 3D-printed pulley assembly showing the inner gear (3DPart1), outer gear (3DPart2), force-carry component (3DPart4), and key component (3DPart3). STL files for all custom 3D-printed components are provided in Supplemental File 1. Please click here to view a larger version of this figure.

Figure 3: Summary scatterplots for low-cost (LC) apparatus data in blue and gold-standard (GS) ergometer data in yellow. Data from all subjects were normalized to peak force, velocity, and power. (A) Force-velocity plot across subjects. (B) Power-velocity plot across subjects. Please click here to view a larger version of this figure.

Figure 4: Statistical Parametric Mapping (SPM) comparison of the full normalized force-velocity and power-velocity curves between the apparatuses. The low-cost apparatus data are shown in blue. The gold-standard ergometer data are shown in yellow. (A) Mean (± SD) normalized force-velocity curves across the common velocity domain. (B) SPM{t} statistic for the force-velocity comparison, where dashed lines indicate the critical threshold (t*). (C) Mean (± SD) normalized power-velocity curves across the common velocity domain. (D) SPM{t} statistic for the power-velocity comparison, where dashed lines indicate the critical threshold (t*). In both (B) and (D), the SPM{t} traces remain within the critical threshold across the entire domain, indicating the low-cost and gold-standard curves are statistically similar across their whole shape. Please click here to view a larger version of this figure.

Figure 5: Summary boxplots for low-cost (LC) apparatus data in blue and gold-standard (GS) ergometer data in yellow. (A) Depiction of idealized Force-Velocity-Power relationships showing measurement locations for five key performance variables depicted in boxplots B-F. Summary box plots for five key performance variables across subjects (N = 15, dot-density circles connected by fine lines). (B) Peak isometric force (F_max), (C) Peak velocity (V_max), (D) force-velocity Curvature (a/F_max), (E) Velocity at peak power (V_opt), (F) Peak power (P_max). Consistent with the hypothesis, there were no statistically significant differences between within-subject pair-wise comparisons for all five of the performance variables generated by the low-cost apparatus as compared to the gold-standard ergometer (Table 1). Please click here to view a larger version of this figure.

Variable	Low-Cost apparatus	Gold-Standard	p-value
Vmax​ (Predicted) (mm/s)	980.58 ± 524.84	1152.50 ± 451.04	0.36
Fmax​ (Predicted) (N)	104.05 ± 36.25	102.99 ± 29.08	0.93
Curvature (a/Fmax​)	0.69 ± 0.70	0.89 ± 1.25	0.29
Peak Power (Pmax​) (W)	12.67 ± 9.45	10.81 ± 6.71	0.55
Velocity at Pmax​ (mm/s)	327.39 ± 151.56	357.06 ± 102.67	0.55

Table 1: Paired t-tests for the five key performance variables. The means and S.D. for the low-cost and gold-standard apparatuses, along with the p values, are reported. The paired t-tests revealed no significant differences between measurements collected on the low-cost apparatus, as compared to the gold-standard ergometer.

Variable	Mean Difference	Bound	Low	High	p	Equivalence
Fmax	1.06	20.60	-11.55	13.67	0.01	TRUE
Vmax	-28.49	283.50	-422.72	365.75	0.14	FALSE
Pmax	3.19	2.47	0.25	6.13	0.66	FALSE
Vopt	24.79	79.91	-64.33	113.91	0.15	FALSE
a/Fmax	-0.20	0.18	-0.82	0.42	0.52	FALSE

Table 2: Two one-sided tests (TOST) of equivalence between the low-cost apparatus and the gold-standard ergometer for each of the five force-velocity-power variables. The mean paired difference, equivalence bound (±20% of the gold-standard mean), 90% confidence interval of the difference, TOST p-value, and equivalence conclusion are reported. A variable is considered statistically equivalent when its 90% confidence interval falls within the equivalence bounds (TOST p < 0.05). Only peak force (F_max) was shown to be equivalent between both apparatuses.

Supplemental Coding File 1: Smartphone accelerometer data-processing script. Python script used to import exported smartphone accelerometer files, compute velocity and position from acceleration data, identify velocity peaks, and export selected data points for force–velocity–power analysis. Please click here to download this file.

Supplemental Coding File 2: Hill curve-fitting script. Script used to fit force–velocity data to the Hill equation and extract Vmax and force–velocity curvature values.Please click here to download this file.

Supplemental File 1: STL files for custom 3D-printed apparatus components. ZIP archive containing the stereolithography (STL) files required to fabricate the custom pulley assembly, including 3DPart1 (Inner Gear), 3DPart2 (Outer Gear), 3DPart3 (Key), and 3DPart4 (Force Carry). These components form the ratchet-and-pawl pulley mechanism used in the low-cost apparatus.Please click here to download this file.

Discussion

A low-cost apparatus was validated as an alternative to gold-standard ergometers for measuring force–velocity and power–velocity relationships of human muscles in the classroom.

The validation results support the assertion that a low-cost, and therefore more economically and technologically accessible, apparatus can generate measurements that are statistically indistinguishable from those obtained using a gold-standard ergometer. The ability of the low-cost apparatus to generate high-quality force–velocity and power–velocity relationships supports its utility as a classroom teaching tool. However, the validation results, with only one of the five performance variables satisfying both TOST criteria, suggest that the low-cost apparatus may not be suitable as a research-grade replacement for a gold-standard ergometer.

The primary advantage of the low-cost apparatus is its affordability, with a total cost of less than $300 compared with approximately $30,000 for a gold-standard ergometer. Two additional requirements are access to a 3D printer and a smartphone. However, these technologies are increasingly common and may not represent substantial barriers to implementation in many educational settings.

Beyond cost savings, affordability creates unique pedagogical opportunities. Students can directly investigate how changes in load affect thumb performance, thereby observing the dynamics of Hill’s force–velocity trade-off in real time and connecting abstract concepts such as the Hill equation and power–velocity relationships to physical experience. Such connections between theory and practice improve knowledge retention and enhance conceptual understanding¹⁰.

The cost of a gold-standard ergometer makes this technology inaccessible to many educational environments. Even when experiments are available, students in large classes may observe them rather than actively participate. In contrast, the affordability of the low-cost apparatus allows multiple systems to be deployed simultaneously, enabling more students to participate directly in data collection. This scalability is particularly advantageous in classrooms with limited instructional time and resources. As a result, a greater proportion of students can benefit from the improved understanding and retention associated with hands-on learning.

The flexor pollicis brevis was selected as the target muscle because the thumb is small, yet comparatively strong and durable, making it convenient and safe for repeated testing. The thumb also provides a useful pedagogical framework, as force production can be contextualized through familiar competitive activities such as thumb war. For instance, students can construct a seed-bracket from FVP summary variable results generated in the classroom as means for predicting the outcome of a thumb-war competition to conclude the lesson. The low-cost apparatus may also be capable of measuring greater forces than the gold-standard ergometer. Whereas the gold-standard ergometer is typically limited to approximately 100 N with a force resolution of approximately 0.3 N, the low-cost apparatus has measured forces approaching 300 N, although it has not yet been fully evaluated under high-force conditions. This capability suggests that the apparatus may be adaptable for investigating the force–velocity relationships of other muscles, including those of the foot, arm, and leg. A potential advantage when testing larger muscles, which typically shorten over greater distances during contraction, is that the current apparatus offers a larger excursion than the gold-standard ergometer, determined by the available lifting distance of the suspended load. Increasing the length of the timing belts could extend this excursion even further. 

The ability to measure force–velocity relationships across a broad range of educational settings creates opportunities to explore more advanced concepts in muscle mechanics. Curvature (a/Fmax), for example, has an important yet often underappreciated relationship with the optimal shortening velocity of a muscle (V_opt)⁸. As curvature approaches one and the force–velocity relationship becomes increasingly linear, V_opt shifts toward higher shortening velocities, approaching ½V/V_max. Conversely, a steeper and less linear force–velocity relationship shifts V_opt closer to V = 0 and F_max. The mechanical implications of this relationship are substantial and offer opportunities for discussion of optimizing muscle performance, including evolutionary adaptations and peak-performance strategies. For example, muscles with more linear force–velocity relationships may maintain effective force production across a wider range of shortening velocities, whereas muscles with steeper force–velocity relationships may be optimized for greater force production^3,13. Measurements obtained using the low-cost apparatus demonstrated that the power output of the human flexor pollicis brevis peaked at 0.33 ± 0.07 V/V_max, consistent with the expected value of approximately one-third V/V_max reported by Nelson et al.⁷. Collectively, these relationships highlight the pedagogical value of force–velocity curvature as a parameter linking force, velocity, and power to muscle specialization and performance optimization.

The low-cost apparatus differs fundamentally from the gold-standard ergometer in the manner by which counterforce is generated and experienced by the participant during testing. In the low-cost apparatus, force is applied through a stiff cable attached to a pulley system supporting a smartphone and an external load. Consequently, the system exhibits substantially greater inertia than the gold-standard ergometer, which in the after-load mode operates using a yielding load with minimal inertial effects. The gold-standard ergometer applies a prescribed counterforce selected before each trial and yields when muscle force exceeds that limit. By contrast, the low-cost apparatus generates a dynamic counterforce that depends on the acceleration imparted by the participant. This apparatus behavior is further influenced by pulley radius because larger pulleys possess greater moments of inertia. Three pulley radii were incorporated so that a single set of weights could generate a large range of distinct effective loads. Because pulley radius affects both mechanical advantage and moment of inertia, the same suspended load is experienced as a different effective resistance at each radius. This design increases the number of force–velocity operating points that can be sampled without changing weights between trials. Consequently, participants experience a variable force profile throughout a contraction when using the low-cost apparatus, whereas the gold-standard ergometer maintains a relatively constant force. These differences in inertial effects may explain the larger standard deviations observed for measurements obtained with the low-cost apparatus (Table 1). Despite these mechanical differences, however, the low-cost apparatus produced measurements that were statistically indistinguishable from those obtained using the gold-standard ergometer.

Several factors require careful control during assembly and operation of the low-cost apparatus to ensure both participant safety and data quality. Excess friction within the bearings or uncontrolled oscillation of the suspended loads can compromise data quality. The weights should hang freely without contacting nearby surfaces, including the table, chair, or floor. Ball bearings should rotate smoothly to minimize resistance. Proper isolation of thumb movement is also essential because excessive movement of the hand, arm, or body may introduce unwanted variation. Finally, careful installation of the safety bolt is critical to prevent accidental loading events from transferring excessive forces to the participant's thumb and mitigate injury risks.

The low-cost apparatus has several additional limitations that should be considered when interpreting the data. The apparatus cannot measure true isometric force because the operation requires a nonzero acceleration, whereas true isometric contractions occur without joint movement. Therefore, traditional weight-based loading systems combined with an isometric protocol may be deployed for measuring true isometric F_max. The low-cost apparatus also imposes a relatively large minimum load because of the mass of the smartphone, even when the smallest gear ratio is used. In the present study, the largest gear ratio was used during high-force contractions to improve data quality. The larger gear increases displacement and thereby improves signal detection relative to the inherent noise of smartphone accelerometers¹¹. Because the apparatus cannot be completely unloaded, the power–velocity relationship often does not return to zero at either endpoint (Figure 1, Figure 3) preventing direct measurement of the zero-power boundaries of the curve. Peak-velocity trials will benefit from using the smallest gear ratio and the smartphone as the sole load to minimize force requirements. Additionally, the low-cost apparatus cannot safely measure eccentric contractions. As a consequence, the data-processing workflow was not configured to quantify negative velocity (lengthening) or negative power (energy dissipation).

Estimation of peak force and peak velocity required Hill-type curve fitting with extrapolation to F = 0 and V = 0. The parameter a was constrained between 0.1 and 1.0 to prevent unrealistic V_max estimates arising from the limited availability of low-force, high-velocity data. Unconstrained values of a/F_max were reported to avoid masking curvature differences, albeit at the cost of increased variance. Without constraints, a/F_max values frequently exceeded 1. The selected range is supported by Hill, who reported the relationship shown in Equation 5¹, and by Alcazar et al., who reported a/F_max values clustered around 0.18–0.28⁴. These observations support the lower end of the constrained range, whereas the upper limit of 1.0 was intentionally set to avoid excessive restriction on higher-velocity contractions.

Equation 5:

Minor post-processing corrections were required to compensate for accelerometer drift; however, no smoothing procedures were applied. The Supplemental Coding File 1 identifies periods of consecutive zero acceleration to determine when the smartphone has returned to its resting position and subsequently resets velocity and position to zero. Accurate identification of true contraction peaks still requires user judgment, as noise peaks may occasionally be mistaken for real motion events. These limitations emphasize the importance of careful assembly, controlled experimentation, and informed data processing when using the low-cost apparatus. Such considerations also provide valuable teaching opportunities in experimental design and data interpretation. Future versions of the Python analysis software may include additional tools for automated peak classification and improved signal discrimination. At present, user-dependent peak identification remains a limitation to experimental reproducibility.

The validation presented here demonstrates that the low-cost apparatus produces educationally equivalent data to those obtained using a gold-standard ergometer. Educators may incorporate this apparatus into physiology curricula, particularly when teaching muscle mechanics and the force–velocity–power relationship^14,15. Future apparatus iterations may rely on widely available recycled materials, such as bicycle gears and frame components, to further improve accessibility. Such developments could expand opportunities for force–velocity–power research and education in settings that currently lack access to expensive ergometry equipment and 3D-printing resources.

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
1.25 kg Weights	NEXO Fitness	ASIN: B09JHQ6QB8	Can be substituted with generic weights approx. 1-2 kg number needed: 2
3D printer timing belt	‎3D printer belt 10m	ASIN: B08974S1CC	Can be substituted with generic 3D printer timing belts: 2mm Pitch, 6mm Width, made of Rubber and Fiberglass number needed: 1
3DPart1 - Inner Gear	N/A	N/A	See Supplemental STLs - Print in ABS number needed: 1
3DPart2 - Outer Gear	N/A	N/A	See Supplemental STLs - Print in ABS number needed: 1
3DPart3 - Key	N/A	N/A	See Supplemental STLs - Print 3-5x in ABS number needed: 1
3DPart4 - Force Carry	N/A	N/A	See Supplemental STLs - Print in PLA number needed: 1
Aluminum beam Long	80/20 inc	1020	1 in x 2 in x 32 in number needed: 1
Aluminum beam Short	80/20 inc	1010	1 in x 1 in x 12 in number needed: 2
Aluminum frame Fastener / Safety Bolt	80/20 inc	4108	number needed: 3
Arm brace	FEATOL	ASIN: B0CGX8QZ3C	Can be substituted with generic arm brace number needed: 1
Arm rest cellular polyvinyl chloride block	Azek board	8/4 X Thickness Traditional Only	14 in x 6 1/2 in x 3/4 in number needed: 14
Carabiners	Rewoalzx	ASIN:  B0FB3VYV3N	number needed: 2
Chrome Alloy Steel Ball Bearings	uxcell	UXCELL 6900 2RS - ASIN: B07FVX92MJ	Dimensions: inner diameter 10 mm, outer diameter 22 mm, width 6 mm. number needed: 1
Excel/Spreadsheet	Microsoft	Excel	Can be substituted with any spreadsheet software number needed: 1
Finger brace	Sopito	ASIN:  B0B5C6RWR4	Can be substituted with generic finger brace number needed: 1
IGOR 8 PRO	Wavemetrics.com	8 PRO	Used with Aurora Scientific Apparatus and for a/Fmax calculation. number needed: 1
Large cellular polyvinyl chloride board	Azek board	8/4 X Thickness Traditional Only	6 3/4 in x 7 in x 3/4 in number needed: 2
Metal axle	Generic	ASIN: B0CC8T8X77	Steel Rod 9.97 mm thick, 45 cm long. number needed: 1
Multi-strand steel wire	Catch All Tackle	ASIN: B00JTWDSQS	1.4mm diameter #400lb rating number needed: 2
Phone holder	Tribe	ASIN: B00SXRXUFE	number needed: 1
Phyphox app	Phyphox.org	N/A	Download from Apple App Store or Google Play Store number needed: 1
Python script	N/A	N/A	Can be downloaded: Supplemental Document 1 muscle_velocity_gui.py number needed: 1
Small cellular polyvinyl chloride board	Azek board	8/4 X Thickness Traditional Only	1 in x 3 1/4 in x 3/4 in number needed: 1
Smartphone	Apple	IPhone 11	Can be substituted with any Smartphone with an accelerometer number needed: 1
Steel wire crimps	Catch All Tackle	ASIN: B00JTWDSQS	#400lb rating number needed: 6
Stoppers	Befenybay	ASIN: B0B1C6BPKQ	inner diameter 10 mm, outer diameter 20 mm, width 20 mm number needed: 2