Moment arms
• Mechanical levers
– Force multiplier
– Speed multiplier
• Lever-like systems
– Pulleys
– Sesamoids
• Interaction with muscle architecture
Force and torque
• Muscles are linear actors
– Linear force: F=ma
– Unidirectional (this is a problem)
• Opposing muscles are rotary actors
– Angular torque: t = Ja
– Transformation: t = r X F
F2
J*a
F
m*a
Force-torque transformation
• t=r X F
– 2-D: t = |r| |F| sin(f)
– |F| sin(f) is perpendicular force
– |r| sin(f) is perpendicular distance
– 3-D
F sin(f)
• Moment arm
– Effective mechanical advantage
– |r| sin(f) ie: scalar
F
r sin(f)
f
r
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
Hamstrings & quadriceps paths
r1
10 t
r2
Review: VI/ST during running
• VI: M=172g, Lf=9.9cm, PCSA=17 cm2
• ST: M=100g, Lf=19 cm, PCSA=4.8 cm2
• Large flexor MA correlates with long Lf
Lf/MA ratio
• Isometric operating range
– Muscle length change – joint range of motion
– DL = r * Dq
• Dynamic shortening velocity
– dL/dt = r dq/dt
– Force-velocity relationship
Small MA
Large MA
Is Lf/MA conserved?
• Homeostasis provides uniformity
– Muscle can shorten ~50% (Weber, 1850)
– Lf/MA conservation would preserve op range
• Doesn’t look like it
7
6
5
4
3
2
1
0
Lf/MA survey of human leg & wrist
0
2
4
6
8 10 12 14 16
LF/MA
Generalized moment arm
• Lever-pulley equivalence
• Perpendicular from joint center
to line of force
r
r
r2
– t = r*F (scalar r)
– t = r(q)*F
• Also defines length change
– dL/dq = r
– This can be used to estimate complex systems
F
L
Biological joints are like pulleys
• Contact surfaces
– Bone-bone
– Bone-muscle
– Nonuniform
• Ligament limitations
MA variation
• Direct-line muscles
– Biceps
– Sinusoidal
• Retroarticular
– Triceps brachii, triceps surae
– Tendon rolls over joint
at least some postures
F
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 (F1)
•Muscle-tendon (F2)
•Proximal bone on bone (F3)
•Proximal bond on tendon (F4)
Limb only
•Ground contact (F1)
•Tendon (F5)
•Proximal bone on bone (F3)
+
Tendon only
•Tendon (F5)
•Bone wrap (F4)
•Muscle-tendon (F2)
No friction:
|F5| = |F2|
F1s1=F2s2
Sesamoid
• Ossified tendon
– Bone-wrapping contact
– Knee; toes
Limb+sesamoid
•Ground contact (F1)
•Muscle-tendon (F2)
•Proximal bone on bone (F3)
•Proximal bond on tendon (F4)
Limb only
•Ground contact (F1)
•Tendon (F5)
•Proximal bone on bone (F3)
Sesamoid only
•Tendon (F5)
•Bone contact (F4)
•Muscle-tendon (F2)
Sesamoid center
Joint center
F4 might cause a
torque, so F2≠F5
Sesamoid mechanics
• Paired pulley/lever systems
– Muscle-sesamoid
– Sesamoid-distal bone
• Sesamoid motion is asymmetric
T
dL
T
dL
Equal tangential
displacement
T
dL
2T
dL/2
Equal angular
displacement
T
dL
4T
dL/4
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
Retinaculum
• Soft tissue tunnel
– Ligament captures tendon near joint
– Wrist, ankle
• Wrap around
– Bone in one direction
– Ligament in other
FCU
Muscle-Joint Interaction
• t(f) = r(f) F(f)
– Angle-MA relationship
– Angle-Force relationship
– Angle-Torque relationship 1
0.8
0.6
0.4
MA
Force
Torque
0.2
0
0
50
100
Joint Angle
150
How sensitive is torque to angle?
• MA variation
• Muscle force variation
• Coincidence
How much MA variation?
Human arm (Murray & al 1995)
• A lot
– Few muscles: Bic, BRD
– Especially quadrupeds
– Bone thickness
prevents zeros
• No so much
Patella tendon (Pandy & Shelburne 1998)
– Retinacula & wrapping
– 25-50%
How much force variation?
20-50%?
Architecture
Species
Measurement
technique
120
Count
100
Rat
Predicted force
•
•
•
•
80
Rabbit
60
Mouse
Human
40
@
14D
Frog
@
14D
Fish
20
Cat
Bird
50
75
100
125
150
Muscle length
175
200
Lf, MA, kinematics and performance
• Fix two, and you can “optimize” the third
• eg: Lf
– Ankle velocity during walking: 100°/s
– Soleus moment arm: 2.4 cm
Observed
Rough estimate, based
on Vmax=8L0/s and
neglecting elasticity
Lf, MA, kinematics and performance
• eg: MA
– Ankle velocity during walking: 100°/s
– Soleus Lf: 2.2 cm
Observed
Time scale for optimizing
• Kinematics
– Neural
– Seconds
• Lf
– Protein synthesis
– Days/weeks (longer?)
• MA
– Bone/evolutionary
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