1. Moment arms Mechanical levers – Force multiplier – Speed multiplier Lever-like systems – Pulleys – Sesamoids Interaction with muscle architecture

2. Force and torque Muscles are linear actors – Linear force: F=ma – Unidirectional (this is a problem) Opposing muscles are rotary actors – Angular torque:  = J  – Transformation:  = r X F F m*a F2F2 J* 

3. Force-torque transformation  =r X F – 2-D: t = |r| |F| sin(  ) – |F| sin(  ) is perpendicular force – |r| sin(  ) is perpendicular distance – 3-D Moment arm – Effective mechanical advantage – |r| sin(  ) ie: scalar F r  r sin(  ) F sin(  )

4. Mechanical advantage Ratio of muscle MA to load MA – Muscles always have dis-advantage in force – Muscle shortens less than load moves Flexor/extensor balance 10 t r1r1 r2r2 Hamstrings & quadriceps paths

5. Review: VI/ST during running VI: M=172g, L f =9.9cm, PCSA=17 cm 2 ST: M=100g, L f =19 cm, PCSA=4.8 cm 2 Large flexor MA correlates with long L f

6. L f /MA ratio Isometric operating range – Muscle length change – joint range of motion –  L = r *  Dynamic shortening velocity – dL/dt = r d  /dt – Force-velocity relationship Small MA Large MA

7. Is L f /MA conserved? Homeostasis provides uniformity – Muscle can shorten ~50% (Weber, 1850) – Lf/MA conservation would preserve op range Doesn’t look like it 0 1 2 3 4 5 6 7 0246810121416 LF/MA L f /MA survey of human leg & wrist

8. Generalized moment arm Lever-pulley equivalence Perpendicular from joint center to line of force –  = r*F (scalar r) –  = r(  )*F Also defines length change – dL/d  = r – This can be used to estimate complex systems F r r r2r2 L

9. Biological joints are like pulleys Contact surfaces – Bone-bone – Bone-muscle – Nonuniform Ligament limitations

10. MA variation Direct-line muscles – Biceps – Sinusoidal Retroarticular – Triceps brachii, triceps surae – Tendon rolls over joint at least some postures F

11. Wrap-around tendons Free body diagram – “Internal” forces can not change CoM trajectory – Clever choice of section can simplify analysis Limb+wrap-around tendon Ground contact (F 1 ) Muscle-tendon (F 2 ) Proximal bone on bone (F 3 ) Proximal bond on tendon (F 4 ) Limb only Ground contact (F 1 ) Tendon (F 5 ) Proximal bone on bone (F 3 ) Tendon only Tendon (F 5 ) Bone wrap (F 4 ) Muscle-tendon (F 2 ) + No friction: |F 5 | = |F 2 | F 1 s 1 =F 2 s 2

12. Sesamoid Ossified tendon – Bone-wrapping contact – Knee; toes Limb+sesamoid Ground contact (F 1 ) Muscle-tendon (F 2 ) Proximal bone on bone (F 3 ) Proximal bond on tendon (F 4 ) Limb only Ground contact (F 1 ) Tendon (F 5 ) Proximal bone on bone (F 3 ) Sesamoid only Tendon (F 5 ) Bone contact (F 4 ) Muscle-tendon (F 2 ) F 4 might cause a torque, so F 2 ≠F 5 Sesamoid center Joint center

13. Sesamoid mechanics Paired pulley/lever systems – Muscle-sesamoid – Sesamoid-distal bone Sesamoid motion is asymmetric T4 T dL dL/4 T T dL T 2 T dL dL/2 Equal tangential displacement Equal angular displacement

14. Patella force multiplier Force lost through patella – Different rotation centers – Internal compression Measurable (at least in cadavers) – Bishop 1977, S&D 1980, Huberti & al 1984, Lu & O’Connor 1996 – Patella tendon force ~50% of quadriceps force in flexion Seedhom & Dowon, 1980

15. Retinaculum Soft tissue tunnel – Ligament captures tendon near joint – Wrist, ankle Wrap around – Bone in one direction – Ligament in other FCU

16. Muscle-Joint Interaction  (  ) = r(  ) F(  ) – Angle-MA relationship – Angle-Force relationship – Angle-Torque relationship

17. How sensitive is torque to angle? MA variation Muscle force variation Coincidence

18. How much MA variation? A lot – Few muscles: Bic, BRD – Especially quadrupeds – Bone thickness prevents zeros No so much – Retinacula & wrapping – 25-50% Patella tendon (Pandy & Shelburne 1998) Human arm (Murray & al 1995)

19. How much force variation? 20-50%? Architecture Species Measurement technique Muscle length Predicted force

20. L f, MA, kinematics and performance Fix two, and you can “optimize” the third eg: L f – Ankle velocity during walking: 100°/s – Soleus moment arm: 2.4 cm Observed Rough estimate, based on V max =8L 0 /s and neglecting elasticity

21. L f, MA, kinematics and performance eg: MA – Ankle velocity during walking: 100°/s – Soleus Lf: 2.2 cm Observed

22. Time scale for optimizing Kinematics – Neural – Seconds L f – Protein synthesis – Days/weeks (longer?) MA – Bone/evolutionary

23. Summary Moment arms convert linear muscles to rotary joints – Force-torque – Length-angle Biological joints often minimize MA variation Constraint against which the nervous system selects movement patterns