Purdue - AAE - 340

Purdue - AAE - 340

Purdue - AAE - 340

Purdue - AAE - 340

Purdue - AAE - 340

Purdue - AAE - 340

A1Particle KinematicsLaw of Motion :F=d( mv ) = m Adtmass constantTo use the equation, must be able to mathematically represent both sides of theequationRHSKinematics: describing motion mathematically r , v , Ar = r opv = e v oped r op= ev

Purdue - AAE - 340

B1Fundamental Vibrations ProblemgAssumptions:particle P of mass msmooth surfacelinear springviscous dampingdashpot provides force proportional to speed;always acts in direction opposing motionMotion imparted to system by external force f (t )fo

Purdue - AAE - 340

E1Example: Single Particle SystemA "massless" rigid rod swings in a vertical plane about a frictionlesspivot point C. At the same time, the shaft CK rotates as indicated below.A particle of mass m is attached at the end of the rod; angles and define

Purdue - AAE - 340

increase complexity of model(a/c and s/c are not really particles)System of ParticlesClassify system as comprised of particles m1, m2 , m3 ,mnClassify forces(1) Internal forces - forces between particles in the systemf jkforce exerted on particle

Purdue - AAE - 340

increase complexity of model(a/c and s/c are not really particles)System of ParticlesUseful definition : Center of MassnMRcm mj R jj 1Motion of the system : Try System FBDDerive EOMF M AicmUseful but limited - solution would producetime his

Purdue - AAE - 340

I1Systems of Particles: Integrals of the MotionI. Work and EnergyOperate on Law of MotionWhich one?F mj Aij(a)cm(b)all forces on mjF M Aiexternal forces on system(a) Single ParticleFiv m ApiipiipdvmdtiFivpvpd1 imvdt 2

Purdue - AAE - 340

Intro1IntroductionNewton's Principia (1686) is composed of threebooks:1. De motu corporum (On the motion of bodies)Volume 1 - mathematical explanation of calculus,basic dynamical definitions and implications2. De motu corporum (On the motion of bod

Purdue - AAE - 340

J1Systems of Particles: ObservationsWe have developed tools for use in analysis of problems that includemodels more complex than single particles, that is, a system that ismodeled as a general collection of n particles (no restrictions on model)For s

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 "organize" 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&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 "massless"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 "massless" 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 &,respectively. Th

Purdue - AAE - 340

-is a t h i n , r e c b &A of icnifipUEUSdk71.ibr~ion- tot& 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 & ) i sdfoa s S ~ I O U ~ s e m(,A~s prih of7:P,?"%S~is &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: &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