CHAP4 Force System Resultant

4.7 Equivalent system
1. Equivalent system When the force and couple moment system produce the same “external” effects of translation and rotation of the body as their resultant , these two sets of loadings are said to be equivalent. 2. Principle of transmissibility The external effects on a rigid body remain unchanged,when a force, acting a given point on the body, is applied to another point lying on line of action of the force.
T
M
P
M
2. Vector quantity (1) Magnitude ( N-m or lb-ft) Mo = Fd d = moment arm or perpendicular distance from point O to the line of action of force.
d o
F
θ d r
F
Remark: (1) The couple moment is equivalent to the sum of the moment of both couple forces about any arbitrary point 0 in space.
Mo= rAx( -F )+ rB x F =(-rA+rB) x F =r x F= M
M=r x F
P F o
A
Force on Point A =Force on point O + couple moment on any point p.
Example:
o
F A
F o A
Point O is on the line of action of the force
F o d A o
4.6 moment of a couple
1. Definition ( couple) 偶力矩 Two parallel forces have the same magnitude, opposite distances, and are separated by a perpendicular distance d.
F
X
P
A
M= F d (Free vector) Mo= F d
Point O is not on the line of action of the force
4.8 Resultant of a force & couple system
1. Objective Simplify a system of force and couple moments to their resultants to study the external effects on the body. 2. Procedures for Analysis (1)Force summations FR=F1+F2+……+ΣF (2)Moment summations MR0= ΣMC+r1o*F1+r2o*F2= ΣMC+ ΣM0 MC:Couple moment in the system Mo:Couple moment about pt.O of the force in the system.
d
2. Scalar Formulation (1) Magnitude M=Fd
(2) Direction & sense (Right-hand rule) •Thumb indicates the direction •Curled fingers indicates the sense of rotation 3. Vector Formulation M= r x F |M|=M=|r x F |=r F sinθ =F d
2. Law of operation (1)
(2) (3) 3. Cartesian Vector Formulation (1) Cross product of Cartesian unit vectors.
v v v v v v i ×i = j × j = k ×k = 0 v v v v v v o i × j = i j sin 90 k = k v v v v v v j ×k = i ; k ×i = j
A F o
equivalent
F o
A
Sliding vector
4. Point O is not on the line of action of the force
F A o
Original system line of action
Couple moment
F F A r o -F
Mc= r x F
1. Moment of a force F about pt. O Mo = r x F where r = A position vector from pt. O to any pt. on the line of action of force F .
F
θ
(1) Magnitude
d
o
Mo=|Mo|=| r x F | =| r|| F | sinθ=F r sinθ =F d (2) Direction Curl the right-hand fingers from r toward F (r cross F ) and the thumb is perpendicular to the plane containing r and F.
4.4 Principle of moments
Varignon’s theorem The moment of a force about a point is equal to the sum of the moment of the force’s components about the point . r F
b a
Ma Ө
O
b’ Moment axis a’
Mo= r x F
A
v F
Axis of projection
(1) Moment of a force F about point 0 Mo = r × F Here, we assume that bb’ axis is the moment axis of Mo (2) Component of Mo onto aa´ axis Ma = Ma ua Ma=Mo cosθ =Mo ●ua=( r × F ) ● ua =trip scalar product Here Ma=magnitude of Ma ua= unit vector define the direction of aa´ axis
3.Resultant Moment of Coplanar Force System
M
Fn
RO
F1
do1 do2
don do3
F2
F3
4.2 Cross Product
1. Definition
(1) magnitude of
(2)Direction of perpendicular to the plane containing A & B
CHAPTER 4 Force System Resultant
4.1 Moment of a Force - - - Scalar Formulation
1.Moment A measure of the tendency of the force to cause a body to rotate about the point or axis. • Torque (T) 扭力 • Bending moment (M) 彎曲力矩
F1 r o Mo=r x F F = F1+F2 F2
Mo= r x (F1+F2) = r x F1+ r x F2 = MO1+MO2
4.5 Moment of a force about a specified Axis
1. Objective Find the component of this moment along a specified axis passes through the point about which the moment of a force is computed. 2. Scalar analysis (See textbook) 3. Vector analysis Point O on axis aa’
F d
d
-F
A
F -F
plane A // plane B
B -F
5. Resultant couple moment
F
Apply couple moment at any point p on a body and add them vectorially.
M2 M1
A
B
M1
M2
MR=ΣM=Σ r x F
-F r B F rB o r
Aห้องสมุดไป่ตู้
A
(2) Couple moment is a free vector which can act at any point in space.