Free Body Diagram FBD

1
Lecture 6

• Free Body Diagram (FBD)
• Rectangular Components of a Force
• Dot Product

2
Free Body Diagram (FBD)

• The rigid body (or part ) of interest is
  separated from all other interacting bodies and
  is drawn to scale.
• Draw all forces, known and unknown acting on that
  body as vectors
• magnitude, direction and sense
• Show all dimensions.

3
Example
Determine Fu, the force component along the axis
of bar AB and Fv, the force component along the
axis of bar AC. Note because bars AB and BC are
pinned ended members (two force member), then the
force must act along the length of the member
(tension or compression).
4
Example
Solution
1) Resolve angles and draw FBD of Point A
5
Example
2) Draw force system
3)Use sine law
6
Example
3) Check equilibrium in the x direction Fucos22.6
0 - Fvcos36.90 85.67-85.65 0.02

• Note
• AB is in tension
• AC is in compression

7
Rectangular Components

• Two dimensions-x,y coordinate system
• F Fx Fy Fxi Fyj
• Fx Fcos?
• Fy Fsin?
• Fz sqrt(Fx2 Fy2)

8
Rectangular Components

• Three Dimensions- x,y,z coordinate system
• F Fxi Fyj Fzk
• Fx Fcos?x
• Fy Fcos?y
• Fz Fcos?z
• F sqrt(Fx2 Fy2 Fz2)
• ?x,?y, and ?z are called direction cosines
• 00 lt ? lt 1800

9
Rectangular Components
z
y
x
10
Dot Product

• Also known as the Scalar Product
• see Appendix in text for more detail
• The rectangular component of a force F along any
  arbitrary direction n can be obtained by
  application of a dot product.
• Fx F i (Fx i , Fy j , Fzk ) i
• i i j j k k 1
• i j j i i k k i j k k j 0

11
Dot Product

• In general if en is a unit vector in the
  direction, n the rectangular component Fn of the
  force F is
• F en
• the unit vector en ( cos?xi , cos?yj , cos?zk )
• hence Fn F en
• ( Fxi , Fyj , Fzk ) ( cos?xi , cos?yj ,
  cos?zk )
• perform dot product
• Fxcos?x Fycos?y Fzcos?z

12
Dot Product

• The rectangular component Fn of force F can be
  expressed in cartesian vector form as
• Fnen Fn( cos?xi , cos?yj , cos?zk )
• The angle ? between the line of action of the
  force F and the direction n can be determined
  from the dot product relationship and the
  definition of rectangular component of a force.

13
Dot Product

• Fn Fcos? F en
• or ? cos- (Fen/F)
• In general the angle ? between any two vectors A
  and B or any two lines with unit vectors e1 and
  e2 is
• ? cos- (AB/AB)
• ? cos- ( e1e2)

14
Additional Material

• Read Section 2.5 - 2.9
• Problems Section 2
• 59,61,66,78,79,82