1. Introduction
The problem of transverse vibration of beams is of importance in many engineering problems. Some of the early studies were based on the Euler–Bernoulli model, which takes into account the bending moment and the lateral displacement. Later models were based on adding shear or rotary inertia effect. In [
1,
2], Timoshenko proposed taking into consideration the shear, as well as the rotation effects, which proved to be suitable for non-slender beams and high frequency vibrations. Dimplekumar et al. [
3] discuss the mathematical modeling for the mechanics of a solid using the distribution theory of Schwartz to the beam bending differential equations; the governing differential equations of a Timoshenko beam with jump discontinuities in slope, deflection, flexural stiffness and shear stiffness were obtained in this paper in the space of generalized functions. Mustafa and Messaoudi [
4] and Fatiha [
5] studied stability of the following Timoshenko system with the nonlinear frictional damping term in one equation:
where the functions
,
depend on
and model the vertical displacement of a beam and the rotation angle of a filament, respectively. The shear angle is
, and
L denotes the length of the beam. The physical parameters appearing in the system are
, the mass density per unit length,
, the polar moment of inertia of a cross-section,
E, Young’s modulus of elasticity,
I, the moment of inertia of the cross-section, and
k, the shear modulus. In this paper the beam is assumed to be uniform, that is the physical parameters
and
k are all positive constants, and
is an arbitrary nonlinear function of the damping term
.
We have solved this system by determining its symmetries and performing a Lie theoretic analysis as detailed below.
The classification of group invariant solutions of differential equations by means of the optimal systems is one of the main applications of Lie group analysis to differential equations. The main idea behind the method is discussed in detail in Ovsiannikov [
6], Ibragimov [
7], Olver [
8] and Hydon [
9]. We can always construct a family of group invariant solutions corresponding to a subgroup of a symmetry group admitted by a given differential equation. Since there is an infinite number of such subgroups, it is not possible to list all of the group invariant solutions. An effective and systematic way of classifying these group invariant solutions is to obtain optimal systems of subalgebras of the Lie symmetry algebra. This leads to non-similar invariant solutions under symmetry transformations.
Recall that one calls a list
a one-dimensional optimal system, if it satisfies the conditions: (1) completeness, i.e., any one-dimensional subalgebra is equivalent to some
; (2) inequivalence, i.e.,
and
are inequivalent for distinct
and
. In this paper, we have used all of the basic invariants to determine the conjugacy classes of one-dimensional subalgebras and at the same time shown that these representatives are comprehensive and mutually inequivalent. This is done by using the formula given in Section 3 of [
10]. This formula is more or less a direct consequence of definitions. For the convenience of the reader, a detailed explanation is given in
Section 2. The idea of using invariant functions to determine conjugacy classes is also discussed in [
11,
12].
Here is a brief outline of the paper. In
Section 2, a formula for computing invariants in the adjoint representation is given. In
Section 3, Lie symmetries and their Lie group transformations for the Timoshenko system are presented. In
Section 4, an optimal system of one-dimensional subalgebras of the corresponding Lie algebra is derived. In
Section 5, all possible invariant variables of the optimal system are presented; moreover, the corresponding reduced systems of ordinary differential equations (ODEs) are also given. As an illustration, some invariant solutions are given explicitly by solving the reduced systems of ODEs. Furthermore, all possible non-similar invariant conditions prescribed on invariant surfaces under symmetry transformations are given. A hinged-free beam has been considered in Example 1, with a constant torque control at the hinged end and a linear force control at the free end.
4. Optimal System of One-Dimensional Sub-Algebras of the Nonlinear Damped Timoshenko System
In this section, we give the one-dimensional optimal system for the algebra with basis (
9). In order to find the optimal system, one needs to classify the one-dimensional sub-algebras under the action of the adjoint representation. We follow the algorithm explained by Ibragimov [
7] and Olver [
8].
The non-zero commutators of the Lie algebra
with basis (
9) are given by:
Recall that the adjoint representation is given by:
The Lie algebra
is solvable, and the adjoint table is given in
Table 1 below:
The adjoint group is generated by
. Using the solvability of
, this group consists of the elements:
Therefore, A is given by:
Theorem 2. An optimal system of one-dimensional sub-algebras of with basis (9) is provided by the following generators: Proof. Let X and
be two elements in the Lie algebra
with basis (
9) given by
and
. For simplicity, we will write X and
as row vectors of the coefficients on the form
and
. Then, in order that X and
are in the same conjugacy class, we must have
, where A is given by (15). Therefore, the theorem is proven by solving the system:
for
in terms of
in order to get the simplest values of
.
The results are presented for different cases in a tree diagram, where each of its vertices is an invariant function, and its leaves are given completely. Moreover, one can verify that all of the parameters and γ appearing in each case are invariants. Therefore, the inequivalence and completeness conditions are satisfied.
All joint invariants are obtained by solving the following system of PDEs, which is given by using Formula (
2):
By solving this system, we obtain
, where Fis an arbitrary function of
. Hence, the basic invariants of Timoshenko System (
1) are
and
; this means that the first three vertices of the tree will be these invariants in any order. In our case, we will consider the order
.
The full details for each leaf are given as follows:
Case 1 , , : Let , , , , and to have : the conjugacy class is , with .
Case 2 , , : Let , , , , to have . The conjugacy class is , with , .
Case 3 , , : Let , , and to make : the conjugacy class is , , .
When , , , we need to solve system (18) again to see what are the invariants as well the next vertices of the tree.
Solving system (18) taking into account that gives , so we can consider to be the next vertex of the tree.
Case 4 , , , : Let , , and to make : the conjugacy class is of the form , .
Again, for the case , , , , we resolve system (18) to have , so next vertex can be :
Case 5 , , , , : Let , , and to make : the corresponding conjugacy class is , where .
Case 6 , , , , : Let , and to make : the conjugacy class is , .
Case 7 , , : Let , , , , and to make : the conjugacy class is , .
Case 8 , , : Let , , , and to make : the conjugacy class is , .
Case 9 , , : Let and , to get : the conjugacy class is of the form , .
By substituting in system (18) and solving it, we get . So the next two vertices can be and in any order. We consider the order , .
Case 10 , , , , : Let and to make : the conjugacy class is , .
In case , solving system (18) yields which implies, after considering and as vertices, that one can consider the invariant as a new vertex.
Case 11 , , , , , : Let and to make : the conjugacy class is , with .
Case 12 , , , , , : Let , to make : then the conjugacy class is , .
Case 13 , , , , : Let to have : the conjugacy class is , .
Case 14 , , , , , : Let to have : the conjugacy class is , .
When with , solving the PDEs system (18) gives . So the next invariant vertex is .
Case 15 , , , , , , : Let to have : the conjugacy class is .
Case 16 , , , , , , : the conjugacy class is . ☐