QUIZ 2 NOTES 3

					- STRUCTURAL DYNAMICS QUIZ 2 -

************************************* WE GET ONE CHEAT SHEET, BOTH SIDES *************************************

NO CALCULATOR ---- SHOW UP AT 8:30, IN CLASS, WILL HAVE 85MIN UNTIL 9:55 --- BRING OWN PAPER, PENCILS, ERASERS

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PAST QUIZ PROBLEMS:

Q1.1a - Define "free response", "forced response", "static response"
Q1.1b - When can gravity be ignored in a vertically oscillating system?
Q1.1c - Describe process to determine m,k,c for mass-stifness-damper model problem
Q1.2 - dual-spring mass-damper excitation problem, find EOM
Q1.3 - Given EOM, calculate response for given:
!Q1.3a - F = 0, x0, x'0, c
Q1.3b - F = kA
!Q1.3c - F = ksin(wn*t)

Q2.2 - 2DOF system, 2 masses, driving force. Derive system's differential equations of motion
Q2.3 - Given EOM for 1DOF system, calculate response x(t) with driving force F(t) when:
	F(t) = periodic function represented by Fourier series
*	F(t) = pulse, with system underdamped. Write answer in terms of impulse, step, and ramp responses (derived in class)
	F(t) = kAu(t), constant function for t > 0. Derive answer using convolution integral (assume undamped)

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HOMEWORK PROBLEMS (QUIZ 1):

0.1 - EOMs, non-oscillating
0.2 - Taylor Series expansion
0.3 - particular solutions of 2nd order linear ODEs

1.1 - complex expressions, converting from one form to another
1.2 - Euler's formula from power series expansion
1.3 - Oscillating beam & dual spring system - find spring stiffness
1.4 - Oscillating solid floating in liquid - find natural frequency of system
1.5 - Inverted pendulum & spring - find EOM, equilibrium position(s), use small-angle approx. to derive differential EOM about eq. pos's

2.1 - overdamped single-DOF free vibration "Model Problem" - derive response for x0 & v0
2.2 - damped single-DOF w/ numerical initial conditions - plot response w/ MATLAB
2.3 - spring, damper, 2 pulleys, mass - find differential EOM, viscous damping factor, solve for v(t)=0 with v0, x0, x'0 given
2.4 - see-saw model, derive diff. EOM, small-angle diff. EOM, nat'l frequency, viscous damping factor, max displacement given in'l cond's

3.1 - mass-spring-damper model problem with dry friction - find EOM, system reponse, under/-/overdamped responses
3.2 - damped single-DOF with two harmonic forces - find forced response
3.3 - inverted pendulum w/ spring at base - derive diff. EOM, small-angle EOM
3.4 - underdamped, excited, vertical dual-mass system - find excitation force where masses separate

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HOMEWORK PROBLEMS (QUIZ 2):

4.1 - see-saw with mass, spring-damper, and mass.
4.1a - Derive m2 so that ANGLE OF STATIC EQUILIBRIUM OF SYSTEM IS ZERO
4.1b - Determine total mass moment of inertia of system in terms of a, b, & m1
4.1c - Derive EOM of system
4.1d - Does gravity appear in the EOM? Why or why not?
~4.2 - Given AMPLITUDE (magnitude) of frequency response of mass-spring-damper, derive PHASE ANGLE
4.3 - Viscously damped pendulum:
4.3a - Derive differential EOM of system
4.3b - Use small angle approximation to derive linearized EOM of system
4.3c - Without using SUPERPOSITION, determine response of linear system
4.4 - Periodic excitation f(t) (square wave)
4.4a - Derive Fourier series representation of f(t)
4.4b - With MATLAB, show how more terms leads to better representation of f(t)

5.1 - Cam and follower (ramp/periodic excitation), harmonic excitation F(t), spring, mass, spring-damper
5.1a - Derive differential EOM of system in terms of y(t) and F(t) (periodic and harmonic excitations)
5.1b - Derive Fourier series representation of y(t) (periodic excitation. use exponential form)
5.1c - Determine response x(t) of system
5.2 - Determine time necessary for impulse response of viscously underdamped 1DOF system to reach peak value, find peak value
*5.3 - Use superposition to derive response of viscously damped 1DOF system (sawtooth excitation). Leave answer in terms of impulse, step, & ramp responses (as shown in class)
5.4 - Derive response of viscously underdamped 1DOF system to force w/ step function using convolution
5.5 - Critically damped, 1DOF, mass-spring-damper system
5.5a - Derive impulse response
5.5b - Derive step response (using impulse response)
5.5c - Derive ramp response (using step response)

6.1 - Spring-damper block connected to wheel by spring:
6.1a - Derive differential equations of motion for system
6.1b - Given 'I' of wheel, write eq'ns of motion as a matrix equation
6.1c - Are the mass and stiffness matrices in prev. question positive definite? Prove this
6.2 - Airfoil w/ 2 springs, attached to CoM C and trailing edge. Derive DIFFERENTIAL EOMS for system
6.3 - Double pendulum, massless bars
6.3a - Derive DIFFERENTIAL EOMs for system
6.3b - Use SAA to linearize EOMs, write linearized EOMs in matrix form
6.3c - Find natural frequencies and modes of vibration
6.3d - Plot the modes
6.1_o - mass w/ spring-damper on both sides and pendulum hanging from center
6.1a_o - Derive DIFFERENTIAL EOMs for system
6.1b_o - Use SAA to linearize and write EOMs as a matrix eq'n

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NEED TO KNOW:

 Need to ask someone about these:
 x IS THERE A DIFFERENCE BETWEEN DIFFERENTIAL EOMS & REGULAR EOMS? NO
 - AMPLITUDE AND PHASE ANGLE - IS THERE A GENERIC FORMULA (BESIDES WHAT HE GAVE US FOR QUIZ 1), OR DO WE NEED TO WRITE DOWN WHAT IS IN THE BOOKS (IMPEDANCE FUNCTION, ETC)?
 - DO WE NEED TO KNOW IMPEDANCE FUNCTION? (BOTTOM OF TEXTBOOK P.112 TO TOP OF P.113)
 - TO WHAT EXTENT DO WE NEED TO KNOW G(iw), AKA FREQUENCY RESPONSE? (TOP OF P.113)
 - DO WE NEED TO KNOW FREQUENCY RESPONSE IN COMPLEX FORM? (BOTTOM OF P.113)
 - RESPONSE OF SYSTEM TO CONSTANT FORCE?
 - DO WE NEED TO KNOW Q-FACTOR/QUALITY FACTOR, RESONANCE CONDITION, MAX AMPLITUDE? (P.116)
 - DOES 1DOF MEAN 1 EOM AND 2DOF MEAN 2 EOMS?
 - WHAT DOES "POSITIVE DEFINITE" MEAN FOR A MATRIX? HOW CAN YOU PROVE IT? (HW 6.1c)
 - WHAT ARE MODES OF VIBRATION?
 - HOW TO FIND NATURAL FREQUENCY OF A SYSTEM? (6.3c)
 - HOW TO FIND MODES OF VIBRATION OF A SYSTEM? (6.3c)
 - FOURIER SERIES -> EXPONENTIAL FORM? WHAT OTHER FORMS EXIST? WHY IS EXPONENTIAL FORM SIGNIFICANT?

 Should figure these out as I go:
 - DERIVATION FOR IMPULSE, STEP, AND RAMP RESPONSES?
 - EXPLAIN SUPERPOSITION BETTER?
 - HOW TO DERIVE FOURIER SERIES OF PERIODIC EXCITATION?
 - " " " " HARMONIC EXCITATION? (DO WE NEED TO KNOW THIS)?
 - VISCOUSLY UNDERDAMPED SYSTEM - FIND PEAK TIME AND PEAK VALUE (IS THAT MAGNITUDE)?
 - WHAT IS A MATRIX FORM EOM?
 - WHAT IS A MASS MATRIX? A STIFFNESS MATRIX?

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TODO:

 x Check previous Quiz 2 on canvas - WON'T HAVE P1, BUT WILL HAVE 2 & 3 BUT BEEFIER -- FIGURE OUT WHAT CONCEPTS
 x Go through book, write down topics from ch. 3,4,5
 x See if you can figure out what sections are? (check groupme?)
 x Check book topics against topics on syllabus, see where we cut off (WEEKS 7-9)
 x See if there was something this detailed for Quiz 1 on Canvas
 x Make HW synopsis like last time
 - Learn moving base (p.128)
 - Learn response to constant force (?? assume x is constant, so dx=ddx=0?)s
 - Go over HW's 4,5,6 completed problems (get from GroupMe)
 x Check previous HW 6 on canvas
 - Go over HW's 1,2,3 completed problems (get from GroupMe)
 ~ Check out speedbump problem
 x Finish Ch 1, put on cheat sheet prep
 x Finish Ch 2, put on cheat sheet prep
 - Finish Ch 3, put on cheat sheet prep
 - Finish Ch 4, put on cheat sheet prep
 - Finish Ch 5, put on cheat sheet prep
 - Finish Ch 6, put on cheat sheet prep
 - Figure out questions from Ch. 3 (last ones from section above), ask someone in the class, then continue
 - what is w_d?

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CHAPTER BREAKDOWN:

(QUIZ 1)

 x Ch 1: LOTS OF SHIT
   Ch 2: RESPONSE OF 1DOF SYSTEMS TO INITIAL EXCITATIONS
 x Ch 2, Sec 1: UNDAMPED 1DOF SYSTEMS. HARMONIC OSCILLATOR
 x Ch 2, Sec 2: VISCOUSLY DAMPED 1DOF SYSTEMS
 ` Ch 2, Sec 3: MEASUREMENT OF DAMPING
 x Ch 2, Sec 4: COULOMB DAMPING. DRY FRICTION
   Ch 3: RESPONSE OF 1DOF SYSTEMS TO HARMONIC & PERIODIC EXCITATIONS.
 x Ch 3, Sec 1: RESPONSE OF 1DOF SYSTEMS TO HARMONIC EXCITATIONS
 - Ch 3, Sec 2: FREQUENCY RESPONSE PLOTS
 ` Ch 3, Sec 3: SYSTEMS WITH ROTATING UNBALANCED MASSES
 - Ch 3, Sec 5: HARMONIC MOTION OF THE BASE (???)

(QUIZ 2)

 - Ch 3: RESPONSE OF 1DOF SYSTEMS TO HARMONIC & PERIODIC EXCITATIONS.
 - Ch 3, Sec 9: RESPONSE TO PERIODIC EXCITATIONS. FOURIER SERIES
 - Ch 4: RESPONSE OF 1DOF SYSTEMS TO NONPERIODIC EXCITATIONS
 - Ch 4, Sec 1: THE UNIT IMPULSE. IMPULSE RESPONSE
 - Ch 4, Sec 2: THE UNIT STEP FUNCTION. STEP RESPONSE
 - Ch 4, Sec 3: THE UNIT RAMP FUNCTION. RAMP RESPONSE
 - Ch 4, Sec 4: RESPONSE TO ARBITRARY EXCITATIONS. THE CONVOLUTION INTEGRAL
 - Ch 5: 2DOF SYSTEMS
 - Ch 5, Sec 2: THE EQUATIONS OF MOTION OF 2DOF SYSTEMS
 - Ch 5, Sec 3: FREE VIBRATION OF UNDAMPED SYSTEMS. NATURAL MODES

STILL NEED:
 ~ 3.2 - Frequency response plots
 x 3.3 - Harmonic motion of base
 x 3.9 - Response to periodic excitations - fourier series
 x 4.1 - Unit impulse / impulse response
 x 4.2 - Unit step function / step response
 x 4.3 - Unit ramp function / ramp response
 x 4.4 - Response to arbitrary excitations - convolution integral
 - 5.2 - EOMs of 2DOF systems
 ? 5.3 - Free vibration of undamped systems - natural modes

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FOR CHEAT SHEET:

 Misc:
 - Quadratic formula: -b +- sqrt(b^2 - 4ac) / 2a
 - SAA: sin(th)=th, cos(th)=1, th^2 = 0, th*dth=0
 - Complex: acos(z) + bsin(z) = sqrt(a^2 + b^2)*cos(z-phi), phi = arctan(b/a)
 - Complex/trig properties: exp(i*a) + exp(-i*a) = 2cos(a), exp(i*a) - exp(-i*a) = 2*i*sin(a)
 - Complex/trig properties: exp(a) - exp(-a) = 2sinh(a), exp(a) + exp(-a) = 2cosh(a)
 - Trig: sin(wt) = (exp(iwt) - exp(-iwt))/2i, cos(wt) = (exp(iwt) + exp(-iwt))/2
 - Solution: homogenous + particular = complete
 - Period: T = 2pi/wn
 - Const. coeff homogenous response: x(t) = A*exp(st)
 - C = amplitude, phi = phase angle, wn = natural frequency
 - Washing machine w/ spring & damper: y(t) = x(t) + e*sin(wt) (y is small mass)
 - Washing machine EOM: M*ddx + c*dx + k*x = m*w^2*e*sin(wt)
 - Usually, x(0) = x0 = C*cos(phi)

 Pendulums:
 - EOM: ddTh + (g/L)sin(Th) = 0
 - wn = sqrt(g/L)
 - Linearize: ddTh + (wn^2)*Th = 0 		(SAA, wn = sqrt(c))

 Undamped 1DOF, harmonic oscillation, homogenous
 - EOM: m*ddx(t) + k*x(t) = 0
 - wn = sqrt(k/m) = sqrt(g/L)
 - s1,2 = +- i*wn
 - Magnitude & phase angle: x(0) = x0 = Ccos(phi), dx(0) = v0 = wn*Csin(phi)
 - response 1: x(t) = A1*exp(i*w_n*t) + A2*exp(-i*w_n*t)
 - response 2: A1 = C/2[exp(-i*phi)], A2 = bar(A1) = C/2[exp(i*phi)]
 - response 3: x(t) = C/2[exp(i(wn*t-phi)) + exp(-i(wn*t-phi))] = Ccos(wn*t-phi)

 Viscously damped 1DOF, harmonic oscillation, homogenous
 - EOM: m*ddx(t) + c*dx(t) + k*x(t) = 0
 - c/m = 2*zeta*wn, k/m = wn^2
 - zeta = viscous damping factor = c/(2*wn*m)
 - s1,2 = -zeta*wn += sqrt(zeta^2 - 1)*wn		(generic)
 - Underdamping: 0 < zeta < 1 -- oscillatory decay, s1 = bar(s2) -- s1,2 = -zeta*wn +- i*wd, wd = sqrt(1 - zeta^2)*wn (wd = frequency of damped vibration)
 - Underdamped response: p. 91 (!!!!!!)
 - Critical damping: zeta = 1 -- aperiodic decay, s1 = s2 = -wn
 - Critically damped response: p. 93 (top)
 - Overdamping: zeta > 1 -- aperiodic decay, s1 > -wn, s2 < -wn, use generic eq'n to find s1,2
 - Overdamped response: p. 92, 93 (!!!)
 - response 1: x(t) = A1*exp(s1*t) + A2*exp(s2*t)
 - response 2: x(0) = A1 + A2 = x0, dx(0) = s1*A1 + s2*A2 = v0
 - response 3: A1 = (-s2*x0 + v0)/(s1 - s2), A2 = (s1*x0 - v0)/(s1 - s2)
 - response 4: combine steps 1, 3 for solution

 Viscously damped 1DOF, harmonic excitation
 - EOM: m*ddx(t) + c*dx(t) + k*x(t) = F(t)
 - Force: F(t) = k*f(t) = k*A*cos(w*t), f(t) = A*cos(w*t)	(w = excitation/driving frequency)
 - response 1: x(t) = X(iw)*exp(iwt) (pretend F(t) = k*A*exp(iwt))
 - response 2: derive for dx(t), ddx(t), substitute in, find characteristic eq'n
 - response 3: cancel out exp(iwt)'s, solve for X(iw). Divide X(iw) by A to get G(iw) (frequency response)
 - response 4: x(t) = A|G(iw)|cos(wt - phi) (took Re of exp(iwt). likewise, Im goes with sin)
 - response 5: |G(iw)| = 1/sqrt(Re^2 + Im^2), phi = atan(Im/Re)
 - if F(t) = Acos(wt), x(t) = (A/(wn^2))|G(iw)|cos(wt - phi)
 - If excitation is Acos(wt) and response is in complex form (exp(i(wt-phi))), response is Re(x(t)) -- AKA exp(i(wt-phi)) -> cos(wt-phi). If Asin(wt), Im(x(t)) -> sin(wt-phi)
 - Frequency response in complex form -- p.113 (do we need to know)?

 Harmonic base motion
 - EOM w/ c, k: -k*(x - y) - c*(dx - dy) = m*ddx -> m*ddx + c*x + k*x = c*dy + k*y
 - Assume x(t) is X(iw)exp(iwt) and y(t) is A*exp(iwt)

 Frequency response
 - G(iw) = |G(iw)|*exp(-i*phi), G(iw)*exp(iwt) = |G(iw)|*exp(i(wt - phi))
 - phi = atan[Im(G(iw))/Re(G(iw))]
 - Even if 'y = Y*cos(wt)', make it 'y = Y*exp(iwt)', then take the Re() at the end
 - Peak occurs when d|G(iw)|/d(w/wn) = 0, or at w/wn = sqrt(1 - 2*zeta^2)
 - |G(iw)|_peak = 1/(2*zeta*sqrt(1 - zeta^2))
 - |G(iw)|_0 = 1

 Transmission
 - Transmissibility: |X(iw)|/A (amp. of response over amp. of excitation)
 - Transmitted force: F_tr = c*dx + k*x
 - Full force transmitted (|F_tr| = |F|) at w/wn = sqrt(2). When w/wn > sqrt(2), F_tr increases & damping inreases

 Periodic frequency response:
 - Period: T = (2pi)/w0
 - Fourier series, trig form: f(t) = (1/2)*a0 + sum(p=1,inf, a_p*cos(p*w0*t) + b_p*sin(p_w0*t)), w0 = 2pi/T
 - Fourier coeff. 1: a0 = (2/T)*int(0,T, f(t)*dt)
 - Fourier coeff. 2: a_p = (2/T)*int(0,T, f(t)*cos(p*w0*t)dt), p = 0,1...
 - Fourier coeff. 3: b_p = (2/T)*int(0,T, f(t)*sin(p*w0*t)dt), p = 1,2...
 - Even functions: remove all b_p & sin
 - Odd functions: remove all a_p & cos (including a0)
 - Amplitude of p'th harmonic: c_p = sqrt(a_p^2 + b_p^2)
 - w = p*w0
 - Fourier series, exp. form: f(t) = (1/2)*A0 + Re(sum(p=1,inf, A_p*exp(i*p*w0*t))), w0 = 2pi/T
 - Fourier coeff. 3: A_p = (2/T)*int(0,T, f(t)*exp(-i*p*w0*t)*dt), p = 0,1,2,3...
 - Set excitation to A*exp(iwt)
 - Response 1: x(t) = Re[A*G(iw)*exp(iwt)] = Re[A*|G(iw)|*exp(i(wt - phi))]
 - Response 2: x(t) = (1/2)*A0 + Re[sum(p=1,inf, A_p*|G_p|*exp(i(p*w0*t - phi_p)] = a0/2 + sum(p=1,inf, |G(i*p*w0)|*[a_p*cos(p*w0*t - phi) + b_p*sin(p*w0*t - phi_p)]
 - G_p = G(iw) w/ 'w' replaced with 'p*w0'. Same for magnitude & phase angle

 Impulse
 - Unit impulse function: delta(t-a) = 0 for t != a, takes place over eps, reaches max of 1/eps, int(-inf,inf, delta(t-a)dt) = 1
 - model problem: m*ddx + c*dx + k*x = delta(x)
 - Impulse response constraints: g(0) = 0, dg(0) = 0, dg(0+) = 1/m
 - Impulse response (underdamped): C = 1/(m*wd), phi = atan(inf) = pi/2, g(t > 0) = (1/(m*wd))*exp(-zeta*wn*t)*sin(wd*t), g(t < 0) = 0

 Step
 - definition: mu(t - a) = 0 for t < a, 1 for t > a
 - add step function to impulse response: g(t) = g(t)*mu(t)
 - Step function is int. of impulse function: mu(t - a) = int(-inf,t, delta(tau - a)*dtau) -> OR, s(t) = int(-inf,t, g(tau)*dtau)
 - Step response of MSD model: s(t) = (1/k)[1 - exp(-zeta*wn*t) * (cos(wd*t) + (zeta*wn/wd)*sin(wd*t))]*mu(t) -> m*wn^2 = k, wd^2 = (1 - zeta^2)*wn^2

 Ramp
 - r(t - a) = (t - a)*mu(t - a) (slope always = 1)
 - integral of unit step fuction: r(t - a) = int(-inf,t, mu(tau - a)*dtau)
 - ramp response: R(t) = int(-inf,t, s(tau)*dtau)

 Arbitrary excitations / convolution
 - Impulsive excitation: hat(F(tau))*delta(t - tau) = F(tau)*DELTA(tau)*delta(t - tau)
 - Response: x(t) = int(0,t, F(tau)*g(t - tau)*dtau), g(t) = impulse response (given above)

 2DOF Systems
 - [M][ddx] + [c][dx] + [K][x] = [F] -> [x] = displacement vector, [F] = force vector
 - [x] = [x1; x2], [F] = [F1; F2], [M] = [m1, 0; 0, m2]
 - [M] = [M]', [C] = [C]', [K] = [K]'

 Free Vibration, Natural Modes
 -