Course Overview
Statics (Freshman Fall)
Engineering Mechanics
Dynamics (Freshman Spring)
Strength of Materials (Sophomore Fall)
Mechanism Kinematics and Dynamics (Sophomore Spring )
Aircraft structures (Sophomore Spring and Junior Fall)
Vibration(Senior)
Statics:  F  0 force distribution on a system
Dynamics: x(t)= f(F(t)) displacement as a function of time and applied force
Strength of Materials: δ = f(F) deflection of deformable bodies subject to static
applied force
Vibration: x(t) = f(F(t)) displacement on particles and rigid bodies as a function of
time and frequency
1
Engineering Mechanics: Statics
Chapter 1 General Principles
•
•
•
•
•
Chapter Objectives
Basic quantities and idealizations of mechanics
Newton’s Laws of Motion and Gravitation
Principles for applying the SI system of units
General guide for solving problems
Chapter Outline
•
•
•
•
•
•
Mechanics
Fundamental Concepts
Units of Measurement
The International System of Units
Numerical Calculations
General Procedure for Analysis
2
1.1 Mechanics
• Mechanics can be divided into 3 branches:
- Rigid-body Mechanics
- Deformable-body Mechanics
- Fluid Mechanics
• Rigid-body Mechanics deals with
- Statics – Equilibrium of bodies, at rest, and Moving with constant velocity
- Dynamics – Accelerated motion of bodies
3
1.2 Fundamentals Concepts
Basic Quantities
• Length - locate the position of a point in space
• Mass - measure of a quantity of matter
• Time - succession of events
• Force - a “push” or “pull” exerted by one body on another
Idealizations
• Particle - has a mass and size can be neglected
• Rigid Body - a combination of a large number of particles
• Concentrated Force - the effect of a loading
4
1.2 Newton’s Laws of Motion
•
•
•
First Law - “A particle originally at rest, or
moving in a straight line with constant velocity,
will remain in this state provided that the
particle is not subjected to an unbalanced force.”
Second Law - “A particle acted upon by an
F  ma
unbalanced force F experiences an acceleration
a that has the same direction as the force and a
magnitude that is directly proportional to the
force.”
Third Law - “The mutual forces of action and
reaction between two particles are equal and,
opposite and collinear.”
5
Chapter 2 Force Vector
Chapter Outline
•
•
•
•
•
•
•
•
•
Scalars and Vectors
Vector Operations
Vector Addition of Forces
Addition of a System of Coplanar Forces
Cartesian Vectors
Addition and Subtraction of Cartesian Vectors
Position Vectors
Force Vector Directed along a Line
Dot Product
6
2.1 Scalar and Vector
• Scalar
– A quantity characterized by a positive or negative number, indicated by
letters in italic such as A, e.g. Mass, volume and length
• Vector
– A quantity that has magnitude and direction, e.g. position, force and
moment, presented as A and its magnitude (positive quantity) as A
• Vector Subtraction R’ = A – B = A + ( - B )
Finding a Resultant Force
• Parallelogram law is carried out to find the resultant force FR = ( F1 + F2 )
7
2.4 Addition of a System of Coplanar Forces
Scalar Notation
– Components of forces expressed as algebraic scalars
Fx  F cos and Fy  F sin 
Cartesian Vector Notation
• Cartesian unit vectors i and j are used to designate
the x and y directions with unit vectors i and j
F  Fx i  Fy j
•
Coplanar Force Resultants
F1  F1x i  F1 y j
F2   F2 x i  F2 y j
F3  F3 x i  F3 y j
Coplanar Force Resultants
FR  F1  F2  F3   FRx  i   FRy  j
Scalar notation
FRx  F1x  F2 x  F3 x
FRy  F1 y  F2 y  F3 y
8
Example 2.6
The link is subjected to two forces F1 and F2. Determine the magnitude and
orientation of the resultant force.
Cartesian Vector Notation
F1 = { 600cos30°i + 600sin30°j } N
F2 = { -400sin45°i + 400cos45°j } N
Thus,
FR = F1 + F2
= (600cos30º - 400sin45º)i
+ (600sin30º + 400cos45º)j
= {236.8i + 582.8j}N
The magnitude and direction of FR are determined in the same manner as before.
9
2.5 Cartesian Vectors
• Right-Handed Coordinate System
A rectangular or Cartesian coordinate system is said to be
right-handed provided:
• Rectangular Components of a Vector
– A vector A may have one, two or three rectangular
components along the x, y and z axes, depending on
orientation
– By two successive applications of the parallelogram law
A = A’ + Az
A’ = Ax + Ay
– Combing the equations,
A can be expressed as
A = Ax + Ay + Az
A  A uA
10
2.5 Direction Cosines of a Cartesian Vector
– Orientation of A is defined as the coordinate direction
angles α, β and γ measured between A and the positive x,
y and z axes, 0°≤ α, β and γ ≤ 180 °
– The direction cosines of A is
Ax
cos  
A
Ay
cos  
A
Az
cos  
A
uA  A / A  ( Ax / A)i + ( Ay / A) j + ( Az / A)k
uA  cos  i + cos j + cos k
cos2  + cos2  + cos2 =1
A  A uA
 A cos i + A cos j + A cos k
 Ax i + Ay j + Az k
11
Example 2.8
Express the force F as Cartesian vector.
Since two angles are specified, the third angle is found by
cos 2   cos 2   cos 2   1
cos 2   cos 2 60o  cos 2 45o  1
2
2
cos   1  (0.5)  (0.707 )  ±0.5
  cos 1 (0.5) 60 o or   cos1  0.5  120 o
By inspection, β= 60º since Fx is in the +x direction
Given F = 200N
F  F cos  i  F cos  j  F cos  k
 (200cos 60 )i  (200cos 60 ) j  (200cos 45 ) k
= 100.0 i  100.0 j  141.4k N
Checking:
F  Fx2  Fy2  Fz2

100 .0  100 .0  141 .4
2
2
2
12
 200 N
2.7 Position Vector: Displacement and Force
Position Displacement Vector
– r = xi + yj + zk
F can be formulated as a Cartesian vector
F = F u = F (r / r )
13
Example 2.13
The man pulls on the cord with a force of 350N.
Represent this force acting on the support A as a
Cartesian vector and determine its direction.
Solution:
r 
3m   2m   6m
2
2
2
 7m
14
2.9 Dot Product
Laws of Operation
Cartesian Vector Formulation
1. Commutative law
- Dot product of Cartesian unit vectors
A·B = B·A or ATB=BTA
i·i = 1 j·j = 1 k·k = 1
2. Multiplication by a scalar
i·j = 0 i·k = 1 j·k = 1
a(A·B) = (aA)·B = A·(aB) = (A·B)a
3. Distribution law
A·(B + D) = (A·B) + (A·D)
4. Cartesian Vector Formulation
- Dot product of 2 vectors A and B
A  B = A x Bx  A y By  Az Bz
5. Applications
- The angle formed between two vectors
  cos1 ( A  B / ( A B )) 
00    1800
- The components of a vector parallel and perpendicular to a line
Aa  A cos   A  u
15
Example 2.17
The frame is subjected to a horizontal force F = {300j} N.
Determine the components of this force parallel and
perpendicular to the member AB.
Since
uB 
rB

rB
2i  6 j  3k
 2    6    3
2
2
2
 0.286i  0.857 j  0.429k
Thus
FAB  F cos 
 F .uB   300 j    0.286i  0.857 j  0.429k 
 (0)(0.286)  (300)(0.857)  (0)(0.429)
 257.1N
16
Solution
Since result is a positive scalar, FAB has the same sense of
direction as uB.
FAB  FAB uAB
  257.1N  0.286i  0.857 j  0.429k 
 {73.5i  220 j  110k}N
Perpendicular component
F  F  FAB  300 j  (73.5i  220 j  110k )  {73.5i  80 j  110k}N
Magnitude can be determined from F┴ or from Pythagorean Theorem
F 
2
2
F  FAB 
 300 N    257.1N 
2
2
 155 N
17
Chapter 3 Equilibrium of a Particle
Chapter Objectives
• Concept of the free-body diagram for a particle
• Solve particle equilibrium problems using the equations of
equilibrium
Chapter Outline
•
•
•
•
Condition for the Equilibrium of a Particle
The Free-Body Diagram
Coplanar Systems
Three-Dimensional Force Systems
18
3.2 The Free-Body Diagram
• Spring
– Linear elastic spring: with spring constant or stiffness k. F = ks
• Cables and Pulley
– Cables (or cords) are assumed negligible weight and cannot
stretch
– Tension always acts in the direction of the cable
– Tension force must have a constant magnitude for
equilibrium
– For any angle , the cable is subjected to a constant tension T
Procedure for Drawing a FBD
1. Draw outlined shape
2. Show all the forces Active forces: particle in motion and Reactive forces:
constraints that prevent motion
3. Identify each of the forces
19
Example 3.1
The sphere has a mass of 6kg and is supported. Draw a free-body diagram of the
sphere, the cord CE and the knot at C.
FBD at Sphere
Two forces acting, weight 6kg (9.81m/s2) = 58.9N
and the force on cord CE.
Cord CE
Two forces acting: sphere and knot
For equilibrium, FCE = FEC
FBD at Knot
3 forces acting: cord CBA, cord CE and spring CD
20
21
3.4 Three-Dimensional Systems
Example 3.7
Determine the force developed in each cable used to support the 40kN crate.
FBD at Point A
To expose all three unknown forces in the cables.
Equations of Equilibrium
Expressing each forces in Cartesian vectors,
FB = FB(rB / rB)
= -0.318FBi – 0.424FBj + 0.848FBk
FC = FC (rC / rC)
= -0.318FCi – 0.424FCj + 0.848FCk
FD = FDi and W = -40k
22
Solution
For equilibrium,
∑F = 0;
FB + FC + FD + W = 0
– 0.318FBi – 0.424FBj + 0.848FBk – 0.318FCi
– 0.424FCj + 0.848FCk + FDi - 40k = 0
∑Fx = 0;
∑Fy = 0;
∑Fz = 0;
– 0.318FB – 0.318FC + FD = 0
– 0.424FB
– 0.424FC = 0
0.848FB + 0.848FC – 40 = 0
Solving,
FB = FC = 23.6kN
FD = 15.0kN
23
Chapter 4 Force System Resultants
Chapter Objectives
•Concept of moment of a force in two and three dimensions
•Method for finding the moment of a force about a specified axis.
•Define the moment of a couple.
•Determine the resultants of non-concurrent force systems
•Reduce a simple distributed loading to a resultant force having a specified location
Chapter Outline
•
•
•
•
•
•
•
•
•
Moment of a Force – Scalar Formation
Cross Product
Moment of Force – Vector Formulation
Principle of Moments
Moment of a Force about a Specified Axis
Moment of a Couple
Simplification of a Force and Couple System
Further Simplification of a Force and Couple System
Reduction of a Simple Distributed Loading
24
4.1 Moment of a Force – Scalar Formation
•
•
Moment of a force about a point or axis – a measure of the tendency of the force
to cause a body to rotate about the point or axis
Torque – tendency of rotation caused by Fx or simple moment (Mo)z
Magnitude
• For magnitude of MO, MO = Fd (Nm)
where d = perpendicular distance
from O to its line of action of force
Direction
• Direction using “right hand rule”
Resultant Moment
MRo = ∑Fd
25
4.2 Cross Product
•
Cross product of two vectors A and B yields C, which is
written as C = A x B = (AB sinθ)uC
Laws of Operations
1. Commutative law is not valid
AxB≠BxA
Rather, A x B = - B x A
2. Multiplication by a Scalar
a( A x B ) = (aA) x B = A x (aB) = ( A x B )a
3. Distributive Law
Ax(B+D)=(AxB)+(AxD)
Proper order of the cross product must be
maintained since they are not commutative
26
4.2 Cross Product
Cartesian Vector Formulation
• Use C = AB sinθ on a pair of Cartesian unit vectors
• A more compact determinant in the form as
i
j
k
A  B  Ax
Ay
Az
Bx
By
Bz
•
Moment of force F about point O can be expressed using cross product
MO = r x F
•
For magnitude of cross product,
MO = rF sinθ
Treat r as a sliding vector. Since d = r sinθ,
MO = rF sinθ = F (rsinθ) = Fd
•
27
4.3 Moment of Force - Vector Formulation
For force expressed in Cartesian form,
i
MO  r  F   rx
 Fx
j
ry
Fy
k
rz 
Fz 
with the determinant expended,
M0  (ry Fz  rz Fy )i  (rx Fz  rz Fx ) j  (rx Fy  ry Fx )k
 0
  rz
 ry
rz
0
rx
ry 
rx 
0 
 Fx 
F 
 y
 Fz 
28
 4 12 12
or =
rAB
rAB
rAB
rAB
 0 12 0   4 
 12 0 0   12 



 0
0 0   12 
29
Example 4.4
Two forces act on the rod. Determine the resultant moment
they create about the flange at O. Express the result as a
Cartesian vector.
rA  5 j m
rB  4i  5 j  2k  m
The resultant moment about O is
MO    r  F   rA  F  rB  F
j k i
j
 i
  0
5 0    4 5
 60 40 20  80 40
0 0 5  60   0 2
  0 0 0   40    2 0
5 0 0   20   5 4
 30i  40 j  60k  kN  m
k 
2 
30 
5  80 
4   40 
0   30 
30
4.4 Principles of Moments
• Since F = F1 + F2,
MO = r x F
= r x (F1 + F2)
= r x F1 + r x F2
4.5 Moment of a Force about a Specified Axis
Vector Analysis
• For magnitude of MA,
MA = MOcosθ = MO·ua
where ua = unit vector
• In determinant form,
uax
uay
uaz
Ma  uax  (r  F )  rx
Fx
ry
Fy
rz
Fz
31
Example 4.8
Determine the moment produced by the force F which tends to
rotate the rod about the AB axis.
0.6
 0 
rc   0  , F   0 
 0.3
 300 
0  0   0 
 0 0.3
M  r  F  0.3
0
0.6   0   180
 0
0.6
0   300   0 
0.4
0.4 
1
rB  0.2 , uB  0.2 
0.2
 0 
 0 
M AB  uBT
 0 
1
M   0.4 0.2 0 180 
 80.4
0.2
 0 
32
Solution:
rCD   0.4 0.4 0.2 
T
F
300rCD
rCD
rOC   0.1 0.4 0.3
0.3 0.4   200   100 
 0
M  rOC  F   0.3
0
0.1  200    50 
 0.4 0.1
0   100   100 
 100 
1
MOA  rOAT M
  0.6 0.8 0  50   100
rOA
 100 
33
4.6 Moment of a Couple
•
Couple
– two parallel forces of the same magnitude but opposite direction separated
by perpendicular distance d
• Resultant force = 0
Scalar Formulation
• Magnitude of couple moment M = Fd
• M acts perpendicular to plane containing the forces
Vector Formulation
• For couple moment, M = r x F
Equivalent Couples
• 2 couples are equivalent if they produce the same moment
• Forces of equal couples lie on the same plane or plane parallel to one another
34
Example 4.12
Determine the couple moment acting on the pipe. Segment
AB is directed 30° below the x–y plane.
Take moment about point O,
M = rA x (-250k) + rB x (250k)
= (0.8j) X (-250k) + (0.66cos30ºi
+ 0.8j – 0.6sin30ºk) x (250k)
= {-130j}N.cm
Take moment about point A
M = rAB x (250k)
= (0.6cos30°i – 0.6sin30°k) x (250k)
= {-130j}N.cm
Take moment about point A or B,
M = Fd = 250N(0.5196m) = 129.9N.cm
M acts in the –j direction M = {-130j}N.cm
35
4.7 Simplification of a Force and Couple System
• Equivalent resultant force acting at point O and a resultant
couple moment is expressed as
FR   F
 MR O   MO   M
• If force system lies in the x–y plane
and couple moments are
perpendicular to this plane,
 FR x   Fx
 FR y   Fy
 MR O   MO   M
36
4.8 Simplification of a Force and Couple System
Concurrent Force System
• A concurrent force system is where lines of action of all the forces
intersect at a common point O
FR   F
Coplanar Force System
•
Lines of action of all the forces lie in the same plane
•
Resultant force of this system also lies in this plane
37
4.8 Simplification of a Force and Couple System
•
Consists of forces that are all parallel to the z axis
38
4.9 Distributed Loading
• Large surface area of a body may be subjected to distributed loadings,
often defined as pressure measured in Pascal (Pa): 1 Pa = 1N/m2
Magnitude of Resultant Force
• Magnitude of dF is determined from differential area dA under the
loading curve.
• For length L, FR   wx dx   dA  A
L
A
Location of Resultant Force
• dF produces a moment of xdF = x w(x) dx about O
xFR   xw( x)dx
L
• Solving for
x
x
 xw( x)dx
L
 w( x)dx
L
39
Example 4.21
Determine the magnitude and location of the equivalent resultant force acting
on the shaft.
Solution
For the colored differential area element,
dA  wdx  60 x 2 dx
For resultant force
FR  F ;
2
FR   dA   60 x2 dx
A
0
2
 x3 
 23 03 
 60    60     160 N
 3 0
3 3
For location of line of action,
2
 x4 
 24 04 
2
A xdA 0 x(60 x )dx 60  4 0 60  4  4 
x



 1.5m
160
160
160
 dA
2
A
40
Chapter 5 Equilibrium of a Rigid Body
Objectives
• Develop the equations of equilibrium for a rigid body
• Concept of the free-body diagram for a rigid body
• Solve rigid-body equilibrium problems using the equations
of equilibrium
Outline
•
•
•
•
•
•
•
Conditions for Rigid Equilibrium
Free-Body Diagrams
Equations of Equilibrium
Two and Three-Force Members
Free Body Diagrams
Equations of Equilibrium
Constraints and Statical Determinacy
41
5.1 Conditions for Rigid-Body Equilibrium
•
The equilibrium of a body is expressed as
FR   F  0
 MR O   MO  0
•
Consider summing moments about some other point, such
as point A, we require
M
A
 r  FR   MR O  0
42
5.2 Free Body Diagrams
Support Reactions
• If a support prevents the translation of a body in a given direction, then
a force is developed on the body in that direction.
• If rotation is prevented, a couple moment is exerted on the body.
43
5.2 Free Body Diagrams
Types of Connection
(1)
Reaction
θ
Number of Unknowns
One unknown. The reaction is a tension force which acts away from the
member in the direction of the cable
θ
F
cable
(2)
θ
θ
θ
F
F
One unknown. The reaction is a force which acts along the axis of the
link.
weightless link
One unknown. The reaction is a force which acts perpendicular to the
surface at the point of contact.
(3)
θ
θ
roller
F
One unknown. The reaction is a force which acts perpendicular to the
slot.
(4)
roller or pin
in confined
smooth slot
θ
F
F
θ
θ
One unknown. The reaction is a force which acts perpendicular to the
surface at the point of contact.
(5)
θ
rocker
θ
F
44
5.2 Free Body Diagrams
One unknown. The reaction is a force which acts perpendicular to
the surface at the point of contact.
(6)
θ
smooth
contacting
surface
F
θ
(7)
One unknown. The reaction is a force which acts perpendicular to
the rod.
member
pin
connected
θ
F
F
θ
θ
(8)
Fy
F
θ
smooth pin
or hinge
φ
Fx
Two unknowns. The reaction are the couple moment and the force
which acts perpendicular to the rod.
(9)
Member
fixed
connected
to collar on
smooth rod
M
F
Fy
(10)
F
φ
fixed
support
Two unknowns. The reactions are two components of force, or the
magnitude and direction φ of the resultant force. Note that φ and θ
are not necessarily equal [usually not, unless the rod shown is a
link as in (2)].
Three unknowns. The reaction are the couple moment and the two
force components, or the couple moment and the magnitude and
direction φ of the resultant force.
Fx
M
45
5.2 Free Body Diagram
Weight and Center of Gravity
•
•
•
Each particle has a specified weight
System can be represented by a single resultant force,
known as weight W of the body
Location of the force application is known as the center of
gravity
46
Example 5.1
Draw the free-body diagram of the uniform beam. The beam has a mass of
100kg.
Solution
Free-Body Diagram
• Support at A is a fixed wall
• Two forces acting on the beam at A denoted as Ax, Ay, with moment MA
• Unknown magnitudes of these vectors
• For uniform beam,
Weight, W = 100(9.81) = 981N
acting through beam’s center of gravity, 3m from A
47
5.4 Two- and Three-Force Members
Two-Force Members
• When forces are applied at only two points on a member, the member is
called a two-force member
• Only force magnitude must be determined
Three-Force Members
When subjected to three forces, the forces are concurrent or parallel
48
5.5 3D Free-Body Diagrams
Types of Connection
Reaction
Number of Unknowns
(1)
F
One unknown. The reaction is a force
which acts away from the member in the
known direction of the cable.
cable
One unknown. The reaction is a force
which acts perpendicular to the surface at
the point of contact.
(2)
smooth surface
support
F
One unknown. The reaction is a force
which acts perpendicular to the surface at
the point of contact.
(3)
roller
F
(4)
Three unknown. The reaction are three
rectangular force components.
Fz
ball and socked
Fy
Fx
Mz
(5)
Fz
Mx
single journal
bearing
Fx
Four unknown. The reaction are two force
and two couple moment components which
acts perpendicular to the shaft. Note: The
couple moments are generally not applied
of the body is supported elsewhere. See the
example.
49
5.5 Free-Body Diagrams
Mz
(6)
Five unknowns. The reaction are two force and three
couple moment components. Note: The couple moments
are generally not applied if the body is supported
elsewhere. See the example.
Fz
Fx
single journal bearing
with square shaft
My
Mx
(7)
Five unknowns. The reaction are two force and three
couple moment components. Note: The couple moments
are generally not applied if the body is supported
elsewhere. See the example.
Mz
Fz
Fy
single thrust
bearing
Mx
Mz
(8)
Five unknowns. The reaction are two force and three
couple moment components. Note: The couple moments
are generally not applied if the body is supported
elsewhere. See the example.
Fz
My
single smooth
pin
Fx
Fy
Mz
(9)
Fz
Fy
My
Fx
single hinge
Five unknowns. The reaction are two force and three
couple moment components. Note: The couple moments
are generally not applied if the body is supported
elsewhere. See the example.
Mx
Mz
(10)
Six unknowns. The reactions are three force and three
couple moment components.
Fz
Fx
fixed support
Mx
Fy
My
50
5.7 Constraints for a Rigid Body
Redundant Constraints
• More support than needed for equilibrium
• Statically indeterminate: more unknown
loadings than equations of equilibrium
51
5.7 Constraints for a Rigid Body
Improper Constraints
• Instability caused by the improper constraining by the supports
• When all reactive forces are concurrent at this point, the body is
improperly constrained
52