Finite element method for structural dynamic and stability analyses

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1 Finite eement method for structura dynamic and stabiity anayses Modue-9 Structura stabiity anaysis Lecture-33 Dynamic anaysis of stabiity and anaysis of time varying systems Prof C S Manohar Department of Civi Engineering IISc, Bangaore India 1

2 Dynamic anaysis of a beam coumn P f x, t P EI, m,, c n 2 2 n 2 EI m P ncr P 2

3 Parametricay excited systems Pt f x, t Pt y m s ut av, d k v c v h A m g yg t d m u x yg xg y t t EI, m, c, y x, t 3

4 P P P P Foower forces Line of action of P remains unatered as beam deforms. Static anaysis can be used to find P cr Line of action of P remains tangentia to the deformed beam axis. Static anaysis does not ead to correct vaue of P. cr

5 Probem 1 How to characterize resonances in systems governed by equations of the form ; ; M t X C t X K t X X X X X when the parametric excitations are periodic. Probem 2 How to arrive at FE modes for PDE-s with time varying coefficients? Probem 3 Are there any situations in staticay oaded systems, wherein one needs to use dynamic anaysis to infer stabiity conditions? 5

6 Quaitative anaysis of parametricay excited systems 1 2 ;, 1,2 u t p t u t p t u t u u u u p t T p t i i i The governing equation is a inear second order ODE with time varying coefficients. It admits two fundamenta soutions.

7 1 2 u1t u2t u t c u t c u t u t p t u t p t u t pit T pit i 1tu t T p2tu t T u t u t T Let and be the fundamenta soutions of this equation. Consider the governing equation at Since, 1, 2, we get t T u t T p t T u t T p t T u t T u t T p If is a soution is aso a soution. u t T A u t u t T a u t a u t u t T a u t a u t We are interested in nature of the soution as t. 7

8 u t T Au t im ut? t 2 2 This is equivaent to asking im u t nt? u t T Au t n u t T Au t T A u t 1 n behavior of im A. n n u t nt Au t n T A u t The behavior of im u t nt is controed by the n Intutivey, one can see that this, in turn, depends upon the nature of eigenvaues of A. 8

9 u t Qvt u t T Au t Introduce the transformation u t T Au t Qv t T AQv t Pre-mutipy by Q t t Q Qv t T Q AQv t t Seect Q such that A is diagonaized. t t That is, we wish to find Q such that Q Q & Q AQ are diagona. Consider the eigenvaue probem: A Seect Q to be the matrix of eigenvectors of A. Q v t T v t 2 9

10 n v t T v t, i 1,2 i i i v t nt v t, i 1,2 i i i We have im v t im v t nt if 1, i 1, 2 t i i i n im v t im v t nt if 1, i 1, 2 t i i i n v t is periodic with period T if 1, i 1,2 i v t is periodic with period 2 T if 1, i 1,2 i i i 1

11 Reduction to norma form Consider Mutipy by exp i tt t T v t T exp t T v t v t T v t i i i exp 1 We reate i & i as i exp it i oge i T i i i i i i t T v i t T exp itvi t t exp tv t is a per exp i i i This eads to exp v t exp t t, i 1, 2 i i i v t t t i i i Periodic function iodic function with period T,for i 1,2. 11

12 v t exp t t i i i Coud be Periodic periodic function aperiodic with decay aperiodic with exposion Remarks 1 i og e i, i 1,2 are caed the characteristic exponents or T the Foquet coefficients. can be expressed as t i Any soution u t u t a v t a v t Behavior of u t as is governed by the nature of, j,i=1,2 j 1 i i i u t i Growth or Decay Osciatory behavior i 1,2 is periodic if =, that is when, i 1,2 are pure imaginary. 1 u t is periodic with period T 1 u t is periodic with period 2T i i 12

13 Determination of the characteristic exponents 1 2 u1t u2t 1, u and, u 1 u t p t u t p t u t Let & be two soutions of this equation with u u We have 1 1 u1t u1t u T u T u t T a u t a u t u t T a u t a u t u u T a u a u a u u T a u a u a u u T a u a u a u u T a u a u a A 13

14 u T u1 T 1 A u2 T u2 T Find eigenvaues of A Infer nature of soutions by using the criteria im v t im v t nt if 1, i 1 and 2 t i i i n im v t im v t nt if 1, i 1, or 2 t v t i i i i n is periodic with period T if 1, i 1, 2 v t is periodic with period 2 T if 1, i 1,2 i i i 14

15 If the condition i parametric resonance. 1 occurs, we say that the system has got into Here the motion grows exponentiay with time. Presence of damping does not imit the ampitude of osciations. Ampitudes coud get imited due to noninear effects. This is contrast with resonance in externay driven systems: 2 P x x P cos t; x ; x x t cost cost 2 2 P Pt im x t im cost cost sin t t im im xt Resonance response ampitudes are imited by damping. Noninearity woud aso become important as response grows. 15

16 Extension to MDOF systems Consider n-dof system U t P t U t Q t U t with P t T P t Q t T Q t U t T AU t Recipe Generate a set of n nn nn independent soutions of the governing equation by using a set of n ineary independent ics. Form the A matrix Find eigenvaues of matrix A. Infer the nature of the soution by examining the nature of the eigenvaues. 16

17 Foower forces P P Line of action of P remains tangentia to the deformed beam axis. Work done by P is dependent on path of deformation. Such forces are caed nonconservative forces. What is the critica vaue of P?

18 Static anaysis x P f y Pcos x P f Psin y sin EIy P cos f y P x cos 1,sin EIy P f y P x EIy Py Pf P x y k y k f x with k cos sin y y y( L) f, yl y x A kx B kx f x BCs:, P EI

19 cos sin sin cos y y y( ) f, y Bk cos sin sin cos y x A kx B kx f x y x Ak kx Bk kx BCs:, y A f y y f f A k B k f y Ak k Bk k 1 1 A k 1 B cos k sin k f k sin k k cos k 19

20 For non trivia soutions 1 1 k 1 cos k sin k k sin k k cos k 1 1 k 1 We get 1 cos k sin k k sin k k cos k This means that ony trivia soution is possibe for a vaues of k. Structure's state of rest y is aways stabe for a vaues of P. This defies expectations. Did we miss something? 2

21 Idea The oss of structura stabiity is accompanied by osciations whose ampitude grow in time. Therefore, incude inertia effects in considering stabiity of equiibrium state. Consider the case when P was appied in a conservative manner. When dynamic anaysis was performed, when P P, the response grew ineary in time with natura frequency=. The resuts from static and dynamic anaysis coincided. cr 21

22 Mode with distributed mass P P Reca BCs at x EIy & EIy Py A A P A P A BCs at x EIy & EIy P has zero component aong AA 22

23 iv EIy Py my y t y t EIy t EIy t, exp iv 2 2 k a x expsx BCs:,,,,,,, y x t x i t iv 2 EI P m s k s a s k a k k k k a & 2 a 1 1 x Acosh x B sinh x C cos x Dsin x,,,

24 x Acosh x Bsinh x C cos x Dsin x, exp ,,, Condition for nontrivia soution 2a k 2a cos k a sinh sin This eads to the reation between P and. y x t x i t Write aib y x, t xexp ia bt EI Instabiity when b Pcr P P 5 y(t)response for cr t 24

25 References V V Bootin, 1963, Nonconservative probems of the theory of eastic stabiity, Pergammon, Oxford. M A Langthejm and Y Sugiyama, 2, Dynamic stabiity of coumns subjected to foower oads: a survey, Journa of Sound and Vibration, 238(5),

26 FE anaysis of vehice-structure interactions m s ut av, k v c v m u m m u s unsprung mass sprung mass EI, m, c, y x, t

27 for t t ex it exit D m su cv u y x t, t kv u y x t, t Dt iv 1 2 EIy my cy f x, t x vt at 2,, D f x t mu m s g kv u y x t t cv u y x t, t Dt f 2 D mu y x 2 t, t Dt x, t whee force for t t iv EIy my cy with conditions at t obtained from equations vaid for t t exit Approach: integra and weak formuation exit

28 Guide way uneveness m s ut av, k v c v m u m m u s unsprung mass sprung mass EI, m, c, y x, t 28

29 u1 t u2 t m s1 k v1 c v1 kv2 c v 2 m u1 m u2 EI, m, c, y x, t Vehices and the beam interact 29

30 m s k v1 c v1 kv2 c v 2 m u1 m u2 EI, m, c, y x, t Vehices and the beam interact 3

31 31

32 Preude : Integra and weak formuations for modeing beam vibrations We consider situations in which the system to be anayzed is described in terms of a governing differentia equation.this is in contrast to our studies so far wherein we started with Hamiton's principe in formuating the probem. Consider EIy m x y f x, t y, t, y, t, EIy, t M t, EIy, t,, y x, y x We aim to find an approximate soution to this equation in the form N, y x t a t x x n1 n n As we have seen earier, the substitution of the assumed soution into the governing equation eads to a residue. 32

33 Weighted residua statement wx EIy mx y f x, t dx If y x, t is the exact soution, EIy m x y f x, t. If y x, t is repaced by its approximation, N y x t a t x x EIy m x y f x, n n, t. n1 The above statement impies that the error of representation is zero in a weighted integra sense. By choosing N independent weight functions, we get N independent equations for the unknowns a t, n 1,2,, N n

34 x x Continuity requirements on w x & are different. The requirements on n are more stringent. The weighted integra statement is equivaent to the governing fied equation and does not take into account BCs. The unknowns a t, n 1,2,, N can be determined by considering n N weight functions w x, n 1,2,, N n n N N wn x EI an tn x x mx an t n x f x, t dx n1 n1 To proceed further with the soution, we need to seect the tria functions n x n N th, 1,2,, which possess 4 order derivatives and satisfy the prescribed boundary conditions. There is no such stringent requirements on the weight w x, n 1,2,, N n functions

35 Consider wx EIy mx y f x, t dx and integrate the first term by parts: wxeiy wxeiy dx wx m x y f x, t dx, w xeiy w xeiy w xeiy dx w x m x y f x t dx This is known as the weak form. Notice: differentiabiity requirement on y x n [and hence on tria functions x, n 1,2. N] has come down to 2 and the requirement on w( x) has gone up to 2. The integration by parts has enabed us to trade the differentiabiity requirements between tria functions and the weight functions. 35

36 Consider the terms w xeiy & w xeiy We can identify two types of BCs: natura and essentia. We ca coefficients of the weight function and its derivatives in the above terms as secondary variabes. EIy Thus, & EIy are secondary variabes. Specification of secondary variabes on the boundaries constitute the natura (or force) BCs. The dependent variabes expressed in the same form as the weight function as appearing in the boundary terms are caed the primary variabes. Thus y x, t & y x, t are the primary variabes. Specification of the primary variabes on the boundaries constitutes the essentia (or geometric) boundary conditions. 36

37 Remarks The number of primary and secondary variabes woud be equa. The SVs have direct physica meaning. EIy EIy : bending moment : shear force. Each PV is associated with a corresponding SV Secondary variabe Primary variabe EIy y x t Bending moment : Sope, Shear force: Dispacement, EIy y x t Essentia BCs invove specifying dispacement & sope at the boundaries. Natura BCs invove specifying BM and SF at boundaries 37

38 Remarks Continued On the boundary either a pv can be specified or the corresponding sv can be specified. A given pair of sv and pv cannot be specified simutaneousy at the same boundary. Thus, at a free end we can specify BM to be zero and sope remains unspecfied; simiary, SF can be specified to be zero and dispacement remains unspecified. By denoting: EIy V & EIy M, we write w xv x w x M x w x m x y f x, t dx w x EIy dx 38

39 w x V x w x M x w x EIy dx w x m x y f x, t dx We now require the weight functions to satisfy the essentia BCs of the probem. Reca: the BCs we are considering are y, t, y, t, EIy t M t EIy t w,,, Accordingy, we demand w, Thus we have w x V x w x M x w V w M w M The weak form thus reads w M t w x EIy dx w x m x y f x, t dx 39

40 w M t w x EIy dx w x m x y f x, t dx This is equivaent to the origina differentia equation and the natura BCs. Reca that we have y x, t a t x x n1 with a t, n 1,2,, N to be determined. n N w M t w xei ant nx xdx n1 N N wx mx ant nx f x, t dx n1 We can use w x x,1,2,, N and obtain equations for a t, n 1,2,, N. n n n n 4

41 Remarks (continued) The method eads to symmetric coefficient matrices. The natura boundary conditions are incuded in the weak form and the approximate soution needs to satisfy ony the essentia boundary conditions. J N Reddy, 26, An introduction to the finite eement method, 3 rd Edition, Tata McGraw-Hi, New Dehi 41

42 EIy m x y f x, t y t y t y t EIy t y x,, y x,,,,,,,, 1 x 2 k n x x1 2 x xk 1 x k 1 3 k xn 1 x n 2 xk 1 4 xk 42

43 Consider the k n1 i1 th i i i1 i k 1 k k th eement Let x - where x x x x x For the k y eement we have y EIy m x y f x, t y, t u1t, y, t u2t, y k, t u t, y, 3 k t u t 4 EIy, t F t, EIy, t F t, EIy, t F t, EIy, t F t 1 2 Weighted residua statement k w EIy m y f, t d e 3 e 4 43

44 Weak form k k k w EIy w EIy w EIy d k 4 w m y f, t d k, k k F w w F F w F w 1 k k k w EIy d w m y f t d F F EIy EIy F EIy F EIy 44

45 k,, k F w w F F w F w 1 k i1 and by seecting w, i 1, 2,3, 4 we get k i 4 k 4 j i i j i i i1 i1 1 j j k 3 2 i k w EIy d w m y f t d y x t u t i EIu t d m u t f, td F F F 4 4 ij i ij i j i1 i1 w j F4 j k ; ij j i ij j i, j j j K u t M u t P t, j 1,2,3,4 k K EI d M m d k P t f t d F k 45

46 ; ; ; M m EI & K

47 Assemby Requirements Inter eement continuity of primary variabes (defection and sope in this case) Inter eement equiibrium of secondary variabes (BM and SF here). Imposition of boundary conditions Primary variabes not constrained the corresponding secondary variabes are zero (uness there are appied externa actions) Free end: defection and sope are not constrained BM and SF are zero uness the free end carries additiona forces. Primary variabes are prescribed to be zero the secondary variabes to be specified functions of time determine the reactions Governing equations of motion 47

48 for t t ex it exit D m su cv u y x t, t kv u y x t, t Dt iv 1 2 EIy my cy f x, t x vt at 2,, D f x t mu m s g kv u y x t t cv u y x t, t Dt f 2 D mu y x 2 t, t Dt x, t whee force for t t iv EIy my cy with conditions at t obtained from equations vaid for t t exit Approach: integra and weak formuation exit

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