Moment and couple In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. z y x z y x o F F F r r r k j i F r M ˆ ˆ ˆ = × = r r r k F r F r j F r F r i F r F r M x y y x z x x z y z z y o ˆ ) ( ˆ ) ( ˆ ) ( − + − + − = r F r M o r r r × = M x = − F y r z + F z r y + x y + z + M y = F x r z − F z r x M z = −F x r y +F y r x x y z r r O A F r r z F x F y r x r y F z
21
Embed
Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Moment and coupleIn 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous.
zyx
zyxo
FFFrrrkji
FrM
ˆˆˆ
=×=rrr
kFrFrjFrFriFrFrM xyyxzxxzyzzyoˆ)(ˆ)(ˆ)( −+−+−=
r
FrM o
rrr×=
Mx = − Fyrz + Fzry
+
x
y+
z+
My = Fxrz − Fzrx
Mz= −Fxry +Fyrxx
y
zrr
O
A
Fr
rzFx
Fy
rxry
Fz
Moment about an arbitrary axis
O
OMr
rrFrn̂
λ
λMr 1. Calculate moment
FrM o
Find moment about λ axis
rrr×=
2. Calculate projection of moment on λ axis
nnFrnnMM O ˆ)ˆ(ˆ)ˆ( ⋅×=⋅=rrrr
λ
)ˆˆˆ.(
ˆˆˆ
)ˆ( kjiFFFrrrkji
nFr
zyx
zyx γβα ++=⋅×rr
γβα
γβα
zyx
zyx
zyx
zyx FFFrrr
FFFrrr ==
kji ˆˆˆ γβα ++
Varignon’s Theorem
O
r
3Fr
A
1Fr
2Fr
...)(... 321321 +++×=+×+×+×= FFFrFrFrFrM o
rrrrrrrrrrr
)(∑×= Frrr
- Sum of the moments of a system of concurrent forces about a given point equals the moment of their sum about the same point
RrFrM o
rrrrr×=×=∑ )(
Couples(1)
O
Fr
Fr
−
ArrBrr
rr
A
B
Mr
d
FrrFrFrM BABA
rrrrrrrr×−=−×+×= )()(
-Couple is a moment produced by two force of equal magnitude but opposite in direction.
FrMrrr
×=
- = vector from any point on the line of action of to any point on the line of action of
- Moment of a couple is the same about all point Couple may be represented as a free vector.
- Direction: normal to the plane of the two forces (right hand rule)- Recall: Moment of force about a point is a sliding vector.
Fr F
r−rr
1Fr
2Fr
2Fr
−
1Fr
−2Mr
1Mr
1Mr
2Mr
Fr
Mr
Fr
−
Couples(2)
[Couple from F1]+[Couple from F2] = [Couple from F1+F2]
couples are free vector. the line of action or point of action are not needed!!!
Force – couple systems
No changes in the net external effects
AB
Fr
A
B
Fr
AB
FrF
r
Fr
−
rr
FrMrrr
×=
- = Moment of about point B
- is a vector start from point B to any point on the line of action of
FrMrrr
×= Fr
rr
Fr
Couple
Sample 1A Tension T of magnitude 10 kN is applied to the cable attached to the top A of the rigid mast and secured to the ground at B. Determine the moment Mz of T about the z-axis passing through the base O.
Sample 2Determine the magnitude and direction of the couple M which will replace the two given couples and still produce the same external effect on the block. Specify the two force F and –F, applied in the two faces of the block parallel to the y-z plane, which may replace the four given forces. The 30-N forces act parallel to the y-z plane.
Sample 3A force of 400 N is applied at A to the handle of the control lever which is attached to the fixed shaft OB. In determining the effect of the force on the shaft at a cross section such as that at O, we may replace the force by an equivalent force at O and a couple. Describe this couple as a vector M.
Sample 4If the magnitude of the moment of F about line CD is 50 Nm, determine the magnitude of F.
Sample 5Tension in cable AB is 143.4 N. Determine the moment about the x-axis of this tension force acting on point A . Compare your result to the moment of the weight W of the 15-kg uniform plate about the x-axis. What is the moment of the tension force acting at A about line OB
Summary (Force-Moment 3-D)Force1. Determine coordinate2. Determine unit vector3. Force can be calculate
Angle between force and x-,y-,z-axis1. Force = Fxi + Fyj + Fzk2. Determine amplitude of force F3. cosθx = Fx/F, cosθy = Fy/F, cosθz = Fz/F
Angle between force and arbitrary axis1. Determine unit vectors (nF, n)2. cosθ = nF・ n
Summary (Force-Moment 3-D)
Vector method
Moment about an arbitrary point O1. Determine r and F2. Cross vectorMoment about an arbitrary axis1. Determine moment about any point on the axis MO2. Determine unit vector of the axis n3. Moment about the axis = MO・n
Angle between moment and axis
Same as angle between force and axis
Moment Consider to use vector method or scalar method
Resultants(1)
Select a point to find moment
Replace forces with forces at point O + couples
Add forces and couples vectorially to get the resultant force and moment
Step1 Step2 Step3
∑=+++= FFFFRrrrrr
...321
∑ ×=+++= )(...321 FrMMMMrrrrrr
Resultants(2)2-D
A Fr
B
A
BM=Fd
Fr
FMvv
⊥Force + couple can be replaced by a force F by changing the position of F.
3-D
Rr
Mr
O 1Mr
2Mr RM
vv⊥2
M2 and R can be replaced by one force R by changing the position of R.
RMvv
//1
M1 can not be replaced
Wrench resultant(1)
M2=Rd
Wrench resultant(2)
2-D: All force systems can be represented with only one resultant force or couple
3-D: All force systems can be represented with a wrench resultant
Wrench: resultant couple M parallel to the resultant force Rr r
Sample 6Determine the resultant of the system of parallel forces which act on the plate. Solve with a vector approach.
Sample 7Replace the two forces and the negative wrench by a single forceR applied at A and the corresponding couple M.
Sample 8Determine the wrench resultant of the three forces acting on thebracket. Calculate the coordinates of the point P in the x-y plane through which the resultant force of the wrench acts. Also find the magnitude of the couple M of the wrench.
Sample 9The resultant of the two forces and couple may be represented by a wrench. Determine the vector expression for the moment M of the wrench and find the coordinates of the point P in the x-z plane through which the resultant force of the wrench passes