MATHMOD-Kurt-02.dvi

Proceedings 5th MATHMOD Vienna, February 2006

(I.Troch, F.Breitenecker, eds.)

SIMPLE ELASTIC SYSTEMS, AN INTRODUCTION BASED ON GEOMETRY

K. S hla her, M. S höberl, H. Ennsbrunner, Johannes Kepler University Linz, Austria Corresponding Author: K. S hla her Institute of Automati Control and Control Systems Te hnology Altenbergerstraÿe 69, 4040 Linz, Austria Phone: +43 732 2468 - 9731, Fax: + 43 732 2468 - 9734 email: kurt.s hla herjku.at Abstra t. Elasti systems like strings, beams, membranes or plates are spe ial approximations of the linearized equations of simple elasti ity. Based on the prin iples onservation of mass and balan e of linear momentum the equations of simple elasti ity are derived. Balan e of momentum of momentum is taken into a

ount by the strong onstitutive assumption, the Cau hy stress tensor is symmetri . The equations of motion of simple elasti ity an be rewritten as Lagrangian or Hamiltonian equations with distributed ports, whi h des ribe the energy ex hange with the environment. Sin e these equations are often too omplex, a redu tion pro edure is applied. In the ase of holonomi onstraints the redu tion

an be applied to the Lagrangian or the Hamiltonian model or to the equations su h that the results

oin ide. This fa t is demonstrated for the rigid body and the Euler Bernoulli beam exemplarily.

1

Introdu tion

Simple elasti stru tures like strings, beams or membranes and plates are basi elements for many engineering elds. But one is often not aware that their mathemati al models are approximations of

ertain equations of linearized elasti ity. Therefore, lets take a short look, how these equations are derived. Mathemati al modeling of dynami systems with methods of lassi al physi s requires two sets of equations. The balan e and/or onservation equations express that ertain physi al quantities or their sum are preserved. In general these equations do not depend on the spe ial behavior of the materials. Conservation of mass or balan e of momentum are representatives of this lass. The onstitutive relations des ribe the behavior of the materials, typi al representatives are Hook's law or fri tion relations. Within this setting one assumes that balan e equations are never violated, whereas onstitutive relations are often approximately known only. Mathemati al models of elasti bodies are based on the onservation of mass and the balan e of linear momentum and momentum of momentum, see [2, 5, 6℄. To simplify this ompli ated set of partial dierential equations, one makes the strong onstitutive assumption of the symmetry of the stress. This relation guarantees that balan e of momentum of momentum is fullled, and we have to take onservation of mass and balan e of linear momentum into a

ount only. Additionally in simple elasti ity, one assumes the existen e of the stored energy fun tion to express ertain onstitutive relations. If this fun tion exists and the symmetry of stress is met, then the derived mathemati al models have the stru ture of a Lagrangian or Hamiltonian system for a ertain hoi e of the oordinates. Depending on the appli ation one may linearize the balan e and/or the onstitutive relations. In general one needs three spatial variables and the time to des ribe the motion of an elasti body. The model of stru tures, like beams or plates with a small extension in one dire tion ompared to the others, an be approximated by redu ed models with less spatial variables. This way, the models of beams, plates, et . are derived by the redu tion of the linearized equations of simple elasti ity. Of ourse, if the redu tion pro ess preserves the Hamiltonian or Lagrangian stru ture, then the resulting models will have this stru ture, too. This ontribution gives a short introdu tion into basi elasti ity and shows, how ertain models an be derived systemati ally. We onne ourselves to the time invariant ase and assume that physi s takes always pla e in an inertial spa e with Eu lidean oordinates. Otherwise the balan e law of linear momentum is not valid in the presented form. The next se tion gives a short introdu tion into the basi equations of simple elasti ity. A fundamental property of the derived equations of motion is that one an rewrite them in the Lagrangian or Hamiltonian form, see [1, 3℄. This approa h is presented in

Port-based Modelling and Control

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the third se tion. The fourth se tion presents the linearized equations. Often, the original equations are too omplex, and one tries to onstru t simpler models. This an be a hieved by adding additional

onstraints. How one obtains the redu ed model is presented in the fth se tion. Two simple appli ations of the presented methods, the rigid body and the Euler Bernoulli beam illustrate this approa h. Finally, this ontribution loses with a short summary. 2

Simple Elasti ity

The geometry of the general equations of elasti ity are far beyond this ontribution. Therefore, we onne ourselves to the ase of simple elasti ity, where the existen e of the so alled stored energy fun tion is assumed. Furthermore, we des ribe the motion in an inertial frame with Eu lidean oordinates and trivial metri . This hoi e is essential, sin e the following onsiderations are not valid in general oordinate systems, see e.g [4℄. The Lagrangian and the Eulerian des ription are the most popular ones in ontinuum me hani s. Sin e we onsider elasti bodies, we hoose the Lagrangian des ription, whi h allows us to take into a

ount the onstitutive relations, whi h des ribe the behavior of material. Furthermore, we

onne ourselves to the time invariant ase, but we permit inputs like for e or stress elds. From now on, we use the standard tensor notation to keep formulas as short as possible and apply Einstein's onvention for sums. Whenever the range of an index i = 1, . . . , n is lear, we us the abbreviation ai b i

=

Xn

i=1

ai b i .

We will onsider fun tions, whi h depend on the time t and on the spatial oordinates X I , I = 1, . . . , p. Let xi , i = 1, . . . , q denote the dependent oordinates, then xi = xi (t, X) assigns the fun tions xi (t, X) to the oordinate xi . Where no onfusion o

urs, we use the same symbol for the oordinate xi and for the assigned fun tion xi = xi (t, X). Sin e we will deal with several higher order derivatives of these fun tions, we use the abbreviations ∂I =

∂ , ∂X I

∂t =

∂ , ∂t

∂IJ = ∂I ∂J ,

∂tI = ∂t ∂I ,

et . We need also the jet or derivative oordinates of rst order xit , xiI and higher order xiM··· with the unordered multi index M = m1 , . . . , mk , . . . , mr , mk ∈ {t}∪{1, . . . , q}, where #M = r is the order of the derivative , see [1℄. This notation is motivated by the assignment xiM = ∂M xi (t, X), ∂M = ∂m1 · · · ∂mr . We will also use the onventions xiM = xi , ∂M xi (t, X) = xi (t, X) for #M = 0. Let us onsider a fun tion f (t, X, xM ), 0 ≤ #M ≤ m. The total derivative (dI f ) (t, X, xN ), 0 ≤ #N ≤ m + 1 of f in the dire tion of I is the unique fun tion dI f , whi h meets ∂I f (t, X, ∂M x (t, X)) =

(dI f ) (t, X, ∂N x (t, X)) .

Obviously, the dierential operator dI or the total derivative into the dire tion of I is given by dI = ∂I + xiM,I ∂iM = ∂I +

X

#M≥0

xiM,I ∂iM ,

∂iM =

∂ ∂xiM

with I ∈ {t} ∪ {1, . . . , q}. To keep the following as simple as possible, we assume that all fun tions are su iently often ontinously dierentiable, and that all regions for the integration are su iently ni e. 2.1

Motion and Coordinates

In general we need three oordinate systems for the modeling of an elasti body, the onguration spa e C , where physi s takes pla e, the referen e spa e R, where we do bookkeeping, and the spa e of generalized

oordinates G , whi h is used to parameterize maps from R to C . We hoose C = Rn , R = Rn and assume that C , R are equipped with the Eu lidean oordinates xi , X I , i, I = 1, . . . , n. In addition C is an inertial spa e. The position of a mass point is given by X ∈ B ⊂ R, where B denotes the set of all mass points of the elasti body. A motion is a map xi


=

φi (t, X) ,

i = 1, . . . , n ,

(1)

6-2



whi h assigns the position x of a mass point X at time t. We assume that we an invert φ su h that I −1 I X = φ (t, x) is met. To parameterize the map (1) we introdu e the independent spatial oordinates ¯ ˆ Iˆ, Iˆ = 1, . . . , n ¯ I¯, I¯ = n X ˆ, X ˆ + 1, . . . , n and the generalized oordinates x ¯i , ¯i = 1, . . . , m together with ¯i ¯ x their derivative oordinates x¯J¯. A parameterization of the map (1) is given by the fun tions ψ i X, ¯M¯ , #M ≥ 0, ϕI su h that

ˆ X ¯ , X I = ϕI X,

ˆ X, ¯ ∂M¯ x¯ t, X ¯ φi (t, X) = ψ i X,

(2)

is met for x¯¯i = x¯¯i t, X¯ . We assume, that ϕ is invertible and meets ϕ Xˆ X¯ , X¯ ∩ ϕ Xˆ Z¯ , Z¯ = {}

for X¯ 6= Z¯ and B = ϕ Xˆ X¯ , X¯ for the set valued fun tion Xˆ . Throughout this ontribution we equip C with the trivial metri . Let vi , wi be two elements of the tangent spa e T (C) of C , then their produ t is given by (v, w) = v i gij wj ,

(3)

gij = δij

with the Krone ker symbol δij . The hoi e of the trivial metri is essential to simplify the following. Before we an pro eed with the balan e laws, we introdu e the spatial velo ity V and the velo ity ve tor eld v, V i = ∂t φi (t, X) , v i = V i ◦ φ−1 (t, x) . (4) In addition we derive from (1) the so alled deformation gradient

The inverse of JIi is denoted by J¯iI . 2.2

JIi

=

(5)

∂I φi .

Conservation of Mass and Balan e of Momentum

Let ρ (t, x) denote the mass density in C and let D be an arbitrary subset D ⊂ B su h that we an integrate over D. With the total time derivative dt , and the volume forms dx = dx1 · · · dxn , dX = dX 1 · · · dX n on C and R we derive the identity dt

Z

ρ (t, x) dx = dt

φ(t,D)

Z

ρ (t, φ (t, X)) |J (t, X)| dX =

D

Z

∂t ρR (t, X) dX

D

with ρR (t, X) =

ρ (t, φ (t, X)) |J (t, X)| ,

where |J (t, X)| denotes the determinant of J . Obviously, onservation of mass implies R

D

∂t ρR (t, X) dX =

(6)

0.

To pro eed with the balan e of linear momentum, we make the strong onstitutive assumption of the symmetry of the Cau hy stress tensor σ (t, x), see [2℄. This assumption implies that the balan e of momentum of momentum is met. Let ∂D denote the boundary of D and ∂j ⌋dx, (i−1)

∂i ⌋dx = (−1)

di · · · dxn , dx1 · · · dx

di is omitted, be the surfa e element proje ted into the ith dire tion, then balan e of where the term dx linear momentum is given by dt

Z

v i (t, x) ρ (t, x) dx =

φ(t,D)

Z

f i (t, x) dx +

φ(t,D)

Z

σ ij (t, x) ∂j ⌋dx =

φ(t,∂D)

Z

φ(t,D)

f i (t, x) + ∂j σ ij (t, x) dx ,

with the body for es f . Using (6) we get dt

R

φ(t,D)


v i (t, x) ρ (t, x) dx

=

R

D

ρR (X) ∂t V i (t, X) dX ,

6-3



and derive in a similar manner the for e eld F in Lagrangian des ription and the rst Piola Kir ho stress tensor P , see [2℄, F i (t, X) = P iJ (t, X) =

su h that the relations

f i (t, φ (t, X)) |J (t, X)| σ ij (t, φ (t, X)) J¯jJ (t, X) |J (t, X)| ,

R f i (t, x) dx Rφ(t,D) ij σ (t, x) ∂j ⌋dx φ(t,∂D)

R F i (t, X) dX RD iJ P (t, X) ∂J ⌋dX ∂D

= =

(7)

are met. Summarizing, we may write the balan e of linear momentum in Lagrangian des ription in the form Z

i

ρR (X) ∂t V (t, X) dX =

D

2.3

Z

F i (t, X) + ∂I P iI (t, X) dX .

D

(8)

Equations of Motion

The equations (6), (8) are in omplete, sin e the onstitutive relations are missing. To over ome this problem, we have to parameterize the map (1) by generalized oordinates. The simplest hoi e is, we

hoose the X I for the independent and the xi for the dependent oordinates. Let us assume that (8) holds for every ni e subset D ⊂ B , then we may on lude that ρR xitt = F i + dI P iI (9) iI i is met. For the present we allow that P , F may depend on X, xM , #M ≥ 0. Therefore, we have to use the total derivative dI instead of the partial derivative ∂I like in (8). Multipli ation of (9) with xjt gij and integration over D leads to Z

or

with

ρR xjt gij xitt dX

=

D Z j i ρR xt gij xtt + dI xjt gij P iI dX = D

Z 1 ρR i dt xt gij xjt + dt xiI gij xjJ S IJ dX 2 2 D Z 1 ρR i dt xt gij xjt + dt (CIJ ) S IJ dX 2 2 D

Z

ZD D

xjt gij F i + xjt gij dI P iI dX

xjt gij F i + dI xjt gij P iI dX

= =

Z

D

xit gij F j dX +

Z

∂D

xit gij P jI ∂I ⌋dX ,

(10) The Cau hy Green deformation tensor C , see [2℄, is symmetri by onstru tion, whereas the symmetry of the se ond Piola Kir hho tensor S , see [2℄, is a onsequen e of the symmetry of the Cau hy stress tensor σ. In simple elasti ity one assumes S IJ =S IJ (X, C) and the existen e of the stored energy fun tion eE (X, C) su h that ∂ 2 eE (X, C) = S IJ (X, C) (11) CIJ = xiI gij xjJ ,

P iJ = xiI S IJ .

∂CIJ

is met. This is possible only, if S is symmetri . In this ase the balan e of energy is given by Z

dt (eK + eE ) dX =

D

with the kineti energy density

Z

D

xit gij F j dX

+

Z

∂D

xit gij P jI ∂I ⌋dX

(12)

ρR i eK (X, xt ) = xt gij xjt . (13) 2 Obviously, the oordinates t, X I , xiM , M = m1 , . . . , mr , mk ∈ {t} ∪ {1, . . . , n}, 0 ≤ #M ≤ 2 are

ne essary to model a simple elasti system, and the equations of motion (9) are partial dierential equations of se ond order. In addition, a simple elasti body allows two types of ports dened by the pairs ((xt ) , (F )), ((xt ) , (P )) distributed over B and ∂B . One an use the ports to onne t the body to other systems in a power preserving manner. If sliding of the ports is permitted then the relations xit = x ˜it , F i = −F˜ i and xit = x ˜it , P iI = −P˜ iI must be met, where e refers to the se ond system. Port-based Modelling and Control

6-4


3


The Hamiltonian and Lagrangian Pi ture

To show that the equations (9) are of the Lagrangian type, it is su ient to nd a Lagrangian density l X, xiM , 0 ≤ #M ≤ 1 su h that the equations δi l + gij F j

(14)

= 0

with the variational derivative δi =

X

#M

(−1)

dM ∂iM ,

(15)

dM = dm1 · · · dmr

#M≥0

oin ide with the equations (9). Obviously, the hoi e l

=

eK − eE

with the kineti energy of (13) and the stored energy fun tion of (11) solves this problem be ause of δi l =

−dt ∂it eK

+ dI

1 JK I S ∂i CJK 2

=

−ρR xitt

+ dI

2

xiK

1 2

S

IK

.

Furthermore, if the for e eld F meets the ondition δij F j = −∂i eF (X, x), then it an be in luded in the Lagrangian by l = eK − eE − eF . To derive the Hamiltonian equations, we introdu e the generalized momenta pi by the Legendre transformation pi = ∂it l = ρR gij xjt (16)

and pro eed with the oordinates t, X I , xi , pi , xit , pi,t , xiM¯ , M = m1 , . . . , mk , . . . , mr , mk ∈ {1, . . . , n}, 1 ≤ #M ≤ 2. The Hamiltonian density h is given by h X, xi , pi , xiM = pi ∂it l − l = eK X, g ij pj /ρR + eE ,

g ij = δ ij

(17)

and the Hamiltonian equations take the form

xit = δ¯i h ,

(18)

pi,t = −δ¯i h

with the variational derivatives ∂ δ¯i = ∂ i = , ∂pi

δ¯i =

X

#M

#M≥0

(−1)

dM ∂iM

in the new oordinates. The Hamiltonian equations are also partial dierential equations of se ond order with respe t to the spatial derivatives, but of rst order only with respe t to the time derivatives. The boundary onditions for both, the Lagrangian or Hamiltonian equations, follow from the same

onsideration like above. For a general dis ussion of Lagrangian and Hamiltonian eld theories, the reader is referred to [1℄. 4

The Linearized S enario

To nd the linearized equations of simple elasti ity, we have to onsider the relations (9), (11). Let us

onsider the equation (9) rst. Now it is appropriate to introdu e the Lagrangian strain tensor E , 2EIJ

= CIJ + δIJ

and displa ement oordinates, see [2, 5, 6℄, by xi = δIi X I + ui .

In the ase of small strain one repla es E by the linearized small strain tensor ε, 2εIJ


=

uiI gij δJj + δIi gij ujJ .

(19)

6-5



Furthermore, we set P iJ ≈ δIi S IJ (X, ε). In this ase the equations (9) simplify to ρR uitt

F i + dI δJi S JI

=

(20)

.

It is worth mentioning that nonlinear onstitutive equations are still possible provided the strain remains small. Often one assumes that the stored energy fun tion eE takes the quadrati form eE (X, ε) =

1 εIJ E IJKL (X) εKL 2

(21)

with E = E JIKL = E IJLK = E KLIJ then (11) simplies to S IJ = ∂εIJ eE = E IJKL (X) εKL ,

(22)

whi h is nothing else than Hook's law. Following these onsiderations and the previous se tion, it is straightforward to derive the Lagrangian or Hamiltonian equations for the linearized s enario. 5

Redu tion

Often the equations (9) an be approximated by simpler ones. Examples are beams or plates, where the extension of the stru ture in a ertain dire tion diers signi antly from the ones in the others. The redu tion under onsideration here is based on (2). Let us introdu e new oordinates by the relations

ˆ X ¯ , X I = ϕI X,

ˆ X, ¯ x¯M¯ xi = ψ i X,

(23)

with Xˆ Iˆ , Iˆ = 1, . . . , nˆ , X¯ I¯ , I¯ = nˆ + 1, . . . , n and x¯¯i = x¯¯i t, X¯ . Let Dϕ−1 denote the Ja obian of ˆ X ¯ = ϕ−1 (X). From (23) we derive the relations X, xit = dt¯ψ i ,

xiI = dI˜ψ i Dϕ−1

with I˜ = 1, . . . , n and the total derivatives dIˆ = ∂Iˆ ,

I˜ I

(24)

¯

¯

dI¯ = ∂I¯ + x ¯iM¯ ,I¯∂¯iM

¯ x¯M¯ . Pro eeding with (24) in an analogous with I¯ ∈ {t} ∪ {1, . . . , n¯ } in the new oordinates t, X, manner we derive the relations for higher order relatives. In addition we get from (23) the tangent map ¯ ¯ ¯ ¯ x˙ i = ∂¯iM ψ i x ¯˙ iM¯ = ∂¯iM ψ i dM¯ x¯˙ i .

(25)

¯ ≥ 1. It is worth mentioning, that (25) ontains dierential operators of the type dM¯ for #M Let us hoose for (23) a non invertible map with respe t to x¯, and let us plug in the fun tions of (24) into (9), then (9) annot be met in general. The reason is, that the relations (23) des ribe additional holonomi onstraints. But we an try to solve the equations ρR dt¯t¯ψ i

=

F i (X, ψ) + dI P iI (X, dI¯ψ) + Fci ,

(26)

where the additional for e eld Fc has been introdu ed. The eld Fc is the equation error and represents the for es indu ed by the onstraints. Following the me hani al prin iple, that Fc is orthogonal to the image of the tangent map (25), we derive the relation Z Z X¯

¯) Xˆ (X

¯ ¯ ˆ X ¯ = ∂¯iM ψ i dM¯ x¯˙ i gij Fcj |Dϕ| dXd

0,

ˆ = dX ˆ 1 · · · dX ˆ nˆ , dX

¯ = dX ¯ nˆ +1 · · · dX ¯ n (27) dX

¯ x whi h must hold for any x¯˙ ¯i = z¯i t, X, ¯M¯ . To redu e the equations in the Hamiltonian pi ture, we have to formulate relations for the generalized momenta p¯¯i and require that


6-6


Z

Z

X¯

˙ ¯i

ˆ − p¯¯i x¯ pi x˙ |Dϕ| dX i

¯) Xˆ (X


!

¯ = dX

(28)

0

is met for pi from(16), x˙ i from (25) and arbitrary fun tions for x¯˙ ¯i . Let us assume, there exists a relation ¯ = m1 , . . . , mr , mk ∈ {ˆ ¯ x x ¯it = f i X, ¯M¯ , p¯ , M n + 1, . . . , n}, then we derive the wanted relation from (16) and (24). The dis ussion of the general redu tion pro edure is far beyond this ontribution, e.g. the relations (27), (28) ontain dierential operators if (25) does it. Therefore, we apply this pro edure exemplary to the rigid body and the Euler Bernoulli beam. 5.1

The Rigid Body

Let us onsider the motion of a planar rigid body, n = 2, whi h is inuen ed by a for e eld F . Now, it is well known that the relation bI , X =X I

i

i

I

x = RI (α) X +

¯ δIi¯rI

,

R=

cos (α) sin (α)

− sin (α) cos (α)

(29)

I¯ = 1, 2 with the rotary matrix R parameterizes the motion of the rigid body, see [6℄. Therefore, we I¯ ¯ = m1 , . . . , mr , mk ∈ {t} and derive from (29),

hoose the generalized oordinates α, r, αM¯ , rM¯ , M

(24) the relations

¯

xit = RIi ΩIJ αt X J + δIi¯rtI ,

¯

2

i I J i I xitt = RIi ΩIJ αtt X J + RK ΩK I ΩJ (αt ) X + δI¯rtt ,

Ω=

0 1

−1 0

together with the tangent map x˙ i

=

¯

RIi ΩIJ αX ˙ J + δIi¯r˙ I .

Now, the equation (27) takes the form Z ¯ 2 j j L K L M J¯ RIi ΩIJ βX J + δIi¯z I gij ρR RK ΩK + δJj¯rtt − F j dX = 0 L αtt X + RK ΩL ΩM (αt ) X B Z Z Z j J¯ I J i I J i I J j β αtt X δIJ X ρR dX + RI ΩJ gij δJ¯rtt X dX − RI ΩJ X gij F dX B B B Z Z Z 2 j I j I J j I¯ I¯ i K j +z δI¯gij RI ΩK αtt + RI ΩJ ΩK (αt ) X ρR dX + δI¯rtt ρR dX − F dX = 0, B

B

B

whi h must hold for all fun tions β , z I¯. Thus, the equations of motion of the rigid body are θαtt =

Z

B

together with

Z

I

J

RIi ΩIJ X J gij F j

X δIJ X ρR dX = θ ,

B

Z

B

dX ,

I¯ M rtt

ρR dX = M ,

=

Z

B

Z

¯

δiI F i dX

X I ρR dX = 0 ,

(30) (31)

B

where the point X I = 0 is the enter of gravity. The Lagrangian L of the rigid body equals its kineti energy and follows as L (α, r, αt , rt ) = =


Z 1 ¯ j j J¯ L RIi ΩIJ αt X J + δIi¯rtI gij RK ΩK L αt X + δJ¯rt ρR dX 2 B 1 ¯ ¯ 2 θ (αt ) + M rtI δI¯J¯rtJ 2

6-7


be ause of (31). With the total time derivative dt = ∂t + ational derivatives δα = equations

P

¯ #M

(−1)

δα L +

Z

dM¯ ∂α∂ ¯ M

, δI¯ =

RIi ΩIJ X J gij F j

B

P


P

∂ ∂ I¯ α + r and the vari¯ ¯ ,t ∂r I¯ M,t ∂α ¯ #M≥0 M M

¯ #M

(−1)

dX = 0 ,

δI¯L +

Z

B

¯ M

the derivation of the Lagrangian

dM¯ ∂r∂I¯ ¯ M

i δIi ¯ F dX = 0

is straightforward. Nevertheless the onsiderations from above are required to derive the expressions for the for e eld F . To derive the Hamiltonian ounterpart we use the oordinates α, rI¯, p¯α , p¯I¯, rtI¯, p¯tα and rewrite (28) as Z ¯ ¯ RIj ΩIJ αt X J + δIj¯rtI gji RIi ΩIJ αX ˙ J + δIi¯r˙ I ρR dX

¯

= p¯α α˙ + p¯I¯r˙ I

B

With (31) we get the relation

¯ ¯ (θαt − p¯α ) α˙ + M δI¯J¯rtJ − p¯I¯ r˙ I = 0 ,

whi h must hold for arbitrary α˙ , r˙ I¯. Following the onsideration from above we an al ulate pi a

ording to 1 ¯ 1 pi = ρR gij xjt = ρR gij RIj ΩIJ p¯α X J + δ j I p¯I¯ θ M

.

The Hamiltonian follows as H (α, r, p¯α , pI¯)

= =

Z Z g ij 1 p¯α p¯¯ p¯α L ¯J¯ 1 j j p ΩK pi pj dX = RIi ΩIJ X J + δIi¯ I gij RK X + δ ρR dX L J¯ M 2 B ρR 2 B θ M θ 1 1 ¯¯ 2 (¯ pα ) + p¯¯δ I J p¯J¯ . 2θ 2M I

Now, it is straightforward to derive the Hamiltonian equations αt = δ¯α H ,

ptα = −δ¯α H +

with the variational derivatives

Z

B

∂ δ¯α = , ∂α

RIi ΩIJ X J gij F j dX , ∂ δ¯α = , ∂ p¯α

¯ ¯ rtI = δ¯I H ,

∂ δ¯I¯ = I¯ , ∂r

p¯tI¯ = −δ¯I¯H +

Z

i δIi ¯ F dX

B

∂ ¯ δ¯I = . ∂ p¯I¯

Like in the Lagrangian pi ture, the onsiderations from above are required to derive the expressions for the for e eld F . 6

The Euler-Bernoulli Beam

Beams are examples of two dimensional stru tures with dierent extensions in X 1 and X 2 dire tions. We hoose a simple elasti material with the stored energy fun tion, see (19), (21), eE =

1 2 2 2 a (ε11 ) + (ε22 ) + 2bε11 ε22 + c (ε12 ) , 2

(32)

a, c ∈ R+ , b ∈ R, a > b, . The equations of motion, see (20), follow as ρR u1tt = d1 S 11 + d2 S 12 ,

with

S 11 = aε11 + bε22 ,

ρR u2tt = d1 S 21 + d2 S 22

S 22 = bε11 + aε22 ,

(33)

S 12 = S 21 = cε12 ,

see also (22). Now, we apply the redu tion, see (23) and [6℄, ¯, X1 = X


ˆ , X2 = X

û u1 = u¯1 − X ¯2¯1 ,

u2 = u ¯2

(34)

6-8



and derive the relations, see (24), û û u1t = u ¯1t − X ¯2t¯1 , u2t = u ¯2t , u1tt = u ¯1tt − X ¯2tt¯1 , u2tt = u ¯2tt 1 1 2 1 2 2 ˆ ε11 = u1 = u ¯¯1 − X u ¯¯1¯1 , 2ε12 = u2 + u1 = 0 , ε22 = u2 = 0 .

A

ording to these assumptions, we must require S 22 = 0 to avoid a oni t with ε22 = 0. With these relations we rewrite (33) and get û ˆ u¯2¯¯¯ + F 1 , ρR u ¯1tt − X ¯2¯1tt = a u ¯1¯1¯1 − X c 111

ρR u ¯2tt = Fc2 ,

where we added the for e eld Fc , aused by the onstraints (34). The tangent map for (34) û ˆ ¯1 u u˙ 1 = u ¯˙ 1 − X ¯˙ 2¯1 = u ¯˙ 1 − Xd ¯˙ 2 ,

u˙ 2 = u ¯˙ 2

P

ontains the total derivative d¯1 = ∂¯1 + #M¯ ≥0 uiM¯ ,I¯∂iM¯ . Let us assume, the beam is built up symmetri ally with respe t to the midline and that ρR , a, b, c, h, L ∈ R+ with the height h and the length L of the beam is met. A

ording to the redu tion prin iple from above, the relation Z

0

Z

L

˙1

u ¯

0

Z

h/2

−h/2

L

Z

h/2

−h/2

ˆ Fc1 dX

ˆ ¯1 u¯˙ 2 Fc1 + u ˆ X ¯ u¯˙ 1 − Xd ¯˙ 2 Fc2 dXd

˙2

+u ¯

Z

h/2

−h/2

Fc2

ˆ c1 dX ˆ + d¯1 XF

!

¯ dX

=

0

=

˙2

u ¯

Z

h/2 −h/2

L

ˆ X ˆ Fc1 Xd

0

must hold for an arbitrary hoi e of the fun tions u¯˙ 1 , u¯˙ 2 . Let us assume the boundary term vanishes, then we derive from Z

h/2

h/2

and Z

h/2

−h/2

û û ˆ = h ρR u¯1 − a¯ ρR u ¯1tt − X ¯2tt¯1 − a u ¯1¯1¯1 − X ¯2¯1¯1¯1 dX u1¯1¯1 tt

ˆ ρR u¯1 ¯ − X û ˆ u¯2¯¯¯¯ ˆ = ρR u¯2tt + X ¯2tt¯1¯1 − a u¯1¯1¯1¯1 − X dX tt1 1111

h2 2 h2 2 2 h ρR u ¯tt − u ¯ ¯¯ + a u¯¯1¯1¯1¯1 12 tt11 12

the well known wave equation

h ρR u ¯1tt − a¯ u1¯1¯1 = 0

(35)

for u¯ and the well known beam equation 1









  2    ¯tt − h u¯2 ¯¯  + a h u¯2¯¯¯¯  = 0 h tt11  1111  ρR u 4 |12 {z } 2

2

(36)

≈0

for u¯2 , provided we negle t the term ≈ h2 in the a

eleration. Evaluating the boundary term of the energy relation (12) for X¯ ∈ {0, L}, where L denotes the length of the beam leads to Z

h/2

−h/2

u ¯1t S 11

+

u¯2t S 21

ˆ =u dX ¯1t u ¯1¯1

Z

h/2 −h/2

ˆ − u¯2t u adX ¯2¯1¯1

Z

h/2

−h/2

ˆ X ˆ= aXd

2 1 1 2 2 ah u ¯t ah¯ u¯1 − u ¯t u¯¯1¯1 . 2

Therefore we an pres ribe relations of the form f F u¯1t , F , F = ah¯u1¯1 and fT u¯2t , T , T = a h2 u¯2¯1¯1 between the for e F , torque T and their ollo ated quantities u¯1t , u¯2t at the boundaries. It is worth mentioning that for the presented example these onditions must be hosen stati ally admissible.


2

6-9



The Lagrangian equations an be derived in a straightforward manner from the Lagrangian density ¯l, ¯l u ¯1t , u ¯2t , u ¯1¯1 , u ¯2¯1¯1 , u ¯2t¯1

=

=

1 2

Z

h/2

h/2



2 2 1 2 ˆ 2 2 1 2 ˆ ˆ ρR u¯t − u ¯t¯1 X + u ¯t −a u ¯¯1 − X u ¯¯1¯1 dX





 2 2  h ρR  u¯1t 2 + h u ¯2t¯1 + u ¯2t    −a 2 |12 {z } 2

≈0

 1 h 2   (37) u ¯2¯1¯1 u¯1¯1 +  12 2

with the variational derivatives, see (15), in the new oordinates. These equations are identi with (35), (36). To derive the standard equations, one negle ts the term. Let us now take the Hamiltonian point of view. We determine the generalized momenta p¯1 , p¯2 a

ording to (28) and get the relations p¯1 =

ρR h¯ u1t

,

h2 2 2 p¯2 = ρR h u ¯t − u¯t¯1¯1 . 12

Obviously, it is straightforward to derive p¯2 as a fun tion of u¯2t . But the determination of the inverse map requires the solution of a dierential equation. Therefore, we stop here and onsider the simplied Lagrangian (37). Now, the determination of the generalized momenta p¯1 , p¯2 is straightforward and the Hamiltonian density is given by ¯ h = p¯1 u ¯1t + p¯2 u¯2t − ¯l =

h 2 h2 2 1 2 2 (¯ p1 ) + (¯ p2 ) + a u¯1¯1 + u ¯2¯1¯1 . 2hρR 2 12

With the variational derivatives, see also (18), one derives the Hamiltonian equations as ¯= u ¯1t = δ 1 h

1 p¯1 , hρR

¯ = ah¯ p¯1,t = −δ1 h u1¯1 ,

¯= u¯2t = δ 2 h

1 p¯2 , hρR

p¯2,t = −δ2 ¯h = −ah¯ u2¯1¯1¯1¯1 .

Of ourse, one an start with the Hamiltonian (17) and apply the redu tion pro edure. If one negle ts the terms ≈ h2 in kineti energy, then one derives the same set of equations. 7

Summary

The mathemati al modeling of elasti stru tures an be signi antly simplied by the use of dierential geometri methods. Starting with the fundamental onservation and balan e prin iples, one has to parameterize ertain maps to bring the onstitutive equations into the play. If one assumes the existen e of the stored energy fun tion in the presented manner, then one deals with simple elasti ity. A further

onsequen e of this assumption is, that one an rewrite the equations of motion in a Hamiltonian or Lagrangian manner. This fa t is often used to derive simpler models, where the simpli ation is ar hived by adding holonomi onstraints. Of ourse, one an also linearize the equations of motion. Examplarily, these approa hes has been presented for the rigid body and for the Euler Bernoulli beam su h that the simplied equations of motion are derived by a systemati redu tion pro edure from the general ones. Referen es

[1℄ Gia hetta G., Mangiarotti L. and Sardanashvily G.: New Lagrangian and Hamiltonian Methods in Field Theory, World S ienti , Singapore, 1997. [2℄ Marsden J.E., T.J.R Hughes.: Mathemati al Foundations of Elasti ity, Dover, 1994. [3℄ Olver P.J.: Appli ations of Lie Groups to Dierential Equations. Springer Verlag, New York, 1986. [4℄ S hla her K., Grabmair G., Ennsbrunner H., Stadlmayr R.: Some Appli ations of Dierential Geometry in Me hani s, pages 261-281, CISM Courses and Le tures, Springer, 2004. [5℄ Villaggio P.: Mathemati al Models for Elasti Stru tures, Cambridge University Press, 1997. [6℄ Ziegler F.: Me hani s of Solids and Fluids, Springer Verlag, New York, Vienna, 1991.


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