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### Notes_J%202012%20BP

Course: AAE 340, Spring 2012
School: Purdue
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Word Count: 355

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of J1 Systems Particles: Observations We have developed tools for use in analysis of problems that include models more complex than single particles, that is, a system that is modeled as a general collection of n particles (no restrictions on model) For systems with large n, tools may be difficult to use for various reasons, including the following two major problems: 1 For general system of n particles, cannot...

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of J1 Systems Particles: Observations We have developed tools for use in analysis of problems that include models more complex than single particles, that is, a system that is modeled as a general collection of n particles (no restrictions on model) For systems with large n, tools may be difficult to use for various reasons, including the following two major problems: 1 For general system of n particles, cannot get ALL EOM just by utilizing differential equations involving the center of mass. 2 Relationships we do have for H , W, KE require summations over all particles applied one at a time Impractical for large n General class of "systems of particles" must be decomposed into subclasses. Mechanical tools can be modified for each subcategory to be applied easily and efficiently. J2 Each subcategory can be defined by additional assumptions (or constraints) associated with the model. The additional assumptions (constraints) are useful to reduce the differential equations to forms better suited for analysis. (None of this is helpful unless a subset, or subcategory, will represent a large range of actual problems; analysis; will analysis produce physically reasonable results; good approximations to a wide range of 'real world' problems) A system of particles that can be modeled as a rigid body is one such model (subcategory). This model works very well in problems for which it is reasonable to assume that the distance between individual particles remains constant. The single additional assumption allows differential equations to reduce to an extremely useful form (Applies especially well when number of particles that is, continuous medium) J3 Rigid Body collection of particles such that the distance between each pair of particles remains fixed Utilizing RB model helps beat the two major problems we noted previously. To modify the differential equations, it is necessary to discuss some additional concepts. The utility of the RB model to overcome our two main problems is associated with concepts: 1 Too many EOMs required Reduce the number of EOMs required to some maximum value (concept of degrees of freedom) 2 New representations for H , W, KE are necessary Transformations and mass moments of inertia (new representations to evaluate summations)
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Purdue - AAE - 340
K1Degrees of FreedomThe number of degrees of freedom (DOF) associated with a system isequal to the number of coordinates used to describe its configurationminus the number of independent constraint equations.Note: No. of EOM = No. of DOFExample: 3 n
Purdue - AAE - 340
L1Vector TransformationsExpand the set of mathematical tools to &quot;organize&quot; or describe anydegrees of freedom represented by anglesunit vector relationships and write them in matrix formuse matrices to describe orientationExample:Rod (Rigid Body)6
Purdue - AAE - 340
N1Inertia Matrixl I q is very important s consider it more carefullybar MomentumObservations:1. Base point / coordinatesEach element of I q is a scalar BUT it is calculated frombcomponents of a position vector with a specified base pointI11 2
Purdue - AAE - 340
N12Inertia Matrixl4. Eigenvalues / EigenvectorsGiven: I q in terms of some set of unit vectorsar MomentumaDefinitions:Direction parallel to unit vectors for which the inertia matrix forpoint q is diagonal (that is, all products of inertia are ze
Purdue - AAE - 340
N18Inertia Matrix: ExamplelGiven: a rectangular box ofconstant density ( m 12 kg).Determine principal momentsand directions for point O atthe center of one edge, that is,ar Momentum I cm eApproach:1. Write I q for some convenient point/vector
Purdue - AAE - 340
P1ExamplelThin, rectangular plate rotates aboutthe vertical axis at rate garMomentumAlso rotates about pin at point Owithout friction(i) Derive EOM(ii) Solve for any reaction forcesand/or moments(iii) Any integrals of the motion?Note: = con
Purdue - AAE - 340
A~ e r v o m ~ bm a i n t a i n s n cons)an+ s pin raterge d isK U I ' ~ #r esped Yo .the massless sh&amp;P recession rate ( 8 ) m aintained c o n s W bb4K e . N lcta~ion 3 l e 8 I Swb erive E o M and f ind e x p r e ~ r o r ,e x t e r n a l -tDrgue
Purdue - AAE - 340
Purdue - AAE - 340
340 Office Hours Spring 2012MTWTHFHowellBosanacHowell7:308:309:3010:30Howell11:3012:301:302:30Bosanac3:304:30Prof Howell office hours: MWF 10-12Bosanac office hours: T 2:00-4:00 Th 10:00-12:00Unavailable
Purdue - AAE - 340
Purdue - AAE - 340
The following equations govern motion in a system. The angles and are the variables ofinterest. 4sin2 2 cos 2 4 sin cos 04 sin 7 cos T 05 2 2 sin cos 10sin 0(a) How many EOM are required?Identify the EOMs. How do you know?(b) Are the EOMs linear
Purdue - AAE - 340
Purdue - AAE - 340
Purdue - AAE - 340
S2009(35 points)2. A bead of mass m can slide along a smooth massless rod AB; attached to the bead is aspring of stiffness k and undeformed length Lo. The rod rotates in a horizontal plane suchthat is free. The unit vectors n are inertially fixed.a. S
Purdue - AAE - 340
Two particles (each of mass m) are attached to the ends of a massless T-shaped bar. The T canmove vertically in a smooth slot and twists freely about a vertical axis. The spring (constant k,unstretched length o ) is attached to one end of the T-bar. Let
Purdue - AAE - 340
AssumethattheEOMshavebeenderived:L h sin + g = 0h L sin L 2 cos + K h = 0whereg,L,andKareconstants.Theseequationsarenonlinearandcoupled.Puttheminafirstorderformsuitablefornumericalintegration.
Purdue - AAE - 340
Purdue - AAE - 340
A dumbbell consists of two particles A and B ( of equal mass m) connected by a rigid masslessrod of length L. Particle A is constrained to move on a fixed, frictionless circular track centeredat point O and of radius R. The system moves on a smooth, HOR
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsName_Below is a rigid body from PS10. The mass is m and the rod possesses uniform density .Let unit vectors b be fixed in the rod.b1LQLb3Integrate and determine the I 22 element of the inertia matrix for point Q ass
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsName_Below is a right triangular thin plate. The red dot is the cm. Given uniform density, the4 0 2cm 0 8 0 kg-m2.corresponding inertia matrix is In2 0 4b2b1n345n1(a) If both sets of unit vectors are fixed in
Purdue - AAE - 340
Shown below is a thin, flat plate of non-uniform density. Inertia characteristics associatedwith the center of mass for the body-fixed vector basis c are given as6 I cm 50c52000 kg-m28Determine the principal moments and principal directions (
Purdue - AAE - 340
Assume that part of a spacecraft is shaped like a right triangular prism. Although thecenter of mass is located at the geometric Center of the prism, the vehicle contentsresult in a non-uniform density. The total m s is 2 kg. I f i are fixed in theass
Purdue - AAE - 340
Purdue - AAE - 340
The rod R (mass M) is pivoted f o the end of a &quot;massless&quot;T-shaped bar. The T-barrmrotates freely about a vertical axis at a rateA bead B (mass m) is attached to aspring atld can move along the frictionless rod.4.(a) Identify a set of cmdinates that
Purdue - AAE - 340
The plate from Problem #1 is mounted on a massless shaft that rotates in frictionless bearingsat A and B. The shaft rotates at the constant rate rad/s. Note that the following inertiamatrix is given2 0 1 I B = 0 4 0 kg-meter21 0 2b(a) Determine the
Purdue - AAE - 340
4/13/12AAE 340 Dynamics and VibrationsName_A uniform, thin rod of length L and mass m is pinned at point Q to a massless disk D.Currently, the rod rotates in a vertical plane such that is a constant rate. The disk rotatesabout a vertical axis at the
Purdue - AAE - 340
Purdue - AAE - 340
3. The uniform, this disk is spinning on the &quot;massless&quot; shaft at the constant rate D ,The shaft assembly rotates in frictionless bearings at A and B with respect to theinertial frame. Let 6 and S be unit vectors fixed in the body and &amp;,respectively. Th
Purdue - AAE - 340
-is a t h i n , r e c b &amp;A of icnifipUEUSdk71.ibr~ion- tot&amp; mL ss m4 . The p lde c an?r e e in a verh'eoQ \$ ane about f he i nerfi'*frxediht c 0 . 4 eihear Sprin ( ConstQnt &amp; ) i sdfoa s S ~ I O U ~ s e m(,A~s prih of7:P,?&quot;%S~is &amp;eii c
Purdue - AAE - 340
A uniform, thin rod of length L and mass M is pivoted at one end so it can move inthe vertical plane. A particle of mass m is fixed to the end of the rod. A spring anddashpot are attached to the rod. Orientation of the rod is described by the angle ,me
Purdue - AAE - 340
Four rectangular parallelepipeds R, S, T, U, are oriented with respect to each other asindicated below. Fixed in each is a set of unit vectors defined as parallel to the edges. Eachparallelepiped is square with edge length L.(a) Define unit vectors t1
Purdue - AAE - 340
Purdue - AAE - 340
In a test of the actuating mechanism for a telescoping antema on a s p e thesuppo*shaft rotates about the f d 4 - axis with an angular rate 6.At a certaininstant, L =1.2met,) = 4 Je, = 2 d s , ( = - 1 . 5 d s , and i =.9met/s and all of therates are c
Purdue - AAE - 340
Assume that a certain system can be modtlkd by an eqmrim of the foamsystem rrrpmrc i s thcmh2vibrations t quals -A6saxb b14d that period.~oteond tht damping tinx consrant is 1 see.In about one sentence esch: &amp;fine these twa time quantities. (U
Purdue - AAE - 340
(a) Solve the following initial-value problemwhere y is in meters and t in seconds. Let y0 = - 1 met, y (0) = 4 metis. Expressthe sinusoidal part of your answer in amplitude, phase angle form.(b) Solve the problemwhere u is in units of inches. Let u(0
Purdue - AAE - 340
Shown below is a particle (mass 2 kg) that can move on a smooth inclined surface.Spring and dashpot are attached. Defme h as the distance (in meters) of the particlealong the surface relative to the position of static equilibrium. Assume that thedashpo
Purdue - AAE - 340
Purdue - AAE - 340
The system below consists of a rigid rod R (mass M) and the red particle B of mass m. Particle Bslides freely on the frictionless rod R. The rod is free to pivot in a vertical plane; the frictionlessT-bar also rotates about a vertical axis at the consta
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsProblem Set 1Due: 9/18/2012On the 340 Blackboard site, a review of dimensions is included in a document: SupplementaryMaterial 203 Review Material Dimensional Analysis. After reading the document,consider the following
Purdue - AAE - 340
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsProblem Set 2Due: 1/25/20121. A particle P moves on the Spiral of Archimedes at a constant speed 2 met/s as shownbelow. The equation governing the spiral is r 3 .urn3On2u(a) Let n be inertially fixed unit vectors.
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsProblem Set 3Due: 2/1/12Problem 1: The mass particle in the figure is suspended from a linear spring of constantk, with an unstretched length o . Define as the position of the particle with respect tothe unstretched pos
Purdue - AAE - 340
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsProblem Set 4Due: 2/10/12Problem 1: A particle of mass m is free to move on the smooth surface. Attached is aspring of constant k and dashpot of constant c. An external force acts; it is selected to beof the form f ( t
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsProblem Set 5Due: 2/17/12Problem 1: Below a particle of mass m is fixed to the end of a massless L-shaped rod. Aspring and a dashpot are attached. The rod can pivot in a vertical plane. In the positionindicated, the sys
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsProblem Set 6Due: 2/24/121. A simple pendulum of length L and mass m appears below.Let m = 2 kg and L = 8 meters.gL(a) Assume small oscillations and derive the EOM.At the initial time, (0) 150 , (0) 0 . Determine the
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsProblem Set 7Due: 3/1/12Problem 1: The system below consists of two identical particles at the ends of a rigid,massless L-shaped rod. The short and long arms of the rod have lengths and L,respectively. The rod can pivot
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsProblem Set 9Due: 3/26/12Problem 1: The system below consists of two particles A and B of equal mass mconnected by a rigid, massless rod. One particle is attached by a pin to the end of amassless T-bar. The rod is free
Purdue - AAE - 340
AAE 340 Dynamics and VibrationsProblem Set 10Due: 4/2/12Problem 1: For an aircraft, assume that unit vectors i are inertial (fixed in the ground)and the body-fixed unit vectors are b .(a) The angles yaw, pitch, and roll represent the rotational degre
Purdue - AAE - 340
Instructors Solutions Manualto accompanyFundamentals of AerodynamicsFourth EditionJohn D. Anderson, Jr.Curator of AerodynamicsNational Air and Space MuseumandProfessor EmeritusUniversity of MarylandPROPRIETARY AND CONFIDENTIALThis Manual is the
University of Florida - EEL - 6447
EEL 4930/6447Laser ElectronicsUpdated:1/10/12 16:23 2011 Henry ZmudaSet 0 Introduction1Laser Electronics (EEL 4930/6447) - 3 CreditsSpring Semester 2012Meeting Time/Place:MWF, 3rd (9:35 10:25) Benton 328Instructor:Office:Office Phone:Cell Pho
University of Florida - EEL - 6447
Gaussian Beam Optics,Ray Tracing, and CavitiesRevised: 1/25/12 1:57 PM 2011, Henry ZmudaSet 1 Gaussian Beams and Optical Cavities1I. Gaussian Beams(Text Chapter 3) 2011, Henry ZmudaSet 1 Gaussian Beams and Optical Cavities2Gaussian BeamsReal o
University of Florida - EEL - 6447
Atomic RadiationRevised: 2/9/12 10:44 2011, Henry ZmudaSet 2 Atomic Radiation1Atomic RadiationA laser is a quantum device.Energy levels in an atomic system are discrete, be theymolecular, solid, or semiconductor.( 2)(1)EnergyLevelDiagram 201
University of Florida - EEL - 6447
Laser Oscillation &amp; AmplificationRevised: 2/2/12 17:02 2011, Henry ZmudaSet 3 Laser Oscillation1Threshold ConditionsFrom Slide 50 of Note set 2, our central equation of lasertheory, we recall:&quot;&quot;2g2 %! o ( f ) = g ( f ) 2 A21 \$ N 2 ! N1 '8n #g
University of Florida - EEL - 6447
Laser CharacteristicsRevised: 3/21/12 13:04 2012, Henry ZmudaSet 4 Laser Characteristics1Laser CharacteristicsStill to be answered:1. What is the laser amplitude?2. What if there are transient effects (time dynamics)?3. What exactly is the pumpin
University of Florida - EEL - 6447
Mode Locked LasersRevised: 2/24/12 10:56 2011, Henry ZmudaSet 4a Mode Locked Lasers1Mode Locked Lasers 2011, Henry ZmudaSet 4a Mode Locked Lasers2Mode Locked Lasers 2011, Henry ZmudaSet 4a Mode Locked Lasers3Mode Locked Lasers 2011, Henry Zm
University of Florida - EEL - 6447
Pulsed LasersRevised: 3/21/12 13:28 2012, Henry ZmudaSet 5a Pulsed Lasers1Laser Dynamics Puled LasersMore efficient pulsing schemes are based on turning thelaser itself on and off by means of an internal modulationprocess, designed so that energy
University of Florida - EEL - 6447
Semiconductor LasersRevised: 3/21/12 12:56 2012, Henry ZmudaSet 6 Semiconductor Laser Fundamentals1Semiconductor LasersThe simplest laser of all. 2012, Henry ZmudaSet 6 Semiconductor Laser Fundamentals2Semiconductor LasersThe simplest laser of
University of Florida - EEL - 6447
Laser DiodesRevised: 3/27/12 15:24 2012, Henry ZmudaSet 6a Laser Diodes1Semiconductor LasersThe simplest laser of all. 2012, Henry ZmudaSet 6a Laser Diodes2Semiconductor Lasers1. Homojunction Lasers2. Heterojunction Lasers3. Quantum Well Lase
University of Florida - EEL - 6487
Vector PotentialsSet 1 - Vector Potentials1Maxwell s Equations Time-Harmonic Electromagnetic Fields Homogeneous Medium!!! &quot; E = # j\$ H # M!!! &quot; H = j\$% E + J!&amp;!i E =%! &amp;m!i H =Set 1 - Vector Potentials2The Wave Equation Time-Harmonic Fie
University of Florida - EEL - 6487
Radiation and ScatteringSet 2 Radiation and Scattering1The Near Field: Recall,z( x, y, z)! r! r!A=4!!R!e &quot; j !R#V# J x !, y !, z ! R dv !( x, y , z )()! !R = R = r !r&quot;yxSet 2 Radiation and Scattering2!!e ! j !RA=!V! J x !, y !