Angular kinetics is the study of torques and their rotational effects on masses.
Weight creates an external torque around the ankle causing the person to rotate clockwise. Muscle force creates and internal torque around the elbow joint axis causing the forearm to rotate towards flexion. Lever arm between force vector and axis in red
A little secret: we have studied angular kinetics throughout the semester 1) Muscle Contractions – 3 rd class levers 2) Muscle co-contraction – developing opposing torques with muscles
Angular Kinetics and Co-contraction Simultaneous contraction of muscles on both sides of a joint Agonist muscle must overcome external load and load from co-contracting, antagonist muscle E.g. Biceps Brachii in elbow flexion Triceps torque Biceps torque
Angular Kinetics: Biceps Torque with Co-contraction Ext. force, lever arm = 40 N, 0.30 m Ext. torque = F * dist.= 40 N * 0.30m = 12 Nm Triceps force = 400 N Triceps lever arm = 0.02 m Triceps torque = 8 Nm Total Extensor torque = Ext. torque + Triceps torque = 12 Nm + 8 Nm = 20 Nm Biceps force External force (40 N) Triceps force
Angular Kinetics: Biceps Torque with Co-contraction Ext. torque = 20 Nm Biceps force = ? N Biceps lever arm = 0.02 m Biceps torque > 20 Nm: Slow lift = 25 Nm 25 Nm = Biceps force * 0.02 m Biceps force = 1250 N Biceps force External force (40 N) Triceps force
Angular Kinetics: Force – Velocity Relationship Concentric: Knee joint velocity and muscle torque during the stance phase of running. Torque (and muscle force) are highest in midstance when the joint stops moving (zero velocity). Muscle shortening velocity is low at this time.
The study of torques and their rotational effects on masses
Torque – the turning effect of a force exerted at a distance to an axis
Effects – positive and negative angular accelerations, stabilize object
Masses – the object under consideration – a whole human or animal, a body segment, IN ALL CASES THE MASS IS A LEVER (a rigid object)
Torque Torque is a vector – direction is either: 1) clockwise or counterclockwise 2) anatomical – flexor or extensor, adductor or abductor Torque measured in Newton * meters: 1 Nm = 1 kg m/s 2 * 1 m = 1 kg m 2 /s 2 1 Nm of torque is a small amount for human biomechanics: elbow torque in biceps curl with 25 lbs (~100N) = 30 Nm knee torque in running = 250 Nm
Newton’s Laws of Angular Motion I. Law of Angular Inertia – An object will remain stationary or rotate with constant angular velocity until an external torque is applied to the object Rotational Inertia – resistance Rotational Inertia = Moment of Inertia = I = mr 2 Rotational resistance depends on objects mass and length Long, massive objects are hard to rotate – Why does tight rope walker carry long pole? (see next slide)
Newton’s Laws of Angular Motion Why? To live, of course! In this case living involves not falling off the wire. The pole provides a support brace – it resists rotation due to its mass but mostly due to its length. As the person falls slightly to one side, he torques the pole around its center. The pole resists this torque due to its large Moment of Inertia and it applies a reaction torque back on the person to stabilize him.
Moment of Inertia Most important application of Moment of Inertia: Moment of Inertia for individual body segments – the amount of resistance to a change in rotation within each segment. Affects the rotational motion caused by muscle torques. Larger people – more mass and longer segments – have larger segment I values (7’ Basketballers)
Moment of Inertia Moments of Inertia are low for most people and most body segments except trunk. Trunk offers some resistance to rotation. Related to Low Back injuries. Moments of Inertia (kgm 2 ) Segment Women Men Trunk 0.8484 1.0809 Arm 0.0081 0.0114 Forearm 0.0039 0.0060 Thigh 0.1646 0.1995 Shank 0.0397 0.0369
Newton’s Laws of Motion III. Law of Angular Reaction – When one object applies a torque on a second object, the second object applies an equal and opposite torque onto the first object “ equal and opposite” – equal magnitude and opposite direction Evident in joint or muscle torques
Law of Angular Reaction Muscles operate as springs which have equal and opposite torques on each lever (i.e. body segment). The, “muscle spring,” rotates each segment in the opposite direction Why does only forearm rotate then in biceps curl?
Law of Angular Reaction – Inverse Dynamics & Muscle Torques Gastroc-Soleus force Gastroc-soleus force creates a clockwise torque on foot (blue arrow) and a counterclockwise torque on leg (red arrow) = ankle joint plantarflexion. Exactly like the spring on the skin calipers torques each arm in the opposite direction
Newton’s Laws of Motion II. Law of Angular Acceleration – a torque will accelerate an object in the direction of the torque, at a rate inversely proportional to the moment of inertia of the object: T = I Torque – the rotational effect of a force applied at a distance to an axis
Two Equations for Torque T = I T = F d I = mr 2 F d = Kinematic – Kinetic Equivalents I = F d
Two Calculation Techniques for Torque 1) What is the lever arm dist ? Biceps attached 3 cm from elbow joint. = 60 ° Forearm Arm Biceps force = 4,000 N 0.03 m Sin 60 ° = d1/0.03 d1=0.026 T = 4000 N (0.026 m) = 104 Nm d1 4,000 N Use length triangle 60 °
Two Calculation Techniques For Torque 2) What is the amount of force perpendicular to lever (the rotational effect of the muscle force)? = 60 ° Forearm Arm Biceps force = 4,000 N 0.03 m Cos 30 ° = F1/4000 F1=3464 N T = 3464 N (0.03 m) = 104 Nm F1 4,000 N Use force triangle = 30 °
Law of Angular Acceleration & and Angular Impulse-Momentum Law of Angular Acceleration restated: T = I * T = I * ( f – i)/time T * time = I * ( f – i) - angular impulse-momentum equation T * time = angular impulse = area under torque-time curve = total effect of the accumulated or applied torque; measured in Nms = kgm/s 2 * m *s = kgm 2 /s Angular Impulse Changes Angular Momentum
Angular Impulse in Movement Analyses Use area under torque-time curve to assess the total effect of a muscle torque Area sensitive to magnitude and temporal changes Brace did not change angular impulse in ACL group ACL group had more angular impulse at the hip and less at the knee compared to healthy group 0.26 0.23 * 0.17 Nms/kg 0.13 0.14 * 0.33 Nms/kg
Law of Angular Acceleration and Inverse Dynamics Inverse Dynamics – an analysis that calculates unknown torques inside the human body. These are the muscle torques that combine to create skillful human movement. Torques at joints produced by all muscles crossing the joints.
Inverse Dynamics Analysis Inverse Dynamics – combines position and acceleration data from kineamatic motion analysis and force data from force platforms or other force sensors to calculate internal torques. Most commonly used in locomotion but also in cycling and other activities.
Joint Torques During Walking Joint torques show neuromuscular contributions to movement. Support torque is sum of 3 joint torques and is exactly like GRF.
Inverse Dynamics Analysis Subject walking up the ramp. Video to produce position and acceleration data. Force plate to measure the known external forces. See analysis on next few slides and on board. White line is force plate. Curves are vertical and ant-post GRFs.
Data used in the I.D. analysis in next few slides: Masses and moment of inertias: Subject: 70 kg Foot: 1.7 kg, 0.0023 kgm 2 Leg: 3.44 kg, 0.0044 kgm 2 Accelerations: Foot vertical: 2.47 m/s 2 Foot horizontal: 4.70 m/s 2 Foot rotational: -52.5 rad/s 2 Leg vertical: 1.30 m/s 2 Leg horizontal: 9.23 m/s 2 Leg rotational: -10.3 rad/s 2
Av – large & down: the body weight plus inertial force of accelerating body mass upward push down on the foot. The upward GRF is larger than the downward ankle reaction force – THUS THE FOOT AND PERSON MOVE UPWARD. Ankle Joint Forces Vertical Ankle Joint Reaction Force (JRF) Av: Fv = ma v GRFv – mg + Av = ma v 975 – (1.07) (9.81) + Av = (1.07) (2.47) Av = -961 N Foot Ankle Vertical Direction: -961 N Av mg Note: Weight applied at the center of mass GRFv = 975 N
Ah – small and backward: the foot pushes the body forward & the body pushes back on the foot at the ankle. The forward GRF is larger than the backward ankle reaction force – THUS THE FOOT AND PERSON MOVE FORWARD. Ankle Joint Forces Horizontal Ankle Joint Reaction Force (JRF) Ah: Fh = ma h GRFh + Ah = ma h 162 + Ah = (1.07) (4.7) Ah = -157 N GRFh = 162 N Foot Ankle Horizontal Direction: -157 N Ah
Each force causes a torque in particular direction – in this case all external force-torques in counterclockwise (dorsiflexor) direction FBD for Ankle Joint Torque Fy = 162 N Fz = 975 N Foot Met head (0.572,0.011) Az =- 961 N Ay = -157 N Ankle Unknown Ankle Torque Fz lever arm Fy lever arm Az lever arm Ay lever arm
Ankle Joint Torque T = I 975(0.055) + 162(0.061) + 961(0.045) + 157(0.047) + Ma = (0.0023)(-52.5) Ma = (0.0023)(-52.5) - 975(0.055) - 162(0.061) - 961(0.045) - 157(0.047) Ma = -114 Nm Ankle Joint Torque CM (0.517,0.072) (0.472, 0.119) Fy = 162 N Fz = 975 N Foot CoM (0.516,0.072) mg Foot Met head (0.572,0.011) Az =- 961 N Fy = -157 N Ankle Unknown Ankle Torque Ankle Joint Torque: large and negative – strong push off required by ankle plantarflexors to propel forward and up the ramp.
T = I 975(0.055) + 162(0.061) + 961(0.045) + 157(0.047) + Ma = (0.0023)(-52.5) Ankle Joint Torque Components Vertical GRF torque = 54 Nm Horizontal GRF torque = 10 Nm Vertical Ankle JRF torque = 43 Nm Horizontal Ankle JRF torque = 7 Nm Inertial torque = 0.1 Nm Vertical torque = 119 Nm Horizontal torque = 20 Nm Nearly all muscle torque due to the muscle response to external loads on the body segment (more on this issue a few slides down)
Kv – large & down: the body weight plus inertial force of accelerating body mass upward push down on the knee The upward ankle force is larger than the downward knee force – THUS THE LEG AND PERSON MOVE UPWARD. Knee Joint Forces Vertical Knee Joint Reaction Force (JRF) Kv: Fv = ma v Av – mg + Kv = ma v 961 – (3.44) (9.81) + Kv = (3.44) (1.30) Kv = -922 N Av = 961 N Leg Knee Ankle Note: Ankle JRFs reversed onto leg (the law of reaction) Vertical Direction: Kv mg -922 N
Kh – small and backward: the leg pushes the body forward & the body pushes back on the leg at the knee. The forward ankle force is larger than the backward knee force – THUS THE LEG AND PERSON MOVE FORWARD. Knee Joint Forces Horizontal Knee Joint Reaction Force (JRF) Kh: Fh = ma h Ay + Kh = ma h 157 + Kh = (3.44) (9.23) Kh = -125 N Ah = 157N Leg Knee Ankle Horizontal Direction: Note: Ankle JRFs reversed onto leg (the law of reaction) Kh -125 N
Knee Joint Torque T = I -961(0.112) + 157(0.205) - 922(0.085) + 125(0.155) + 114 +Mk = (0.0044)(-10.3) Mk = (0.0044)(-10.3) + 961(0.112) -157(0.205) + 922(0.085) - 125(0.155) -114 Mk = 20.4 Nm Knee Joint Torque CM (0.584,0.324) Ax = 157N Ay = 961 N Leg Knee Ankle -125 N (0.472, 0.119) (0. 669, 0.479) Mk Trq. = 114 Nm Knee joint torque low and positive (extensor) in direction. Walking uphill had larger ankle vs. knee extensor torques. Some (i.e. horizontal) external joint forces torqued the leg in the desired direction – aided & so reduced muscle effort. -922 N Ky
Joint Torques in Old & Young Adults Old adults have larger hip torques and lower knee torques. Shows altered motor strategy with age. Level Walking Stair Ascent
Several Muscles Combine to Produce Torque at Each Joint These muscles create the torques at each joint. Each muscle torque is the combined effect of all the extensor and flexor muscles at each joint. For example, an extensor knee torque occurs when the quadriceps produce more extensor torque than the flexor torque produced by the hamstrings and gastrocnemius. The co-activating muscles have an overall extensor effect in this case.
Joint Torques in Obese and Lean Adults 1) Hip torques equal 2) Obese less knee torque at slow speed and same torque at same speed as lean 3) Obese more ankle torque at both speeds Obese Lean
Inverse Dynamic Analysis Inverse dynamic analysis calculates unknown joint torques inside the human body (also called muscle torques). Torques at joints produced by all muscles crossing the joints and show how each muscle group contributes to a particular movement. Joint torques are interpreted as the motor pattern of a movement – they show the neurological strategy used in a movement
Work Done By A Torque Avg lever arm = 0.25 m Avg Muscle torque = 10 Nm 40 N
While torque is not work, it can do work: Work = Torque *
= angular displacement = 0.78 rad
Work = 10 Nm * 0.80 rad = 8.0 J
(check with linear calculation:
Work= mgh: 40 N(h f ) – 40 N(h i )= 8.0 J
h f – h i = 0.20 m)
Joint Power Produced By Joint Torques Elbow joint angular velocity, torque and power Power = Torque * Positive power – concentric contraction, positive work, increase energy Negative power – eccentric contraction, negative work, decrease energy
Joint Power Produced By Joint Torques Calculate work from power curve: Work is area under the power curve or a portion of the curve: Power = Watts = T /s = Nm/s = kgm 2 /s 2 / s = kgm 2 /s 3 * s (for area) = kgm/s 2 * m = force * distance = WORK
Joint Power Produced By Joint Torques Knee power, torque, and angular velocity during stance phase of running. Knee flexes during brief flexor torque then longer extensor torque – low positive power & work then large negative power & work Knee extends during long extensor torque then shorter flexor torque – large positive power & work then low negative power & work
Joint Power Produced By Joint Torques Knee power, torque, and angular velocity during stance phase of running. Peak torque at zero velocity – at maximum knee flexion, maximum quadriceps stretch – muscle force maximized early in movement. Peak power at mid levels of torque and velocity – both torque and velocity contribute to power – muscle work maximized in middle of movements.
Joint Power Produced By Joint Torques Knee power & torque in STAIR ASCENT. Positive powers dominate by concentric contractions. Torque and velocity in same direction.
Joint Power Produced By Joint Torques Knee power & torque in STAIR DESCENT. Negative powers dominate by eccentric contractions. Torque and velocity in opposite directions.
Work Done By Joint Torques Positive work equal between groups in ascent. Negative work not equal between groups in descent.
Work Done By A Torque Joint torques during stair descent Old adults have larger hip torque and this torque performs more work: 0.41 vs. 0.24 J / kg Young adults have larger knee torque and this torque performs more work: 0.81 vs. 0.56 J / kg Positive work – concentric contraction – increase energy
Angular Kinetics Summary Torque Torque Torque Torque is more important in terms of loads on the body and loads produced by muscles than is weight. Torque causes all human rotations – external torques from various weights and forces (e.g. GRF) and internal torques from muscles combine to move animals around the environment.
Five Years From Now… Please come back and visit me in five years to tell me how much you understand about biomechanics.
Rotational Inertial Torques Inertial torques are caused by angular acceleration of body segment Solid line – joint or muscle torque in running Dashed line – inertial torque due to mass and length of body segments Inertial torques are very small – body segments offer little resistance to rotation - swing phase rotations are easy.
We hypothesize a generalized biomechanical principle that lower extremity muscles dissipate less mechanical energy in gait tasks that lower the center of mass compared to the mechanical energy they produce in gait tasks that raise the center of mass.
Subjects: 18 healthy males and females, age: 23 yr, mass: 71 kg.
Running velocity was constrained at 3.35 m/s (8 min/mile pace)
3-dimensional lower extremity joint powers and work were calculated through inverse dynamics. This work quantified muscular contributions to energy changes through the entire stride (termed: Joint Work).
Total work per stride was calculated from the change in subject’s total energy over the complete gait cycle in each gait (termed: d Energy)
One way ANOVA on 3 levels of incline with repeated measures and a few specific t-tests, p<.05
I love biomechanics, don’t you?
Do We have Any Idea What We Are Doing? Negative and positive joint work were identical in shoulder ab- and ad-duction and both were equal to change in energy.
Hip Knee Ankle Swing Stance Power (W) -500 500 Power (W) -500 500 Descent Level Ascent Power (W) -500 500
Joint Work & d Energy in Three Gaits Joint work: Descent = -108 J Level = 22 J Ascent = 159 J d Energy: Descent = -163 J Level = 12 J Ascent = 178 J * Joint Work ≠ d Energy, p<.001 * * *
Stance Phase Kinematics and Resultant GRFs We propose that the fundamental mechanism causing these results is the rate of acceleration in each movement. Muscles, through their lengthening contractions, dominate energy dissipation in movements with low accelerations and velocities. As the rate of acceleration increases other non-muscular tissues contribute to energy dissipation. Ascent Descent
Source of Mechanical Energy Generation Actin Myosin Power Stroke
Sources of Mechanical Energy Dissipation Art K: “The Jiggle Effect,” or “Mystery Work”
Level running had small bias towards positive joint work suggesting an equivalent result: muscles do more positive then negative work in locomotion.
Results may also partially explain the higher metabolic cost in ascending vs. descending gaits (i.e. more muscle effort) in addition to the decreased efficiency of shortening contractions used in ascent.
Data supported the hypothesized biomechanical principle that lower extremity muscles dissipate less mechanical energy in gait tasks that lower the center of mass compared to the mechanical energy they produce in gait tasks that raise the center of mass.