[email protected] engr-36_lec-03_vector_math.ppt 1 bruce mayer, pe engineering-36:...
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[email protected] • ENGR-36_Lec-03_Vector_Math.ppt1
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Engineering 36
Chp 2: Vector
Mathematics
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt2
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Unit Vector, u, from Geometry
For some structural elements such as cables and slender rods the LoA for the Force associated with the member is CoIncident with the element Geometry
Recall The Unit Vector Force DeComp Formulation
kjiu zyxˆcosˆcosˆcosFˆFF
Use Geometry to find u and hence the Direction CoSines
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt3
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Unit Vector, u, from Geometry
Consider a Force Vector F, with LoA running from Pt-A to Pt-B with CoOrds as shown
F
B
A
BBB zyx ,,
AAA zyx ,, Define DIRECTION
Vector AB
kzz
jyy
ixxAB
AB
AB
AB
ˆ
ˆ Or using Δ Notation
zzjyixAB ABABAB ˆˆˆ
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt4
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Unit Vector, u, from Geometry
Next Calculate the Geometric Length, AB, of The Direction Vector AB using Pythagorus
F
B
A
BBB zyx ,,
AAA zyx ,,
“Normalizing” AB to its Length will produce a vector of unit Length; i.e., u
222 ABABAB
AB
zyx
LABAB
Now AB has the same Direction as u, but a different length
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt5
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Unit Vector, u, from Geometry
Normalize AB to Produce û
F
B
A
BBB zyx ,,
AAA zyx ,,222
ABABAB
AB
zyx
kzjyix
AB
AB
L
ABu
ˆˆˆ
ˆ
Note That this Calc also yields the Direction CoSines ABz
ABy
ABx
Lz
Ly
Lx
cos
cos
cos
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt6
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: û from AC
During Construction a building wall is temporarily held in place by ropes staked to Ground as shown in the Space Diagram at right
Find the Unit Vector for the Force exerted on the stake by Rope AC
Solution Plan• Define CoOrd System• Calc û as AC/AC
ACF
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt7
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: û from AC
Locate CoOrd Triad at Pt-A
Calculate the Δ’s
z
y
x ftz
fty
ftx
AC
AC
AC
1127
8
16
Then AC
ftkjiAC ˆ16ˆ8ˆ16
Calc Distance AC
ftft
ftAC
24576
16816
2
2222
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt8
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: û from AC
Calc û from AC & AC
z
y
x Finally û
ft
ftkji
ACACu
24
ˆ16ˆ8ˆ16
ˆ
Also the Space ’s
kjiu ˆˆˆˆ3
2
3
1
3
2
8.13132arccos
53.7031arccos
8.13132arccos
z
y
x
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt9
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: cartesian F by MATLAB The Cord AB exerts
a force, F, of 12 lbs on the Pipe
Express the Force as a vector, F, in Cartesian form using MATLAB
By Physics: since this a cord/cable the Force LoA is CoIncident with the Cord/Cable Geometry
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt10
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: cartesian F by MATLAB Math used
Use MATLAB to perform these Operations• file H13e_P2_105.m
Note the use of the DOT product in the construction of the Anon Fcn that Calcs the Magnitude of any Vector (or us “norm”)
Space Angle notation Equivalency
zzjyixAB ABABAB ˆˆˆ
ABABLAB ABAB u
ABFuF
CoOrds B-Pt &A -Pt Find
z
y
x
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt11
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Vector Mathematics
VECTOR Parameter Possessing Magnitude And Direction, Which Add AccordingTo The Parallelogram Law • Examples: Displacements,
Velocities, Accelerations, Forces
SCALAR Parameter Possessing Magnitude But Not Direction • Examples: Mass, Volume, Temperature
Vector Classifications• FIXED Or BOUND Vectors Have Well Defined
Points Of Application That Cannot Be Changed Without Affecting An Analysis
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt12
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Vectors cont.• FREE Vectors May Be Moved In Space
Without Changing Their Effect On An Analysis
• SLIDING Vectors May Be Applied Anywhere Along Their Line Of Action Without Affecting the Analysis
• EQUAL Vectors Have The Same Magnitude And Direction
• NEGATIVE Vector Of a Given Vector Has The Same Magnitude but The Opposite Direction
Equal Vectors
Negative Vectors
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt13
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Vector Addition
Parallelogram Rule For Vector Addition
Examine Top & Bottom ofThe Parallelogram• Triangle Rule For
Vector Addition
B
B
C
C
QPQP
PQQP • Vector Addition is
Commutative
• Vector Subtraction →Reverse Direction ofThe Subtrahend
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt14
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Vector Addition cont.
Addition Of Three Or MoreVectors Through RepeatedApplication Of The Triangle Rule
The Polygon Rule ForThe Addition Of ThreeOr More Vectors• Vector Addition Is Associative SQP
Multiplication by a Scalar• Scales the Vector LENGTH
SQPSQP
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt15
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Recall Vector/Force Components
Using Rt-Angle Parallelogram to Resolve Force Into Perpendicular Componentsyx FFF
yxyx FF jiFFF
Define Perpendicular UNIT Vectors Which Are Parallel To The Axes
Vectors May then Be Expressed as Products Of The Unit VectorsWith The SCALAR MAGNITUDESOf The Vector Components
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt16
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Adding by Components
Find: Sum, R, of 3+ VectorsSQPR
yyyxxx
yxyxyxyx
SQPSQP
SSQQPPRR
ji
jijijiji
Plan: • Resolve Each Vector Into Components• Add LIKE Components To Determine
Resultant• Use Trig to Fully Describe Resultant
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt17
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Adding by Comp. cont.
The Scalar Components Of The Resultant R Are Equal To The Sum Of The Corresponding Scalar Components Of The Given Forces
yyyyy
xxxxx
VSQPR
VQPPR
x
yyx R
RRRR 122 tan
Use The Scalar Components Of The Result to Find By Trig the Magnitude & Direction for R
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt18
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: Component Addition
Find the Sum for the Vectors with Magnitudes and Directions as Shown
Solution Plan• Resolve The 4 Vectors
Into Rectangular Components
• Determine the Sum Components By Adding the Corresponding Vector Components
• Calculate Sum Result, R, Magnitude & Direction1004 V
1501 V802 V
1103 V Do
Usi
ng M
ATLA
B
Com
man
d W
indow
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt19
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
MA
TL
AB
Sessio
n
>> V1 = 150*[cosd(30) sind(30)]V1 = 129.9038 75.0000 >> V2 = 80*[cosd(110) sind(110)]V2 = -27.3616 75.1754 >> V3 = 110*[cosd(270) sind(270)]V3 = 0 -110 >> V4 = 100*[cosd(360-15) sind(360-15)]V4 = 96.5926 -25.8819 >> VR = V1 + V2 + V3 + V4VR = 199.1348 14.2935 >> [alpha,Rm] = cart2pol(VR(1),VR(2))alpha = 0.0717Rm = 199.6471 >> alphaDeg = 180*alpha/pi alphaDeg = 4.1055 >> Rmag = norm(VR)Rmag = 199.6471 >> a = (180/pi)*atan2(VR(2),VR(1))a = 4.1055
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt20
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: Component Add
Resolve V’s Into Rectangular ComponentsVector No. Mag x-Comp y-Comp)
1 150 129.9 75.0
2 80 -27.4 75.2
3 110 0.0 -110.0
4 100 96.6 -25.9
= 199.1 14.3
j.i.jiR 3141199 yx RR
The Resulting Vector Sum in Component form
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt21
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: R by Mag & Dir
As Use Geometry & Trig to Find R in Magnitude & Direction Form
NRNR 6.199cos
1.1991.199cos
1.41.199
3.14tan
4.16.199 NR
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt22
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Resultant of Two Forces
Force: Action Of One Body OnAnother; Characterized By Its • Point Of Application• Magnitude (Intensity)• Direction
Experimental Evidence Shows That The Combined Effect Of Multiple Forces May Be Represented By A Single Resultant Force
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt23
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Concurrent Force Resultant
CONCURRENT FORCES Set Of Forces WhoseLoA’s All Pass Through The Same Point• A Set Of Concurrent Forces Applied To A
body May Be Replaced By a Single Resultant Force Which Is The Vector Sum Of The Applied Forces
SQPR
FQP
VECTOR FORCE COMPONENTS Two, or More, Force VectorsWhich, Together, HaveThe Same Effect As An Original, Single Force Vector
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt24
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Resultant cont.
The Resultant Is Equivalent To The Diagonal Of A Parallelogram Which Contains The Two Forces In Adjacent Legs• As Forces are VECTOR Quantities
2D Vector3D Vector
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt25
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Particle Equilibrium When The Resultant Of All Forces Acting
On A Particle Is Zero, The Particle Is In Equilibrium:• Recall Newton’s 1st Law: If The Resultant
Force On A Particle Is Zero, The Particle Will Remain At Rest Or Will Continue At Constant Speed In A Straight Line
Particle acted on by 2 Forces• Equal Magnitude• Same Line of Action• Opposite Direction 00 yx FFFR
Particle acted upon 3+ Forces• Graphical Soln → Closed Polygon• Algebraic Solution
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt26
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: Resultant
a) the tension in each of the ropes for a = 45o,
b) the value of a for which the tension in rope 2 is a minimum.
A barge is pulled by two tugboats. If the resultant of the forces exerted by the tugboats is 5000 lbf directed along the axis of the barge, then determine
SOLUTION:• Find a graphical solution by applying the
Parallelogram Rule for vector addition. The parallelogram has sides in the directions of the two ropes and a diagonal in the direction of the barge axis and length proportional to 5000 lbf.
• The angle for minimum tension in rope 2 is determined by applying the Triangle Rule and observing the effect of variations in a.
• Find a trigonometric solution by applying the Triangle Rule for vector addition. With the magnitude and direction of the resultant known and the directions of the other two sides parallel to the ropes, apply the Law of Sines to find the rope tensions.
R = 5 kip
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt27
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: Cable Tension
Graphical Solution• Parallelogram Rule With Known Resultant
Direction & Magnitude, Known DIRECTIONS For Sides– By Scaling With a Ruler (cf. Engr-22)
lbf2600lbf3700 21 TT
105sin
lbf5000
30sin45sin21 TT
lbf2590lbf3660 21 TT
• Analytical Solution– Triangle Rule with
Law of Sines
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt28
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example: Min Tension
Minimum Tension In Rope 2 Is Determined By Applying The Triangle Rule And Observing The Effect Of Variations in α.• By Geometric Construction
Minimum Rope-2 Tension 2 OccursWhen T1 & T2 Are Perpendicular 30sinlbf50002T lbf25002 T
30coslbf50001T lbf43301 T
3090 60
30º
LoA for T 1
• Analysis Simplified by Rt-Triangle
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt29
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Hull Drag Example
A model Sailboat Hull undergoes a drag test. Three cables align its bow on the channel centerline. For a given speed, the tension is 40 lbs in cable AB and 60 lbs in cable AE• Find the drag force and
the tension in cable AC.
SOLUTION PLAN• Choosing the hull-eye
(the point of concurrency) as the free body, draw a free-body diagram.
• Express the condition for equilibrium for the hull by writing that the sum of all forces must be zero
• Resolve the vector equilibrium equation into two component equations. Solve for the two unknown cable tensions.
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt30
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Hull Drag Example Choosing the hull-eye as the
free body, draw a free-body diagram. Analyze Geometry.
25.60
75.1ft 4
ft 7tan
56.20
375.0ft 4
ft 1.5tan
0 DAEACAB FTTTR
Express the condition for equilibrium for the hull by noting that the sum of all forces must be zero
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt31
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Hull Drag Example Resolve the vector equilibrium
equation into X-Y component equations. Solve for the two unknown cable tensions
iF
jT
ji
jiT
ji
jiT
DD
AE
ACAC
ACACAC
AB
F
TT
TT
lb 06
9363.03512.0
56.20cos56.20sin
lb 84.19lb 73.34
26.60coslb 4026.60sinlb 40
j
i
jiR
609363.084.19
3512.073.34
000
AC
DAC
T
FT
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt32
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Hull Drag Example
j
i
R
609363.084.19
3512.073.34
0
AC
DAC
T
FT
609363.084.1900
3512.073.3400
ACy
DACx
TF
FTF
lb 66.19
lb 9.42
D
AC
F
T
This equation is satisfied only if EACH COMPONENT of the resultant is equal to ZERO
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt33
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
WhiteBoard Work
Let’s WorkThis NiceProblem
= 200 NW=400 N
Find θ so that the Cart will NOT Roll BackWards (down) or be Pushed-Up the
15%-Grade Ramp
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt34
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt35
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Bruce Mayer, PERegistered Electrical & Mechanical Engineer
Engineering 36
Appendix
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt36
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
WhiteBoard Work
Let’s WorkThis NiceProblem
1F
2F
kN 3 BAAB TT
Show that F2 = −F1
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt37
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-03_Vector_Math.ppt38
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
MA
TL
AB
Co
de
% Bruce Mayer, PE% ENGR36 * 15Jul12% file = H13e_P2_105.m%clear; clc; % Clear Out Memory & Screen%% CoOrds for Pts A & B by Geometry & TrigA = [5, 3*cosd(20), -3*sind(20)]; % ftB = [0 0 6]; % ft% find the AB vector = [delX*i, delY*j, delZ*k],ABv = B-A %% Make Anon Fcn to find the Magnitude of any Vector%* use the DOT product to find Mag^2 of the VectorMagV = @(V) sqrt(dot(V,V))disp(' ')% Find Magnitude of vector AVbABm = MagV(ABv) % in ft% find uAB as ABv/ABmuAB = ABv/ABm;Fm = 12; % the Magnitude of F in lbs% find the Answer by Fm*uABFv = Fm*uAB;% Extra Credit: Find the Space Angles for Fv%* AKA the CoOrd Direction AnglesQ = acosd(uAB);%% Display ResultsFvDisp = sprintf('F components in lbs: Fx = %f Fy = %f Fz = %f',Fv(1), Fv(2), Fv(3));disp(FvDisp)disp(' ')QDisp = sprintf('SpaceAngles in °: Qx = %f Qy = %f Qz = %f',Q(1), Q(2), Q(3));disp(QDisp)%% do a quick check on the result for Fdisp(' ')disp('check answer by finding the Mag of Fv using MagV fcn')FmChk = MagV(Fv);FmChkDisp = sprintf('FmChk = %f', FmChk);disp(FmChkDisp)%disp(' ')disp('Find Vector Mag using built-in MATLAB command NORM')ABmByNORM = norm(Abv)