General Flashcards
Fourier Transform
This is a way to describe an oscillating function in terms of angle and intensity by changing the function to be a function of angular frequency (rad/sec) or wavenumber (k=seconds to complete a wave).
It uses eulers identity to switch the imaginary parts of the frequency into cos and sine terms. This is then converted into polar coordinates in the intensity phase space.
Use the table to convert regular functions in fourier equations that are used to analyze a frequency.
In the manual the period is given so 2 pi over the angular frequency give syou seconds and the functions for several common signal inputs.

Even and odd functions
f(-x)=f(x) means the function is even (x2 cos)
f(-x)=-f(x) means that it is odd (x3 sin)

Orthogonal functions
This is when the integral of f(x)*g(x)=0
Taylor Series
Taylor Series represent using a series of bisected functions to model an original function f(x). They converge when |x-a|

Fourier transform of a cosine wave
This will yield one line because each and every interval is on the same frequency which means your peak will show that one frequency.
Fourier transform of a series of signals with t(duration)=tau
This will yield several bumps because thetime domain shows 2 intensities within one 360 degree rotation.
Integers, rational, and irrational numbers
Integers are whole numbers
Rational numbers are numbers that can be written as a ratio of two integers
irrational numbers cannot be expressed as a ratio of two integers. They include numbers that repeat or do not terminate Ex: (3)^.5/2
This is irrational because 3^.5 has an operator
Real numbers: include all of the above
factorial
n!=(n)*(n-1)*…(2)*(1)
Complex Numbers (Add, Subtract, multiply, divide)
Add/subtract: combine like terms.
Mult.: FOIL
Divide: Multiply by the conjugate of the denominator
Using conjugates
if A=a+ib and A#=a-ib then A*A#=a2 + b2
Sets
A set is a collection of objects or entities. If X belongs to a set S then X ∈ S where S={n0,n1…n} or a set could be defined by a function S={X:X3>27 }
Joining Sets
∈
C ∪ D = {X:X∈C or X∈D) This is the union of sets C and D
C ∩ D = {X:X∈C and X∈D) This is the intersection of sets C and D
And if all of a set (set A) are part of a larger set (set B) then AcB where the c is the inclusion symbol
Ø
This is the null set symbol. It means that there are not any elements within a set.
Exponent Laws
n√a=a(1/n)
am/an= am-n am*an= am+n
(am)n=am*n
(a*b)m= am*bm
Simplify n √am
n √am= a(1/n)*m = a (m/n)
Simplify n √ab
n √ab= a(1/n) * b(1/n)
How to take the root of a function with a divisor
Multiply the denominator through and divide the whole term by the diviseor with the root operator used.
Rule for binomial expansion
if we have a binomial given by (A+X)n then the expansion is given by B*An-m * Xm
Where: m=term # starting from 0 to n+1
B=(The exponent*coefficient of the term m-1)/m
Ex: Term 3 of (A+X)7 is given by A7+7A6X+(7*6/2)A5X2…
Determinants
Determinants are the values of matrices. The order of a determinant is given by the sqaure root of the number of elements which are the numbers inside of the matrix.
determinant nomenclature
the position of any value is given by aij where i= row # and j = collumn #
Finding determinants of second order matrix
If the determinant is second order then the determinant is given by:
Product of primary diagonal-Product of secondary diagonal
Where: Primary diagonal are the values given by a11 and a22 and the secondary diagonal is given by the values a21 and a12
Minor of a matrix
This is the reduced matrix that can be used to find the determinant of higher order matrices. The order is given by crossing out the values that intersect at the value of interest and finding the determinant of the resulting matrix multiplied by the value of interest.

Sign of determinant parts
If the leading coefficient is in the position ij where i+j is
even then the coefficient is positive
odd then the coefficient is negative
Solving Linear Equations with two unknowns
when given a set of equations like
a1x+b1y=c1 and a2x+b2y=c2
Multiply the the first equation by -a2 and the second by a1 so that when added the first terms cancel. This enables you to have only one unknown to be solved for.
y=(-a2c1+a2c2)/(a1b2-a2b1) and x=(c1b2-c2b1)/(a1b2-a2b1)
Solving linear equations with 3 unknowns
If we have three equations we can create three matrices; C=A*b where A=3x3 coefficient matrix, b = 3x1 matrix of xyz (or any three unknowns) and C= 3x1 matrix of constants
Then b= A-1*C

Scalar Product
a dot b= ax*bx+ay*by+az*bz=|a|*|b|*cos(alpha)
Where: a dot b represents the magnitude of the shared a and b vector space
Cross Product
axb is the cross product of a and b and represents the magnitude of the distance between the tips of a nd b in the direction that is normal to the plane created by a and b.
axb is given by the determinate of a and b
axb=
|ax ay az|
|bx by bz|
i j k |
How to solve third order algebraic equations when one root is given
When one root is given you can divide the third order equation by the root to make it a quadratic which is then solved via the quadratic formula.
Long division for equations
To do long division of an equation you always eliminate the higher order term.

Basic logarithm function
The answer to a log is the exponent
logbn=a -> ba=n and n=blogb^n
logb1=?
=0 because b0=1
logbb=?
=1 because b1=b
expand: logbm*n
logbm*n= logbm+logbn
expand: lobb(m/n)
logb(m/n)=logbm-logbn
logb(1/n)=?
logb(1/n)=-logbn because logbn-1=-1*logbn
logbm(1/n)=?
logbm(1/n)=(1/n)*logbm
radians to degrees formula
radians= degrees * (pi/180) AKA there are 180o in pi radians
30o to radians
30o=pi/6 radians
pi/3 radians to degrees
60o
45o to radians
pi/4
275o to radians
5pi/4 radians
3pi/2 to degrees
270o
330o to radians
-pi/6 or 11pi/6
5pi/3 to degrees
300o
120o to radians
2pi/3 radians
rpm to rad/sec
rpm*(360/rev)*(pi rad/180)*(min/60 sec)=rad/sec
sin, cos, tan of 17pi/6
17pi/6= 12pi/6+5pi/6
sin(5pi/6)=1/2
cos(5pi/6)= -3.5/2
tan(5pi/6)= -1/3.5
sin, cos, tan 5pi/3
sin (5pi/3)= -3.5/2
cos(5pi/3)= 1/2
tan(5pi/3) = -3.5
polar coordinate transform
(x, y) = (r, θ)
where: r=(x2 + y2) and θ=tan-1(y/x)
x=rcosθ and y = rsinθ
Law of cosines
for a triangle with angles A,B,C opposite to sides a,b,c
c2 = a2 + b2 - 2abcos(C)
Law of sines
a/sin(A)=b/sin(B)=c/sin(C)
Using law of cosines
Law of cosines can be used to find:
Distances between objects starting at one point and travelling in different directions.
diagonals of a parallelagram
General Equation for a line
Ax+By+C=0
m=-A/B and
b=-C/B
slope intercept line equation
y=mx+b where:
m=(y2-y1)/(x2-x1)
two point line formula
y-y1/x-x1=(y2-y1)/(x2-x1)
Slope relationship between perpendicular lines
m1=-1/m2
angle between two lines where m1>m2
tan(α)=(m1-m2)/(1+m1*m2)
general equation for conic sections
ANY second order equation
Ax2 + Bxy + Cy2 + Dx + Ey + F
criterion for ellipse, parabola, and hyperbola
ellipses occur when (B2-4AC)<0
parabolas occur when (B2-4AC)=0
Hyperbolas occur when (B2-4AC)>0
ellipse equation
(x-h)2/a2 + (y-k)2/b2 = 1
where C=(h,k) and a,b are the radius in the x and y directions
Limits
Limits are the value a function approaches at some independent variable value.
If, the function exists at the limit then it is continuous.
L’Hospital’s Rule
Suppose we have two functions, f(x) and F(x) where both go to zero as x approaches A. The limit for f(x)/F(x) is given by f’(x)/F ‘ (x)
Formal definition of a derivative
The derivative of a function, y=f(x) is the instantaneous rate of change or
dy/dx=lim (Δx->0) (Δy)/(Δx) and can be proven by subbing y+Δy for y and x+Δx for x within a function given by y=f(x) and then solving for Δy/Δx and taking the limit as Δx approaches 0
mulitplication rule for derivatives
d/dx u*v= u dv/dx + v du/dx
power rule for derivatives
y=n xm
y’ = (m*n)xm-1
d/dx un = ? where u = f(x)
d/dx un = nun-1 du/dx via the chain rule
d/dx (vu) where both v and u =f(x)
d/dx(vu) = u*vu-1 dv/dx + vu lnv du/dx
d/dx (u/v)=?
d/dx (u/v) = [v(du/dx) - u(dv/dx)] / v2
d/dx (logcu) = ? where u=f(x) and c=constant
d/dx (logcu) = 1/u logce * du/dx
How to treat derivatives
Derivatives act as differentials so they can be inversed or have parts cancel (eg (dy/dt)/(dx/dt)=dy/dx
Implicit differentiation
if f(x,y)=c then to find dy/dx
- ) differentiate f(x,y) where d/dx(xy)= y + x(dy/dx)
- ) isolate dy/dx
partial derivatives
These are written with the funky d symbol (∂) or f ‘x and are formally defined by
if z=f(x,y) then ∂f/∂x = lim Δx->0 [f(x+Δx,y) - f(x,y)]/ (Δx)
where: if the variable is not in the denominator it is treated like a constant
chain rule for partial derivatives
if z=f(x,y) and x=f(t) ; y =f(t) then ∂z/∂t = [(∂z/∂x * ∂x/∂t) + (∂z/∂y * ∂y/∂t)]
Meaning of f ‘ and f ‘’ being greater than zero
if f ‘ >0 then the slope of f at whatever x is positive so f(x+dx)>f(x)
if f ‘’ >0 then the slope of f at x is increasing therefore f ‘ (x+ dx)> f ‘ (x)
Procedure for finding maxima/minima
- ) find the first derivative
- ) Find the values of x so that the first derivative (f ‘ )=0 . These are the function’s critcal values
- ) find the second derivative f ‘’
- ) Substitute the critical values from 2 into f ‘’
if f ‘’ (x)>0 then at x there is a local minima
if f ‘’ (x) <0 then at x there is a local maxima
Solving for local maxima/minima when f’‘=0
To evaluate if f(x) is a local maxima or minima when f ‘’ is 0 you must find f ‘ (x+1) and f ‘ (x-1) and use those values to evaluate what f(x) is
Integration
This is the process of finding the function where the function’s derivative is given
d( ∫ f(x) dx) = f(x) dx
∫=the function whose differential is
anti-derivative: Power rule
∫ axb dx = [axb+1/(b+1)] + c
definite integral definition
∫ab f(x) dx = F(b)-F(a)
Integration by Parts
When the integrand can be split into two functions; one that represents a u where u=f(x) and another that represents dv where dv=g(x)dx and g(x) is easily integrated.
Under these conditions: ∫ udv = uv - ∫ vdu
Common functions for integration by parts
Generally allow ln to be u (du=dx/x) and x, ex, or other easily integrated functions be dv
How to use dy in a double integral
When using dy the slice is taken perpendicular to the y axis.
- ) with f(x) solve for x so that way x=f(y)
- ) determine the bounds and integrate ∫ f(y) dy
Formula for generating a solid around the x axis
V= ∫ab pi y2 dx
where y= f(x)
Laplace Transformation
These involve the transformation of derivatives where ∫0inf e -st F(t) dt = f(s) and can be written f(s) = L {F(t)} = L{F} = F(s)
∫ e x n =?
∫ e x n dn = (1/x) e x n + c
Finding Fr for coplanar and parallel forces
Fr= ΣF
dr = ΣFi*di / ΣFi where d is the perpendicular distance to some point O
Fr for concurrent force systems?
Fr = Σ Fi where each F is a vector so vector addition is used. This is a system where all the forces intersect at one point.
Non Concurrent force systems: Fr and dr = ?
In a nonconcurrent system the forces do not intersect at a common point and the body is not static. A moment around the center of gravity is given by
Σ Fi * di = Fr * dr
where the direction of Fr is determined by vector addition
Couple
A couple is a torque that is produced by two parallel but opposite forces. The resulting moment is given by M= F*d where d is the total distance between the parallel forces and F is the force of one of the forces
The magnitude of the couple is not a function of location. AKA the couple has the same magnitude regardless the center of the moment.
Solving static structures: Method of Joints
The idea of this method is that each joint must be in equilibrium therefore if we isolate each and every joint and solve for the internal forces then each will be in equilibrium. Procedure:
- ) Draw FBD with everything internal in tension
- ) Solve for the reaction forces (use logic not tension)
- ) choose a joint where 2 unknowns exist and solve.
- ) rinse and repeat
Equations of Equilibrium
This says that for a static body:
ΣFx = ΣFy = ΣFz = ΣM0 = 0
When solving a structure we can assume that the internal reactions cancel each other out and only treat the system by analyzing the external forces with these equations.
Method of Sections
This “cuts” the structure and analyzes the internal forces at the cut. Procedure:
- ) Draw FBD
- ) solve for Rxns
- ) “cut” the structure and draw internal forces in tension. Do NOT cut more than three members at once.
- ) ΣMo = 0 where o is the intersection of two of the forces which leaves the third unknown and the external forces which makes solving easy.
- ) ΣFx = ΣFy =0 solve for the other two unknowns
What is the tension in a cable/rope equal to?
Tension in a cable or a rope is constant throughout the cable/rope. The direction does NOT matter.
Finding moments in 3-space
- ) break forces into their three components
- ) ΣMx = Fz*dz + Fy * dy It is the same as 2-space but everything that is not in the direction of the axis is considered to be causing a moment about the said axis.
Friction
Friction is the contact resistance between bodies in contact with one another and is proportional to the normal force. It acts parallel to the interface across the surface and is summated at the point where the normal force acts.
Fmax= N * μs
Fmotion=N * μk where μs >μk
Total force of two interacting solids
R = N + Ff where Ff = μ * N and the angle of R is given by tan ϕ = Ff /N= μ and ϕ is the angle between Ff and N not to the horizontal
Belt Friction
This is the idea that a belt going over a solid has unequal tensions where the greater tension is in the direction of impending motion.
T2max = T1 *e^(μs*β) where β is the angle of contact in radians
Centroids solving for CG
x̄ = [Σ W*x]/W and similar with the y bar
Where: x represents the distance of a homogenous chunk of the object from any defined x axis and W is the weight of that chunk.
Centroid of areas
x̄ = ∫ x dA/∫dA
centroid of lines
This is fundamentally similar to the centroid of an area but uses a line integral given by x̄ = ∫ x dL/∫dL where L refers to the line coordinates
This is most common for arcs where dL=rdθ and x=rcosθ and y=rsinθ
Theorem of Pappus
This says that when you rotate a line about an axis 360o that the resulting surface area is given by A=L 2 pi X where X is equal to the cg or centroid
Theorem of Guldinus
This states that the volume of an area revolved around an axis is given by V = 2*pi*x*D where x is the centroid or cg of the area
Strain and Elongation
When a material is put into a stress state it experiences deformation by a total amount given by δ . and the strain is given by δ/L = ε
Twisting or angular deformation is given by γ = δ/L = tan ϕ where ϕ is the angle from vertical to the deformation
Modulus of elasticity
E= σ/ε = (P/A)/(δ/L) and represents the per unit stress over per unit strain within the elastic region of deformation
Poissons ratio
This the ratio of lateral to longitudinal strain
Coefficient of thermal expansion
δT = L α (ΔT)
Solving statically indeterminate structures
These are structures where the number of unknowns exceeds the number of equations of equilibrium so we need to use the material properties of the material to solve.
- ) Draw FBD for a selected member
- ) ID known and unknowns. Write all equations to constrain via equilibrium then for material properties
- ) Use material properties to solve for an unknown. Plug into Equations of Equlibrium.
Twisting
When a shaft is placed under some torque T it experiences a shear stress τ where dF= τ dA and the internal resisting moment is given by τ r dA = dMT
finding shear from a torque
J= ∫ r2 dA = the polar moment of inertia across the cross sectional area
Jmax= pi*d4/32
and τ = Tr/J
angle of twist
This is the angle of rotation caused by a torque.
It is related to the shear force by the shear modulus G.
τ / γ =G where γ is the shear strain
G= TL/θJ where J is from 0 to R
Shear strain from twisting
γ = C θ /L where θ is in radians
The angle of twist for compound shafts
This is the summation of individual twists. J does not depend on the material but G is material dependent
Beam analysis
“walk the beam” These are usually subject to several loads and moments.
- ) solve reactions
- ) “walk the beam” to find moment and stress diagrams
Sign convention for beam analysis
Shear is positive when walking the beam left to right and the shear points upwards.
Moments are positive when the forces on either side of the point make the beam bend like a smiley face.
Shear diagrams at x=0 and x=L
The shear is ALWAYS 0 at x=0 and x=L because otherwise the beam would be mobile and not static
Relating moments to shear
∫AB dM = ∫AB V dl where V=V(l) and is determined by the shear diagram
This means that the change in the bending moment from A to B = the area under the shear curve from A to B
Flexure Formula
This is the formula for understanding the tension/compression within a bent beam. It says that in a uniform beam that σy/y = constant and that the total moment Mt = σmax/ymax ∫ y2 DA = Iy σmax/ymax
Position of the neutral axis
This influence the moment about the x axis and represents the line of the beam where σ=0 It is given by the centroid of the cross section.
second moment of area formula
Ix = ∫-l/2l/2 y2 dA where the bounds are measured from the centroid
Parallel Axis Theorem
This says that the moment of inertia about any axis parallel to the neutral axis is equal to Ib = INA + A*D2 where D is the distance from the NA to the reference axis
General procedure for moment of inertia of composites
- ) Divide each cross-section into several chunks with known Icg
- ) located the neutral axis via finding the centroid (C = ∫ y dA/∫dA)
- ) Calculate INA about each centroid.
- ) Use the parallel axis theorem to relate each component to the overall cross-sectional I where holes are treated as negative values and the others summate.
What are the units of moment of inertia?
L4 when solving for anything with moment of inertia try to use inches and centimeters
Units of moments and torque
L*F
Units of shear in shear diagrams
This is simply put in terms of pounds. It represents the total shear throughout the cross section
Units of G/E
This is in F/L2 because strain is unitless (L/L)
Types of supports that create counter moments
Fixed connection to walls create opposite moments. These also will have shear at their open end.
Dynamics
This is the study of motion of particles and bodies. Particles are point objects where it is assumed to be concentrated at one point. Bodies are systems of particles that form an object where the relative motion of different parts of the object must be considered.
Newton’s Three Laws of Motion
- ) A particle acted upon by a balanced force system has no acceleration. AKA an object in motion stays in motion unless acted upon by an external force
- ) F=ma
- ) Action and reaction forces are always equal and opposite
Kinematic equations
Assuming that a is a constant
S= Vot + at2/2
Vf = Vo + at
Vf2 = Vo2 + 2a*s
How to use the kinematic equations
When a body is under constant acceleration and the problem is trying to describe the motion of a particle or body in terms of time and position identify the known variables (to, tf, Vo, Vf, a) and the unknown variables.
Use the kinematic equations to solve.
Relating s, V, and a
given s = f(t) then s ‘ = V(t) and s ‘’ = a(t)
Also s = ∫ V(t) = ∫∫ a(t)
Change of position with derivatives
ds = (VdV)/f(V)
Position as a function of force when F=f(t)
x = (1/m) ∫∫f(t) dt dt + C1*t + C2
Angular Velocity
ω = θ/t where θ is in radians and the change in position is given by rθ
Relating velocity and angular velocity
V = r * ω
Angular acceleration
This is the change of angular velocity given by d2 θ/dt2 = α
Also α*dθ = ω*dω
Do the kinematic equations apply to circular motion?
Yes. Replace V with ω; a with alpha; S with theta
flight of a particle given its initial conditions
y:

centripital and centrifugal acceleration
Centripital acceleration is an = V ω = rω2 = V2/r and is directed towards the center of rotation
Centrifugal acceleration is at = dV/dt = rα + 2ω dr/dt and is directed in the direction tangent to the path of the object at that moment
centripetal and centrifugal forces
the centripetal F; Fn = WV2/gr = W*an/g
The centrifugal F; Ft= (W/g)*dV/dt = (W/g) * at
Ideal bank angle
tanθ = V2/gR + tanθf for sliding down the curve and -μ for sliding up
where: θf = friction angle = tan-1 μ
θ = ideal bank angle where an object travelling at V through a curve with R will neither slide up or down.
Forces of a rotation that is not about the object’s center of mass
ΣFn = (W/g)*ro* ω2
ΣFt = (W/g)*ro * α
ΣMz = α Iz
where: ro=the distance from the center of rotation to the center of mass of the object in rotation
Iz = Io + m*ro2
alpha= rotational acceleration
ro ω2 = an and ro * α=at
Radius of Gyration
k is a distance and it can regarded as the distance from the axis of inertia to the center of gravity for an object that is rotating around a center that is not outside of the object.
k = (I/m).5
Rotational dynamic equilibrium
This is a state where V is constant and an object is rotating.
The first step is to draw FBD and find the resultant force on the said object.
In this state (W/g)ro α d -Izα = 0 where d is the distance from the CG to the point of rotation
Fundamental work energy relationship
The work done on a translating object where the force is in the direction of motion is equal to the change in kinetic energy of that object. This is given by
F*S=(W/2g)(Vf2-Vi2)
Work energy with variable force
Work = kS2/2 where k is the spring constant and S is the spring’s displacement.
Spring constants in parallel
if springs are in parallel then kf= Σ ki
Spring constants in a series
For springs in a series (1/kf) = Σ(1/ki)
Frequency of oscillation: springs
ω = +-[2k/m].5 and f (1/s) = +-(1/2pi)[2k/m].5
Period of oscillating springs
T = 1/f = 2pi[m/2k].5
Conservative Systems
These are systems that lack friction or other dissipative forces. Therefore
KE + PE = K + U = c
and (d/dt)(KE+PE) = 0
Solving translating systems
- ) FBD
- ) ID known forces, unknown forces, and what you are attempting to solve for
- ) ΣF = ma = external forces
Oscillations: Steps to solve via energy
- ) FBD
- ) if the system is conservative then
PE = W*h + kx2/2 = Wg(L-Lcosθ) + (k/2)(bsinθ)2
KE = (1/2)mV2= (1/2) m (ωL)2 where ω= dθ/dt
- ) d/dt(PE+KE)=0 and for small θ sinθ=θ and cos=1
- ) Simply and (1/2pi)*(coefficient of θ) = f
Momentum
Momentum is p=m*V and because dV/dt=a then F=dp/dt where p is a vector
Impulse
This is a force that acts on an object for a short duration of time such that F(t)dt = impulse force.
∫t0t F(t) = (V-V0)m = ∫vv mdV
Force vs. time curves
If force = F = f(t) then (Σ ∫F(t))/m = dV over the period
Conservation of Momentum
IF there are no external forces acting on a system then the internal components of a system have the property where m1 * V1 = m2 * V2
It generally works when dt~0 but there is a dV
Conservation of Momentum: applications
conservation of momentum should be used when there is an impulse and a change in mass/velocity that occurs over dt~0
Ex: a bullet hitting a pendulum
How to solve a bullet hitting a pendulum
- ) apply the conservation of momentum to find v
- ) assuming it is a conservative system then KE1 + PE1 = KE2 + PE2
Impacts
These are when two objects collide and it is assumed that the time over which the collision takes place ~0. Therefore conservation of momentum and impulse physics applies.
Elastic impact
This assumes that there is ~0 loss of energy due to deformation and impact is over a time interval =0
Coefficient of restitution
this is a constant that is used to adjust for deformation upon impact of otherwise elastic objects such that
e = (relative V before)/(relative V after) = separation V/approach V where the relative nature of the velocity is described as the absolute value of their convergence/divergence
Coefficient of restitution when collision is non-parallel
When collisions are non parallel but both are spheres you can break the problem down into the x and y componenets where the y component does not change unless an external force acts upon it. Therefore the coefficient of restitution only applies to the x velocities.
Solving Vi and Vf with e and nonparallel collisions
- ) Use conservation of momentum in the x and y directions. Vy does not change.
- ) e = abs(V2/V1) (summate for each object)
- ) separate e so that one of the two terms is the unknown in the conservation of momentum.
- ) You should now have two equations and two unknowns. Solve
- ) Use the resulting vs initial velocities to find changes in momentum, KE, or other
Solving plane surfaces below water
Use pressure prisms and remember that the Fr (fluid) does NOT go through the CG of the plane.
Solving manometer/barometer problems
- ) Choose an A and B point
- ) move from point A to B where when moving down you add P= γh
- ) Remeber that equal heights are of equal pressures
Center of pressure
When an object is submerged the center of pressure measured from the fluid surface is given by Xc = ∫ (X2 dA) / ∫(X dA)
or through using the parallel axis theorem: Xc-Xo=Io/AXo where Xo is the CG measured from the free surface
Tension in a thin-walled pressure cylinder
There are generally two stresses acting on the walls. There is the hoop stress σ1 and the longitudinal stress, σ3.
Note: at the boundary of the pressure and the wall there is a pressure that acts opposite to the internal pressure. On the outer edge there is no pressure.
Hoop stress formula
If the cylinder’s wall is >(1/10)*diameter then σ = pr/t where t is the thickness, p is the pressure, and r is the radius
Thick walled cylinders
These are cylinders where the wall is thicker than (1/10)*diameter so that the hoop stress within the material varies with radial position.
f= [(ri2 + ro2)/(ro2-ri2)] = inner σ/internal p = σ/p
Buoyancy force
This is the force of the displaced fluid. It acts through the center of gravity of the displaced fluid in the up direction
Steps to solve: dams and gates
- ) FBD
- ) find all relevant weights, hydrostatic forces, and buoyant forces and the places that they are acting on the other object.
- ) summate forces to find reactions
- ) summate moments to find positions
Bernoulli’s theorem
if a fluid is both inviscid (frictionless) and incompressible
then E= (P/γ) + z + V2/2g
Energy Equation
The energy equation is the equivalent to bernoullis when there is head loss or energy is added to the system. To use identify to place where the system has less energy to add loss. Head loss and power is in terms of feet of energy.
(P1-P2)/γ + (z1-z2) + (V12 - V22)/2g = HL - Hp
Head loss due to friction
Frictional head loss is proportional to the length of the pipe. Inversely proportional to the diameter to some power because SA = f(d2). Varies as a function of the roughness of the pipe and flow turbulence. Varies with velocity like solids but is independent of pressure.
hf = f (L/d) (V2/2g) where L is the pipe length, f is the friction factor, V is the average velocity ( ∫ V dA/ ∫dA)
Orifice coefficients
This is a result of the vena contracta effect where the change in momentum causes the discharge of flow to have a smaller diameter than the outlet. The smallest diameter in comparison to the orifice diameter is related by the coefficient of contraction.
How to use the coefficient of contraction
Cc = smallest area of jet/ area of orifice
Q for fluids
Q is the volumetric flowrate given in ft3/s
velocity from a orifice
V = Cv [2gh].5 where Cv is the velocity coefficient (usually .98) and h is the distance from the free surface to the discharge.
This only works if there is not a turbine or other thing being moved by the flow.
Minor losses
These are losses due to changes in the geometry of the flow (conservation of momentum requires external forces)
HL = Σ ksys (V2/2g)
Viscosity
The viscosity of a fluid is defined by its internal resistance to flow.
dynamic viscosity
This is the ratio of applied shear to the change in velocity throughout a fluid.
μ = τ /(dV/dy) where y is the vertical distance from the stationary surface to the point of interest within the fluid.
Kinematic viscosity
This is ν = μ/ρ and has units of L2/s. It represents the viscosity of a fluid without relating it to the density of the fluid.
Newtonian fluids
These are common fluids where viscosity is independent of shear.
non-newtonian fluids
for a non-Newtonian fluid the viscosity depends on the rate of applied shear such that it becomes more viscous as the shear increases
Reynolds number (pipes + open channels)
Pipes: Re = ρVd/μ = Vd/ν
Open channels: VL/ν where L is the characteristic length
The reynolds number is related to whether a flow is considered turbulent or laminar which dictates whether the flow lines of the fluid move in parallel or at random.
pump head for a pump moving a material to a resivor with a higher z
Work/ γ * Q = (z2-z1)+(P2-P1)/γ + (V22-V12)/2g +hl
Where: hl = f(l/D)(V2/2g) where V is the V exiting the pump and throughout the pipe of length l and diameter d
hl can also be minor losses with are equal to the summation of the loss coefficients times V2/2g
What is Q
Q is the volumetric flow rate it is given in cubic feet/sec
How to solve multi-pipe problems
- ) draw diagram and ID end states for P, V, Z, l, and D + unknowns
- ) write energy equation from 1->2 , 1-> 3… where hL = f1(l1/d1)(V12/2g) + f2(l2/d2)(V22/2g) + minor losses. This is added to the second state of the fluid.
- ) Conservation of matter says that Qin = Qout
- ) Solve
Pitot Tube
This is a L-shaped tube that measures the stagnation pressure of a moving flow. If we consider h to be the difference in heights of a fluid in a pitot tube and a static pressure tube (vertical tube) then Vflow= [2gh].5
Venturi Tube
This is a tube for measuring the velocity of a flow that uses a differential pressure across a contraction.
For the formula A1>A2 and P is measured through h of the fluid where h=P/γ
Flow over weirs
weirs are notched openings in vertical gates.
H=height from the crest to the level upstream surface= energy head of weir.
To solve weirs be aware that the velocity is proportional to the height of the flow and that the vena contracta effect influences the flow.
Sluice gates
These are gates where the flow is moving out of the bottom of the tank, under the gate.
Solving sluice gates
- ) Draw FBD
- ) label end states where 1 is on the top of the resivoir and 2 is on the top of the outer flow and datum is given at the ground.
- ) apply Bernoulli’s equation or the energy equation
Force on an object redirecting fluid
Fx = ρ (ΔVx ) with an equal for Fy
V2 is found through conservation of mass
Current; symbol, units, meaning
Current, I, is a measure of the flow of electric charges in a conductor.
It is given in amperes which is related to the number of mols of electrons that are flowing through the cross-sectional area of a wire
Charge: Units, symbol, meaning
Charge, Q, is the total current over a one second time interval
Amps= Coloumbs/second
Q=coloumbs which are proportional to the mols of electrons
Electromotive force (EMF)
This is given in units of Volts, V, and is related to the difference in potential energy. It causes for current to flow.
Resistance: units, meaning, symbol
Restitance, R, is the discrete resistance to current and arises from the physical characteristics of the material that current is flowing through.
It is measured in ohms, Ω
Capacitance; what is it, units, and symbol
Capacitance is a measure of how well two conductive materials separated by an insulator hold charge. It is measured is Farads, F
Inductance
Inductance is a measure of property of a coil to create an EMF that opposes the net flow of current through the coil. It is given by the symbol L and is measured in henrys H
Energy: what is it, symbol, and units
Energy is considered to be the amount of work a system is capable of doing on another object. This is measured is Joules where 1 joule = 1 N*m because Work = F X d
Power: What is it, units, and symbol
Power is the rate of doing work. It is measured in Watts where 1 W = 1 J/s
Horsepower
This is a type of power where 1 hp = 746 W = 2546 btu/hr
Efficiency: what is it, units, and symbol
For electrical systems efficiency is given by μ = (Power delivered to load)/(Power Generated) = Po / Pi it is measured as a percent
For mechanical pumps and things μ = (Power produced - loss)/(power produced)
Regulation: What is it, Units, and symbol
This is efficiency for voltage defined as:
(No load V - Full load V)/(Full V) and given as a percentage
Gram-calorie
This is the energy required to raise the temperature of 1 gram of water 1 degree centigrade measured in joules and dependent of the specific heat of the material in question.
what units are milli
10-3
units of Mega
M = 106
Micro units
μ = 10-6
Units of pico
p = 10-12
kilo
k = 103
Giga
G = 109
Coulomb’s Law
This says the mutual force acting between two particles is given by the electrostatic force:
F = kq1q2/d2 where k ~ 9*109 Nm2/C2
q is the charge of each particles in coulombs where oppositely charged particles attract
Electric Field intensity/magnitude
This is the idea that is we were to have some charged particle and then we place a positive charge in its vicinity that it would experience a force/coloumb of positive charge given by
E = F/q1 =N/C
Electric field direction
This will depend on the charge of the particle/s but if only one particle is present the direction is radial to the charge. If more than one is present you can break it into two+ systems and add vectors.
Conductors/Insulators
These are concepts for different materials that behave differently when current flows through them. Metals are conductors because their large shells and abundant valence electrons allows them to transfer charges easily. Insulators are generally materials that have small orbitals and less discrete ways to transfer charge.
Ionic substances for charge
Ionic substances when in solution or as a molten solid are great conductors because the charge can “bounce” between atoms but as solids they are very poor conductors because charges are within the tight inner bonds.
Semiconductors
These are materials that act as insulators when the voltage across the material is less than the threshold value which prevents charge from flowing through the system. Above the threshold value this shifts.
Factors influencing resistance
These include things like heat, humidity, and other
Faraday’s Law for electrolysis
This is a law relating the amount of dissolved or precipitated material in an electrolytic cell as a function of the charge flowing through the system.
If you find: (atomic mass/valency) that is the total mass of a substance that would change if 96500 coulombs passed through the system.
then: (mass/valency/96500) = (mass change / # coloumbs)
and: Q (coloumbs) = I * t
Faraday: unit, what is it
This is a convenient way to group coloumbs (charge). it says that one faraday = 96500 coulombs
Calculating resistivity of a circular conductor
R= ρ L/A where:
ρ = CM*Ω / ft ; CM = circular mils (1 mill = .001 in) ; A= d2 where d is the wire diameter in cm ; L = wire length in feet
change in resistance due to T
R2 = R1 (1+ α dT) where α = temperature coefficient
Rankin
oR = oF + 459.67
Resistance at absolute zero
electrons do not flow at absolute zero (in theory)
Kelvin
oK = oC + 273.15
farenheight to celsius
oF = oC*(9/5) + 32
oF is always greater than Celsius
resistance in series
REQ = Σ Ri
Use this to combine resistors in parallel and simplify circuits
Resistance in parallel
1/Req = Σ 1/Ri or Req = 1/Σ(1/Ri)
Conductance
This is simply the inverse of resistance given in siemens, G
G=1/R
G in series and in parallel
Conductance acts opposite to resistance so
Gseries = 1/(Σ 1/Gi) and Gparallel = ΣGi
Calculating conductance
For parallel plates separated by a diaelectric
C (farads) = K*∈*A*(n-1)/d
where: K = diaelectric constant
∈ = 8.85 * 10-12 C2/N*m2
d = plate seperation
A = area of ONE plate
n = # of plates
Equivalent capacitence
Capacitence in a series: Ceq = 1/ (Σ(1/Ci)) because charge is being distributed linearily among the capacitors like how charge is distrubuted among resistors
Capacitence in parallel: CEQ = Σ Ci This is because the charge can be transfered between lines
Equivalent inductance
In a series: Leq = ΣLi because charge flows through each inductor like resistors
In parallel: Leq = q/(Σ(1/Li))
Calculating inductance for a coil
L (henrys) = N2μ A/l
where μ = constant of magnetic permeability
l = avg length of form
A= cross sectional area of hypothetical cylinder within the coil
DC circuits
Within a closed DC circuit current flows in one direction from the positive to negative battery terminal
Ohms law
V = I*R where voltage is the potential energy difference across the resistance. I is the charge flowing through the resistance and R is the amount of resistance
Kirchoff’s Rules for closed electrical systems
Σ Iin = ΣIout where I is the current flowing in and out of a node which is just an intersection of wires. This represents the conservation of charge/matter.
AND
Σ Vrise = Σ Vdrop where V is the voltage found through “walking the circuit” This represents the conservation of energy
Current division rules
This represents the distribution of current throughout parallel resistors and is given by I1 = R2*I/(R1+R2) and I2 = R1*I/(R1+R2)
Where I1 and I2 represent the current flowing through the resistor of the same subscript. I = ΣI1 + I2 and represents the total current distributed through the parallel resistors
Voltage division rules
This is for resistors in series where V1 = R1 * V/(R1 + R2) and vice versa for V2 where V is the total voltage drop
Voltage across parallel resistance is equal among each of the resistors.
AC current
This is alternating current where V=f(t) and V(t) = Vpsin ωt
where ω = radians/sec and represents the angular frequency of the AC
Vp= peak voltage at which V(t) is oscilating between
Converting from frequency to angular frequency
ω (rad/sec) = f*2pi = (1/s)*rad
Root mean square voltage
This is the weighted average voltage given by
VRMS = [(1/T) ∫0t V(t)2 dt].5 = Vp/2.5 = .707 Vp because the integration is based on a 2pi cycle where T is the period (s) Vp = peak voltage
DC V conversion
Vdc = (1/T) * (∫0t V(t) dt) = 2Vp/pi = .637 Vp
where t is chosen to equal 1/2 a full cycle (ωt = pi)
Using AC in diagrams
Just plug V(t) in for V in all of the other formulas
What does Irms represent?
This represents the effective current moving through a resistor. This represents the current that would effectively flow through a resistor at time=t for a DC current
Lead and lag
The current and voltage over time in sin/cos space are related by the phase angle which represents which is “ahead” of the other in time.
if for voltage the phase angle > 0 then it is a phase lead
Defining a function (V -> Vrms ->phase space)
If given V= Vp *sin(ωt+θ) then Vrms = .707 Vp and can be represented in polar as Vrms (θ) which translates to a rectangular vector given by
a+bi = (Vrms *cosθ)+ (Vrms *sinθ)*i
Impedence
Now that we have a bunch of imaginary numbers plain logic doesn’t work so we have Z=impedance = V/I where V=Vrms and I=Irms as complex numbers (remember in DC that R=V/I)
Capacitance with impedance
Capacitance causes the voltage to lag the current by 90o where
I = ωCVp cos(ωt) = Vp/(1/ωC) * sin(ωt+90)
Voltage across a capacitor
Vp sin(ωt) = ∫0t i(t)/c d(t) where C is measured in farads
Lenz’s Law
V(t) = -L d(i(t))/dt
how to use Lenz’s law
Seperate d(i(t)) from dt and solve for i(t) by integrating ∫ d(i) = ∫ V(t)/L dt
Power average DC
Pavg = V2/R (watts) where V = Vdc
Power Avg AC
Pavg = VrmsIrms
Real, reactive and apparent power
Power =VI which has real and non-real (imaginary) components which can be written in vector space.
Apparent power = S = magnitude
Real power = VIcosθ in watts
reactive power = VIsinθ in VARS
Loop method
When there are many meshes you can assign each a loop current (I1, I2…) and find the total voltage drop through the loop. When using this method always move counter clockwise and when wires are shared use V=R(ΣI)
Parallel resonance (RLC)
The resonant frequency, fo, represents the frequency at which Z=Zo =Zmax
fo = 1/2pi[LC].5 where Zo = L/CR =Q2R and It = V/Zo
Series resonance
fo (hz) = 1/2pi[LC].5
Bandwidth
This is the frequencies for AC where P and I are >.707 max
Filters
These are networks designed to discriminate frequencies of signals. A pass band creates constant amplitudes whereas a reject band has a low amplitude response.
attenuator
This is a network that is designed to isolate one network from another preserve a match between networks.
Three Phase Power
This is fundamentally based on the concept that if you have three voltages each 120o from one another the time weighted average is equal to VRMS and can be used to power systems like motors without as much vibration.
Delta circuit
This is where there is a three phase delta load connected to a delta power source where Z is between nodes.
In these systems the Vline = VAB=VBC=VCA but the phase currents (ICA, ICB, IAB) are equal to 1/([3].5) * IL
and in a balanced system P = [3].5 VL*IL
Finding line currents in three phase power with phase currents
Line currents are the currents delivered to the loads (A,B,C) and are denoted by either IL or IA,B,C… where IB = IBC- IAB and so on where the rule is that the inner subscripts of the numbers cancel each other out
total power in three phase delta system
PT = summation of all lines (VAB2/ZAB)
Wye three phase circuit
This is a circuit with a Y configuration that is similar to a delta configuration but it is not balanced so it has a N wire (neutral) at the core.
Currents and loads in a Y system
In a wye system line currents are equal to one another and the neutral.
Phase voltage in a Wye system
Vcoil = VL/[3].5 where VL is measured from the node to nuetral.
Power in a wye system
This is the same as a delta system
P = [3].5 VL IL cosθ
Magnetic circuits
This similar to an electric circuit where
Resistance = Reluctance
Magnetic force = voltage
flux flow = current
magnetic reluctance
This is like resistance for a magnetic circuit and is given by R = L/μA where μ is the magnetic permeability constant of the core of the material, L is the mean length, and A is the cross-sectional area
Current density
This is a measure of the density of magnetic field lines given by
B (teslas) = ϕ/A = μH
Field intensity for magnets
H = t/L = t/2pi r
Transformers
These are electromagnetic devices that consist of one or more coils that is used to “step up” or “step down” voltages for various uses. They are generally rated in KVA or VA where a 100 KVA transformer can deliver 100KW of power if pf = 1
Turns rule for transformers
This says that V1/V2 = I2/I1 = N1/N2 = a
Impedence loading through transformers
If we have a transformer with I1 and Zi represents the impedence from the coils with a Z2 = load related to the other side of the transformer with I2 then it is said that Zi = a2 Z2 where a = N1/N2
This is also referred to how much of the load is reflected into the primary.
Short Circuit tests for determining transformer character
A transformer is loaded with the various electrical character for which it is rated and then on the primary an ammeter, wattmeter, and voltmeter are attached. An ammeter is attached to the secondary.
copper losses = wattmeter rating = losses from within the primary coil
Equivalent impedence = Zeq = V/I1 where I1 and V are measured by the ammeter and voltmeter respectively
Equivalent Resistence = Req = W/I12
equivalent reactance = X = [Z2 - R2].5
Losses within a transformer
These are similar to any coil where there are
- ) copper losses which refers to the heat that is created in the copper
- ) core loss related to eddy currents and hysteresis
Per-Unit Notations
This is definition oftentimes applied to electrical equipment to normalize quantities to a common base. This basically says that there are relative values of electrical properties within the equipment
Base current (amps) = base KVA/ base KV
Base Impedence = base voltage/ base current = (base voltage KV)2*10/ base KVA
Ways to test transformer properties
- ) short circuit test
- ) open circuit test
General menaing of current
When stated to find current this means the line current.
general meaning of voltage
This means finding the volts to nuetral and KVA per phase
Autotransformer
This is when there are more than one coils serially connected usually in a way that there is a winding abc with a load parallel to bc.
What does N1 equal in an autotransformer?
N1 = winding abc (the full winding) and N2 = the winding parallel to the load usually winding bc
a = ? for an autotransformer
a = V1/V2 - I2/I1 + N2/N1 and I through the second winding is given by It-I2
Rotating electrical machines working principle
All rotating electrical machines operate on the principle of electromagnetic induction such that whenever there is a change in the magnetic flux there is an induced electrical force.
Faraday’s Law
This says that whenever there is a change in magnetic flux an emf is produced and is given by
e= - d(Nϕ)/dt where Nϕ is the magentic flux through a closed loop that has current flowing through it.
Magnetic Flux
This is measured by ϕ and is given by B X A where B is the magnetic field and A is the loop such that if the loop is perpindicular to the main field that ϕmax is achieved
Right hand rule for F, B, and I
For a loop with current, I, within a magnetic field, B the force on the loop is given by your thumb, your pointer is in the direction of the magnetic field, and I is the direction of your middle finger such that F = B X I
Torque for an electromotive moter
This is given by T = ϕmaxI sin ωt where ωt is a term relating the rotation of the coil to radians because F = B X I where B is given by ϕ
DC motors function
This is relatively similar to an AC motor but there is a commuttor attached to the loop so that way the direction of current switches ever pi radians. This has the effect of keeping the net current unidirectional and is in the function of a abs(sin ωt) where ω is proportional to the angular frequency of the coil
How to reduce jittering in a DC motor
With one loop the DC current oscilates between 0 and max every pi radians. With multiple loops you can have lead/lag @ 120o or other to stabilize the avg voltage across the loop
Commercial generator operation
This generally has many distributed N-S poles throughout a cylinder that maximizes the flux throughout the entire rotating and prevents the coil from being stagnate because it will never reach a stable equilibrium where it can be perturbed with changing phase.
AC generator function
This is very similar to a DC generator but has slip rings not a commutor so that way voltage/current oscilates in both positive and negative space
frequency of an AC generator
f = pN/2 where p is the number of poles, N is the rev/sec of the mechanically driven loop (gas powered)
power factor
This is the ratio of usable power to total power.
Synchronous motors have a leading power factor
Time domain analysis in electrical systems: idea
This is the idea of using diff eq to solve for various parameters of RLC circuits where the beginning and end states are known and we are solving for the middle period.
Generally: t0 = the time when a switch or something is turned on
t1= steady state when either there is not a fluctuation in the electrical parameters or there is nothing occurring. This is usually at infinity
What are the three primary physcial components in an time domain analysis problem?
These are usually RLC circuits which include a resistor, inductor, and capacitor.
Only the resistor does not instantly change with t. Inductors and capacitors store charge so that when there is a change in the state of the system they do not instantaneouslt shift to the final state.
Inductors
These are coils that having current flowing through them. The moving charges induce a magnetic field with the positive end in the direction of the entering current
Energy stored within an inductor
The energy stored within an inductor is given by W = (1/2)LI2 where W=joules, L=inductance (Henrys), and I = current (amps)
Voltage across an inductor
This is related to time and inductance by VL = L di/dt where the positive voltage is in the direction of the current entering the coil. If there is not a change in the current across an inductor than the induced voltage is 0.
Voltage across an inductor with time
Because VL = di/dt L if we have i drop to 0 in dt~0 then VL= infinity. In the situation where the source is removed then the inductor become a source where current is generated as given by lenz’s law to oppose the change of current within the inductor
Capacitors
These are effectively a series of parallel plates that accumulate charge proportional to the surface area of the interfacing plates.
Power within a capacitor
This is given by W = (1/2) CV2 where W = joules, C = capacitence (Farads), V = Vc across the capacitor
current flow through a capacitor
This is given by the relationship ic = C (dVc/dt) therefore in a DC circuit where dV/dt = 0 the capacitor acts like an open circuit and there will not be any current flowing across the capacitor
Total/Complete Response
This is the description of an RL or RC circuit within a changing state. It is said that there are two sources of energy, the natural response refers to the internal response of either a capacitor or a inductor. The forced response is the induced response given by an external forcing like a battery or generator. This is also called the steady state response because it is the function that describes the system if closed at t=infiinity.
how to solve a transient state problem
Because there are generally two forces that are influencing the system; the natural and forced responses we can independently analyze each response then add them together to find the complete response.
dRL circuits at t = 0+ (closing switch)
via Kichoff’s voltage law L (di/dt) + Ri = V where i = i(t)
in= K e-R<span> </span>/L where K is determined from the conditions at t=0 and in(t) is the natural response. K = V/R
iss = V/R = steady state
i(t) = (V/R) (e-R/L * t + 1)
RC circuits
These are circuits that have resistance and a capacitor connected to a source. Because electrons are matter they do not instantly accumulate onto the capacitor therefore when connected to a source there is a natural and forced response
complete response of an RC circuit
by KVL: Vr + Vc = Vs
it = (V/R) e-1/RC * t Vc= (1/C) Ve-1/RC * t Vr= - V e-1/RC *t
Systems
This can be described as a combination of diverse interacting elements that are integrated to achieve one objective.
Systems engineering perspective
This is the idea of looking at all of the interacting elements and evaluating if the system reaches its objective. Systems engineers do not care about design only outcomes.
open loop automatic control system
This is a system where the output has no effect upon the input.
This is basically y=f(x) where y, the output, only depends on the input, x
closed loop (feedback) control systems
This is where the outcome of an input is fed back into the component to alter future outputs.
It is similar to saying y = f(x,t) where the output, y, is not only a function of the input, x but also is influenced by time. In this specific instance with time the prior y values are used to change future y values.
This means that y(x1, t1) =/ y(x1, t2) even though the input is the same
On-off controllers
This is a closed system where the system is turned on/off if the error signal is a certain value. This is like a thermostat.
Step controllers
This is a type of on-off controller where the error signal varies the magnitude of the output. For example in a furnace, if the error signal returns that it is very cold then it may overshoot the desired temperature.
Servomechanisms
These are closed loop systems where the output is measured in comparison to a reference and the power to the system not only varies in magnitude but information to change the output. For example the autopilot on a plane tracks the course of the plane in reference to the ideal and uses that to create an output that varies velocity.
Standard block diagram symbols/nomenclature
See chart given but a few notable symbols include
v = the independent input
b = feedback variable
r = reference input = f(b, v) and is the signal the component uses
e= r-b = actuating signal = the error
c = controlled variable; the measured variable; the output
Gv = reference input elements
G = dynamic unit; both control systems and components
H = feedback elements; the components
Differential equation method of system analysis
This is the method of describing the feedback system through implicit differentiation and series expansion. These are best for quantized systems where products or discrete values that can be returned into the function.
Transfer functions of feedback control systems
This basically uses a laplace transformation of a differential equation to interpret the magnitude, phase, and phase of a system.
Semiconductors
Semiconductors in a pure sense include silicon and germanium. Both have 4 valence electrons and are covalent
Intrinsic semiconductors
This includes germanium and silicon and represent a group of materials when, at a certain temperature, the covalent bonds break down and the valence electrons are excited into the conductance band and leave a hole. The combination of these is called an electron hole pair
Extrinsic semiconductors
This is the idea of creating impurities in the crystal lattice that either enhances the ability for electrons to shift position or stifles it. These are the extrinsic semiconductors.
N-type semiconductors
These are extrinsic semiconductors that are doped with elements that have 5 valence electrons so that the fifth electron is in the conductance band. Because these semiconductors carry more electrons they are considered to be negative thus the name N-type.
P-type semiconductor
These are semiconductors that are doped with a trivalent atom so that there are electron vacancies that can then be filled. An example of this is placing Al within silica or germanium
minority carriers
This is the idea that at certain temperatures and under certain conditions the materials meant to insulate (silica and germanium) may break down and begin to conduct these are known as minority carriers
The P-N junction
This is the junction between the positive and negative material. These are unidirectional interfaces of flow.
diffusion current
This is the idea that at the interface of the positive and negative charges that there is diffusion of electrons from the negative to the positive.
Depletion region
This says that because of diffusion across the P-N boundary that the negative material will gain many positive charges and the P material will fill its vacancies. They become saturated and there is a depletion region where flow is not occuring. The potential across this is significant.
Semiconductor diodes
These are simple devices with a P-N junction
forward bias
This is defined as the positive charge of the battery pushing current into the positive end of the terminal. It requires a significant voltage for current to flow.
Reverse bias
This is when the negative end of the battery is connected to the negative side of the semiconductor so that the depletion area becomes larger and only the minority carrier carry material. This current is called the reverse saturation current, Is.
Period or Series on the Periodic table
These are the horizontal rows on the periodic table. The row indicates the number of energy orbitals for the row.
Group or Family
These are vertical rows on the periodic table. Groups indicate the number of valence electrons for all elements in that row.
What does the atomic number represent?
These are the total number of protons within an atom.
Atomic mass meaning
This is the weighted average of the proportions of isotopes of that atom commonly found in nature.
Isotopes
These are molecules with the same number of protons but different numbers of nuetrons which changes their mass.
STP
0 oC ; 273 oK ; 32 oF @ 760 mm Hg; 1 atm; 2116.2 psfa
mole/pound Molecular Volume
This is the molecular volume of a gas at STP. For any gas @ STP this is 22.4 liters/mole of gas.
In imperial this is 359 ft3/pound of gas.
This also says that if we have a 22.4 liter container of 1 mole of gas @ STP it will exert 1 atm of pressure
Gay-Lussac Law
P1/T1 = P2/T2
This says that the pressure of a given mass of gas within a fixed volume is linearily related to the temperature of the gaseous volume
Avogadro’s Number
This says there are 6.024 * 1023 molecules/mol
Ideal Gas Equation
PV = nRT
where P = abs pressure; V = gaseous volume; T=oK
n= moles; R=.082 l * atm/mole*K
Graham’s Law
This says that the rate of diffusion (r) between two gasses is inversely proportional to the square root of the gas density/molecular weight
r1/r2 = [ρ2/ρ1].5 = [m2/m1].5
Dalton’s Law of Partial Pressures
This says the sum of the partial pressures of each gas is equal to the total pressure of the mixture
PT = Σ Pi
Synthesis Reactions
These are reactions that produce one product from 2+ reactents
Decomposition Reactions
These have one reactant that decomposes into several products
Single Displacement Reactions
These are reactions where one more reactive atom replaces a less reactive atom to create a lower PE state
Ex: Zn + 2HCl -> ZnCl2 + H2
Ion Exchange Reactions
These are reactions where there is a double displacement of atoms. It commonly represents the neutralization process between acids and bases.
AB + CD -> AD + CB
Ex: NaOH + HCl -> NaCl + HOH which works because NaCl is in a low PE state as an ionic compound and HOH is a strong hydrogen bond
Reversible Reactions
These are reactions that go either direction and do NOT go to completion because there is a constant system perturbation as long as all products and reactents remain in close proximity.
Equilibrium condition of a reaction
This is when the rate of forward reaction = rate of reverse reaction. Also known as dynamic equilibrium
Law of Le Chatelier
This says that when a system in dynamic equilibrium is subjected to a stress that the system shifts so the stress is relieved and dynamic equilibrium is restored.
How does pressure influence the dynamic equilibrium of a gaseous reaction?
An increase in pressure will favor the smaller volume product/s or the side of the reaction with less separate molecules.
Dynamic Equilibrium with a change in concentration/quantitu
With a shift in the concentration of materials the other side of the reaction is favored
How does dynamic equilibrium shift with temperature?
With an increase in temperature the endothermic product is favored, with a decrease in temperature the exothermic reaction is favored.
How do catalysts shift dynamic equilibrium?
Catalysts lower the PE barrier to reacting therefore a catalyst equally increases the rate of both the forward and backwards reaction.
Equilibrium Constant Formula
For some reaction given by WA+ZB <=> XC + YD at a position where R1 = R2 and [] = molarity of the substance (moles/L)
Keq = [C]x [D]y/ [A]w [B]Z
Keq meaning
K represents the proportion of products and reactants at equilibrium
If K>1 then the forward reaction (exothermic) is favored
If K=1 then neither product or reactants are favored
If K<1 there are more reactants than products and the reverse reaction is favored
What is Q
Q = K but when the reaction is not at dynamic equilibrium
Ionization Constants
These are Keq for the ionization of different acids. The larger the ionization constant the more stronger the acid or base is.
HAa + H2O <=> H3O+ + Aa-
The solubility Product
This shows the solubility of slightly soluble salts where the large the solubility constant, the more soluble the salt
ABs <=> A+aq + B-aq
Solids within K calculations
The molarity of a solid is 1
K for H2O <=> H+ + OH- ?
This is Kw aka the water ionization constant which is equal to 10-14
This means that within 107 liters of water there is one mole of OH and H because Kw = 10-7 * 10-7
pH formula
pH = -log[H+] thus a smaller pH is a more acidic solution
Relationship between pH and pOH?
pH+pOH=14
What does a pOH of 1 mean
This means that at equilibrium, the substance has a molarity of 10-2 for OH or 10-2 moles OH/L meaning that it is a very basic liquid
Percantages in a solution
This is the number of units of weight of material is in 100 units of weight of the solution. It is a form of PPM.
Molarity
M = moles of solute/litre of solution
1M means that in a solution with a total final volume of 1 litre that 1 mole of solute has been added.
Normality
N = # of equivalents/litres of solution = #of equivalents * molarity
where: # of equivalents is given by the number of moles of a reactent in the equation and the molarity is the other reactent or product in the equation. It is most common for titrations
Molality
This is the moles of solute/litres of solution
Alpha Decay
This is the spontaneous emission of an alpha particle (4He) which decreases the reactent by two protons and 2 nuetrons
Beta Decay
This is the spontaneous emission of a beta particle (a nuetron that shifts to an electron) This causes the particle to add a proton increasing its mass number and atomic number by 1
Mass relationship to readioactive decay
If we have a mass at t=0 of mo then m after some t is given by m = mo e-δt where δ = decay constant
t1/2 = ?
t1/2 = ln2/δ = .693/δ and represents the amount of time that it takes for half of a radioactive species to decay
Definition of one calorie
This is the amount of energy needed to raise the temperature of 1 gram of water by 1 oC
The heat of fusion
This is the energy required to break the bonds of a solid as it is transitioning to a liquid even though the temperature does not shift.
Heat of fusion for water
it takes 79.7 calories to melt 1 gram of ice
Heat of Vaporization
This is the amount of energy needed to change the phase of a liquid to a gas. For water this is equal to 539.6 calories/gram. That also means that in the transition from a gas to a liquid that 539.6 calories are released.
Bond Energy
This is the amount of energy needed to break the bond of two or more atoms.
Using Hess’s Law the bond energies of each reactent can be used to find the total heat released in a reaction
Hydrocarbons
These are compounds solely composed of carbon and hydrogen. They are non-polar, covalent, and insoluble with low melting and boiling points.
The general formula for hydrocarbons is R-H where R= the hydrocarbon with an added hydrogen, H
Aliphatic compounds
These are carbon compounds where carbon atoms are arranged in straight or branched chains
aromatic compounds
These are carbon compounds where the carbon atoms are arranged into rings
Homoogous series
These are a family of hydrocarbons with the same general structure. They also share the same general formula and similar properties.
Methane series: Alkanes formula and structure
Cn H2n+2 Methane is CH4
Saturated hydrocarbons
These are hydrocarbons that are bonded singly to themselves and the H atoms.
Unsaturated hydrocarbons
These are hydrocarbons where the carbon atoms do not have all four bonds with other things aka they have double bonds.
Ethylene Series: Alkenes-olefin series
Formula: CnH2n Ethelyne: C2H4 These are unsaturated hydrocarbons and the two carbon atoms have a double bond
Acetylene Series: Alkynes
General formula: CnH2n+2 Acetylene: C2H2
These are considered unsaturated because they have a triple bond between the carbons making them non-polar
Benzene Series
Formula: CnH2n-6 Benzene: C6H6 These are also unsaturated because each carbon is bonded to one other thing
Substituted Hydrocarbons
These are hydrocarbons where the hydrogen is replaced by a functional group which determines its properties.
Alcohols
General Formula: R-OH
methyl alcohol: CH3-OH
Soaps
These are broadly a metallic salt of a fatty acid. They are made by boiling hydrocarbons with a strong base like NaOH or KOH to form soap
Carbohydrates
These are C, H, O compounds where the H:O ration is 2:1
Common carbs
Glucose, sucrose, starch, and cellulose
Valence
This refers to the number of electrons an atom may gain or lose to create orbital stability
Oxidation
This is the loss of electrons. It occurs at the anode (+ side) of an electrolytic cell.
AN OX is Positive
Reduction
This is the gain of electrons during electrolysis that occurs at the cathode (negative end) of the battery.
RED CAT is Negative
How to balance a redox reaction
- ) Write half reactions (with electrons) for each of the elements in the reaction
- ) multiply by coefficients to that electrons balance
Oxidizing/Reducing Agents
These are the elements that oxidize or reduce the other element. The oxidizing agent is reduced during the reaction
Standard oxidation Potentials
This is a chart with a large number of half reactions of oxidation. A large electromotive potential represents that upon being oxidized the element is in a lower PE state thus it is more favored.
Using Standard Oxidation Potentials
To use the E0 values you summate the associated value with the half reactions created when balancing the reactions.
Electrolysis
This is the process of decomposing or recomosing metals through electroplating
One Faraday
This is the amount of electrical charge to liberate one gram equivalent weight of an element from solution. One faraday is equal to one mole of electrons.
A coulomb
This is 6.24 * 1018 electrons or other elementary charges. One faraday is equal to 96490 coulombs
Boyles Law
This says that pressure and volume of a gas at a constant temperature are inversely related by a linear relationship given by:
P1 * V1 = P2 * V2 furthermore if dT/dt = 0 dP/dt = 0 and dV/dt = 0
Charle’s Law
If the pressure of a gas is held constant then dT/dV = c meaning that any change in temperature has a direct linear change in volume.
General Energy Equation
For an ideal, frictionless, steady flow system with dt~0
PE1 + KE1 + U1 + FW1 + Qa = PE2 + KE2 + U2 + FW2
KE = Kinetic Energy = .5 gV2
PE = Potential Engery = z
U = internal heat
FW = work done on/by the system = PV
Qa = work added to the system
Qa = ?
This is the change in work
Qa = ΔU + W
Is work path dependent?
Yes. Work = W = F x d meaning that if d changes as a vector the work done by/on the system changes.
Work for a piston
W = ∫ P dV where P = f(V) for the volume of the piston being closed
Specific Heat
This is the amount of energy needed to only increase the temperature of a substance without phase change
If dQ heat is tranfered to a substance with a mass of m the temperature will change by some dT related to the internal character of the material.
In other words C = dQ/mdT
How does internal energy relate to specific heat
Q = m*c * ∫ dT
k for specific heat is?
k = CP/Cv where CP = specific heat when dP/dt = 0
Cv=specific heat when dV/dt = 0
Enthalpy
Enthalpy = H
H = U + PV and ΔH (for gases) = m* ∫ Cp dT
Entropy
For a reversible tansfer of heat dS = dQ/T
and ΔS = ∫ dQ/T = m C ln (T2/T1)
Volumetric analysis of gases
For a mixture of gases at some temperature and pressure
Bx + By + Bz = 100% where B = V(component)/VT
Gravimetric analysis
Gx + Gy + Gz = 100%
G = m(component)/mT = V(component)*rho / VT * rhoT
Heating Value of a fuel
This is the amount of heat given up by the products of combustion when cooled to the original temperature that the reactants started.
Air within combustion
Assume that air is .77 N2 and .23 O2
Perfect gas assumption
This assumes that the gaseous atoms have negligible or no volume and has no attractive forces between individual atoms.
Vapor
These are gases that are in a state where they can reverse into a liquid state. When vapors are heated to temperatures that far exceed boiling point they are “superheated” and may form a perfect gas.
Saturation temperature
This is a thermodynamic term for describing the boiling point for a substance at some pressure.
Quality of a wet steam mixture
This is defined as the % of mass that is in a saturated gas state.
Throttling calorimeter
This is a device used to measure the quality of a wet steam mixture. Steam’s enthalpy is measured at two points of a tube.
x = h1 - h2 / hfgl
Mollier chart
This is a chart that has various enthalpy values for wet steams
Relative Humidity
ϕ = Pw/Pg
The relative humidity represents the ratio of the partial pressure of water vapor being the vapor in a superheated state to the partial pressure of saturated vapor at that same temperature.
humidity ratio
This the ratio of mass of water to the mass of dry air and given by γ
This is related to relative humidity by = γPa /.622 Pg
How do steam power plants work?
A boiler converts chemical energy into heat which provides high temperature to a prime mover like a turbine or steam engine.
Refrigeration
This describes the process of heat being absorbed from a low temperature region which is commonly done through evaporation and decompressing vapors to convert the latent heat into phase changes.
Conduction
This is the most common form of heat transfer and is described by
Q = kA(T1 - T2) /X where
Q = heat transferred (btu/hr)
A = area through which the heat flows
X = the distance of the heat flow
k = constant that is material dependent
maximum nozzle flow is obtained by?
This is a function of the exit pressure for the nozzle. If P2 < Pc then the flow out of the nozzle will be less than the maximum attainable value.
Pc = critical pressure and represents the optimal exit pressure for the nozzle
If the flow through the nozzle has exit pressures equal to or less than the stagnation pressure then flow is maximized
Metallic Structures and alloys
This takes the opinion that metals are crystalline structures that are composed of small composites of metals.
Metal strength with particle size
Generally metals with smaller grains are stronger
Ferrous Alloys
These are alloys that share basic physical characteristics to iron steel
Engineering metals
These are non-ferrous metals like magnesium, zinc, nickel, and aluminum which are metals that recently came into the engineering toolbox
Auxiliary metals
These are materials that are often capture in small quantities for use in specialized areas.
Precious Metals
These are metals that have commoditized value in the international market.
What are the three most common packing structures for a metal?
They are generally face centered cubic, body centered cubic, or hexagonal
Strong metals crystalline packing structure
These will generally be body-centered cubic at room temperature and include materials like chromium, iron, moly, tantalum, tungsten, and vanadium
Crystal packing structure of ductile materials at room temp
These are most commonly found as face centered cubic like aluminum, copper, gold, iridium, lead, nickel, palladium, platinum, rhodium, and silver
What metals are hexagonally packed?
Beryllium, cadmium, magnesium, titanium, and zinc
slip in metals
This is the crystalline cleavage of metals along discrete planes. It is proportional to shear. After slip has occurred it is a self-stopping mechanism that results in definite work hardening.
Phase diagrams for metals
These are generally eliptic curves where the upper limit is the liquidus the middle has liquids and solids and the base is the solidus line.
Lever rule
This says that when in a mixed state of liquid and solid that the %liquid = (S-X)/(S-L) where T = C and X is the specific composition being analyzed
How to understand alloys
Alloys are characteristically similar to geologic phase diagrams where there is immiscibility, phase changes, eutectics, and perieutectics.
Annealing
This is the process of heating a metal to its critical point to heal any slip or internal deformation. It increases the malleability of the metal
Steel
This is a type of ferrous alloy with ~2% carbonaceous material
Stainless steels
These have a layer of chromium oxide rich alloy (~10%) on the outer edge which prevents rust.
allotropic changes
These are reversible shifts in material properties like structure, eletrical properties, and magnetic properties which occur in iron-carbon alloys at critical points/temperatures.
Corrosion
This is the result of chemical reactions between metals and the enviroment
Corrosion influences (6)
Corrosion can be caused by
acidity, oxidizing, electrolysis, film formation, agitatition, or temperature
Mechanisms of corrosion
Generally acidity and oxidation creates the corrosive enviroment but the rate of corrosion is determined by electrolysis, film formation, agitation, and temperature
Film formation
This is the development of an oxidized film that separates the base metal from the corrosive environment. Agitation physically removes this layer and increases the rate at which corrosion occurs.
How does temperature influence corrosion?
Generally in chemical corrosion increased temperature increases the extent of the reaction and increases the corrosion
galvanic action
This is when there are two metals with different electronegativities that are connected by a conductive medium. The higher electro-chemical potential element will act as the anode and these atoms will go into solution.
Stress corrosion
This is a form of material failure due to corrosive and dynamic stresses
Fretting corrosion
This is basically corrosion related to the shear of two interfacing metals which is eliminated with proper lubrication
Material testing via radiography
Radiography of an alloy can reveal where there are variations in density
Material testing via accoustics
These tests act very similar to seismeic scans where changes of material properties across boundaries are reflected and velocity is a function of density
Materials testing via magnetics
This uses changes in the ferromagnetic reluctance because of internal heterogeneity to understand the material’s internal structural flaws.
How to measure the tensile strength of a metal
These are generally destructive tests that use a pendulum to collide with metal to understand the energy transfer.
Thermal conductivity
This is the ability for heat to flow through materials expressed in
Cal/cm2 /cm /sec/ oC
Relative thermal conductivity
These are relative measures of how well heat flows through a material using copper as the base
Electrical Conductivity
This is related to the atomic structure and atomic number of materials in a metal.
Conductivity = R = rho L/A ohms
R = resistance, rho = resistivity NOT density (ohm-cm), L = cm, A = cm2
Factors influencing thermal and electrical conductivity
In general less valence electrons in covalent or metallic states create lower thermal and electrical conductivity. Impurities, strain, and increased temperature increases the disorder of the lattice and increases the resistance to heat or electric conductivity and this defines some alloys
Thermal Expansion
There are two forms of thermal expansion: Linear expansion is the expansion along a central axis and Volumetric expansion is the change in total volume
Linear and volumetric thermal expansion equation
Lf = L0 (1+α(ΔT))
Vf = V0(1+ αv (ΔT))
αv ~3*α
Denseness
This is independent of weight and refers to the lack of porosity in a material
Porosity
This is the quality of a material having voids which enable fluids to be transferred through the material
Fusibility
This is the relative ease for a metal to be melted. In general softer metals are more easy to fuse
Volatility
This is the ease at which a substance is vaporized
Is volatility and fusibility directly related?
They are not meaning that fusibility and volatility are dpendent on the specific material.