Muscle Function and Energetics of Locomotion — Equations

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A consolidated reference of the equations used across Weeks 5–8 for muscle mechanics, musculoskeletal lever systems, the energetics of locomotion, and body-size scaling. Please see lecture notes on the units, and carefully check your work for consistency of units when solving problems.

Muscle force, length, and velocity

Equation Name Purpose
$F_{max} = F_{active} + F_{passive}$ Total isometric force Measured force during maximally stimulated isometric contraction is the sum of active (cross-bridge) force and passive (titin + connective tissue) force.
$F_{active} = F_{max} - F_{passive}$ Active force Active force is computed by subtracting separately measured passive force from the total; used to construct the active F–L curve point-by-point.
$(V + b)(F + a) = b(F_0 + a)$ Hill force–velocity equation The classic hyperbolic relation between shortening velocity $V$ and load $F$. $F_0$ = maximum isometric tension (at $V = 0$); $V_0$ = maximum shortening velocity (at $F = 0$); $a$ = heat-of-shortening coefficient; $b = a(V_0/F_0)$.
$P = F \cdot V$ Mechanical power Instantaneous power output; computed point-by-point from the F–V curve. Peak power occurs at intermediate force and velocity (~0.2–0.3 $V_{max}$).
$F_{tot} = F_{FV} \cdot F_{FL} \cdot F_{act}$ Combined muscle model Standard multiplicative muscle model (e.g., OpenSim): total force is the product of velocity-, length-, and activation-dependent factors.

Muscle architecture and scaling

Equation Name Purpose
$\sigma = F / \text{PCSA}$ Specific tension (stress) Force normalized by physiological cross-sectional area; ~18–30 N/cm² across vertebrate skeletal muscle, used to compare intrinsic capability across muscles.
$F_{max} = \text{PCSA} \times \sigma_{spec}$ Maximum isometric muscle force Maximum force a muscle can produce equals its PCSA times specific tension; PCSA = muscle volume / fiber length.
$\text{PCSA} = \dfrac{\text{Volume}}{\text{fiber length}}$ Physiological cross-sectional area Cross-section of muscle perpendicular to the fibers; determines maximum force capacity (proportional to sarcomeres in parallel).
$F_{max} \propto \text{PCSA}$ Force capacity scaling Maximum force is proportional to the number of sarcomeres in parallel — i.e., the PCSA.
$\text{Displacement}, V_{max} \propto L_{fiber}$ Displacement and velocity capacity Maximum shortening distance and maximum shortening velocity are proportional to the number of sarcomeres in series — i.e., the fiber length.
$\text{Work}, P \propto \text{Volume} = \text{PCSA} \times L_{fiber}$ Work and power capacity Maximum work and power are proportional to muscle volume (or, equivalently, mass via conserved muscle density ~1.06 g/cm³).
$V_{myofibril} + V_{SR} + V_{Mt} \approx 1$ Cellular volume-fraction constraint The sum of myofibrillar, SR, and mitochondrial volume fractions is constrained within a fixed muscle-cell volume; high specialization in one component comes at the expense of the others.

Lever systems and effective mechanical advantage

Equation Name Purpose
$T = F \times D$ Torque Rotational effect of a force around a fulcrum, where $D$ is the perpendicular distance from the force’s line of action to the fulcrum.
$F_1 D_1 = F_2 D_2$ Lever balance Torques on either side of a fulcrum balance at equilibrium.
$F_{muscle} = F_g \times \dfrac{R}{r}$ Limb lever equation Muscle force required to resist the ground reaction force, where $r$ is the muscle moment arm (skeletal morphology) and $R$ is the GRF moment arm (set by posture).
$\text{EMA} = r / R$ Effective mechanical advantage Ratio of muscle moment arm to GRF moment arm. Higher EMA → lower required muscle force per unit body weight; scales positively with body mass across vertebrates.
$T_h = F_g \times R_h$ Joint torque (inverse dynamics) Torque at a joint from the GRF and its moment arm to that joint.
$W_h = T_h \times \Delta\theta$ Joint work Net work at a joint, computed by integrating joint torque through joint angular displacement.

Work loops and energy cycling

Equation Name Purpose
$W = \oint F \, dL$ Work loop (net mechanical work) Net work done by a muscle equals the area enclosed by its force–length trajectory over a contraction cycle. Counterclockwise = positive (motor), clockwise = negative (brake), no enclosed area = strut/spring.
$L_{\text{MTU}} = L_{\text{CE}} + L_{\text{SEE}}$ Muscle-tendon unit length partition At every instant, MTU length equals the sum of contractile-element (fascicle) length and series-elastic-element (tendon) length.
$F_{\text{CE}} = F_{\text{SEE}}$ Series-element force balance In a series arrangement, the contractile element and tendon transmit the same force at every instant.
$F_{\text{SEE}} = k_{\text{tendon}} \cdot \Delta L_{\text{SEE}}$ Tendon stiffness relationship Tendon force is approximately proportional to tendon stretch over the operating range; stiffness $k$ increases with tendon cross-sectional area.
$\tau_{exo} = k_{rot} \times \Delta\theta_{ank}$ Exoskeleton torque Passive elastic torque produced by an ankle exoskeleton, with rotational stiffness $k_{rot}$.

Forces and dynamics of locomotion

Equation Name Purpose
$\sum F_{vertical} = Mg + F_{legs} = 0$ Vertical force balance Average vertical force from the legs equals body weight across an integer number of strides.
$F_{avg}/W_b = T_{step}/T_c = L_{step}/L_c$ Weyand step-cycle equation Predicts average vertical GRF (in body weights) from the ratio of step duration to stance duration.
$F_{peak} \propto 1/t_c$ Peak force vs. contact time As speed rises, contact time shrinks and peak vertical GRF must rise to maintain weight support.
$\text{COM work} \propto s^4$ Kuo step-length scaling Center-of-mass work rate at step transitions scales approximately as the fourth power of step length.

Cost of transport and metabolic rate

Equation Name Purpose
$\dot{V}O_2 = \dot{V}_E (F_IO_2 - F_EO_2)$ Fick principle (respirometry) Oxygen consumption equals ventilation rate times the difference between inspired and expired O2 fractions; the basis of indirect calorimetry.
$\text{CoT} = \dot{V}O_2 / \text{speed}$ Mass-specific cost of transport Energy used per unit distance traveled, per unit body mass (mL O2 kg⁻¹ m⁻¹ or J kg⁻¹ m⁻¹).
$\dot{E}_{\text{metab}}/W_b = C \cdot (1/T_c)$ Kram & Taylor metabolic rate equation Mass-specific metabolic rate per body weight equals the cost coefficient $C$ divided by stance contact time $T_c$; $C \approx 0.189$ J/N is nearly constant across mammals.
$E_{\text{cot}}/W_b = C \cdot (1/L_c)$ Kram & Taylor cost-of-transport equation CoT per body weight equals the cost coefficient $C$ divided by step length $L_c$; larger animals have longer $L_c$, so lower CoT.
$1 \text{ mL } O_2 \approx 20.1 \text{ J}$ Oxycaloric coefficient Average energetic equivalent of aerobic oxygen consumption; converts respirometry data to joules.

Body-size scaling

Equation Name Purpose
Strength $\propto L^2$, Mass $\propto L^3$ Isometric scaling Under geometric similarity, strength (proportional to muscle cross-section) grows with the square of linear dimension, while mass grows with the cube — larger animals are relatively weaker.
$\text{CoT} \propto M_b^{-0.25}$ CoT vs. body mass scaling Across runners, mass-specific cost of transport decreases as body mass to the −0.25 power on log-log axes (Taylor et al. 1970).
$\text{fAS} = \dot{V}O_{2max} / \text{BMR}$ Factorial aerobic scope Ratio of maximum to baseline metabolic rate; athletic species sit on a separate scaling line 2–4× higher than non-athletic species.
$\text{fAS}_{athl} = 17.66 \cdot M_b^{0.184}$ Athletic-species fAS scaling Allometric fit for athletic mammals (Weibel et al.); aerobic scope rises with body mass even within the athletic group.
$\text{fAS}_{nonathl} = 8.29 \cdot M_b^{0.100}$ Non-athletic-species fAS scaling Companion fit for non-athletic mammals; lower intercept and shallower slope.
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