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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 infinitesimal generator generates a contraction semigroup on a Hilbert space, whose norm is directly linked to the Hamiltonian.
1. Introduction
The micro and nanoscale physical phenomena have different properties from macro-scale [Citation1–Citation3]. Carbon nanotubes (CNTs) are allotropes of carbon. They have diameters as small as 1 nm and lengths up to several centimetres. CNTs have amazing mechanical and electrical properties such as high electrical conductivity, chemical stability, high stiffness and axial strength [Citation4]. These excellent properties have led to wide practical application of CNTs in NanoElectroMechanical Systems (NEMS). Due to novel properties 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 [Citation5]. For example, Li et al. investigate dynamics and stability of transverse vibrations of nonlocal nanorods [Citation6]. Nonlocal longitudinal vibration of viscoelastic-coupled double-nanorod systems is studied by Karlicic et al. [Citation7]. Heidari investigates controllability and stability analysis of a nanorod [Citation2].
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 [Citation8,Citation9].
This information is of great interest in analysing and simulating complex network system. Over the last years, many researchers worked on port-Hamiltonian systems, extending the theory and/or solving applied control problems, see, e.g., Jeltsema and Doria-Cerezo [Citation10], Macchelli and Melchiorri [Citation11], and Ramirez et al. [Citation12]. For an overview and more details we refer the reader to [Citation9].
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 [Citation13] 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 [Citation7] the mathematical modelling of vibration in nanorods.
We consider a nanorod with length and cross-sectional area
which is depicted in .
In our case, the cross-sectional area is constant along the -coordinate, but in general it could have arbitrary shapes along this
-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
-direction. An infinitesimal element of length
is taken at a typical coordinate location
. Further, we take that the force
is the resultant of an axial stress
acting internally on
, where
is assumed to be uniform over the cross-section. The stress resultant
may vary along the length, and is also a function of time, i.e.,
. Using our assumptions, we find that
In addition, an axially distributed force 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
-direction is
where is the mass of an infinitesimal element and
is the displacement in the
-direction. Substituting
and simplifying (2) gives
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 () at all other points in the domain. The nonlocal constitutive equation for an elastic medium is as follows
where is the elastic modulus,
is the nonlocal parameter (length scales) [Citation7] and
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
, see [Citation14],
By substituting Equations (1) and (5) into (4), the stress resultant for the nonlocal theory is obtained as
where the last term, , is the strain rate in the nanorod. Finally, we consider an external force
in which the parameter is the stiffness coefficient of the viscoelastic layer and the last term represents uniform damping, see [Citation7]. 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
and next use (3), to get the following equation of motion
which is mentioned in [Citation7].
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 [Citation8], 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 our model can be written as
with ,
, and
bounded operators on the Hilbert space
,
a symmetric
matrix and
an anti-symmetric
matrix both consisting only of
,
, and
‘s, and
and
self-adjoint and non-negative.
The state that we choose is given by
Equation (3) implies
where we used (10) and (7).
Using Equations (10) and (11), the time derivative of is written as
where we used the assumption that the parameters are constants. Using Equation (6), this becomes
Using (12) and the above equality, we find that
where . We write the matrix of the right-hand side as a product,
From this we see that our model (15) can be written in the form (9) with
It is easy to see that these expressions satisfy the conditions stated below (9). If we assume that is a classical solution of (9), then
satisfies the following equality
For our model this becomes
We see that the first term represents the change of Hamiltonian () through the boundary, whereas the integral term represents internal damping.
The above power balance is very standard for port-Hamiltonian systems, see [Citation8,Citation9,Citation15]. However, there are a few differences between the form (9) with ,
,
and
given in (16)–(17) and the form studied by Jacob and Zwart [Citation8]. The most obvious one is that
is non-invertible. Moreover, our
is not invertible. First results for port-Hamiltonian systems with a non-invertible term in front of the time derivative can be found in [Citation16], but only for ordinary differential equations. In Villegas [Citation17, Chapter 6] port-Hamiltonian systems with
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
4. Existence of solutions
We study the existence of solutions under the assumption that the rod is fixed, i.e.,
As state space, we take all states with finite energy but satisfying the constraints
As inner product on , we take the inner product associated to the Hamiltonian (19), i.e.,
where the latter inner product is the standard inner product on and
is the upper four by four block of
, i.e.,
Lemma 4.1 is a closed subspace of
.
(25)
Proof. Since all physical parameters in (24) are positive, the norm associated to the inner product (23) is equivalent to the standard norm on . This directly implies that if the sequence
converges in
, then the first, third, and fourth component converge in
. Hence it remains to show that the second component converges as well. By (22) and the convergence of
, the third component of
, we have that
On the other hand, by assumption , and combining this with (25) gives that
is absolutely continuous with
and
. Using this equation once more together with the fact that
gives that
satisfies the condition of
, and thus
is closed subspace of
. ⁏
From this lemma, we find that with the inner product (23) is a Hilbert space. Next, we define the (candidate) infinitesimal generator associated with our p.d.e. We refer to Chapter 5 and 6 of [Citation8] for more detail on semigroup theory.
For we define
where
Since is defined implicitly, it is important to know that it is well defined, i.e., the outcome
is uniquely defined. This is shown next.
Lemma 4.2 The operator with domain
is well-defined.
Proof. So what we have to show is that the needed to define
is unique. Let us assume that there are two, i.e.,
are such that
and
Since
is linear, we see that this implies that
. So if we can show that for an arbitrary
the condition
implies that
, then we have shown that
is well-defined.
Assume that there exists is such that
. Using (26) and (22) this implies that
,
and
. Since
this implies that
. ⁏
Theorem 4.3. The operator defined in (26) and (27) generates a contraction semigroup on
.
(28)
Proof. Using Lumer-Phillips Theorem, see e.g. [Citation18, Theorem II.3.15] or [Citation8, Theorem 6.1.7], we have to show two properties of , namely for all
and for all there exists an
such that
We begin by showing (28).
Using the definition of and the inner product on
where the last equality is in . Since
, we have that
see (16) and (17). Since and
, we find that
where we have used the boundary conditions of . So we see that (28) holds. Next, we show (29).
Let be given. Then, we have to find
and
such that
where and
. Furthermore, we have the conditions, see (22)
By considering Equation (32) and Equation (33), we have
Since , we find
From (30) and (31) it follows that
Combining this with (34) and using (36) we find the following differential equation in and
where are positive constants, and
are a linear combination of
. The solution of (38) is given by
where and we used the first boundary condition of (34). To satisfy the second boundary condition, we have to solve
Since and
are positive constants, this is solvable and so the
and
are given by (39) with
Note that these functions lie in and
satisfies the boundary conditions of (34). Given these solutions, the functions
and
follows from (36) and (37), respectively. Equation (30) gives
. Summarizing we see that
is surjective, and so
generates a contraction semigroup on
. ⁏
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 is zero at
and
for all time, see also (21). Using Equation (3) and (7) this implies that
We use these boundary conditions to solve (see (6))
for . We find, see e.g. [Citation19, Section 7.5]
with (Green’s function)
where .
Choose now the state
Using (3), (6) and (7) we find (for )
where is the mapping defined by (41) and (42), i.e.
Hence our model can be written as
This we can write in the port-Hamiltonian format (9) with the identity,
and
Note that since for all
,
is a self-adjoint bounded, strictly positive operator. Using this and the fact that the physical parameters are positive, we find that
is a coercive operator on
. As in [Citation8, Chapter 7] we choose as our state space
with inner product
where the latter is the standard inner product on .
For we define
with domain
From Lemma 7.2.3 and Theorem 7.2.4 of [Citation8] the following theorem follows.
Theorem 5.1. For , the operator
with domain
as defined in (49) and (50) generates a contraction semigroup on the state space
. If
, then
generates a unitary group on
.
Note that we have written the domain of in the standard form, verify e.g. [Citation8, Equation (7.22)] or [Citation15]. However, since
maps
-functions onto
-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 . Using (6) we see that for
our model can be written as
Since are positive constants, and
is a positive operator, the operator
is positive. Using Theorem 5.1 and Theorem 2.2 of [Citation20], we see that under the same boundary conditions as formulated in (50) this model generates a contraction semigroup on the state space
.
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. [Citation21]. 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
whereas the Hamiltonian associated to the formulation in Section 5 equals
Although they have the same unit [J] and are equal in the first two terms, for they differ in the last term(s). Best to see this is by noticing that there is an
in last term of
, whereas this missing in
. Since the last term in
comes from (6), in which the
is absent, we conclude that
and
measure different quantities for
.
When there is no damping, then both Hamiltonians are constant. Thus, along solutions, we will have that
where 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 [Citation8] 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
where . Since this is a Sturm-Liouville operator, existence, uniqueness, and other properties of the differential equation
are well known, see, e.g. [Citation19, Section 7.5]. For instance, the solution map will again be a strictly positive operator, and so Equation (45) still hold (with another ). 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 throughout this paper. For many port-Hamiltonian systems the step from constant to spatial varying (physical) parameters is easy, see the examples in, e.g. [Citation8]. 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,
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., , 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 [Citation18, 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 standard port-Hamiltonian systems, all boundary conditions could somehow be treated in the same theorem. Here the situation is different. For instance, when the boundary conditions are replaced by no force at the boundary, i.e.,
, 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 boundary conditions on
. In the formulation of Section 5, the expression of
changes, since the differential equation for
has to be solved under the condition
.
Another topic which we like to study in the future is the port-Hamiltonian formulation 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.
Nomenclature
Table
Acknowledgments
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.
Disclosure statement
No potential conflict of interest was reported by the authors.
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