Lecture # 2-3D Vectors, Free Body Diagram, and Moments PDF

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Summary

Lecture # 2-3D Vectors, Free Body Diagram, and Moments
Agenda
3D Vectors
2D and 3D Position Vectors
FreeBodyDiagrams
EquationsofEquilibrium
Moment of Forces
Moment of a Couple
Cross Product...

Description

ENG TEC 3SD3- Statics and Dynamics Lecture # 2- 3D Vectors,Free Body Diagram, and Moments Dr. Ahmed Mostafa P.Eng., P.E., CEng MIEI., CPEng MIEAust., IntPE.(Canada). CEO– CMTE Inc. Adjunct Professor Ryerson University&Lakehead University E-mail: [email protected] Summer 2021

Agenda  3D Vectors  2D and 3D Position Vectors  Free Body Diagrams  Equations of Equilibrium  Moment of Forces  Moment of a Couple  Cross Product

Vector Resolution in 3D

y

F

Fx= F Cos  Fy= F Sin 

 Fy



Fx= F Sin  Fy= F Cos 

 x

Fx

F= Fx i+Fy j F

Fx2  Fy2

A= A’ + Az A’= Ax + Ay A= Ax + Ay + Az ( all three components of A act in positive i, j, and k

y

F

Fx= F Cos  Fy= F Sin 

 Fy



Fx= F Sin  Fy= F Cos 

 Fx F

Fx2  Fy2

Cos = Fx/F Cos = Fy/F

x F= Fx i+Fy j

, and  are measured between the tail of A and the positive x, y and z

A= A’ + Az A’= Ax + Ay A= Ax + Ay + Az ( all three components of A act in positive i, j, and k

Fr=800N

x

2D Position Vector Fixed vector which locates a point relative to another point

y B (2,2)

r

X =(Bx-Ax) =2 Y= (By-Ay)=2 r=

x2  y2

y

A (0,0)

x

x

We can now find all the angles

r= xi+yj

It is important in formulating a Cartesian force vector directed between two points in space.

3D Position Vector Fixed vector which locates a point in space relative to another point

A

B

Free Body Diagrams 







A free body diagram (force diagram or FBD) is a graphical illustration used to visualize the applied forces, movements, and resulting reactions on a body in a given condition. They depict a body or connected bodies with all of the applied forces and moments, as well as reactions, that act on that/those body(ies). The body may consist of multiple internal members, for example, a truss, or be a compact body such as a beam. A series of free bodies and other diagrams may be necessary to solve complex problems.

Free Body Diagrams

Free-Body Diagram All forces acting on A are shown 󰇟B 󰇟Beer eer et a. Book󰇠

Free-Body Diagrams – Truss Case

All forces on the truss shown (External and reactions)

Free-Body Diagrams TBC

TDC TAC

TCA

TCD

TDE

TCB W= 200#

Free-Body Diagrams

Cables and Pulleys

   

Cable supports tension Neglect weight No stretch Constant tension throughout its length

Springs

Smooth Contact Surface

Equations of Equilibrium |

In equilibrium only if the two horizontal forces are equal in magnitude and have opposite sides. -100 + 100 = 0 Therefore, the first equation of equilibrium: Sum of horizontal forces should be equal to zero.

F

x

0

Equations of Equilibrium

In equilibrium only if the lifting force has equal magnitude and opposite side with the total weight. +200 – 2x100 = 0 Therefore, the second equation of equilibrium: Sum of vertical forces should be equal to zero.

F

y

0

Static Equilibrium 

The state of an object in which the forces applied counteract each other so that the object remains stationary.

3D Force System

Moment of Force  

The turning effect of a force (torque) is known as the moment. It is the product of the force multiplied by the perpendicular distance from the line of action of the force to the pivot or point where the object will turn.

The moment of a force about a point is equal to the sum of the moments of the force components about the same point.

M=F*d

M=F*d

Assume counterclockwise is positive

Moment of a Force M=F*d

𝑎

󰇛𝑏󰇜

In which of the above cases the moment is higher?

Moment of a Couple

Equations of Equilibrium |

In equilibrium only if the two horizontal forces are equal in magnitude and have opposite sides. -100 + 100 = 0 Therefore, the first equation of equilibrium: Sum of horizontal forces should be equal to zero.

F

x

0

Equations of Equilibrium

In equilibrium only if the lifting force has equal magnitude and opposite side with the total weight. +200 – 2x100 = 0 Therefore, the second equation of equilibrium: Sum of vertical forces should be equal to zero.

F

y

0

Equations of Equilibrium

In equilibrium only if the two moments of the weights are equal in magnitude and have opposite sides. 50x8 – 200x2 = 0 Therefore, the third equation of equilibrium: Sum of moments should be equal to zero.

M

z

0

Cross Product...