Download Introduction to the Engineering Design Process

Scalars
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A scalar is any physical quantity that can be completely
characterized by its magnitude (by a number value)
Mathematical operations involving scalars – follow rules of
elementary algebra
In text represented with letters in italic type
Examples: mass, volume, length
Vectors
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Possess a magnitude, direction, and sense
Represented graphically by an arrow (tail, tip or head) SHOW
– Magnitude: proportional to the length of arrow
– Direction: defined by angle between reference axis and line of
action of the arrow
– Sense: indicated by the arrowhead
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In text, vector symbolized by boldface type, A, and its
magnitude (always positive) by |A | or simply A
In slides, vector symbolized by boldface type, A, and
magnitude by regular type, A
In handwritten work, vector represented by letter with an
arrow over it, its magnitude by letter enclosed in absolute
value symbol or by letter itself SHOW
Obey parallelogram law of addition
Examples: position, force, moment
Multiplication and division of a vector by
a scalar
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Product of a vector, A, and a scalar, a
– Vector, aA
– Magnitude, aA
– Sense of aA is the same as A provided a is positive, it is opposite
to A if a is negative
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Division can be converted to multiplication and then laws of
multiplication applied, A/a = (1/a) A, a≠0
Product is associative with respect to scalar multiplication
a(bA) = (ab)A
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Product is distributive with respect to scalar addition
(a + b)A = aA + bA
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Product is distributive with respect to vector addition
a(A + B) = aA + aB
Vector addition
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Parallelogram law SHOW
– Join the tails
– Draw parallel dashed lines from the head of each vector to the
intersection at a common point
– Resultant, R, is the diagonal of the parallelogram (extends from the tails
of the two vectors to the intersection of the dashed lines)
– Special case – two vectors collinear – parallelogram law reduces to an
algebraic or scalar addition, R = A + B
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Triangular construction (head-to-tail fashion) SHOW
– Adding B to A
– Connect the tail of vector B to the head of vector A
– Resultant, R, extends from the tail of vector A to the head of vector B
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Vector addition is commutative, A + B = B + A
Vector addition is associative, (A + B) + D = A + (B + D)
Vector subtraction
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Difference between two vectors A and B
– Subtraction defined as a special case of addition
– R’ = A – B = A + (-B) = A + (-1)B
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SHOW
Resolution of a vector
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Vector may be resolved into two “components” having known
lines of action by using the parallelogram law
– SHOW
– Starting at the head of R, extend dashed line parallel to a until it
intersects b, likewise dashed line parallel to b until it intersects a
– Two components A and B are then drawn such that they extend
from the tail of R to the points of intersection
– R is resolved into the vector components A and B
Problems in Statics involving force (a
vector quantity)
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Find the resultant force, knowing its components
Resolve a known force into its components
Procedure for analysis – addition of
forces
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Apply parallelogram law
– Sketch vector addition using parallelogram law
– Determine interior angles from geometry of the problem (recall
sum total of interior angles of parallelogram = 360°)
– Label known angles and known forces
– Redraw half portion of constructed parallelogram to show
triangular head-to-tail addition components
– Unknowns can be determined from known data on triangle and
use of trigonometry
– If no 90° angle, law of sines and/or law of cosines may be used
SHOW
– EXAMPLES (pgs 29-32)
Vector components parallel to x and y
axes (Cartesian components)
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F can be resolved into its vector components Fx and Fy parallel to the x and y
axes, F = Fx + Fy
Unit vectors i and j designate directions along the x and y axes
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SHOW
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i and j vectors have a dimensionless magnitude of unity
Their direction will be described analytically by a “+” or “-” depending on whether
they are pointing along the positive or negative x or y axis
F = Fxi + Fyj (Cartesian vector form)
F’ = F’x(-i) + F’y(-j) = - F’x(i) - F’y(j)
The magnitude of each component of F is always a positive quantity, represented
by the scalars Fx and Fy
The magnitude of F is given in terms of its components by the Pythagorean
theorem,
F  Fx2  Fy2
The direction angle θ, which specifies the orientation of the force, is determined
from trigonometry,
F
  tan 1 y
Fx
From this point forward i and j will simply be written in regular type since by
definition they are vectors
Addition of vectors in terms of their
(Cartesian) components
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F1 = (F1xi + F1yj), F2 = (F2xi + F2yj), F3 = (F3xi + F3yj)
FR = F1 + F2 + F3
FR = (F1xi + F1yj) + (F2xi + F2yj) + (F3xi + F3yj)
FR = F1xi + F2xi + F3xi + F1yj + F2yj + F3yj
FR = (F1x + F2x + F3x) i + (F1y + F2y + F3y) j
FR = FRxi + FRyj
– FRx = F1x + F2x + F3x or FRx = ∑ Fx
– FRy = F1y + F2y + F3y or FRy = ∑ Fy
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EXAMPLES (pgs 40-43)
Rectangular components of a 3-D vector
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Assume right-handed coordinate system
A = Ax + Ay + Az SHOW
Cartesian vector notation, A = Axi + Ayj + Azk
Magnitude of Cartesian vector
– A = (Ax2 + Ay2 + Az2)1/2
– SHOW
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Direction of Cartesian vector
– SHOW
– cos α = Ax/A, cos β = Ay/A, cos γ = Az/A
Unit vector
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Unit vector has a magnitude of 1 and specifies a direction
If a unit vector uA and a vector A have the same direction →
A can be written as the product of its magnitude A and the
unit vector uA, A = A uA
– uA (dimensionless) defines the direction and sense of A
– A (has dimensions) defines the magnitude of A
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A unit vector having the same direction as A is represented by
uA = A/A
Using the unit vector to obtain the
direction cosines
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A unit vector in the direction of A (A = Axi + Ayj + Azk)
uA = A/A = Ax/A i + Ay/A j + Az/A k
uA = A/A = uAxi + uAyj + uAzk
(Ax/A = uAx = cos α, Ay/A = uAy = cos β, Az/A = uAz = cos γ)
uA = A/A = cos α i + cos β j + cos γ k
(uA has a magnitude 1 and recalling that A = (Ax2 + Ay2 + Az2)1/2)
2
2
2
1 = (cos α + cos β + cos γ)
1/2
Squaring both sides
2
2
2
1 = cos α + cos β + cos γ
Equation can be used to determine one of the coordinate direction
angles if the other two are known
Addition and subtraction of 3-D
Cartesian vectors
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A = (Axi + Ayj + AZk), B = (Bxi + Byj + BZk)
R = A + B = (Ax + Bx) i + (Ay + By) j + (Az + Bz) k
R’ = A - B = (Ax - Bx) i + (Ay - By) j + (Az - Bz) k
FR = ∑ F = ∑ Fxi + ∑ Fyj + ∑ Fzk
EXAMPLES (pgs 52-55)
Position vector
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If r extends from point A (xA, yA, zA) to point B (xB, yB, zB)
– SHOW
– rAB = (xB – xA) i + (yB – yA) j + (zB – zA) k
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The i, j, k components of the position vector rAB may be
formed by taking the coordinates at the head of the vector
(point B, (xB, yB, zB)) and subtracting the corresponding
coordinates of the tail of the vector (point A, (xA, yA, zA))
Force vector directed along a line
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The direction of a force can be specified by two points
through which its line of action passes
Formulate F in a Cartesian vector form, realizing it has the
same direction as the position vector r directed from point A
to point B (A and B are points on a cord along F)
F = Fu = F (r/r)
Procedure for analysis
– Determine position vector r directed from A to B, and compute
its magnitude r
– Determine the unit vector u = r/r which defines the direction of
both r and F
– Determine F by combining its magnitude F and direction u
F = Fu
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EXAMPLES (pgs 65-68)
Dot product
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Dot product of vectors A and B, A∙B, “A dot B”
– A∙B = AB cos θ (where 0°≤θ≤180°), SHOW
– The result is a scalar, not a vector
– The dot product is also referred to as the scalar product of
vectors
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Applicable laws of operation
– Commutative law: A∙B = B∙A
– Multiplication by a scalar: a (A∙B) = (aA)∙B = A∙(aB) = (A∙B) a
– Distributive law: A∙(B + D) = (A∙B) + (A∙D)
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Cartesian vector formulation
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i∙i = (1)(1) cos 0° = 1
i∙j = (1)(1) cos 90° = 0
i∙k = (1)(1) cos 90° = 0
Similarly, j∙j = 1, k∙k = 1, j∙k = 0
Dot product of two general vectors
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A∙B = (Axi + Ayj + AZk)∙(Bxi + Byj + BZk)
= AxBx (i∙i) + AxBy (i∙j) + AxBZ (i∙k)
+ AyBx (j∙i) + AyBy (j∙j) + AyBz (j∙k)
+ AzBx (k∙i) + AzBy (k∙j) + AzBz (k∙k)
= AxBx + AyBy + AzBz
Applications of dot product
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Determining the angle formed between two vectors or
intersecting lines
(recall A∙B = AB cos θ)
θ = cos-1 (A∙B/AB), 0°≤θ≤180°
A∙B is computed from A∙B = AxBx + AyBy + AzBz
if A∙B = 0 → θ = 90° → A is perpendicular to B
Applications of dot product (continued)
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Components of a vector parallel and perpendicular to a line (aa’)
– SHOW
– Parallel
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For Ap (projection of A onto line aa’) → Ap = A cos θ
Direction of line specified by the unit vector u (u=1)
Ap = A cos θ = (A)(1) cos θ = A∙u [and Ap = (A∙u)(u)]
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The scalar projection of A along a line is determined from the dot product
of A and the unit vector u which defines the direction of the line
If Ap is positive, then Ap has a sense which is the same as u
If Ap is negative, then Ap has the opposite sense to u
– Perpendicular
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A = Ap + An → An = A - Ap
An → An = A sin θ, θ = cos-1 (A∙u/A)
Or by Pythagorean Theorem An = (A2 – Ap2)1/2
EXAMPLES (pg 76-80)