Force Composition and Force Resolution

Force Composition and Force Resolution

Force Composition

• Definition: Vector addition of multiple linear forces to produce a single, representative resultant force.
• Purpose: Determines the combined action of multiple internal and/or external forces acting simultaneously.
• Applications:
  • Determining normal or typical motion.
  • Identifying poor movement patterns or poor motor control.
• Types of Linear Forces:
  • Co-linear Forces: Forces that are in a straight line (and co-planar), producing a resultant that acts linearly or translatory (e.g., tug-of-war).
  • Non-co-linear Forces: Two or more vectors acting on the same object at angles to each other.

Methods for Representing/Adding Non-Linear Vectors

1. Parallelogram Method

• a + b = R
• Complete the parallelogram, maintaining the magnitudes of the two vectors.
• Draw the resultant force from the initial origin to the opposite end of the parallelogram.
• Advantage: Useful for two vectors.
• Disadvantage: Can become complex with more than two vectors.

2. Polygon Method

• a + b + c = R
• Start with a single vector (order doesn't matter due to commutative property: a + b = b + a).
• Place the tail of the second vector to the tip of the first, maintaining magnitude and direction.
• Continue until all vectors are used.
• Draw the resultant from the tail of the first vector (initial origin) to the tip of the last vector.

Common Mistakes with Force Composition

1. Magnitude not maintained.
2. Direction not maintained.
3. Tail-to-tip arrangement not followed in polygon method.
4. Resultant not drawn correctly (tail-to-tip or to the end of the completed parallelogram).

Force Composition: Functional Movement

• Representation of forces generated by muscles (e.g., rotator cuff muscles).
• Determination of the action of combined muscle forces.

Force Composition: Dysfunctional Movement

• Example: Torn supraspinatus affecting combined cuff muscle forces.
• Application to both internal and external forces.

Applications/Discussion

1. Muscle Actions and Testing

• Example: Anterior and posterior deltoid actions (flexion and extension, respectively).
  • Manual muscle test (MMT) for full deltoid in shoulder abduction.
  • Force composition demonstrates how anterior and posterior deltoid contribute to abduction.
• Example: Pectoralis major (clavicular and sternal heads).
  • Force composition to explain adduction.

2. Opposing Forces

• Determining the resulting action from multiple forces (e.g., G1, W1 pulling down, HF pulling up and to the left).

Force Resolution

• Definition: Opposite of force composition; resolving a resultant force into two component forces.
• Component forces are at right angles to each other, with the resultant in between.
• All three parts (two components and the resultant) have the same point of application.
• One component acts parallel to the segment/lever (Fx), typically providing joint compression/stabilization or joint distraction.
• The other component acts perpendicular to the segment/lever (Fy), typically providing the majority of rotary movement at the joint.
• Direction of vectors depends on direction of resultant.
• Lines connecting the tips form a rectangle.

Naming Conventions for Components

• Many different names exist depending on the area of study or application.
• Parallel Component (Fx):
  • Parallel component
  • Tangential component
  • Radial component
  • Unit vector designation
  • Other vector designations depending on the free body diagram (FBD)
• Perpendicular Component (Fy):
  • Perpendicular component
  • Normal component
  • Transverse component
  • Unit vector designation
  • Other vector designations depending on the FBD

Class Designations

• Tangential Component: Parallel to the lever or segment; primary function of compression or distraction.
• Normal Component: Perpendicular to the lever or segment; primary function of rotary movement.

Internal and External Forces

Internal Forces

• Example: Biceps force resolved into normal (Fy) and tangential (Fx) components.
  • Normal force contributes to joint angular motion.
  • Tangential force creates compression (A-B) or distraction (D) force on the joint; can act as a stabilizing force.

External Forces

• Example: Gravity.
  • Normal force acts on segment to try to produce motion opposite to muscle forces.
  • Tangential component may be compressive or distractive.

Tangential Component and Moment Arms

• Basic examples: tangential force going directly through the joint axis results in effect on joint is from linear force only (no moment arm).
• Typically, tangential components (internal and external) do not go directly through the joint: having a small moment arm allowing for some rotation at the joint as well as compressive/distractive forces.
• QLf and its components are examples.

Applications/Discussion

1. Internal and External Forces – What is Really Happening?

• Forces can influence or act on joints and tissues in multiple ways (not just movement).
  • Examine actions/functions of forces (QLf, FxQLf, FyQLf, GLf, FpLf) and their influences on the joint and other structures.

2. Effects on the Bar

• Determine the effects on the bar from given forces.

3. Making it ‘Harder’ or ‘Easier’

• Hand placements to resist or test knee extension.

4. Practice

• Resolve resultant vectors into their components (HF, G1, W1).

Trig Review

• Standard coordinate system relationships:
  • cos: related to the adjacent side of the right triangle (typically associated with the Cartesian system x-component, although not always).
  • sin: related to the opposite side of the right triangle (typically associated with the Cartesian system y-component, although not always).
  • tan: the association between the opposite and adjacent sides.

Right Triangle Relationships

• Typically used when the angle is known and one side magnitude is known:
  • cos \theta = \frac{Fx}{QLf} (adjacent/hypotenuse)
  • sin \theta = \frac{Fy}{QLf} (opposite/hypotenuse)
  • tan \theta = \frac{Fy}{Fx} (opposite/adjacent)

Right Triangle Relationships

• Typically used when the two sides are known but trying to calculate the angle:
  • \theta = cos^{-1}(\frac{Fx}{QLf}) [adjacent/hypotenuse]
  • \theta = sin^{-1}(\frac{Fy}{QLf}) [opposite/hypotenuse]
  • \theta = tan^{-1}(\frac{Fy}{Fx}) [opposite/adjacent]

Pythagorean Theorem

• Fx^2 + Fy^2 = QLf^2

Note: For exams, quizzes, etc., the equations as presented just above will be given. Memorization of the cos, sin, tan equations will not be required.

Example

• F is resolved into 2 components: Fx & Fy
• It is known that F = 63 and \theta = 25°
• What to know the magnitude of Fx and Fy

Rearrange the drawing to reflect trig/math rules:

• cos \theta = \frac{Fx}{F} (adjacent/hypotenuse)
• sin \theta = \frac{Fy}{F} (opposite/hypotenuse)
• cos(25) = \frac{Fx}{63}
• Fx = 63 * cos(25)
• Fx = 57.1 (rounded)
• sin(25) = \frac{Fy}{63}
• Fy = 63 * sin(25)
• Fy = 26.6

Magnitudes can be verified by:

• Fx^2 + Fy^2 = F^2
• (57.1)^2 + (26.6)^2 = F^2
• F = \sqrt{(57.1)^2 + (26.6)^2}
• F = 63

Notice that the Pythagorean Theorem could have been used to calculate one of the sides once the other side was known.

What if F, Fx, and Fy were known but you wanted to find the angle, θ?

• tan \theta = \frac{Fy}{Fx} (opposite/adjacent)
• \theta = tan^{-1}(\frac{Fy}{Fx})
• \theta = tan^{-1}(\frac{26.6}{57.1})
• \theta = 24.98° (Not exactly 25° because of rounding)

Glossary of Terms