© Sharif University of Technology - CEDRA

© Sharif University of Technology - CEDRA

In the Name of AllahAdvanced Engineering DynamicsAli Meghdari, Ph.D., ProfessorEmail: meghdari@sharif.eduHome Page: http://meghdari.sharif.eduSundays & Tuesdays: 7:30-8:45 AMOFFICE HOURS: Sundays: 1:30-3:00 PM.,Tel: 6616-5541© Sharif University of Technology - CEDRA

• TEXT BOOK: Advanced EngineeringDynamics, By: Jerry H. Ginsberg, CambridgeUniversity Press, 2nd Ed., 1995, ElectronicsVersion 2008, and Lecture Notes.• REFERENCES:Engineering Mechanics: Dynamics, By: J.L.Meriam & L.G. Kraige, John-Wiley & Sons, 4thEd., 1998.Advanced Dynamics; Modeling & Analysis, By:A.F. D’Souza & V.K. Garg, Prentice-Hall, 1984.Dynamics, By: T.R. Kane & D.A. Levinson,McGraw-Hill, 1985.© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

GRADING:HomeworkQuiz:Mid-Term ExamFinal Exam:(10 % of the Final Grade)*(20% of the Final Grade)(30% of the Final Grade)(40% of the Final Grade)* Homework will be assigned every other session, and solutions will be postedonline. Short pop quizzes will be given sometimes during the semester.© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

Dynamic Forces:Failing to acknowledge the importance of dynamicforces and loads can have severe consequences…• New Ferrari: $1,000,000• Son borrows Father's newcar to try out......© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Dynamic Forces:Dynamics are important in recreation as well.© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Applications• Satellite orientation and control• Launch vehicles• Weapons systems© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Applications• Micro/Nano Sensors – accelerometers, gyroscopes, rotation sensors…Draper Labs Comb Drive Tuning Fork GyroVibrating Wheel Gyro: BerkeleySensors and Actuators Center© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Applications• Predicting loads on:– Airplanes– Automobiles– MEMS / NANO Systems– Manufacturing Tools– Robots and Manipulators– Bones and Muscles– and just about anything…© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

TOPICS:1. A Quick Review of Cartesian Tensors2. Introduction, and Review of Undergraduate Dynamics3. Kinematics: Coordinate Transformations, CurvilinearCoordinates, Generalized Coordinates, Euler’s Angles,Moving Reference Frame, General 3-D Motion.4. Particle Dynamics5. Inertia Tensors6. Rigid Body Dynamics: Eulerian Equations of MotionMid-Term Exam: (4 rd week of Farvardin, 1392)© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

A Quick Review of Cartesian TensorsTensors: Mathematical quantity used to describe a physical variablespecified in a particular coordinate system by itscomponents.Rank or Order: (i.e. r th -order tensor)In a 3-Dimensional ordinary physical space (Euclidean Space, E),the number of components of a tensor is “ 3 r ”, where “ r ” is orderof the tensor.‣ If r=0 3 0 = 1-component, Tensor is “Scalars”, (i.e. mass, speed)‣ If r=1 3 1 = 3-components, Tensor is “Vectors”, (i.e. velocity, force)‣ If r=2 3 2 = 9-components, Tensor is “Dyadics”, (i.e. stress, strain)© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

Index Notation:A ( Ax, Ay, Az): means vector A in (x,y,z) coordinates.A ( A1 , A2, A3): means vector A in (x 1,x 2,x 3) coordinates.A Axi Ayj Azk A1e1 A2e2 A3e3AieiRange Rule:An unrepeated letter index is understood to take on thevalues 1,2,3. (Range Index or Free Index in one single term).Ex:F ma Fi maiF FF123 ma1 ma ma; force balance23; force balance; force balanceininini. e. f 0 i,j 1,2,3j ijx direction . y direction . z direction . Note: Homogeneity in free indices is required. ( )i© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

Summation Rule:A repeated letter index (twice & no more) is understood to imply asummation over 1,2,3. It is called Dummy or Summation Index.Ex:Dot-Product of Two Vectors:AB A B CiijA B1111AA B C2jB2A2AB32BC3j3i1A3A BiB C3ijA BiiAmBm( i,j 1,2,3)Ex:Stress Dyadic; a 2 nd order tensor with 9-components. Considerstress at a point of a body: xxyxzxxyyyzyxz 11yz 21 zz 31© Sharif University of Technology - CEDRA122232132333 ijBy: Professor Ali Meghdari

Ex:Cross-Product:RS TeST111eST222eST333( S T S T ) e ( S T S T ) e ( S T S T ) e233213113212213or;R R1 e1 R2e2 R3e3 Riei ( ijkSjTk) ei( i,j,k Dummy Indices , as :1,2,3)where:R1e1 ( 1 jkSjTk) e1 ( S2T3 S3T2) e1 ( 11kS1Tk 12kS2TK 13kS3Tk) 111112113SSS111TTT123 121122123SSS222TTT123 131132133SSS333TTT123 e1e1R 1, R 2, R 3, all together contain 27-components, where 3-negative,3-positive, all other 21 are zeros.© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

Special Tensor Quantities:‣Permutation Symbol ijk110( i,j,k) (1,2,3),(2,3,1),(3,1,2) ( i,j,k) (3,2,1),(2,1,3),(1,3,2) all others,(Repeated Indices ) ‣Kronecker Delta mn10m n m nEx: Dot-Product of Unitary Orthogonal Base Vectors:e1.e1 1 ee e e 1.2 0i.j ijetc . If a a ie b i , and b e j j , then:a.b a e . b e a b e . eiijjijijaibjijx 1e 3e 1x 3e 2x 2a b a ( b )ijijaibi© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

Ex: Cross-Product of Unitary Orthogonal Base Vectors:e1 ee1 ee1 eetc.123 0 e3 e2ei ej ijkekwhere : ijk011forforfori jijk kji a a ie ib j eTherefore, if , and , then:ja baiei bjeja bijei eja bijijkeka bijka bijekb )( i,j,k dummy indices© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

A Vector Function: is a vector “ r = r(q j) ” in which its magnitudeand/or direction in a reference frame {A} depends on n-scalarvariables “ q j” in frame {A}.rr1 ( qj) e1 r2( qj) e2 r3( qj) e3 ri( qj) ei( j 1,...,n)A Scalar Function: is the coefficient of “ e i” in the vector function “ r ”and the reference frame {A}, being “ r i(q j) ” which is also unique.Gradient of a Scalar Function:Consider a Scalar Function (i.e. pressure, density) as f(x i), then:grad f =ffx1e1fx2e2fx3e3fxiei( "Del"isavector operator)Sometimes: x1e1x2e2x3e3xiei, ieiff , iei© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

Divergence of a Vector Function:Consider a Vector Function as: A = A m(x i)e m(m=1,2,3 and i=1,2,3)Also:DivA A (x A (x111A2x2e1x32eA3 ) xA2xAxmm3eA3) ( A em,mmmi mA ( ei) ( Amem) ei em im Ai, i Am,mxixixixixmACurl of a Vector Function:Consider a Vector Function as: V = V k(x j)e k11A2eA2A e3A3) (k=1,2,3 and j=1,2,3)CurlV VexV111ex2V22exV333V (x32Vx23) e1V (x13Vx31) e2V (x21Vx12) e3 (xjej) ( Vkek) Vxkjej ekVkijkei ijkxjk,j© Sharif University of Technology - CEDRAVeiBy: Professor Ali Meghdari

Laplacian of a Scalar Function:Consider a Scalar Function as f(x j), then; the LaplacianOperator is:2 (xiei) (xjej)xixjij2x212x222x232xxii, iiThen:2f2xf212xf222xf232 fxxiif, ii( idummy index )© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

Example:Show that;A.( BC) B.(C A) C.(AB)let:DB C ijkBjCkeiD eiiDi ijkBjCkbut:A.DA DiiA iijkBjCk ijkA BijCkBjjkiCkAiBj( CA)jB.(CA)Also;(it is OK to change the order of indices in a cyclic form)A. D A D A B C C A B C ( AB) C.(AB)iiijkijkkkijijkk© Sharif University of Technology - CEDRABy: Professor Ali Meghdari

© Sharif University of Technology - CEDRA