ARTICLE TEMPLATE
Port-Hamiltonian modelling of nonlocal longitudinal vibrations in a viscoelastic nanorod
H. Heidaria and H. Zwartb
aSchool of Mathematics and Computer Sciences, Damghan University, Damghan, P.O. Box
36715-364, Iran;bDepartment of Applied Mathematics, University of Twente, P.O. Box 217,
7500 AE Enschede, The Netherlands and Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
ARTICLE HISTORY Compiled August 21, 2019 ABSTRACT
Analysis of nonlocal axial vibration in a nanorod is a crucial subject in science and engineering because of its wide applications in nanoelectromechanical systems. The aim of this paper is to show how these vibrations can be modelled within the framework of port-Hamiltonian systems. It turns out that two port-Hamiltonian descriptions in physical variables are possible. The first one is in descriptor form, whereas the second one has a non-local Hamiltonian density. In addition, it is shown that under appropriate boundary conditions these models possess a unique solution which is non-increasing in the corresponding “energy”, i.e., the associated infinites-imal generator generates a contraction semigroup on a Hilbert space, whose norm is directly linked to the Hamiltonian.
KEYWORDS
Descriptor port-Hamiltonian; viscoelastic; nonlocal vibration
1. Introduction
The micro and nanoscale physical phenomena have different properties from macro-scale [1–3]. Carbon nanotubes (CNTs) are allotropes of carbon. They have diameters as small as 1 nm and lengths up to several centimeters. CNTs have amazing mechanical and electrical properties such as high electrical conductivity, chemical stability, high stiffness and axial strength [4]. These excellent properties have led to wide practical application of CNTs in NanoElectroMechanical Systems (NEMS). Due to novel prop-erties and vast applications of CNTs in industry, there is a lot of research on static, buckling and vibration analysis of CNTs using the local and the nonlocal models [5]. For example, Li et al. investigate dynamics and stability of transverse vibrations of nonlocal nanorods [6]. Nonlocal longitudinal vibration of viscoelastic coupled double-nanorod systems is studied by Karlicic et al, [7]. Heidari investigates controllability and stability analysis of a nanorod [2].
Many electrical, mechanical and electromechanical systems can be suitably modelled in port-Hamiltonian (pH) framework. This modelling exposes fruitful information on physical characteristics of the system such as the relation between the energy storage,
dissipation, and interconnection structure [8, 9]. This information is of great interest for analyzing and simulating complex network system. Over the last years many re-searchers worked on port-Hamiltonian systems, extending the theory and/or solving applied control problems, see e.g. Jeltsema and Doria-Cerezo [10], Macchelli and Mel-chiorri [11], and Ramirez et al. [12]. For an overview and more details we refer the reader to [9].
To the best of our knowledge, in spite of a large amount of research on vibration of nanorod and pH systems, there is only little research on pH modelling of vibration of nanorods. In [13] we studied the problem, but there a pH formulation was found using non-physical variables. Therefore, pH modelling of vibration of an elastic nanorod using physical variables is considered in this paper. The rest of paper is organized as follows. In Section 2, a short review on nonlocal theory and governing equations are given. Section 3 presents the first port-Hamiltonian formulation. This is in descriptor form, the existence of its solutions is done in Section 4. In Section 5, a second pH formulation is given. The relation between the two formulations is discussed in Section 6. We end the paper with the conclusions and discussion on future works.
2. Model formulation
In this section, we recall from [7] the mathematical modelling of vibration in nanorods. We consider a nanorod with length ` and cross-sectional area A which is depicted in Fig. 1.
In our case, the cross-sectional area is constant along the x-coordinate, but in general it could have arbitrary shapes along this x-coordinate. We assume that the material of a nanorod is elastic and homogeneous. Also, we consider the free longitudinal vibration of the nanorod in the x-direction. An infinitesimal element of length dx is taken at a typical coordinate location x. Further, we take that the force N is the resultant of an axial stress σxx acting internally on A, where σxx is assumed to be uniform over the
cross-section. The stress resultant N may vary along the length, and is also a function of time, i.e., N = N (x, t). Using our assumptions, we find that
N (x, t) = Z
A
σxx(x, t)dA = σxx(x, t)A. (1)
In addition, an axially distributed force eF is assumed, having dimensions of force per unit length, which results from external sources, either internally or externally
applied. The equilibrium of forces in the x-direction is −N + N +∂N ∂xdx − ˜F dx = ∂ 2w ∂t2 dm, (2)
where dm = ρAdx is the mass of an infinitesimal element and w is the displacement in the x-direction. Substituting dm = ρAdx and simplifying (2) gives
∂N
∂x = ˜F + ρA ∂2w
∂t2 . (3)
Next we model the stress-strain relation. Based on nonlocal Eringen’s theory, it is assumed that the stress at a point is related to the strain (xx) at all other points in
the domain. The nonlocal constitutive equation for an elastic medium is as follows
σxx− µ ∂2σxx ∂x2 = E xx+ τd ∂ ∂txx , (4)
where E is the elastic modulus, µ is the nonlocal parameter (length scales) [7] and τd is the viscous damping coefficient of the nanorod. We remark that we assume all
parameters to be constant. We consider the following standard relation between the strain and w, see [14],
xx =
∂w
∂x. (5)
By substituting equations (1) and (5) into (4), the stress resultant for the nonlocal theory is obtained as N − µ∂ 2N ∂x2 = EA ∂w ∂x + τd ∂2w ∂x∂t , (6)
where the last term, ∂x∂t∂2w, is the strain rate in the nanorod. Finally, we consider an external force
˜
F = a2w + b2∂w
∂t (7)
in which the parameter a is the stiffness coefficient of the viscoelastic layer and the last term represents uniform damping, see [7]. In the following sections we show that the equations (3), (6), and (7) can be written in a port-Hamiltonian form. In some papers, one can find one (scalar) equation describing the motion. To write the equations (3), (6) into one equation, we have first differentiate equation (6) with respect to x and next use (3), to get the following equation of motion
˜ F + ρA∂ 2w ∂t2 − µ ∂2 ∂x2 ˜ F + ρA∂ 2w ∂t2 = EA ∂ 2w ∂x2 + τd ∂3w ∂x2∂t , (8) which is mentioned in [7].
3. Descriptor port-Hamiltonian formulation
As many physical models, our model can also be written in a port-Hamiltonian form. However, it is not the standard formal as for instance studied in [8], but there will appear a non-invertible operator in front of the derivative of the state, i.e, it is of descriptor form. Hence we show that for a suitable state z our model can be written as
Edz dt = P1
∂
∂x(Hez) + P0(Hez) − R0(Hez) (9) with E, He, and R0 bounded operators on the Hilbert space L2((0, `); Rn), P1 a
sym-metric n × n matrix and P0 an anti-symmetric n × n matrix both consisting only of
−1, 0, and 1’s, and E∗He and R0+ R∗0 self-adjoint and non-negative.
The state z that we choose is given by
z = w ρA∂w∂t µρA∂t∂x∂2w ∂w ∂x N . (10) Equation (3) implies ˙ z2= ρA ∂2w ∂t2 = ∂N ∂x − ˜F = ∂N ∂x − a 2w − b2∂w ∂t (11) = ∂ ∂xz5− a 2z 1− b2 ρAz2, (12)
where we used (10) and (7).
Using equations (10) and (11), the time derivative of z3 is written as
˙ z3 = µρA ∂3w ∂t2∂x = µ ∂ ∂x( ˙z2) = µ ∂2N ∂x2 − a 2µ∂w ∂x − b 2µ∂2w ∂t∂x, (13)
where we used the assumption that the parameters are constants. Using equation (6), this becomes ˙ z3 = N − (EA + a2µ) ∂w ∂x − (EAτd+ b 2µ)∂2w ∂t∂x = z5− (EA + a2µ)z4− EAτd+ b2µ µρA z3. (14)
Using (12) and the above equality, we find that
E ˙z = 0 ρA1 0 0 0 −a2 −b2 ρA 0 0 ∂ ∂x 0 0 −EAτd−µb2
µρA −EA − µa
2 1 0 0 ρAµ1 0 0 0 ∂x∂ ρA1 −ρAµ1 0 0 z, (15)
where E = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 !
. We write the matrix of the right-hand side as a product,
0 1 0 0 0 −1 −b2 0 0 ∂ ∂x 0 0 −EAτd− µb2 −1 1 0 0 1 0 0 0 ∂x∂ −1 0 0 a2 0 0 0 0 0 ρA1 0 0 0 0 0 µρA1 0 0 0 0 0 EA + µa2 0 0 0 0 0 1 .
From this we see that our model (15) can be written in the form (9) with
P1 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 , He= a2 0 0 0 0 0 ρA1 0 0 0 0 0 µρA1 0 0 0 0 0 EA + µa2 0 0 0 0 0 1 , (16) P0 = 0 1 0 0 0 −1 0 0 0 0 0 0 0 −1 1 0 0 1 0 0 0 0 −1 0 0 , and R0= 0 0 0 0 0 0 b2 0 0 0 0 0 EAτd+ µb2 0 0 0 0 0 0 0 0 0 0 0 0 . (17)
It is easy to see that these expressions satisfy the conditions stated below (9). If we assume that z is a classical solution of (9), then He(t) := 12
R`
0 z(t) T
E∗Hez(t)dx satisfies
the following equality ˙ He(t) = 1 2(Hez) TP 1Hez ` 0− 1 2 Z ` 0 (Hez)T(R0+ R∗0)Hezdx. (18)
For our model this becomes He(t) = 1 2 Z ` 0 a2w(x, t)2+ ρA∂w ∂t(x, t) 2+ µρA∂2w ∂t∂x(x, t) 2+ (EA + µa2)∂w ∂x(x, t) 2dx, (19) ˙ He(t) = 1 2 ∂w ∂t(x, t)N (x, t) ` 0 − Z ` 0 b2∂w ∂t(x, t) 2+ τ dEA + µb2 ∂2w ∂t∂x(x, t) 2dx. (20)
We see that the first term represents the change of Hamiltonian (He(t)) through the
boundary, whereas the integral term represents internal damping.
The above power balance is very standard for port-Hamiltonian systems, see [8, 9, 15]. However, there are a few differences between the form (9) with E, P1, P0 and R0
given in (16)–(17) and the form studied by Jacob and Zwart [8]. The most obvious one is that E is non-invertible. Moreover, our P1 is not invertible. First results for
found in [16], but only for ordinary differential equations. In Villegas [17, Chapter 6] port-Hamiltonian systems with P1 non-invertible is treated. We will not follows this,
but take a more direct approach. So, in the next section we show that the model (15) possesses a unique solution which is non-increasing in the Hamiltonian He(t).
4. Existence of solutions
We study the existence of solutions under the assumption that the rod is fixed, i.e., ∂w
∂t(0, t) = ∂w
∂t(`, t) = 0. (21)
As state space we take all states with finite energy but satisfying the constraints Zt:= zt= z1 z2 z3 z4 ∈ L2((0, `); R4) | z2 is absolutely continuous (22) dz2 dx ∈ L2(0, `), z2(0) = z2(`) = 0, and µ dz2 dx = z3 .
As inner product on Zt, we take the inner product associated to the Hamiltonian (19),
i.e.,
hzt, ˜zti1 := hzt, He,1z˜ti, (23)
where the latter inner product is the standard inner product on L2((0, `); R4) and H e,1
is the upper four by four block of He, i.e., E∗He
He,1= a2 0 0 0 0 ρA1 0 0 0 0 µρA1 0 0 0 0 EA + µa2 . (24)
Lemma 4.1. Zt is a closed subspace of L2((0, `); R4).
Proof. Since all physical parameters in (24) are positive, the norm associated to the inner product (23) is equivalent to the standard norm on L2((0, `); R4). This directly implies that if the sequence {zt,n, n ∈ N} converges in Zt, then the first, third, and
fourth component converge in L2((0, `); R). Hence it remains to show that the second component converges as well. By (22) and the convergence of z3,n, the third component
of zt,n, we have that z2,n(x) = Z x 0 1 µz3,n(τ )dτ → Z x 0 1 µz3(τ )dτ, x ∈ [0, `]. (25) On the other hand, by assumption z2,n→ z2, and combining this with (25) gives that
z2 is absolutely continuous with z2(0) = 0 and µdzdx2 = z3. Using this equation once
more together with the fact that z2,n(`) = 0 gives that z2 satisfies the condition of Zt,
From this lemma we find that Zt with the inner product (23) is a Hilbert space.
Next we define the (candidate) infinitesimal generator associated to our p.d.e. We refer to Chapter 5 and 6 of [8] for more detail on semigroup theory.
For zt∈ D(A) we define
Azt= 0 ρA1 0 0 0 −a2 −b2 ρA 0 0 d dx 0 0 −EAτd−µb2
µρA −EA − µa
2 1 0 0 ρAµ1 0 0 zt z5 =: A1 zt z5 , (26) where
D(A) = {zt∈ Zt| there exists z5∈ H1(0, `) s.t. A1
zt
z5
∈ Zt}. (27)
Since A is defined implicitly, it is important to know that it is well-defined, i.e., the outcome Aztis uniquely defined. This is shown next.
Lemma 4.2. The operator A with domain D(A) is well-defined.
Proof. So what we have to show is that the z5 needed to define A is unique. Let us
assume that there are two, i.e., z5, ˜z5 ∈ H1(0, `) are such that A1(zz5t) and A1 zt ˜
z5 ∈ Zt.
Since A1 is linear, we see that this implies that A1 z5−˜0z5 ∈ Zt. So if we can show
that for an arbitrary z5 ∈ H1(0, `) the condition A1 z05 ∈ Zt implies that z5 = 0,
then we have shown that A is well-defined.
Assume that there exists z5 ∈ H1(0, `) is such that A1 z05 ∈ Zt. Using (26) and
(22) this implies that dz5 dx ∈ H 1(0, `), dz5 dx(0) = dz5 dx(`) = 0 and µ d2z 5 dx2 = z5. Since µ > 0
this implies that z5 = 0.
Theorem 4.3. The operator A defined in (26) and (27) generates a contraction semi-group on Zt.
Proof. Using Lumer-Phillips Theorem, see e.g. [18, Theorem II.3.15] or [8, Theorem 6.1.7], we have to show two properties of A, namely for all zt∈ D(A)
hAzt, zti1+ hzt, Azti1 ≤ 0, (28)
and for all g ∈ Zt there exists an f ∈ D(A) such that
(I − A)f = g. (29)
Using the definition of A and the inner product on Zt
hAzt, zti1+hzt, Azti1= hAzt, He,1zti + hHe,1zt, Azti
= hA1(zz5t) , He,1zti + hHe,1zt, A1( zt z5)i = h A1(zzt5) 0 , He(zzt5)i + hHe( zt z5) , A1(zzt5) 0 i,
where the last equality is in L2(0, `); R5). Since A1(zzt5) ∈ Z
t, we have that A1(zzt5) 0 = 0 ρA1 0 0 0 −a2 −b2 ρA 0 0 d dx 0 0 −EAτd−µb2
µρA −EA − µa
2 1 0 0 ρAµ1 0 0 0 dxd ρA1 − 1 ρAµ 0 0 zt z5 = 0 1 0 0 0 −1 −b2 0 0 d dx 0 0 −EAτd− µb2 −1 1 0 0 1 0 0 0 dxd −1 0 0 He zt z5 = P1 d dx + P0− R0 He zt z5 ,
see (16) and (17). Since P0+ P0T = 0 and R0+ R∗0 ≥ 0, we find that
hAzt, zti1+hzt, Azti1 ≤ hP1 d dxHe( zt z5) , He( zt z5)i + hHe( zt z5) , P1 d dxHe( zt z5)i = 1 ρAz2z5 ` 0 = 0,
where we have used the boundary conditions of z2. So we see that (28) holds. Next we
show (29). Let g = g1 g2 g3 g4
∈ Zt be given. Then we have to find ft = f1 f2 f3 f4 ! ∈ D(A) and f5∈ H1(0, `) such that f1− 1 ρAf2 = g1, (30) a2f1+ s2f2− df5 dx = g2, (31) s3f3+ (EA + µa2)f4− f5 = g3, (32) − 1 µρAf3+ f4 = g4, , (33)
where s2 = 1 + b2 and s3 = 1 +EAτd+µb 2
(22)
µdf2
dx = f3, f2(0) = 0 = f2(`). (34)
By considering equation (32) and equation (33), we have s3 EA + µa2 − 1 µρA 1 f3 f4 = f5+ g3 g4 . (35) Since s3+EA+µa 2 µρA 6= 0, we find f3 = µρA µρAs3+ EA + µa2 f5+ g3− (EA + µa2)g4 (36) f4 = 1 µρAs3+ EA + µa2 [f5+ g3+ µρAs3g4] . (37)
From (30) and (31) it follows that
(a 2 ρA+ s2)f2− df5 dx = −a 2g 1+ g2.
Combining this with (34) and using (36) we find the following differential equation in f2 and f5 d dx f2 f5 = 0 a12 a21 0 f2 f5 + ˜ g1 ˜ g2 , (38)
where a12, a21 are positive constants, and ˜g1, ˜g2 are a linear combination of g1, · · · , g5.
The solution of (38) is given by f2(x) f5(x) = cosh(λx) qa12 a21 sinh(λx) qa 21 a12 sinh(λx) cosh(λx) 0 f5(0) + Z x 0 cosh(λ(x − τ )) qa12 a21 sinh(λ(x − τ )) q a21 a12 sinh(λ(x − τ )) cosh(λ(x − τ )) ˜ g1(τ ) ˜ g2(τ ) dτ, (39)
where λ =√a12a21 and we used the first boundary condition of (34). To satisfy the
second boundary condition, we have to solve 0 = f2(`) = r a12 a21 sinh(λ`)f5(0)+ Z ` 0 cosh(λ(` − τ ))˜g1(τ ) + r a12 a21 sinh(λ(` − τ ))˜g2(τ ) dτ.
(39) with f5(0) = −1 sinh(λ`) Z ` 0 r a21 a12 cosh(λ(` − τ ))˜g1(τ ) + sinh(λ(` − τ ))˜g2(τ ) dτ.
Note that these functions lie in H1(0, `) and f2 satisfies the boundary conditions
of (34). Given these solutions, the functions f3 and f4 follows from (36) and (37),
respectively. Equation (30) gives f1. Summarizing we see that I − A is surjective, and
so A generates a contraction semigroup on Zt.
5. Second Hamiltonian formulation
In this section we show that there is a second port-Hamiltonian formulation for the nanorod. Therefore we use the boundary conditions already in the formulation. So we assume that w is zero at x = 0 and x = ` for all time, see also (21). Using equation (3) and (7) this implies that
∂N
∂x(0, t) = 0, ∂N
∂x(`, t) = 0, for all t ≥ 0. (40) We use these boundary conditions to solve (see (6))
N − µ∂
2N
∂x2 = f,
for f ∈ L2(0, `). We find, see e.g. [19, Section 7.5]
N (x) = Z `
0
g(x, ζ)f (ζ)dζ (41)
with (Green’s function)
g(x, ζ) = γ sinh(γ`) ( cosh(γx) cosh(γ(ζ − `)) x < ζ cosh(γ(x − `)) cosh(γζ) x > ζ , (42) where γ2 = µ−1.
Choose now the state
z = w ρA∂w∂t ∂w ∂x . (43)
Using (3), (6) and (7) we find (for τd= 0)
˙ z = 1 ρAz2 a2z1− b 2 ρAz2+ ∂N ∂x ∂2w ∂x∂t = 1 ρAz2 a2z1− b 2 ρAz2+ ∂ ∂xG(EA ∂w ∂x) ∂ ∂x h 1 ρAz2 i ,
where G is the mapping defined by (41) and (42), i.e.
(G(f )) (x) = Z `
0
g(x, ζ)f (ζ)dζ. (44)
Hence our model can be written as
˙ z(t) = 0 1 0 −1 −b2 ∂ ∂x 0 ∂x∂ 0 a2 0 0 0 ρA1 0 0 0 EA · G z(t) . (45)
This we can write in the port-Hamiltonian format (9) with E the identity,
P1 = 0 0 0 0 0 1 0 1 0 , P0 = 0 1 0 −1 0 0 0 0 0 , R0= 0 0 0 0 b2 0 0 0 0 , (46) and Hz = a2 0 0 0 ρA1 0 0 0 EA · G z. (47)
Note that since g(x, ζ) = g(ζ, x) > 0 for all x, ζ ∈ [0, `], G is a self-adjoint bounded, strictly positive operator. Using this and the fact that the physical parameters are positive, we find that H is a coercive operator on L2((0, `); R3). As in [8, Chapter 7]
we choose as our state space Z = L2((0, `); R3) with inner product hf, giZ =
1
2hf, H(g)i, (48)
where the latter is the standard inner product on L2((0, `); R3). For z ∈ D( ¯A) we define ¯ Az = 0 1 0 −1 −b2 ∂ ∂x 0 ∂x∂ 0 a2 0 0 0 ρA1 0 0 0 EA · G z (49) with domain D( ¯A) = {z ∈ Z | z2∈ H1(0, `), z2(0) = z2(`) = 0, G(z3) ∈ H1(0, `)}. (50)
From Lemma 7.2.3 and Theorem 7.2.4 of [8] the following theorem follows.
Theorem 5.1. For b ∈ R, the operator ¯A with domain D( ¯A) as defined in (49) and (50) generates a contraction semigroup on the state space Z. If b = 0, then ¯A generates a unitary group on Z.
Note that we have written the domain of ¯A in the standard form, verify e.g. [8, equation (7.22)] or [15]. However, since G maps L2-functions onto H2-functions, the last condition in (50) is always satisfied, and thus could be removed.
In the formulation (45), and thus Theorem 5.1, we have assumed that τd= 0. Using
(6) we see that for τd6= 0 our model can be written as
˙ z(t) = [P1 ∂ ∂x + P0− R0] (Hz(t)) − 0 ∂ ∂x 0 G τd ρA 0 − ∂ ∂x 0 (Hz(t)) =: [P1 ∂ ∂x + P0− R0] (Hz(t)) − GRSG ∗ R(Hz(t)) . (51)
Since τd, ρ, A are positive constants, and G is a positive operator, the operator S is
positive. Using Theorem 5.1 and Theorem 2.2 of [20], we see that under the same boundary conditions as formulated in (50) this model generates a contraction semi-group on the state space Z.
6. Relation between the two formulations
In Sections 3 and 5 we have shown that the model of the nanorod as presented in Section 2 allows for two port-Hamiltonian formulations. These formulations are both leading to a well-posed differential equation, and so it is only natural to ask for the relation between these two. Let us begin by stating that it is not exceptional to have more than one Hamiltonian, see e.g. [21]. In the study of partial differential equations, the knowledge of conserved quantities, e.g. Hamiltonian is very useful for gaining insight in the system. Thus knowing more than one Hamiltonian is seen as a positive fact.
For the formulation found in Section 3 the Hamiltonian is given by
He(zt) = 1 2 Z ` 0 a2w(x)2+ ρA∂w ∂t(x) 2+ µρA∂2w ∂t∂x(x) 2+ (EA + µa2)∂w ∂x(x) 2dx,
whereas the Hamiltonian associated to the formulation in Section 5 equals
H2(z) = 1 2 Z ` 0 a2w(x)2+ ρA∂w ∂t(x) 2dx+ 1 2EA Z ` 0 Z ` 0 ∂w ∂x(x)g(x, ζ) ∂w ∂x(ζ)dζdx.
Although they have the same unit [J] and are equal in the first two terms, for µ 6= 0 they differ in the last term(s). Best to see this is by noticing that there is an a2 in last term of He, whereas this missing in H2. Since the last term in H2 comes from (6), in
which the a2 is absent, we conclude that H
e and H2 measure different quantities for
µ 6= 0.
solu-tions, we will have that Z ` 0 µρA∂ 2w ∂t∂x(x, t) 2+(EA + µa2)∂w ∂x(x, t) 2dx− EA Z ` 0 Z ` 0 ∂w ∂x(x, t)g(x, ζ) ∂w ∂x(ζ, t)dζdx = c,
where c is a constant, only depending on the initial condition. As said above this relation does not follow from an equality like (6), but is a property of the complete model.
In [8] there is no example with two Hamiltonians, and so it surprising that the model of the nanorod has two. Looking at the derivation of the model once more, we notice that the first model cannot be derived when the parameters are spatially depending, see the third equality in (13). The second model has a natural extension to spatially depending coefficients by replacing the left-hand side of (6) by
N − ∂ ∂x µ∂N ∂x ,
where µ = µ(x) > 0. Since this is a Sturm-Liouville operator, existence, uniqueness, and other properties of the differential equation
N (x) − ∂ ∂x µ(x)∂N ∂x (x) = f (x)
are well-known, see e.g. [19, Section 7.5]. For instance, the solution map will again be a strictly positive operator, and so equation (45) still hold (with another G). So we feel that the bi-Hamiltonian property only holds in the constant parameter case.
7. Conclusions and further work
Concluding we see that we have derived two different port-Hamiltonian formulations corresponding to the same differential equation. Since the original model was build under the assumption of constant parameters, we have kept this assumption through-out this paper. For many port-Hamiltonian systems the step from constant to spatial varying (physical) parameters is easy, see the examples in e.g. [8]. However, for the model of the nanorod this is less obvious. For the model derived in Section 5 this is possible if one replaces the left-hand side of (6) by, see also the discussion in the previous section, N − ∂ ∂x µ∂N ∂x .
For the model derived Section 3 this is much less clear. However, this should only be done, when the correct nanorod model for spatial dependent coefficients has been derived.
If damping is present, i.e., b > 0, then the time-derivative of both Hamiltonians is non-positive. We believe that in this case both semigroups are strongly stable, i.e, the solutions converge to zero as time goes to infinity. A possible proof could be to apply
[18, Corollary V.2.22]. To check if the system is exponentially stable, the eigenvalues need to be calculated/estimated.
We have only studied the nanorod under one set of boundary conditions. For stan-dard port-Hamiltonian systems all boundary conditions could somehow be treated in the same theorem. Here the situation is different. For instance, when the bound-ary conditions w(0, t) = w(`, t) = 0 are replaced by no force at the boundbound-ary, i.e., N (0, t) = N (`, t) = 0, then for the formulation in Section 4 the boundary conditions must be removed from the state space, see (22), and enter the domain (27) as bound-ary conditions on z5. In the formulation of Section 5, the expression of g changes, since
the differential equation for N has to be solved under the condition N (0) = N (`) = 0. Another topic which we like to study in the future is the port-Hamiltonian formu-lation when one of the boundary conditions is non-zero, i.e., for instance when there is a control at the boundary. Since we had to put the boundary conditions into the state space (Section 3) or use it to reformulate our problem (Section 5), this problem is non-trivial. It is by no means clear that it will lead to a boundary control system, like standard port-Hamiltonian p.d.e.’s do.
Acknowledgements
We want to express our thanks to Serge Nicaise and Marius Tucsnak for their useful comments which really helped our research on this problem further.
References
[1] H. Heidari, H. Zwart, and A. Malek, Analysis of the three dimensional heat conduc-tion model in nano- or micro-scale, Proceedings of the 19th Internaconduc-tional Symposium on Mathematical Theory of Networks and Systems (MTNS) (2010), pp. 1877–1882.
[2] H. Heidari, Dynamical analysis of an axially vibrating nanorod, Int. J. Appl. Math. 29 (2016), pp. 263–270.
[3] A. Malek, H. Heidari, and M. Vali, Artificial magnetic nano-swimmer in drug delivery, 22nd Iranian Conference on Biomedical Engineering (ICBME) (2015), pp. 331–336. [4] V.N. Popov, Carbon nanotubes: Properties and application, Mater. Sci. Eng. 43 (2004),
pp. 61–102.
[5] M. Simsek, Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory, Physica E 43 (2010), pp. 182–191. [6] C. Li, C.W. Lim, and J.L. Yu, Dynamics and stability of transverse vibrations of nonlocal
nanobeams with a variable axial load, Smart Mater. Struct 20 (2011), pp. 15–23.
[7] D. Karlicic, M. Cajic, T. Murmu, and S. Adhikari, Nonlocal longitudinal vibration of viscoelastic coupled double-nanorod systems, Eur. J. Mech. A-Solid 49 (2015), pp. 183– 196.
[8] B. Jacob and H. Zwart, Linear port-Hamiltonian Systems on Infinite-Dimensional Spaces, Vol. 233, Birkh¨auser Basel, 2012.
[9] A. van der Schaft and D. Jeltsema, Port-Hamiltonian Systems Theory: An Introductory Overview, Now Publishers Incorporated, Delft, The Netherlands, 2014.
[10] D. Jeltsema and A. Doria-Cerezo, Port-Hamiltonian formulation of systems with memory, Proc IEEE 100 (2012), pp. 1928–1937.
[11] A. Macchelli and C. Melchiorri, Modeling and control of the Timoshenko beam. The dis-tributed port Hamiltonian approach, SIAM J. Control Optim. 43 (2004), pp. 743–767. [12] H. Ramirez, B. Maschke, and D. Sbarbaro, Irreversible port-Hamiltonian systems: A
gen-eral formulation of irreversible processes with application to the CSTR, Chem. Eng. Sci. 89 (2013), pp. 223–234.
[13] H. Heidari and H. Zwart, Port-Hamiltonian formulation of nonlocal longitudinal vibration in nanorod, Submitted (2018).
[14] S. Narendar and S. Gopalakrishnan, Axial wave propagation in coupled nanorod system with nonlocal small scale effects, Compos.: Part B 42 (2010), pp. 2013–2023.
[15] Y. Le Gorrec, H. Zwart, and B. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators, SIAM J. Control Optim. 44 (2005), pp. 1864–1892.
[16] C. Beattie, V. Mehrmann, H. Xu, and H. Zwart, Port-Hamiltonian descriptor system, Math. Control Signals Syst. 30 (2018).
[17] J.A. Villegas, A port-Hamiltonian Approach to Distributed Parameter Systems, Ph.d. thesis, University of Twente, 2007.
[18] K.J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics Vol. 194, Springer-Verlag, New York, 2000, with con-tributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
[19] A.W. Naylor and G.R. Sell, Linear Operator Theory in Engineering and Science, 2nd ed., Applied Mathematical Sciences Vol. 40, Springer-Verlag, New York-Berlin, 1982.
[20] H. Zwart, Y. Le Gorrec, and B. Maschke, Building systems from simple hyperbolic ones, Syst. Control Lett. 91 (2016), pp. 1–6.
[21] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), pp. 1156–1162, Available at https://doi.org/10.1063/1.523777.
Nomenclature
Symbol Meaning Unit
A cross sectional area m2
a2 stiffness coefficient of the light viscoelastic layer g m s−2 b2 damping coefficient of the light viscoelastic layer g s−1
E elastic modulus g m−1s−2
xx strain
-˜
F axially distributed force g s−2
µ nonlocal parameter m2
N resultant force of axial stress g m s−2
ρ mass density g m−3
σxx axial stress g m−1s−2
τd viscous damping s
