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Applications of Noether`s Theorem to the Equations of Motion of Inclined Sagged Cables



W. Chatanin, A. Loutsiouk and S. Chucheepsakul
 
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ABSTRACT

Noether`s theorem is the principal systematic procedure for finding conservation laws for the complicated systems of differential equations that derived by the variational method. The non-linear equations of motion of inclined sagged cables are studied by Lie group method. The infinitesimal generators, which are the main tool for this theory, are calculated. After applying the infinitesimal criterion of invariance to linear combination of these generators, the family of all variational symmetries are derived and then used to find conservation laws for the equations of motion of inclined sagged cables by using the Noether`s theorem.

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  How to cite this article:

W. Chatanin, A. Loutsiouk and S. Chucheepsakul, 2008. Applications of Noether`s Theorem to the Equations of Motion of Inclined Sagged Cables. Journal of Applied Sciences, 8: 3284-3288.

DOI: 10.3923/jas.2008.3284.3288

URL: https://scialert.net/abstract/?doi=jas.2008.3284.3288
 

INTRODUCTION

In the study of differential equations, conservation laws have many significant uses. An important problem is how to find conservation laws for given differential equations. Emmy Noether, 1918 was the first to combine the methods of variational calculus with the theory of Lie groups and to formulate a general approach for constructing conservation laws for Euler-Lagrange equations when their variational symmetries are known. "Noether proved the remarkable result that for systems arising from a variational principle, every conservation law of the system comes from a corresponding symmetry property. For example, invariance of a variational principle under a group of time translations implies the conservation of energy for the solutions of the associated Euler-Lagrange equations and invariance under a group of spatial translations implies conservation of momentum (Olver, 1993). A thorough study of the Noether`s theorem with many references can be found in Olver (1993). Numerous examples and applications of the Noether`s theorem are presented in Ibragimov (1994). Algorithm of finding an admitted Lie group is demonstrated in Meleshko (2005). Chatanin et al. (2008) applied Lie group theory to the non-linear equations of motion of inclined unsagged cables to construct invariant solutions to the problem. In this study, the Lie group theory is applied to the non-linear equations of motion of inclined sagged cables. By using the Noether`s theorem, conservation of energy for the solutions of the associated Euler-Lagrange equations is obtained.

MATERIALS AND METHODS

Physical model and equations of motion: The configuration in the global coordinates (X,Y) describes the position of the inclined sagged cable as shown in Fig. 1. The angle θ shows the inclination of the cable with respect to X axis. After the disturbance of an external excitation, the cable configuration is changed into the dynamic configuration which is displaced from the initial configuration. The displacement vectors in X and Y directions are represented by respectively. The horizontal span XH is fixed and the cable`s vertical span YH is varied to attain specified values of θ.

In deriving the equation of motion by the virtual work-energy principle the cable is considered to be perfectly flexible, homogeneous, linearly elastic with negligible torsional, bending and shear rigidities. Therefore, the strain energy is due only to stretching of the cable axis.

Fig. 1: Configurations of an inclined sagged cable

The total strain of the cable at the displaced state, with assumption of moderately large vibration amplitudes, can be expressed as follows:

(1)

In this study, the initial strain e is assumed to have a small value and is neglected. The prime (`) is used to denote differentiation with respect to From Srinil et al. (2004) and Rega et al. (2004), the governing equations of motion are:

(2)

(3)

Where:

EA is the cable axial stiffness,

is the horizontal component of the cable static tension Ts, wc is the cable weight per unit length, g is the gravity constant. Here, the curve that defines shape of the cable is

This non-linear system describes the coupled longitudinal and vertical displacement`s dynamics and is valid also for slightly sagged horizontal cables if one assumes Ts ≈ H rendering ρ = 1. It contains quadratic and cubic nonlinear terms due to the cable`s axial stretching even in the absence of initial sag (taut string case). A method for solving the non-linear equations of motion for horizontal and inclined elastic sagged cables by using a numerical technique is demonstrated in Srinil et al. (2005).

Calculation of infinitesimal generators: An infinitesimal generator X for the problem is written in the following form:

(4)

The main aim is to determine all functions α, β, γ and η that correspond to the one-parameter symmetry groups of the Eq. 2 and 3. Since the governing equations form a system of second order PDE, the second prolongation of the infinitesimal generator is needed. The second prolongation of X is given by:

(5)

Where:

(5`)

and

According to theorem 2.31 Olver (1993):

(6)

(7)

where, the notation |F = 0 means that the second prolongation of X is applied to the solutions of Eq. 2 and 3. Substituting expressions (5,5`) and (2-3) into Eq. 6 and 7 and then equating the coefficients of all independent monomials to zero, we obtain the determining equations. After solving the determining equations, we get the functions α = 0, β = k1, γ = k2t+k3, η = k4t+k5, where k1,...,k5 are arbitrary constants. Thus, the infinitesimal generators have the form:

(8)

The infinitesimal generator contains five arbitrary constants. Consequently, the Lie algebra derived from the governing equations is spanned by the following five linearly independent generators:

Conservation laws: Consider a system of the second order partial differential equations:

Fi(x, u, u(1), u(2)) = 0
(9)

where, x = (x1, x2,...,xn) are the independent variables, u = (u1, u2,...,um) are the dependent variables, are the first and second order derivatives, respectively.

Definition: Equation

Di[Ci(x,u,u(1))] = 0
(10)

is called a conservation law for Eq. 9 if it is satisfied by all solutions u (x) of Eq. 9. The vector C = (C1, C2,...,Cn) is called a conserved vector and Di is the total derivative with respect to xi. Consider a variational integral:

∫L(x,u,u(1))dx
(11)

and its Euler-Lagrange equations:

(12)

where, the Lagrangian L of the system involves the independent variables x = (x1, x2,...,xn), the dependent variables u = (u1,u2,...,um) and the first-order derivatives of u with respect to xi.

Infinitesimal criterion of invariance: The following condition will be necessary and sufficient for a connected group of transformations to be a symmetry group of the variational problem.

Theorem: A connected group of transformations G is a variational symmetry group of the variational integral (11) if and only if:

pr(1)X(L)+LDi(ξi) = 0
(13)

for every infinitesimal generators:

(14)

Noether`s theorem: (Ibragimov, 2006) Let the variational integral (11) be invariant under the group with the generator (14). In other words, let the invariance test (13) be satisfied. Then the vector field C = (C1,C2,...,Cn) defined by:

(15)

is a conserved vector for Eq. 12, i.e., C satisfies the conservation law Di(Ci) = 0.

RESULTS AND DISCUSSION

Since the equations of motion of the inclined sagged cables are derived by the variational method presented in Srinil et al. (2004), we can apply Noether`s theorem to this system to obtain conservation laws for the system as follows:

The Lagrangian of the equations of motion of the inclined sagged cables is given by:

(16)

Where:

m is mass per unit length and ε0 is the initial static strain of the cable centerline. The infinitesimal generators for the problem have the form:

The Lie algebra corresponding to the governing equations is spanned by the following five linearly independent generators:

The conservation law for the equations of motion of the inclined sagged cables is:

Letthe generator X be a linear combination of five generators X1,X2,...X5.

The first prolongation of the generator X is given by:

Applying the infinitesimal criterion of invariance (13) to X, we gets:

This gives:

It follows that λ2 = λ4 = λ5 = 0 and we get that the family of all variational symmetries is given by λ1X13X3, where, λ1 and λ3 are arbitrary constants.

For the variational symmetries

the associated conserved vectors are given by:

If we let λ3 = 0 and λ1 = 1, the variational symmetry of translation in time

are obtained. This gives the law of conservation of energy, Dt(Ct)+Dx(Cx) = 0, where, the associated conserved vectors are:

CONCLUSION

Noether`s theorem is a powerful method for finding conservation laws for complicated systems of differential equations arising from variational principles. The equations of motion of inclined sagged cables were derived by the variational method in Srinil et al.(2004). Thus, Noether`s theorem can be applied to obtain the conservation laws, which play an important role in the analysis of basic properties of the solutions of this system. For each variational symmetry, there is a corresponding conservation law. Applying the infinitesimal criterion of invariance, the researchers find all variational symmetries. In particular, the variational symmetry of translation in time is obtained and yields the conservation of energy law.

ACKNOWLEDGMENTS

The research is supported by Development and Promotion for Science and Technology Talents Project of Thailand (DPST). The first author would like to express deep gratitude to Prof. Dr. Sergey V. Meleshko from Suranaree University of Technology for his valuable suggestions.

REFERENCES
1:  Chatanin, W., A. Loutsiouk and S. Chucheepsakul, 2008. Applications of Lie group analysis to the equations of motion of inclined unsagged cables. Applied Math Sci., 46: 2259-2269.
Direct Link  |  

2:  Ibragimov, N.H., 1994. CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1: Symmetries Exact Solutions and Conservation Laws. 1st Edn., CRC Press Inc., USA., ISBN: 0-8493-4488-3.

3:  Ibragimov, N.H., 2006. A Practical Course in Differential Equations and Mathematical Modelling. 2nd Edn., ALGA Publications, Sweden, ISBN: 91-7295-988-6.

4:  Meleshko, S.V., 2005. Methods for constructing Exact Solutions of Partial Differential Equations. 1st Edn., Springer, USA., ISBN: 0-387-25060-3.

5:  Olver, P.J., 1993. Application of Lie Group to Differential Equations. 2nd Edn., Springer, New York, ISBN: 0-387-95000-1.

6:  Rega, G., N. Srinil, W. Lacarbonara and S. Chucheepsakul, 2004. Resonant nonlinear normal modes of inclined sagged cables. J. Euromech., 457: 7-10.
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7:  Srinil, N., G. Rega and S. Chucheepsakul, 2004. Three-dimensional non-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cables. J. Sound Vib., 269: 823-852.
CrossRef  |  

8:  Srinil, N., S. Chucheepsakul and G. Rega, 2005. Internally resonant nonlinear free vibrations of horrizontal/inclined sagged cables. Proceedings of the 5th International Conference on Multibody Systems,Nonlinear Dynamics and Control, September 24-28, 2005, California, USA., pp: 1-9.

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