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Vector Algebra. Course Content. Introduction to the Course Biomechanical Concepts Related to Human Movement Anatomical Concepts Related to Human Movement Applications in Human Movement. Biomechanical Concepts. Basic Kinematic Concepts Vector Algebra Basic Kinetic Concepts. Vector Algebra.

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course content
Course Content
  • Introduction to the Course
  • Biomechanical Concepts Related to Human Movement
  • Anatomical Concepts Related to Human Movement
  • Applications in Human Movement
biomechanical concepts
Biomechanical Concepts
  • Basic Kinematic Concepts
  • Vector Algebra
  • Basic Kinetic Concepts
vector algebra4
Vector Algebra
  • Introductory Concepts
  • Vector Composition
  • Vector Resolution
vector algebra5
Vector Algebra
  • Introductory Concepts
  • Vector Composition
  • Vector Resolution
vector algebra introductory concepts
Vector Algebra: Introductory Concepts
  • Definitions
  • Vector representation
  • Muscle force vectors
definitions
Definitions
  • What is vector algebra?
  • What is a scalar quantity?
  • What is a vector quantity?
vector representation
Vector Representation

+y

+y

-x

-x

+x

+x

+z

+z

-y

-y

90°

180°

 = -40°

270°

vector representation9

Force Vector

Line of force

(or pull).

Length represents

magnitude.

Tail represents

point of force

application.

Arrow head represents

direction.

Vector Representation
  • A vector quantity is represented by an arrow.
slide12
Muscle

Force

Vectors

  • Point of application
  • Direction
  • Magnitude
  • Line of force

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide13
Muscle

Force

Vectors

  • Biceps brachii

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide14
Muscle

Force

Vectors

  • Brachialis

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide15
Muscle

Force

Vectors

  • Deltoid

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide16
Muscle

Force

Vectors

  • Pectoralis major

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide17
Muscle

Force

Vectors

  • Pectoralis major

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide18
Muscle

Force

Vectors

  • Pectoralis minor

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

vector algebra19
Vector Algebra
  • Introductory Concepts
  • Vector Composition
  • Vector Resolution
vector composition
Vector Composition
  • Process of determining a resultant vector from two or more vectors
  • New vector called the resultant (R)
vector composition graphical solution chaining

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.

Vector Composition: Graphical Solution (Chaining)
  • Select a vector to start with and draw it, maintaining direction and magnitude.
vector composition graphical solution chaining22

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.

Vector Composition: Graphical Solution (Chaining)
  • Chain the tail of the next vector to the head of the first, maintaining direction and magnitude from original vector.
vector composition graphical solution chaining23

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.

Vector Composition: Graphical Solution (Chaining)
  • Continue to chain vectors in this manner until they are all chained.
vector composition graphical solution chaining24

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.

Vector Composition: Graphical Solution (Chaining)
  • Draw in the resultant vector by connecting the tail of the first vector in the chain to the head of the last vector in the chain.
vector composition graphical solution chaining25

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.

Vector Composition: Graphical Solution (Chaining)
  • The head of the resultant vector will be the end that is connected to the head of the last vector.
vector composition graphical solution chaining26

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.

Vector Composition: Graphical Solution (Chaining)

Vector P = 50 N

What is the magnitude of the resultant vector?

order of chaining does not matter

R

B

D

C

A

Hamilton & Luttgens. (2001).

Fig 10.2. p. 267.

Order of chaining does not matter.

If A=50 N of force, what would you estimate the magnitude of R to be?

70°

How would you state the direction of R?

magnitude of r is dependent on direction of components not just magnitude
Magnitude of R is dependent on direction of components, not just magnitude.

If F=300 N of force, what would you estimate the magnitude of R to be?

How would you state the direction of R?

slide30

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-6. p. 64.

slide31

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-12. p. 69.

If Q=50 N of force, what would you estimate the magnitude of R to be?

How would you state the direction of R?

slide32

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-13. p. 69.

vector algebra33
Vector Algebra
  • Introductory Concepts
  • Vector Composition
  • Vector Resolution
vector resolution
Vector Resolution
  • Taking a resultant vector and breaking it down into 2 or more component vectors
there is an infinite of combinations of component vectors for any given r
There is an infinite # of combinations of component vectors for any given R.
  • 8 = 4 + 4
  • 8 = 3 + 1 + 2 + 2
  • 8 = 10 + (-2)
  • 8 = 1.5 + 6.5
so how do we know which components to resolve for37

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-33. p. 79.

So, how do we know which components to resolve for?
  • 2D (3D conceptually)
  • Orthogonal
  • Horizontal & Vertical
  • Exceptions
    • Muscles
    • Other
vector resolution graphical solution

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-33. p. 79.

Vector Resolution:Graphical Solution
  • Draw a rectangle which includes R as the diagonal of the rectangle.
slide39

Hamilton & Luttgens. (2001). Fig 10.1. p. 266.

If Vr was 200 m/s, what is the magnitude of Vv and Vh?

Vv or Vy

Vh or Vx

Why might you want to do this?

resolving muscle force vectors
Resolving Muscle Force Vectors

Direction of resolution is in direction of interest.

In this case, movement of shoulder girdle isvertical(elevation & depression) and horizontal (protraction & retraction).

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

resolving muscle force vectors42
Resolving Muscle Force Vectors
  • Draw line of pull.
  • Draw vertical component.
  • Draw horizontal component.
  • Complete rectangle to assure proper magnitudes of components.

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide43

What are the linear effects produced by this muscle?

  • Draw line of pull.
  • Draw vertical component.
  • Draw horizontal component.
  • Complete rectangle to assure proper magnitudes of components.

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide44

Draw line of pull.

  • Draw vertical component.
  • Draw horizontal component.
  • Complete rectangle to assure proper magnitudes of components.

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide45

If the resultant force is 100 N, how much force is acting to elevate the scapula? To retract the scapula?

  • Draw line of pull.
  • Draw vertical component.
  • Draw horizontal component.
  • Complete rectangle to assure proper magnitudes of components.

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

mechanical axis of a bone
Mechanical Axis of a Bone
  • The longitudinal axis of the bone
slide48

Draw in the normal component first.

Fnormal

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide49

Draw in the tangential component second.

Fnormal

Ftangential

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide50

Complete the rectangle to make sure that you have the lengths of your component vectors correct.

Fnormal

Ftangential

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide51

How would you express the direction of the resultant muscle force? The components?

Fnormal

Ftangential

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide52

What are the linear effects produced by this muscle?

Fnormal

Ftangential

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide53

If the resultant muscle force is 500 N, what is the magnitude of the components?

Fnormal

Ftangential

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.

slide54

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-30. p. 77.

  • Draw a line to represent the mechanical axis of the bone.
slide55

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-30. p. 77.

  • Draw in the normal component first.

Fnormal

slide56

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-30. p. 77.

  • Draw in the tangential component second.

Fnormal

Ftangential

slide57

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-30. p. 77.

  • Complete the rectangle to make sure that you have the lengths of your vectors correct.

Fnormal

Ftangential

slide58

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-30. p. 77.

Fnormal

How would you express the direction of the resultant muscle force? The components?

0

Ftangential

slide59

Fnormal

Ftangential

Fnormal

Ftangential

Component magnitudes vary, depending on magnitude & direction of R.

vector resolution other
Vector Resolution: Other

Fw,perpendicular

Fw,parallel

slide61

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-28. p. 75.

Fv

slide62

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-29. p. 76.

Differences in normal component?

slide63

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-29. p. 76.

Differences in tangential component?

Differences in muscle insertion angle?

slide64

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-31. p. 77.

slide65

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-32. p. 78.

slide66

From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-36. p. 82.

value of vector analysis
Value of Vector Analysis
  • Helps us understand forces and their effects!
for the next lecture day
For the next lecture day:
  • Lecture Topic #2
    • Subtopic C – Basic Kinetic Concepts