Distributed Port Hamiltonian Formulation ofFlexible Beams Under Large Deformations

Gou Nishida* and Masaki Yamakita**

Abstract— In this paper, a formulation of flexible beamsunder large deformations for distributed parameter port Hamil-tonian systems is presented. This model is one example ofsystems that have complex energy variables. For such a model,a unified modeling method is introduced with multivariablerepresentation. First, a Stokes-Dirac structure is related to thecalculus of variations by using a jet bundle formalism. Next,the flexible beams model is represented as the port Hamiltoniansystem. Finally, the model is compared to a conventional modeland two reduced models.

I. INTRODUCTION

Port Hamiltonian systems have been developed as a gen-eral control model for passivity [12]. The system clarifiesthe power port such that the product of two energy variablesis equal to its power (for example currents and voltages,or forces and velocities). These systems can be connectedthrough the boundary ports with the total power is preserved.Such a power-conserving property is represented by a Diracstructure [2], [3].

Recently, the framework has been extended to a distributedparameter system with a Stokes-Dirac structure [2]. Thestructure means the change of the interior energy is equalto the power supplied to the system through its boundary.The internal energy variables can be stabilized by a dampinginjection as a boundary control.

Some mathematical extensions of the Stokes-Dirac struc-ture have been considered. In the Timoshenko beam models,Hodge-star operators have been introduced into Stokes-Diracstructure [7]. And this concept have been generalized as thehigher order structures [5]. As more general representation,the constant Stokes-Dirac structure for multi variable systemswas presented with a constant differential matrix and itsadjoint [8].

In this paper, a formulation of flexible beams under largedeformations for distributed parameter port Hamiltonian sys-tems is presented. This model is one example of systemsthat have complex energy variables. For such a model, aunified modeling method is introduced with multivariablerepresentation.

First, the Stokes-Dirac structure is related to the calculusof variations by using a jet bundle formalism. In otherwords, we introduce a situation of field Euler-Lagrangiansystems and the higher order energy variables are regarded

*G. Nishida is with the Department of Mechanical Control Systems,Tokyo Institute of Technology, Meguroku Oookayama 2-12-1, Tokyo, Japan<nishida@ac.ctrl.titech.ac.jp>

**M. Yamakita is with the Department of Mechanical Control Systems,Tokyo Institute of Technology / RIKEN, Meguroku Oookayama 2-12-1,Tokyo, Japan <yamakita@ac.ctrl.titech.ac.jp>

as independent variables on the fiber. This structure are basedon the same concept of [8] basically, however it shows therelation to variational problems clearly. The formulation isnot limited by differential forms of both flows and efforts. Inthis study such differential forms are regarded as belongingto a base manifold of independent variables. In this senseit is a pure extension of studies [2], [5]-[8]. Furthermoretreatments of multivariable systems are discussed. Next,the equation of flexible beams is represented as the portHamiltonian system. Finally, this model is compared to oneconventional model and two reduced models.

The reasons why we introduce this method can be consid-ered as follows.

For applications of control there are two advantages. First,the distributed parameter port Hamiltonian systems provideus with the concept of the power port which stabilize infinitedimensional systems using the simple control, that is, thedamping injection. The beam models discussed later consistof complex energy variables. Generally, it is difficult tofind such a pair of energy variables with modeling meth-ods based on observation except simple systems. Secondly,the port Hamiltonian systems can be connected with eachother through the boundary ports. This means that we cantreat a hybrid network structure of various physical systemsconsisting of both lumped parameter systems and distributedparameter systems [7], [9]. For example, we can consider thatsubsystems with large variations are modeled by distributedparameter systems and the others are regarded as lumpedparameter systems.

From the theoretical point of view, the following three ben-efits can be expected. First, this approach clarifies the relationbetween Euler-Lagrange equations that are calculated by avariational principle and infinite dimensional systems thatare related to passivity. And this correspondence provides aunified modeling method. If the systems have the Lagrangianand it is known, such power ports are given from this conceptautomatically. This situation corresponds with many physicalsystems. Secondly, the integration by parts formula on thevariational complex yields the freedom of formulations. Thenit is possible to express more extensive systems. Thirdly, sucha model indicates an interconnection of energy structuresin the case of multivariable systems. The structures thatconsists of Hodge star operators only doesn’t affect thewhole boundary structure [5], [7]. (Note that the operatorcorresponds to an identity operator in case of variationalcomplexes.) Then we can identify the structure of indirectconnection to boundary ports immediately.

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TA1.2

0-7803-9354-6/05/$20.00 ©2005 IEEE 589

II. MATHEMATICAL PRELIMINARIES

In this section some required mathematical concepts: a jetbundle formalism, vertical forms and variational complexesare quoted from the results [13], [14]. Note that the basicconcepts of distributed parameter port Hamiltonian systemsare not explained in this paper at all. The detailed definitionsfollow the original paper [2].

Let us consider a fiber bundle (M, �, X). The k-th orderpartial derivatives of f will be denoted by ∂Jf(x) =∂kf(x)/∂xj1∂xj2 · · · ∂xjk with J = (j1, · · ·, jk) is a multi-index of order k = �J . Let σf be a smooth section of abundle (M, �, X) defined by uα = fα(x), α = 1, · · · , lwhere x = (x1, · · ·, xm) are m independent variables andu = (u1, · · ·, ul) are l dependent variables. The v-th prolon-gation u(v) = f (v)(x) : X → U (v) := U × U1 × · · · × Uv

is defined by uαJ = ∂Jfα(x) ⊂ Uk. Now we introduce v-th

jet space M (v) = X × U (v). Let A be a space of smoothfunctions P (x, u(v)) called differential functions.

Total derivatives Di can be thought of as a kind of vectorfield on the infinite jet space.

Di =∂

∂xi+

∑J

uJ,xi

∂

∂uJ. (1)

In particular, Di acts on the basic forms by Di duJ =d(DiuJ) = duJ,i.

Vertical k-forms are a finite sum

ω =∑

PαJ duα1

J1∧ · · · ∧ duαk

Jk, (2)

where PαJ ∈ A . Since only the differentials duα

J appear inthese forms, the analogue of the differential of the ordinaryde Rham complex is the vertical differential

dω =∑ ∂Pα

J

∂uβK

duβK ∧ duα1

J1∧ · · · ∧ duαk

Jk. (3)

We consider an equivalence relation on the space ofvertical forms, with [ω] = ω + div η, ω, η ∈ ∧k. The spaceof equivalence classes is the space of functional k-forms∧k∗ = ∧k/ div(∧k)p. The natural projection from

∧k to∧k∗

is denoted by an integral sign∫

ω dx stands for [ω]. Thisdefinition gives the integration by parts formula.

∫ψ ∧ Diη dx = −

∫(Diψ) ∧ η dx (4)

where ψ ∈ ∧k, η ∈ ∧l and Di is the total derivative.Let ω =

∫ω dx be a functional k-form corresponding to

the vertical k-form ω. The variational differential δ of ωis the functional (k + 1)-form corresponding to the verticaldifferential of ω:

δω =∫

dω dx . (5)

An v-th order variational problem means the problemof finding the extremals of a functional, is referred to asthe Lagrangian L =

∫ L(x, u(v)) dx over some space of

functions u = f(x). Then its variational differential is thefunctional 1-form

δL =∫

dL dx =∫ { l∑

α=1

∑J

∂L∂uα

J

duαJ

}dx

=∫ { l∑

α=1

(∑J

(−D)J∂L∂uα

J

)duα

}dx

=∫ {

E(L) · du}dx (6)

where E is the Euler operator. The relation E(L) ≡ 0 yieldsEuler-Lagrange equations.

If we interpret the differentials duα as infinitesimal vari-ations in the uα, with corresponding variations duα

J =DJduα in the derivatives, the above computation (6) is thesame as the traditional determination of the Euler-Lagrangeequations from the definition of the variational derivatives.This interpretation leads to a natural correspondence betweenthe standard Stokes-Dirac structure with differential formsand the definition on a variational complex.

III. FURTHER ASPECTS OF STOKES-DIRAC STRUCTURES

This section shows two extensions of the Stokes-DiracStructure. First, a higher order Stokes-Dirac structure onvariational complexes of jet bundles will be presented withthe definitions mentioned in the previous section. Next, wewill discuss on the multivariable structure.

A. Stokes-Dirac structures on variational complexes

Let Z be an n-dimensional smooth manifold with asmooth (n−1)-dimensional boundary ∂Z . Here the space∧1∗ is the center of topic because it is related to the calculusof variation yielding Euler-Lagrange equations.

From (6), the integrand of functional dL in∧1∗ can be

defined by a product of differential functions ∂L/∂uαJ ∈ A

and differentials duαJ ∈ ∧1. The differentials duα can be

regard as functions A and used to define flows. And effortsare equal to functions A .

Definition 3.1: Let F and E be linear spaces as follows:

F := ∧1(Z) × ∧1(Z) × ∧0(∂Z) ,

E := ∧0(Z) × ∧0(Z) × ∧0(∂Z) .(7)

A pairing between flows f and efforts e is defined by

〈〈 (f1, e1), (f2, e2) 〉〉

:=∫

Z

(e1

p ∧ f2p + e1

q ∧ f2q + e2

p ∧ f1p + e2

q ∧ f1q

)dx

+∫

∂Z

(e1

b ∧ f2b + e2

b ∧ f1b

)dx (8)

where f = (fp, fq, fb) ∈ F and e = (ep, eq, eb) ∈ E .

Then let us consider the Stokes-Dirac structure that iswritten by elements of the two spaces.

590

Theorem 3.1: The linear subspace

D ={(f, e) ∈ F × E|[

fp

fq

]=

[0 Dx

Dx 0

][ep

eq

],

[fb

eb

]=

[1 00 −1

][ep|∂Z

eq|∂Z

]}(9)

satisfies a Dirac structure with the pairing (8), where Dx is atotal differential operator concerning with spatial variables.

Proof: The statement follows from the proof of theoriginal [2].

First D ⊂ D⊥ is showed. Indeed, if we substitute (9) for

the right side of (8), the sum of these terms is equal to zero.Then D ⊂ D

⊥.Next, we will show D

⊥ ⊂ D. Let us consider a conditionof (f1, e1) ∈ D

⊥ such that (8) is zero for all (f2, e2) ∈ D.Then we consider∫

Z

(e1

p ∧ Dxe2q + e1

q ∧ Dxe2p + e2

p ∧ f1p + e2

q ∧ f1q

)dx

+∫

∂Z

(e1

b ∧ e2p − e2

q ∧ f1b

)dx = 0 . (10)

We can compute as follows:∫

Z

{Dxe2

q ∧ (e1p − f1

b ) + (e1q + e1

b) ∧ Dxe2p

+ (f1p + Dxe1

b) ∧ e2p + e2

q ∧ (f1q − Dxf1

b )}

dx = 0 .

(11)

Then if (11) holds for every (f2, e2) ∈ D, then each wedgeproducts have to be zero. This means that (f1, e1) ∈ D also.Namely, this implies that D

⊥ ⊂ D. As a result we haveD = D

⊥.

Note that exterior differential operators d in the definitionof the standard Stokes-Dirac structure are replaced withtotal differential operators Dx on spatial variables in thevariational complex.

Remark 3.1: If we consider a more general situation, then(9) can be extended to a higher order structure concernedwith DJ and −(−D)J . In this meaning (9) is a special caseof it. But in this application we will only use first orderdifferential operators.

Theorem 3.2: The linear subspace

D ={(f, e) ∈ F × E|[

fp

fq

]=

[0 ∓I±I 0

] [ep

eq

], fb = 0, eb = 0

}(12)

satisfies Dirac structure with the pairing (8), where I = idZ

is an identity operator on the manifold.

Proof: It is easy to verify that D ⊂ D⊥ with substituting

(12) for the right side of (8).Similarly, to prove D

⊥ ⊂ D, let us consider a conditionof (f1, e1) ∈ D

⊥ such that (8) is zero for all (f2, e2) ∈ D.We have∫

Z

{e2

q ∧ (f1q ∓ e1

p) + e2p ∧ (e1

q ± f1p )

}dx = 0 . (13)

If (13) holds for every (f2, e2) ∈ D, then (f1, e1) ∈ D mustbe satisfied. Then D = D

⊥.

Corollary 3.3: The structure (9) does not have the bound-ary ports. It corresponds to the ∗-type structure [5]. Then,(9) is in relation to the internal connection of the energyvariables in multivariable systems.

Remark 3.2: The reason why we consider the quotientspace related to divergence terms (e.g. a conservation lowcorresponding with Noether’s theorem of field theory [15,pp.4-5]) is because these can be evaluated by integrationon the boundary and then variations on the boundary canbe eliminated by some boundary conditions in case ofthe variational calculus. On the other hand, if we regardthese factors as a distributed port Hamiltonian system ofconservation laws [2, §4.2], then we can write down anothersystem with boundary ports.

B. Treatments of multivariable systems

The minimum structures (9) and (12) can be expanded tothe multivariable system.

Proposition 3.4: For any integer number m > 0, thefollowing multivariable systems with disturbances can beconsidered generally.

⎧⎨⎩

F = A · E + Γb · Θ + G · Fd + Γd · Λ ,[Fb

Eb

]= B · E , Ed = −G∗ · E ,

(14)

where the energy variables are defined as follows:

F = [f1, · · · , f2m]� , E = [e1, · · · , e2m]� ;Fb = [fb,1, · · · , fb,m]� , Eb = [eb,1, · · · , eb,m]� ;Fd = [f1, · · · , f2m]� , Ed = [e1, · · · , e2m]� ;Θ = [dθ1, · · · , dθ2m]� , Λ = [δλ1, · · · , δλ2m]� .

(15)

The non-diagonal operator matrix A corresponds with theboundary energy structure with the flows F and the effortsE and the real-valued matrix B means the boundary portswith the energy variables Fb and Eb. The diagonal operatormatrix G defines the distributed energy structure with theenergy variables Fd and Ed, where G∗ is the dual operatormatrix of G. As the disturbance, the pair of the non-diagonalreal-valued matrix Γb and the variables Θ is a boundarydisturbance and the pair of the diagonal real-valued matrixΓd and the variables Λ is a distributed disturbance. All ofthe matrices are 2m × 2m.

Proof: By a linearity of operators involved in Stokes-Dirac structures, the proof is lead from the fact that multivari-able systems can be decomposed into the minimum structuredirectly. On the other hand, the disturbance structure decom-position have been given in [6].

Corollary 3.5: If G = Γd = 0, then (14) sarisfies a Diracstructure.

Proposition 3.6: Let us consider a multi variable Stokes-Dirac structure (14) which has n pairs of energy variables.

591

Let G = Γd = 0. If both the i-th row and the j-th row in Ainclude the same pattern of operators and the correspondingboth the i-th column and the j-th column include the samepattern of operators also where 1 ≤ i < j ≤ n , thenboth the j-th row and the j-th column can be eliminated bymodification (fa

i , eai ) = (fi, ei + ej). Conversely, an effort

ei = e1 + e2 can split into two pairs (fai , ea

i ) = (fi, e1) and(fa

k , eak) = (fi, e2).

Proof: If the i-th row is equal to the j-th row in Ain terms of operator patterns, this means that fi is equal tofj . On the other hand, if the i-th column is equal to thej-th column in A in terms of operator patterns, this meansthat aki = akj for any k �= i, j. Then, by addition the i-thcolumn to the j-th column we obtain fa

k = aki(ei +ej). Theconverse case can be showed by the same way.

IV. APPLICATIONS FOR FLEXIBLE BEAMS

In this section, distributed port Hamiltonian formulationsof flexible beams are presented. And some features inmodeling are discussed successively.

First, the equation of flexible beams under large overallmotions on the plane is introduced by [1]. This modelcarries the advantage that drastic simplification of the inertiatemporal part is obtained by the linear uncoupled inertiaoperator. Next a classical approach based on small strains issummarized. This model was introduced as a conventionalmethod to compare with the previous approach in [1]. Finallysimplified beam models can be considered. These are theTimoshenko beam model and the Euler-Bernoulli model.These reduction procedures are discussed in comparison withthe model under large deformations.

A. Flexible beams under large overall motions on the plane

Let Aρ be a mass per unit length and let Iρ be a massmoment of inertia of a cross section. Let EA, GA andEI be an axial, a shear and a flexural stiffness of beams,respectively.

Let us consider the equations of motion

⎧⎨⎩

Aρ

[ytt

wtt

]− ∂x

(Λ CΓ

)=

[00

]

Iρ θtt − EI θxx − ΞΛ CΓ = 0(16)

where t is the time, x ∈ Z := [0, L] is the spatial coordinatealong the equilibrium position, (x+y) is the axial position, wis the shearing position, θ is the rotation of the cross sectionalong the undeformed length of the beam and the matrices:Λ, C, Γ1, Γ2 and Ξ , are given by

Λ :=[cos θ − sin θsin θ cos θ

], C :=

[EA 00 GA

],

Γ :=[Γ1

Γ2

]= Λ�

[1 + yx − cos θ

wx − sin θ

],

Ξ := [−wx 1 + yx] . (17)

The kinetic energy T and the potential energy U areexpressed as

T =12

∫

Z

[Aρ yt

2 + Aρ wt2 + Iρ θt

2]dx , (18)

U =12

∫

Z

[EAΓ1

2 + GAΓ22 + EI θx

2]dx . (19)

Next, the variational differential (5) of the Lagrangian den-sity L = T − U is obtained as the functional 1-form:

δL =∫

Z

[Aρ yt dyt + Aρ wt dwt + Iρ θt dθt + ΞΛ CΓ dθ

− (Λ CΓ

)� [dyx

dwx

]− EI θx dθx

]dx . (20)

From (20) the energy variables p, ε and the co-energy vari-ables ν, σ are defined by the following.

π1 = Aρ yt , ν1 = −yt ;π2 = Aρ wt , ν2 = −wt ;π3 = Iρ θt , ν3 = −θt ;ε1 = θ , σ1 = ΞΛ CΓ ;ε2 = yx , σ2 = −(

Λ CΓ)1;

ε3 = wx , σ3 = −(Λ CΓ

)2;

ε4 = θx , σ4 = −EI θx

(21)

where ( · )i means an extracted i-th element. The rate ofchange of these energy variables and the rate of change ofthe Hamiltonian with respect to the energy variables canbe connected to the Stokes-Dirac structure, respectively, bysetting

fπi = −∂πi

∂t, fεj = −∂εj

∂t, (22)

eνk= νk , eσl

= σl . (23)

The following proposition is the main result of this section.

Theorem 4.1: Consider the space of power variables F ×E with the pairing (8). Then, define the following linearsubspace D of F × E :

D ={(fπ1 , · · · , fπi , fεj , · · · , fεj ,

eν1 , · · · , eνk, eσl

, · · · , eσl) ∈ F × E |⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

fπ1

fπ2

fπ3

fε1

fε2

fε3

fε4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 Dx 0 0

0 0 0 0 0 Dx 0

0 0 0 -I 0 0 Dx

0 0 I 0 0 0 0

Dx 0 0 0 0 0 0

0 Dx 0 0 0 0 0

0 0 Dx 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

eν1

eν2

eν3

eσ1

eσ2

eσ3

eσ4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,

⎡⎢⎢⎢⎢⎢⎢⎣

fb1

fb2

fb3

eb1

eb2

eb3

⎤⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎣

eν1|∂Z

eν2|∂Z

eν3|∂Z

−eσ2|∂Z

−eσ3|∂Z

−eσ4|∂Z

⎤⎥⎥⎥⎥⎥⎥⎦

}.

(24)

Then D = D⊥, that is, D is a Dirac structure.

Proof: If a multi variable system satisfies a Stokes-Dirac structure, it can be decomposed to some small struc-tures. Then we only have to check whether all the mini-mum structures satisfy a Stokes-Dirac structure [5]. FromCorollary 3.5, the structure (24) includes three Stokes-Diracstructures (9): {fπ1 , fε2 , eν1 , eσ2}, {fπ2, fε3 , eν2 , eσ3} and{fπ3 , fε4, eν3 , eσ4}. The structure {fπ3, fε1 , eν3 , eσ1} is the

592

I-type of the extended Stokes-Dirac structure (12), thendoesn’t make its boundary port.

B. Classical approach based on small strains

The approach using a floating flame leads to a compli-cated expression for the kinetic energy. Then this results inequations of motion with highly coupled nonlinear terms.

We introduce the infinitesimal strain assumption: cos θ ≈1, sin θ ≈ θ. Let φt be an angular velocity of the floatingframe. The other parameters are defined in the same mannerof the previous section. The following equations governingthe motion of the beam are obtained.⎧

⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Aρ

[ytt − φttw − 2φtwt − φt

2(x + y)]

−EAyxx = 0

Aρ

[wtt + φtt(x + y) + 2φtyt − φt

2w]

−GA (wxx − θx) = 0Iρ (θtt + φtt) − EI θxx − GA (wx − θ) = 0

. (25)

The kinetic energy T and the potential energy U of thesystem are given by

T =12

∫

Z

[Aρ yt

2 + Aρ wt2 + Iρ (θt + φt)2

+ 2Aρ φt

{−wyt + (x + y)wt

}

+ Aρ φt2{(x + y)2 + w2

}]dx , (26)

U =12

∫

Z

[EAyx

2 + GA (wx − θ)2 + EI θx2]dx . (27)

The variational differential of (26) and (27) lead to thefollowing definitions of the energy variables

π1 = Aρ (yt − wφt) , ν1 = −yt ;π2 = Aρ

{wt + (x + y)φt

}, ν2 = −wt ;

π3 = Iρ (θt + φt) , ν3 = −θt ;π4 = y , ν4 = Aρ φt

{wt + (x + y)φt

};

π5 = w , ν5 = Aρ φt (−yt + wφt) ;ε1 = (wx− θ) , σ1 = GAθ ;ε2 = yx , σ2 = −EAyx ;ε3 = (wx − θ) , σ3 = −GA wx ;ε4 = θx , σ4 = −EI θx .

(28)

In the result some manipulations yield the system represen-tation

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

fπ1

fπ2

fπ3

fπ4

fπ5

fε1

fε2

fε3

fε4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 -I 0 0 Dx 0 0

0 0 0 0 -I Dx 0 Dx 0

0 0 0 0 0 I 0 I Dx

I 0 0 0 0 0 0 0 0

0 I 0 0 0 0 0 0 0

0 Dx -I 0 0 0 0 0 0

Dx 0 0 0 0 0 0 0 0

0 Dx -I 0 0 0 0 0 0

0 0 Dx 0 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

eν1

eν2

eν3

eν4

eν5

eσ1

eσ2

eσ3

eσ4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)

where the boundary port is the same as (24), then it is omittedfor brevity.

Now the following proposition can be considered onshadowed terms in (29).

Both the sixth row and the sixth column with respect to thepair (fε1 , eσ1) can be eliminated by the setting (εa

3 , σa3 ) =(

(wx − θ) dx,−GA (wx − θ))

by Proposition 3.6, becausethe sixth row and the sixth column are equal to the eighthrow and the eighth column, respectively.

Remark 4.1: By making a comparison between (24) and(29) it is clear that these include the same number of Stokes-Dirac structures. Then the boundary structures correspond toeach other except for ε3 = wx dx since a I-type structuredoesn’t cause a boundary port. The essential difference ofeach representation is the I-type structures. In other wordsthe difference expresses a concept of the reduction of beammodels. Indeed (24) is simpler than (29) clearly.

C. Simplified models

First by introducing in (16) the following infinitesimalstrain assumption:

[Γ1

Γ2

]= Λ�

[1 + yx − cos θ

wx − sin θ

]≈

[yx

wx − θ

](30)

or by eliminating the effect φt in (25), we obtain theTimoshenko beam model. In the same way as the previoussection with φt = 0, we define the energy variables:

π1 = Aρ yt dx , ν1 = −yt ;π2 = Aρ wt dx , ν2 = −wt ;π3 = Iρ θt dx , ν3 = −θt ;ε2 = yx dx , σ2 = −EAyx ;ε3 = (wx− θ) dx , σ3 = −GA (wx − θ) ;ε4 = θx dx , σ4 = −EI θx .

(31)

If a buckling never arise, a longitudinal motion of thebeam (the structure corresponding to {fπ1, fε2 , eν1 , eσ2} of(29) ) can be separated as a wave equation from a transversalmotion of the beam (see [4], [7]).

Next, the Euler-Bernoulli model is obtained by assuming ashear deformation is negligible in addition to the Timoshenkobeam model. By assuming (wx − θ) → 0 and GA → ∞, weobtain

Aρ wtt + EI wxxxx = 0 . (32)

The total energy of the system is given by

H =12

∫

Z

[Aρ wt

2 + EI wxx2]dx . (33)

If the Euler-Bernoulli beam model is regarded as the reduc-tion of the Timoshenko beam model, the following energyvariables are defined by picking out from (31).

π2 = Aρ wt , ν2 = −wt ;ε4 = wxx , σ4 = −EI wxx .

(34)

But it is obvious that the simple reduction of the Hamiltonianformulation doesn’t give a proper representation from a partof the structure (29). In this case, it is known that thisrepresentation can be achieved by introducing a higher orderStokes-Dirac structure [5].

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Now let us consider another approach. The followingproperty corresponds with the integration by parts formula(4). We show the concrete calculation based on Hamilton’sprinciple for the potential energy U of the Euler-Bernoullimodel (33) as follows:

κ δU =∫

Z

wxxdwxxdx

=∫

∂Z

wxxdwx −∫

Z

wxxxdwxdx (35)

=∫

∂Z

wxxdwx −∫

∂Z

wxxxdw +∫

Z

wxxxxdwdx

where κ = 1/EI . A part of the Euler-Lagrange equation canbe obtain from the last term of the last line of (35). Indeed,if the integrals on ∂Z in (35) are vanished by setting theboundary conditions to zero, then the result E(L) ≡ 0 can begiven by δU = EI wxxxx and the variational δT = Aρ wtt

of the kinetic energy.From the last term of the second line of (35), the following

energy variables are defined using the freedom of integrationfunctional by parts.

π2 = Aρ wt , ν2 = −wt ;ε5 = wx , σ5 = EI wxxx .

(36)

Then we have[fπ2

fε5

]=

[0 Dx

Dx 0

] [eν2

eσ5

],

[fb1

eb1

]=

[eν2

−eσ5

]. (37)

Here, if we adopt the formal port Hamiltonian representa-tion of conservation laws [2], the following relation can beconsider from the first term of the second line of (35). Theenergy variables of conservation laws are defined as follows:

πc = wxx , νc = −EI wx ;εc = wx , σc = −EI wxx .

(38)

Then we have[fπc

fεc

]=

[0 Dx

Dx 0

] [eνc

eσc

],

[fbc

ebc

]=

[eνc

−eσc

]. (39)

Now, since {fbc , ebc} in (39) are fixed to zeros by theboundary conditions, actually (39) can be considered as aninternal gyrator related to each of the higher order energyvariables.

Remark 4.2: In (37) if we consider the damping injectionwt = −kwxxx to the port (fb1 , eb1) for k > 0, it is similar toshear force feedback [11]. (On the other hand, the feedbackwxt = −kwxx to (fb2 , eb2) for the standard higher orderStokes-Dirac structure [5] is corresponding to direct strainfeedback [10].) Though the originals of the direct strainfeedback and the shear force feedback hold the exponentialstability in terms of functional analysis, port Hamiltoniansystems are guaranteed for asymptotically stability only. Butthe similarity from another mathematical formalism is aninteresting result.

V. CONCLUSIONS

In this paper, we studied one of formulations of dis-tributed parameter port Hamiltonian systems. We presentedthe Stokes-Dirac structure is related to the calculus of vari-ations by using jet bundle formalism. And the equation offlexible beams is represented as the port Hamiltonian system.Finally this model is compared to one conventional modeland two reduced models.

The one-to-one correspondence between the field Euler-Lagrangian equations and the distributed parameter portHamiltonian systems became obvious lately. But we avoidedexplaining the details and we only presented the transfor-mation procedure of concrete examples. We will present infurther detail in another paper.

We assumed that the system exists on the contractibleregion. If the domain is a complex shape, we can decomposeit to some contractible regions. Then a network structure ofthese dependent regions can be considered for practical use.Some topic can be considered as future works in relation tonumerical analysis.

REFERENCES

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[2] A.J. van der Schaft and B.M. Maschke, “Hamiltonian formulation ofdistributed-parameter systems with boundary energy flow”, Journal ofGeometry and Physics, Vol.42, pp.166-194, 2002.

[3] T. Courant, “Dirac manifolds”, Trans. American Math. Soc., 319,pp.631-661, 1990.

[4] G. Golo and A.J. van der Schaft and S. Stramigioli, “Hamiltonianformulation of planar beams”, 2nd IFAC Workshop on Lagrangianand Hamiltonian Methods for Nonlinear Control, Seville, pp.169-174,2003.

[5] G. Nishida and M. Yamakita, “A Higher Order Stokes-Dirac Structurefor Distributed-Parameter Port- Hamiltonian Systems”, in Proc. 2004American Control Conference, Boston, MA, pp.5004-5009, 2004.

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[7] A. Macchelli and C. Melchiorri, “Modeling and control of the Tim-oshenko beam. The distributed port Hamiltonian approach”, SIAMJournal on Control and Optimization, 43(2), pp.743-767, 2004.

[8] A. Macchelli, A.J. van der Schaft and C. Melchiorri, “Port HamiltonianFormulation of Infinite Dimensional Systems I. Modeling”, Proc. 43rdIEEE Conference on Decision and Control, Bahamas, pp.3762-3767,2004.

[9] A. Macchelli, A.J. van der Schaft and C. Melchiorri, “Port HamiltonianFormulation of Infinite Dimensional Systems II. Boundary Controlby Interconnection”, Proc. 43rd IEEE Conference on Decision andControl, Bahamas, pp.3768-3773, 2004.

[10] Z. H. Luo, “Direct Strain Feedback Control of Flexible Robot Arms:New Theoretical and Experimental Results”, IEEE Trans. on Auto-matic Control, Vol.38, No.11, Nov., pp.1610-1622, 1993.

[11] Z. H. Luo and B. Z. Guo, “Shear Force Feedback Control of Single-Link Flexible Robot with a Revolute joint”, IEEE Trans. on AutomaticControl, Vol.42, No.1, Jan., pp.53-65, 1997.

[12] A.J. van der Schaft, L2-Gain and Passivity Techniques in NonlinearControl, 2nd revised and enlarged edition, Springer Communicationsand Control Engineering series, Springer-Verlag, London, 2000.

[13] P.J. Olver, Applications of Lie Groups to Differential Equations:Second Edition, Springer-Verlag, New York, 1993.

[14] D.J. Saunders, The Geometry of Jet Bundles, Cambridge Univ. Press,Cambridge, 1989.

[15] J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, 1967.

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