static torque.pdf

Torques ESS 3093 Biomechanics

Torques and Moments of Force


Class Objectives

Introduce Levers and Mechanical Advantage Define Torque and moment arm

Moment arm as a radius vector Torque as a vector cross product

Static Equilibrium Including Torque Define Center of Mass Estimate the Location of the Center of Mass


Levers and Leverage


Mechanical Advantage

armreistance

armeffortMA =


Levers Trade Force for Distance If Mechanical Advantage is > 1,

Effort force < Resistance force Effort movement > Resistance movement

If Mechanical Advantage is < 1,

Effort force > Resistance force Effort movement < Resistance movement

What does that tell you about range of motion? Speed?


Classes of Levers

1st Class: Direction reversed

2nd Class Direction maintained MA > 1

3rd Class: Direction maintained MA < 1

Effort

Effort

Effort

Resistance

Resistance

Resistance


Torque The turning effect caused by a force Moment of force


Mathematical Definition of Torque T = r F

T = torque (Nm) r = moment arm (m)

Distance from the axis of rotation to the line of action of the force

Perpendicular to the line of action of the force F = force (N) Units: Newton x meters (Nm)

Axis of rotation

Moment arm

Force

x

90

Moment arm


Torque Direction

Direction + Torque = counter clockwise - Torque = clockwise

Right hand rule


Internal and External Torque

Internal Torque External Torque


Moment Arm

Definition: Distance from the axis of rotation to the line of action or the force AND Perpendicular to the line of action of the force

Axis

Moment arm Force


Moment Arm: General Case 1. What is the moment arm to the muscle

force, OR 2. What is the component of muscle force

to the bone?

Moment arm?

Muscle Force

Axis Bone


Moment Arm: General Case Method 1 Use trig to find the length of the moment arm from the axis to the line of action of the force Moment arm = d sin

Torque = Fm d sin

Moment arm?

Muscle Force (Fm)

Axis, Joint center

Bone

d


Moment Arm: General Case Method 2 Use trig to find the component of muscle force which acts to the bone F = Fm sin

T = d Fm sin Muscle

Force

Axis Bone

Component of Muscle Force to the bone


Example 1

A force of 200 N is exerted 0.25 m from the axis of rotation. What is the resulting torque?

r

F


Example 2

What is the torque about the elbow produced by the following conditions: a 900 N force pulling on the forearm at an angle of 110 deg from the horizon at a point 3 cm to the right of the elbows

axis of rotation with the forearm horizontal


Fm = 900 N

110 deg

d = 0.03 m

rm = ?

T = rm Fm rm = d sin 110

T = Fm d sin 110

T = 900 N x 0.03 x sin 110

T = 25.4 Nm


The muscle tendon moment arms vary with joint angle.

Arthur Jones


Moment arm as a vector Torque as a vector cross product


Simple Torque Calculation

T = r F +rx +Fy = +T (CCW) T = rx Fy

+rx

+Fy

Convention

+Y

+X

+ rotation


Simple Torque calculations

T = r F +ry +Fx = -T (CW) T = -(ry Fx)

+ry

+Fx Convention +Y

+X

+ rotation


Torque as a Vector Cross Product

Tz = (rx Fy) (ry Fx)

Tz= (xpa-xar) Fy (ypa-yar) Fx

Fy, Fx

rx = xpa xar ry = ypa - yar

Axis of rotation (ar), x,y

Point of application (pa) x,y


Torque as a Vector Cross Product

Tz = (rx Fy) (ry Fx)

Tz= (xpa-xar) Fy (ypa-yar) Fx

Fy

rx = xpa xar ry = ypa - yar

Axis of rotation (ar), x,y

Point of application (pa) x,y Fx

rx

ry


Fm = 900 N

110 deg

d = 0.03 m

rm = ?

F = 900 cos 110, 900 sin 110 r = 0.03, 0

T = (rx Fy) (ry Fx)

T = 0.03 x 900 sin 110 - 0

T = 25.4 Nm

FMx = 900N x cos 110 = -307.8 FMy = 900N x sin 110 = 845.7 Rx = 0.03 Ry = 0 Tz = (rx Fy) (ry Fx) Tz = (0.03 x 845.7) (0 x -307.8) Tz = 25.4 0 T = 25.4


Static Equilibrium

F = 0 T = 0

Combination allows solutions of indeterminate problems


A person is holding a 420N (94lb) dumbbell 0.4m from the elbow in a static position as shown.

The arm weighs 45N and the center of mass of the arm is 0.15m from the elbow.

What is the torque generated by the elbow flexors if the muscle tendon moment arm is 0.05m?

What is the force generated by the elbow flexors to maintain this static position (assume Fm is vertical)?

Tm = ?

Fm = ?

Fj = ?

F j


Tdumb = r x F Tdumb = 0.4 x -420 N Tdumb= -168 Nm Tarm = r x F Tarm = 0.15 x -45 Tarm = -6.75 Nm Sum of Torques = Tm + -168Nm + -6.75Nm = 0 Tm -174.75 = 0 Tm = 174.75 Nm Tm = r x F 174.75 Nm = 0.05 m x Fm Fm = 3495 N Ftotal=-420 -45+3495 + Fj = 0 Fj = -3030 N

F j


Food for thought: The torques produced by free weights vary as the moment arms of these weights change during the movement

Wdb

Wdb

Dumbbells dont get heavier, but the torque gets larger as the elbow is flexed

T = r x F


Application: Low Back Pain Lower back pain (LBP) is

the most costly musculoskeletal disorder in industrial nations

80% will suffer LBP

LBP can be completely debilitating to temporarily annoying


Biomechanics of Lifting The disk between L5/S1 incurs the greatest moment in lifting and is one of the most vulnerable tissues to force-induced injuries.


Lifting Calculate the forward

bending torque about L5/S1 axis: Lw = 0.25 m Lp = 0.4 m


Lifting T = 0 T = Tb + Tw + Tp = 0 Tb = -Tw - Tp Tb = -(Fw Lw) - (Fp Lp) Tb = -(-450*.25) - (-200*.4) Tb = 192.5 Nm

If the moment arm of the erector spinae is .05 m then what is the magnitude of the Fm?

T = F x moment arm F= T/moment arm = 192.5Nm/.05m = 3850N This generally represents the force on the disc

as well as muscle and connective tissue


Center of Mass Mean position of the mass of a body A point about which a bodys mass is

evenly distributed The CoM need not be within body


Example


Center of Mass: Balance Point


CoM by Mass Moments Mass Moment = Mass Displacement

m1

x1


CoM by Mass Moments MMs of all components = MMCoM

m1 m2

x1 x2


CoM by Mass Moments MMs = m1x1 + m2x2 = MtotalxCoM MtotalxCoM = m1x1 + m2x2 = (m1 + m2)xCoM xCoM = (m1x1 + m2x2)/(m1 + m2) Works for any number of masses (not just 2)

m1 m2

x1 x2 xCoM

Total Mass


Segmental Center of Mass

Thigh CoM: 43% of the length from the hip to the knee Mass: 14.6% of body mass

Leg CoM: 43% of the length from the knee to the ankle Mass: 4.3% of body mass

Foot CoM: 50% of the length from the ankle to the 2nd metarsal head Mass: 1.4% of body mass


CoM by Mass Moments

43% LThigh

14.6% BM

Thigh CoM: 43% of the length from the hip to the knee Mass: 14.6% of body mass

Leg CoM: 43% of the length from the knee to the ankle Mass: 4.3% of body mass

Foot CoM: 50% of the length from the ankle to the 2nd metarsal head Mass: 1.4% of body mass

4.3% BM 1.4% BM

43% LLeg 50% Lfoot


2D components of CoM


CoM in Two Dimensions?

The center of mass of the thigh is located 43% of the segment length from the hip joint.

What is the location of the CoM of the thigh segment with the hip located at (0.5, 1.0) and the knee located at (0.275, 0.61)?


(0.275, 0.61)

(0.5, 1.0) X direction

CoMx = Xprox - 0.43(Xprox-Xdist)

= 0.5 0.43(0.5-0.275)

= 0.403

Y direction CoMy = Yprox - 0.43(Yprox-Ydist) = 1.0 0.43(1.0-0.61)

= 0.832

CoM will be at (0.403, 0.832)

static torque.pdf

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