Weak formulation and Lagrange equations of motion

Transcription

1 Chapter 4 Weak formulatio ad Lagrage equatio of motio A mot commo approach to tudy tructural dyamic i the ue of the Lagrage equatio of motio. Thee are obtaied i thi chapter tartig from the Cauchy equatio of motio after reformulatig the origial problem i a o-called weak form. 1 The, we itroduce thi formulatio via irtual Work Law. Furthermore, we itroduce the Lagrage equatio of motio for the retricted cae i which the diplacemet i give a a liear combiatio of pecified fuctio; i thi cae we alo how the equivalece of the Lagrage equatio of motio ad thoe obtaied uig the Galerki method. Fially, we deal with the geeral cae. 4.1 irtual Work Law ad weak formulatio Coider a virtual diplacemet, δu, i.e., a arbitrary diplacemet that i compatible with cotrait of the tructure. If we take the dot product of the Cauchy equatio of motio Dt time the virtual diplacemet, δu, we obtai δu d = ρf δu d + Dt Notig that = ρf + div T (4.1) div T δu d (4.) div T δu d = T δu d T : grad (δu) d (4.3) 1 A differet approach aumig the Lagrage equatio of motio for a elatic olid, e.g., a derivig from a total eergy priciple may alo be egive. A idicated i Chapter, we avoid the ue of the termiology irtual Work Priciple i order to emphaize that thi i ot a idepedet potulate. 45

2 46 oe obtai δu d = ρf δu d + t δu d + Dt T : grad (δu) d (4.4) Equatio 4.4 i the irtual Work Law. 4. Retricted formulatio For the ake of clarity, we itroduce the Lagrage equatio of motio for the retricted cae i which the diplacemet, u = x x 0, i give a a liear combiatio of pecified fuctio, i.e., 3 A a coequece, we have Combiig Eq. 4.4, 4.5, ad 4.6 yield ( k u(ξ α, t) = q k (t)ψ k (ξ α ) (4.5) δu(ξ α, t) = δq k (t)ψ k (ξ α ) (4.6) Dt ψ kd ρf ψ k d + t ψ k d+ T : grad ψ k d ) δq k = 0 (4.7) Notig that δq are arbitrary oe obtai Dt ψ k d = ρf ψ k d + t ψ k d T : grad ψ k d k = 1,,..., (4.8) Thi i a ytem of N differetial equatio for N ukow that may be itegratated to olve the problem. It may be oted that Eq. 4.8 i equivalet to Dt ψ k d = ρf ψ k d + div T ψ k d (4.9) 3 Typically, the um pa over a fiite umber, N, of fuctio. If the fuctio ψ form a complete et of fuctio, a N goe to ifiity the appoximate repreetatio give i Eq. 4.5 ted to a exact expreio. Fuctio that are commoly ued i practice are: (i) the atural mode of vibratio ad (ii) fiite-elemet iterpolatio fuctio. However, other fuctio (e.g., power, ξ 1p ξ q ξ 3r ) may be ued a well. Note that the volume 0 (image of the material volume i the ξ α -pace) i time idepedet. Therefore, Eq. 4.5 i meaigful eve for large diplacemet (thi i oe of the advatage of formulatig the problem i term of material co-ordiate).

3 47 i.e., < Dt ρf div T, ψ k >= 0 (4.10) < a, b >= a b d (4.11) deote the ier product betwee two vector fuctio. Thi demotrate the equivalece betwee the Lagrage equatio of motio ad thoe obtaied uig the Galerki method. Therefore, i the followig to refer to the retricted equatio a the Galerki equatio of motio. Commet Note that a typical defiitio of the hape fuctio ψ k (ξ α ) i that propoed by the Fiite Elemet Method (FEM). I thi cae, oce the body i ub-divided i elemet, uch fuctio are choe i order to be equal to 1 i a ode, ad zero i all the other urroudig ode. Thee hape fuctio are alo deoted a tet fuctio ad are deoted a ˆψ k (ξ α ), a the correpodig Lagrage variable are dooted a ˆq k. I Figure 4.1 thi repreetatio i applied for the cae of a beam-like tructure. It i worth to poitig out that that i thi cae the ew variable ˆq k aume aturally the role of compoet dipacemet at ode. Figure 4.1: A exact ulutio u(ξ, t) depicted together it FE approximatio obtaied by uig a bae fuctio the F.E. hape fuctio ˆϕ m for a clamped beam

4 Galerki Equatio of Motio I thi ectio, we obtai a explicit expreio for Galerki equatio of motio. For the ake of implicity, we aume that the body force i coervative, i.e., Ω i the potetial eergy. Alo, we will ue the relatiohip which i eaily obtaied by otig that i valid for ay q r. D Dt (ρj) = f = Ω (4.1) (ρj) = 0 (4.13) (ρj) q r = 0 (4.14) Coider the firt term i Eq Note that v = Dx Dt = x t = q ψ (4.15) ξ α The, the acceleratio i give by Dv Dt = v t = q ψ (4.16) ξ α ad therefore, Dt ψ k d = M k q (4.17) M k = ρψ k ψ d (4.18) Next, coider the body-force term. Uig Eq. 4.1 ad 4.13, we have ρf ψ k d = ρ Ω x d = ρω d = U (4.19) U = ρω d (4.0) i the total potetial eergy.

5 49 Next, coider the urface-force term. Note that, i aeroautic, the force per uit area, t, i that that the fluid exert o the olid. We will ue the followig otatio e k = t ψ k d (4.1) ad refer to e k a geeralized aerodyamic force. Fially we wat how that, for elatic olid udergoig ietropic deformatio, we have T : grad ψ k d = ρe d (4.) I order to how thi, ote that, recallig the ymmetry of the tre teor, we have Next, ote that T : grad ψ k d = Ideed, ote that ε αβ = g α g β, Thu, τ αβ [ψ kα/β + ψ kβ/α ] d (4.3) ψ kβ/α + ψ kα/β = ε αβ (4.4) g α = ψ ξ α q (4.5) ε αβ = g α g β + g α g β = ψ k ξ α g β + ψ k ξ β g α (4.6) Fially, combiig Eq. 4.3 ad 4.4, ad recallig that τ αβ = ρ e (4.7) we have T : grad ψ k d =, E i the elatic eergy. ε αβ ρ e ε αβ ε αβ d = Combiig the equatio above, the equatio of motio may be writte a ρed = E (4.8) M k q + E + U = e k, k = 1,,... (4.9) Note that, i geeral, the trai teor i quadratic i q ; o eve if the tre teor i a liear fuctio of the trai teor (liear elaticity), the elatic eergy will be of fourth order i q.

6 50 If oly the quadratic term i q are retaied i the expreio of E (the liear oe are zero, becaue we aume that the referece cofiguratio i tre free), the E = K k q (4.30) ad we have M k q + K k q = U + e k (4.31) It could be uefull to give the explicit form of the matrix K k for the cae of a liearly elatic olid. Ideed, if oe coider Eq ad a liear kiematic for the deformatio, i.e., i term of cartheia compoet ε = 1 (u i,j + u j,i ) (4.3) oe ha E := 1 τ ε d = 1 C km ε ε km d = 1 8 Thu, uig the relatio give by Eq. 4.5 oe ha C km (u i,j + u j,i ) (u k,m + u m,k ) d E = 1 [ C km q r (t) ( ψ r i,j + ψ r ) ] [ j,i q (t) ( ψ k,m + ψ 8 m,k) ] d r = 1 q (t)q r (t) 1 ( ψ r 4 i,j + ψ r ) ( j,i Ckm ψ k,m + ψ ) m,k d (4.33) r = 1 q (t)q r (t) K r r km with K r = 1 4 ( ψ r i,j + ψ r ) ( j,i Ckm ψ k,m + ψ ) m,k d (4.34) km which i the formal expreio of the tiffe matrix for the liearly elatic olid. Ideed, E = q 1 q 4 ( ψ i,j + ψ ) ( j,i Ckm ψ k,m + ψ m,k) d = K q (4.35) km Note that thi matrix i ymmetric becaue of the phyical property of the liearly elatic olid give by the Eq

7 51 It could alo be uefull to give the explicit form of the matrix K k for the pecial cae of a hookea elatic olid: if oe alo coider a liear kiematic (Eq. 4.3), the elatic eergy give by Eq become [ ] λ 1 E = M (u k,k + u k,k ) + µ (u i,j + u j,i ) 4 d k ( ) [ λ = q m div ψ m + µ ( ) ] q r ψri,j + ψ rj,i d (4.36) 4 M m the Eq. 4.5 ha bee ued. The, { ( ) E = λ q m div ψ q m div ψ + (4.37) M m [ µ ( ) ] ( ) q r ψri,j + ψ rj,i ψi,j + ψ j,i d r = q m m m λdiv ψ mdiv ψ + µ ( ) ( ) ψmi,j + ψ mj,i ψi,j + ψ j,i d = K m q m m Comparig the above equatio with Eq ad uig abolute otatio yield K m = [λ div ψ m div ψ + µ ym (grad ψ m ) : ym (grad ψ )] d (4.38) M which exhibit the depedece of the tiffe matrix upo the fuctio ψ for a hookea liear elatic olid: from thi equatio the ymmetry of the tiffe matrix i apparet. Of particular iteret are the free-vibratio equatio (i.e., the equatio whe the force are idetically equal to zero): M k q + r K k q = 0 (4.39) 4.4 Lagrage Equatio of Motio: geeral formulatio Let u tart agai from the virtual work law 4 δx d = f δx d + Dt ρ t δx d T : δe d (4.40) We wat to geeralize the reult of the lat ectio by aumig that x i give by x = x(t, ξ α ) = x(q r (t), ξ α ) (4.41) 4 Note that the lat term i Eq. 4.4 ha bee developed i the followig coiderig that (δu) (δx) = ( T (δu) + (δu))/ = δ(( T u + u)/) = δe.

8 5 i.e., ame a before, except that the ξ α depedecy i ot ecearily liear o q r. The, ad δx = r v = r x δq r (4.4) x q r (4.43) Therefore, Dt δx d = r = r Dt x dδq r [ ( D ρ v x ) v D ( )] x dδq r (4.44) Dt Dt Notig that (ee Eq. 4.43) v q r = x (4.45) ad we have Alo, ad Combiig ρ D Dt ( v x ) d = ρv D ( ) x Dt Dt v v = 1 v (4.46) ρ D 1 v d = d Dt q r dt q r ρ v d (4.47) ( ) D x = v (4.48) Dt d = ( d δx d = dt T = ρv v d = ρ v T T q r ρ v d (4.49) ) δq r (4.50) d (4.51)

9 53 i the kietic eergy. Next coider the body-force itegral ad aume that f = Ω (4.5) Ω i the potetial eergy. Thi yield ρf δx d = ρ Ω x dδq r = ρ Ω dδq r = U δq r (4.53) U = ρω d (4.54) Next coider the work doe by the tre vector, t, t x d = t x dδq r = e r δq r (4.55) with Equatio 4.56 i a geeralizatio of Eq e r = t χ r d (4.56) χ r := x (4.57) Fially coider the work doe by the iteral tree. Note that, ice hece τ αβ δε αβ d = δϵ αβ = ϵ αβ δq r (4.58) E = ρ e ε αβ ε αβ dδq r = E δq r (4.59) ρe d (4.60) Combiig with Eq. 4.4 oe obtai [ ( ) d T T ] δq r = U δq r + e r δq r E δq r (4.61) dt q r Fially, coiderig that δq r motio are arbitrary, oe obtai the deired Lagrage equatio of ( ) d T T + U + E = e r (4.6) dt q r

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