J. N. R E DDY ENERGY PRINCIPLES AND VARIATIONAL METHODS APPLIED MECHANICS

Transcription

1 J. N. E DDY ENEGY PINCIPLES AND VAIATIONAL METHODS IN APPLIED MECHANICS T H I D E DI T IO N

2 JN eddy - 1 MEEN 618: ENEGY AND VAIATIONAL METHODS A EVIEW OF VECTOS AND TENSOS ead: Chapte 2 CONTENTS Physical vecto Axioms of mathematical vecto Scala and vecto poducts Tiple poducts of vectos Components of vectos Summation convention Second-ode tensos Tansfomation of components The del opeato and calculus of vectos and tensos Cylindical and spheical coodinates systems J. N. eddy VECTOS&TENSOS -

3 JN eddy - 2 A EVIEW OF VECTOS AND TENSOS Much of the mateial included heein is taken fom the instucto s two books exhibited hee (both published by the Cambidge Univesity Pess) J. N. eddy VECTOS&TENSOS - 2

4 JN eddy - 3 PHYSICAL VECTOS Physical vecto: A diected line segment with an aow head. Examples: foce, displacement, velocity, weight Unit vecto along a given vecto A: A The unit vecto, ê A ( A 0) A is that vecto which has the same P A ê A Q diection as A but has a magnitude that is unity. J. N. eddy VECTOS&TENSOS - 3

5 JN eddy - 4 MATHEMATICAL VECTOS ules o Axioms Vecto addition: (i) A + B = B+A (commutative) (ii) (A + B)+C = A+(B+C) (associative) (iii) A+0=A (zeo vecto) (iv) A+( A) = 0 (negative vecto) Scala multiplication of a vecto: (i) α(βa)= αβ (A) (associative) (ii) (α + β)a = α A+ βa (distibutive w..t. scala addition) (iii) α (A+B)=αA+αB (distibutive w..t. vecto addition) (iv) 1. A=A. 1 A B B A A+B=B+A J. N. eddy VECTOS&TENSOS - 4

6 JN eddy - 5 VECTOS (continued) Wok done Magnitude of the foce multiplied by the magnitude of the displacement in the diection of the foce: F u WD F cosu F u WD F u cos J. N. eddy VECTOS&TENSOS - 5

7 JN eddy - 6 VECTOS (continued) Inne poduct (o scala poduct) of two vectos is defined as A B A B cos AB cos A = A A = A B = B A A J. N. eddy VECTOS&TENSOS - 6

8 JN eddy - 7 VECTOS (continued) Moment of a foce Magnitude of the foce multiplied by the magnitude of the pependicula distance to the action of the foce: M F, M sinf M F F e M F O P θ M O θ sin sin J. N. eddy VECTOS&TENSOS - 7

9 JN eddy - 8 VECTOS (continued) Vecto poduct of two vectos is defined as AB A B sin e ˆ AB sin AB AB A AB B J. N. eddy VECTOS&TENSOS - 8

10 JN eddy - 9 PLANE AEA AS A VECTO The magnitude of the vecto C = A B is equal to the aea of the paallelogam fomed by the vectos A and B. C = A B B ê θ S A S S nˆ In fact, the vecto C may be consideed to epesent both the magnitude and the diection of the poduct of A and B. Thus, a plane aea in space may be looked upon as possessing a diection in addition to a magnitude, the diectional chaacte aising out of the need to specify an oientation of the plane aea in space. epesentation of an aea as a vecto has many uses in mechanics, as will be seen in the sequel. ˆn J. N. eddy VECTOS&TENSOS - 9

11 JN eddy - 10 SCALA TIPLE PODUCT The poduct A. (B C) is a scala and it is temed the scala tiple poduct. It can be seen fom the figue that the poduct A. (B C), except fo the algebaic sign, is the volume of the paallelepiped fomed by the vectos A, B, and C. B C A C B A. (B C) J. N. eddy VECTOS&TENSOS -10

12 JN eddy - 11 EXECISES ON VECTOS 1. If two vectos ae such that what can we conclude? 2. If two vectos ae such that what can we conclude? 3. Pove that ABCABC AB0 AB0 4. If thee vectos ae such that ABC 0 what can we conclude? 5. The velocity vecto in a flow field is v 2 i ˆ 3 ˆ j (m/ s). Detemine (a) the velocity vecto nomal to the plane 3 ˆ v n n i-4ˆ k passing though the point, (b) the angle between v and v n, (c) tangential velocity vecto on the plane, and (d) The mass flow ate acoss the plane though an aea A m if the fluid density is 10 kg/ m and the flow is unifom. J. N. eddy VECTOS&TENSOS -11

13 JN eddy - 12 COMPONENTS OF VECTOS Components of a vecto e ˆ e z ˆ3 e ˆ e ˆ y 2 ˆ e x ˆ1 e z x 3 A x x 1 y x 2 1, 0, 0, , 0, 1, ,,, ,, A A A A x x y y z z A A A n ˆ n n n x x y y z z n n n z x 3 A A 1 A 2 A 3 i x x 1 y x 2 A A i J. N. eddy VECTOS&TENSOS -12

14 JN eddy - 13 SUMMATION CONVENTION Omit the summation sign and undestand that a epeated index is to be summed ove its ange: A A A A i1 A A (summation convention) i i i i Dummy index A A A Dummy indices Scala poduct i i j j A ˆ B ˆ AB e e AB i i j j i j i j AB AB i j ij i i 0, if i 1, if i ij i j J. N. eddy VECTOS&TENSOS -13 j j

15 JN eddy - 14 Vecto poduct SUMMATION CONVENTION (continued) AB AB sin A B AB AB AB i i j j i j i j i j ijk k AB A A A B B B ijk 2 i j j k k i i j ijk k 0, if any twoindicesaethesame 1, if i j k, and they pemute inanatual ode 1, if i j k,and they pemute opposite toanatual ode ijk i j k i j k 1 J. N. eddy VECTOS&TENSOS -14

16 JN eddy - 15 SUMMATION CONVENTION (continued) Contaction of indices: The Konecke delta ij modifies (o contacts) the subscipts in the coefficients of an expession in which it appeas: A A, AB AB AB, i ij j i j ij i i j j ij ik jk Coect expessions: F ABC, G H ( 23AB) PQF i i j j k k i i j j k Fee indices Incoect expessions: A BC, A B and F ABC i j k i j k i j k J. N. eddy VECTOS&TENSOS -15

17 JN eddy - 16 SUMMATION CONVENTION (continued) One must be caeful when substituting a quantity with an index into an expession with indices o solving fo one quantity with index in tems of the othes with indices in an equation. Fo example, conside the equations It is coect to wite p abc and c deq i i j j k i i k but it is incoect to wite pi bc j j ai which has a totally diffeent meaning bc a i pi bc p p p i j j ai a1 a2 a3 J. N. eddy VECTOS&TENSOS -16 j j p

18 JN eddy - 17 SUMMATION CONVENTION (continued) The pemutation symbol and the Konecke delta pove to be vey useful in establishing vecto identities. Since a vecto fom of any identity is invaiant (i.e., valid in any coodinate system), it suffices to pove it in one coodinate system. The following identity is useful: - Identity: ijk imn jm kn jn km J. N. eddy VECTOS&TENSOS -17

19 JN eddy - 18 EXECISES ON INDEX NOTATION Execise-1: Check which one of the following expessions ae valid: (a) abc( df); (b) abc( df) m s m m s m s s (c) a bc( d f); (d) x x i j i i i m m (e) a 3; (f)? i ij jk ki Execise-2: Pove A A A ABCijkABC i j k B B B C C C Execise-3: Simplify the expession ( AB)( CD) Execise-4: Simplify the expession A( BC) Execise-5: ewite the expession in vecto fom ABC D J. N. eddy VECTOS&TENSOS ê mni i j m n j

20 JN eddy - 19 SECOND-ODE TENSOS A second-ode tenso is one that has two basis vectos standing next to each othe, and they satisfy the same ules as those of a vecto (hence, mathematically, tensos ae also called vectos). A second-ode tenso and its tanspose can be expessed in tems of ectangula Catesian base vectos as T S S ee ˆˆ S ee ˆˆ; S S ee ˆˆ S ee ˆˆ ij i j ji j i ji i j ij j i A second-ode tenso is symmetic only if SS T Sij Sji Second-ode identity tenso has the fom I ij ˆ i e j J. N. eddy VECTOS&TENSOS -19

21 JN eddy - 20 SECOND-ODE TENSOS We note that ST T S tensos) because (whee S and T ae second-ode ST S ee ˆˆ T ST STee ˆˆ ij i j kl k l ij kl i j k l ij jl i l TS T ee ˆˆ S TS STee ˆˆ ij i j kl k l ij kl i j k l jl ij i l We also note that (whee S and T ae second-ode tensos and A is a vecto) ST S ee ˆˆ T ST ST ee ˆˆ ij i j kl k l ij kl i j k l ij kl jkp i p l SA Sˆ e A SA SA ij i j k k ij k i j k ij j i SA See ˆˆ A SA SA ee ˆˆ ij i j k k ij k i j k ij k jkp i p J. N. eddy VECTOS&TENSOS -20

22 JN eddy - 21 CAUCHY STESS TENSO Stess tenso is a good example of a second-ode tenso. The two basis vectos epesent the diection and the plane on which they act. The Cauchy stess tenso is defined by the Cauchy fomula (to be established): t 1 x 1 t 3 σ ê 1 x 3 ê 3 T ˆ ˆ o ti ij nj t n n σ ê 2 t2 x 2 t i j ji ji j 13 t ee ˆˆ ee ˆˆ t t ˆ ˆ ˆ 2 e112 e222 e332 i i ji j i ij i j J. N. eddy VECTOS&TENSOS t

23 JN eddy - 22 CAUCHY S FOMULA t at1 a1t2 a2t3 a3ρ vf ρ va anˆ a ˆ ˆ ˆ 1e1a2e2a3e3 0 a ( nˆ) a, a ( nˆ) a h a ˆ ˆ 3 ( e3n) a, v a 3 F 1 F 2 t ( 2) F 3 a 2 = t2 h t t1( 1nˆ) t2( 2nˆ) t3( 3nˆ) ( af) 3 As h 0, we obtain 2 x 1 t ˆn t Inteio of the body x 3 ê 3 ê ê 2 1 t t1( 1nˆ) t2( 2nˆ) t3( 3nˆ) t nˆ nˆ i i 3 ˆn On the bounday a t 1 t 2 a 1 ( 1) t = t 1 t t ( 3) a 3 = t ˆn 3 1 ( nˆ ) x 3 x 2 t t 3 ˆn x 2 x 1 J. N. eddy VECTOS&TENSOS -22

24 JN eddy - 23 HIGHE-ODE TENSOS A n th -ode tenso is one that has n basis vectos standing next to each othe, and they satisfy the same ules as those of a vecto. A n th -ode tenso T can be expessed in tems of ectangula Catesian base vectos as T T ˆˆ e e ijkp i j k p ; C ε n subs eee ˆˆˆ eee ˆˆˆ base vectos The pemutation tenso is a thid-ode tenso J. N. eddy VECTOS&TENSOS -23 n ijk i j k The elasticity tenso is a fouth-ode tenso C ijkl i j k l

25 JN eddy - 24 TANSFOMATION OF TENSO COMPONENTS A second-ode Catesian tenso S (i.e., tenso with Catesian components) may be epesented in baed ( x and unbaed Catesian coodinate 1, x2, x3) ( x1, x2, x3) systems as S s ee ˆ ˆ s ij i j mn m n The unit base vectos in the unbaed and baed systems ae elated by and, j ij i i ij j ij i j Thus the components of a second-ode tenso tansfom accoding to s s ij im jn mn J. N. eddy VECTOS&TENSOS -24

26 THE DEL OPEATO AND ITS POPETIES IN ECTANGULA CATESIAN SYSTEM JN eddy - 25 Del opeato: i xi x x x Laplace opeato: 2 xixi x x x Gadient opeation: F F ê i, whee F isa scala function x i Gad F defines both the diection and magnitude of the maximum ate of incease of F at any point. J. N. eddy VECTOS&TENSOS -25

27 THE DEL OPEATO AND ITS POPETIES IN ECTANGULA CATESIAN SYSTEM JN eddy - 26 F F nˆ, whee nˆ isa unit vecto nomal n to the suface F constant F F We also have nˆ and nˆ F F n Divegence opeation: Gi G i jg j, whee Gisavecto function x x i The divegence of a vecto function epesents the volume density of the outwad flux of the vecto field. i J. N. eddy VECTOS&TENSOS -26

28 THE DEL OPEATO AND ITS POPETIES IN ECTANGULA CATESIAN SYSTEM JN eddy - 27 Cul opeation: Gj G i jg j ijk k, x ε x i whee Gisavecto function. The cul of a vecto function epesents its otation. If the vecto field is the velocity of a fluid, cul of the velocity epesents the otation of the fluid at the point. i J. N. eddy VECTOS&TENSOS -27

29 JN eddy - 28 CYLINDICAL COODINATE SYSTEM 0 cos sin x sin cos 0 y z z 0 x cos sin sin cos 0 y z z A A A A z z Del opeato in cylindical coodinates 1 z z J. N. eddy VECTOS&TENSOS -28 x z θ y, z z x ê z y

30 JN eddy - 29 CYLINDICAL COODINATE SYSTEM 1 ( A ) A A z A z 1 1 z Az A A A z 1( A) A A z z A A 1 A Az A A ee ˆ ˆ ee ˆ ˆ A ee ˆ ˆ ee ˆ ˆ ee ˆ ˆ z 1 A 1A A ee ˆ ˆ ee ˆ ˆ J. N. eddy VECTOS&TENSOS -29 z z z Hee A is a vecto: A A A A Veify these elations to youself A z z z ee ˆ ˆ A z ee ˆ ˆ z z z z z z

31 JN eddy - 30 SPHEICAL COODINATE SYSTEM sin cos sin sin cos x cos cos cos sin sin y sin cos 0 z x sincos coscos sine ˆ y sinsin cossin cos z cos sin 0 A A A A Del opeato 1 1 sin J. N. eddy VECTOS&TENSOS -30 x z x y z 2 2 Line paallel to ê θ y, sin, cos sin z θ x ê ê cos y Line paallel to ê

32 JN eddy - 31 SPHEICAL COODINATE SYSTEM sin sin sin A 1 ( sin ) A A 1 A A sin sin 1 (sin A ) A 1 1 ( ) 1 ( A ) A A A A sin sin A A 1 A A 1 A A sin A A sin 1 A 1 A 1 A A ee ˆ ˆ ee ˆ ˆ cos ee ˆ ˆ A sin 1 A A sin cos ee ˆ ˆ A sin J. N. eddy VECTOS&TENSOS -31

33 JN eddy - 32 EXECISES ON VECTO CALCULUS Establish the following identities (using ectangula Catesian components and index notation): 1. ( ) n 2. ( ) n n2 3. ( F ) 0 4. ( FG) A A A ( ) ( ) AB ABBA A( A) ( AA) AA. 1 2 J. N. eddy VECTOS&TENSOS -32

34 JN eddy - 33 Execise: Check appopiate box Quantity Vecto Scala Nonsense ( f ) ( F) ( f ) ( F) ( F) ( f ) f scala; F vecto J. N. eddy VECTOS&TENSOS -33

35 JN eddy - 34 INTEGAL THEOEMS involving the del opeato d nˆ ds Ad nˆ Ads Ad nˆ Ads y ˆj y ˆi n yˆ j nˆ n xˆ i ds x (Gadient theoem) (Divegence theoem) (Cul theoem) nˆ n ˆin ˆj x J. N. eddy VECTOS&TENSOS -34 n n x y n x x y y n

36 J. N. eddy VECTOS&TENSOS -35 EXECISES ON INTEGAL IDENTITIES n n ˆ ˆ. volume ( ).... ( ) d d d d n d d n d d n n d d n n Establish the following identities using the integal theoems: JN eddy - 35