Static equilibrium. Biomechanics 2 Static Equilibrium. Free-Body diagram Internal forces of structures. Last week: Forces and Moments

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1 Static equilibrium Biomechanics 2 Static Equilibrium Free-Body diagram Internal forces of structures Last week: Forces and Moments Force F: tends to change state of rest or motion Moment M: force acting over an arm, tends to change state of rest or rotation Newton s definition (IV): An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line This week: Static equilibrium Static = no acceleration, i.e. no change of rest or motion Newton s second law: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed a = ΣF/m ΣF = ma = 0 Equilibrium = no net force and no net moment ΣF = 0 Decomposing the force F in x- and y component: ΣF x = 0 ΣF y = 0 ΣM = 0 All these forces come from interactions Newton s third law: When two bodies interact, the force of the first body on the second is equal and opposite to the force of the second one on the first

2 Forces always involve interaction When two bodies interact, the force of the first body on the second is equal and opposite to the force of the second one on the first Smooth incline Which forces are acting on the engine? Smooth incline Which forces are working on the truck and engine?

3 Isolation of mechanical system (Free-Body Diagram) 1. Clearly define particular body or mechanical system to be analysed 2. Isolate or cut away body (or system) from all surrounding bodies in a diagram showing the complete external boundary 3. Represent all forces acting on the body Known forces From other contacting bodies Don t forget gravity! Represent known forces by vectors showing correct magnitude, direction, sense Represent unknown forces by vectors with symbols for magnitude and direction. Incorrect assumptions about vector sense will reveal themselves in answer as negative numbers 4. Indicate coordinate axes on diagram

4 Types of forces Free-Body diagram (I)

5 Free-Body diagram (II) Free-Body Diagram (III) Muscle, tendon, ligament Only tensile forces Directed along organ axis Muscle force proportional to cross-sectional area Bone Compression, tension, bending Do not isolate through bone!!! Articular joint Compression only No tension No friction (no shear force) Normal force Rotate as hinge

6 Free-Body Diagram (IV) Equilibrium conditions Sum forces zero ΣF x = 0 ΣF y = 0 Sum moments is zero ΣM o = 0 Special cases Two-force members Forces must be: Equal in magnitude Opposite in direction Collinear F muscle on arm F joint on arm F earth on arm F w on arm Three-force members Lines of action must be concurrent Except when forces parallel Graphical solution: closed polygon (three force rule) Often, systems can be reduced to three-force members by combining known forces

7 Step 1 Find intersection of the two forces with known lines of action Step 2 Line of action of third force must intersect with the intersection point in step 1 to ensure zero sum of moments. Now the directions of all forces are known Step 3 Use force triangle to find magnitude of forces Three-force rule Typical situation: 1 force with direction and magnitude known (gravity) 1 force with direction known but magnitude unknown (muscle) 1 force with direction and magnitude unknown (joint force) R E H Alternative equilibrium equations & statical determinacy ΣF x = 0 ΣM A = 0 ΣM B = 0 Line AB NOT perpendicular to x-direction ΣM A = 0 ΣM B = 0 ΣM C = 0 Points A, B and C NOT on one line However you turn it, there are only three equations Statically indeterminate body or system More external supports or constraints than needed to maintain equilibrium Supports/constraints that can be removed without disturbing equilibrium are redundant Otherwise produce statically determinate system by linking supports that are (almost) parallel (example: muscles) Statically determinate body or system Support/constraints adequate: minimum number to maintain equilibrium Equilibrium equations sufficient to determine unknown forces Three unknowns can be found (force, couple, distance, angle) Improper constraints/support Not enough support/constraints to maintain equilibrium

8 Static equilibrium Make statically determinate system Knee Cartilage repair Rehabilitation Tibio-femoral force? Patella-femoral force? Step 1: Tibio-femoral force Normally, there are many separate muscles spanning any joint To make a statically determinate system, take all these muscles together in a single muscle force A Free-Body diagram Step 2: Patello-femoral force B Force intersection C Force polygon F p F pf F p

9 Summary This week Static equilibrium No acceleration Sum forces zero Sum moments zero Free body diagram Rigid bodies Next week Deformable bodies Distributed forces inside Stresses and strains