Applications of Lie Group Analysis to the Equations of Motion of Inclined Unsagged Cables

Transcription

1 Applied Mathematical Sciences, Vol., 008, no. 46, Applications of Lie Group Analysis to the Equations of Motion of Inclined Unsagged Cables Waraporn Chatanin Department of Mathematics King Mongkut s University of Technology Thonburi, Bangkok, Thailand s @st.kmutt.ac.th Anatoli Loutsiouk Department of Mathematics King Mongkut s University of Technology Thonburi, Bangkok, Thailand loutsiouk.ana@kmutt.ac.th Somchai Chucheepsakul Department of Civil Engineering King Mongkut s University of Technology Thonburi, Bangkok, Thailand somchai.chu@kmutt.ac.th Abstract Analytical solutions to the equations of motion of inclined unsagged cables have been obtained by using the Lie group theory. The infinitesimal generators corresponding to the problem are calculated and then used to construct some invariant solutions. Invariant solutions can be found as solutions of a reduced system of differential equations that has fewer independent variables. Consequently, the system of second - order partial differential equations is reduced to a system of ordinary differential equations and some of these systems can be solved analytically. Mathematics Subject Classification: 35Q80, 35Q7,70G65, 35A5 Keywords: exact solutions, inclined unsagged cables, Lie group theory, invariant solutions

2 60 W. Chatanin, A. Loutsiouk and S. Chucheepsakul 1 Introduction Lie group theory was originally designed for constructing exact solutions of differential equations. This theory was originated by a great mathematician of 19th century, Sophus Lie (Norway, ). Group analysis provides two basic ways for constructing exact solutions of differential equations, group transformation of known solutions and construction of invariant solutions. The symmetry Lie groups and invariant solutions for the wave propagation in gas and the transonic equation were obtained by Ames et al. [10]. After their work, Torissi and Valenti [5] applied the infinitesimal group analysis to a second - order nonlinear wave equation describing a non - homogeneous process. Bluman and Kumei [] studied the wave equations with two and four-parameter symmetry groups and the corresponding invariant (similarity) solutions. Peters and Ames [4] applied Lie group method to the nonlinear dynamic equations of elastic string. Group invariant solutions for a second - order hyperbolic wave equation involving an axially transported speed and nonlinear terms were constructed by Fung et al. [9]. Ozkaya and Pakdemirli [1] investigated exact solutions of transverse vibrations of a string moving with time-dependent velocity. In this paper, the Lie group theory is applied to the nonlinear equations of motion of inclined unsagged cables. In section 3, a method for seeking the infinitesimal generators admitted by the equations is demonstrated and the construction of some invariant solutions is presented in section 4. Physical Model and Equations of Motion.1 Cable Model The configuration ( x, ỹ) in the global coordinates (X, Y ) describes the position of the inclined unsagged cable as shown in Figure 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 ũ( x, t) and ṽ( x, t) respectively. The horizontal span X H is fixed and the cable s vertical span Y H is varied to attain specified values of θ.. Equation of Motion 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 the stretching of the cable axis. The total strain of the cable at

3 Equations of motion of inclined cables 61 X H x x t H x x t o y y x, t o u x, t X y y x, t v x, t Y H Y H Figure 1: Configurations of an inclined unsagged cable. the displaced state, with assumption of moderately large vibration amplitudes, can be expressed as follows ē = e +(1+ỹ ) 1 [ũ +ỹ ṽ + 1 (ũ +ṽ )]. (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 x. From [6] and [3], the governing equations of motion are ρü = {u + Ē ρ 3 (u + y v )+ Ē ρ 3 (u + y u v + 1 (1 + u )(u + v ))}, () ρ v = {v + Ē ρ 3 (y u + y v )+ Ē ρ 3 (u v + y v + 1 (y + v )(u + v ))}, (3) where the dot denotes differentiation with respect to t, x = x X H, y = ỹ X H, u = ũ X H, v = ṽ X H, t = t gh X H w c, ρ = 1+ỹ, ỹ = x tan θ, ỹ = tan θ, E = EA H, EA is the cable axial stiffness, H = Ts ρ is the horizontal component of the cable static tension T s,

4 6 W. Chatanin, A. Loutsiouk and S. Chucheepsakul w c is the cable weight per unit length, g is the gravity constant. A method for solving the nonlinear equations of motion for horizontal and inclined elastic sagged cables by using a numerical technique is demonstrated in Srinil [7]. Here, the unsagged suspended cables are studied. Because the cable is unsagged we can neglect the second derivative of y, and so the simplified non-linear equations of motion of inclined unsagged cables take the form ρu tt {u xx + E ρ 3 (u xx + ψv xx +3u x u xx + ψu x v xx + ψu xx v x + v x v xx )} =0, (4) ρv tt {v xx + E ρ 3 (ψu xx + ψ v xx + u x v xx + u xx v x +3ψv x v xx + ψu x u xx )} =0, (5) where ψ = y = constant. 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 T s 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). 3 Calculation of Infinitesimal Generators An infinitesimal generator X for the problem is written in the following form X = α(x, t, u, v) x + β(x, t, u, v) + γ(x, t, u, v) t u + η(x, t, u, v) v. (6) The main aim is to determine all functions α, β, γ, and η that correspond to the one-parameter symmetry groups of Equations (4) and (5). 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 pr () X = X + γ x + γ t + γ xx + γ xt + γ tt + η x u x u t u xx u xt u tt v x +η t + η xx + η xt + η tt, (7) v t v xx v xt v tt

5 Equations of motion of inclined cables 63 where γ x = D x (γ αu x βu t )+αu xx + βu xt η x = D x (η αv x βv t )+αv xx + βv xt γ xx = D x (γ αu x βu t )+αu xxx + βu xxt η xx = D x (η αv x βv t )+αv xxx + βv xxt γ t = D t (γ αu x βu t )+αu xt + βu tt η t = D t (η αv x βv t )+αv xt + βv tt γ tt = D t (γ αu x βu t )+αu xtt + βu ttt η tt = D t (η αv x βv t )+αv xtt +βv ttt (7 ) and D x = + u x x + u u xx +v x u x + u xt u t + u xxx u xx + u xxt u xt + u xtt u tt v v xx v x + v xt v t + v xxx v xx + v xxt v xt + v xtt v tt +... D t = + u t t + u u xt u x + u tt u t v t + v v xt v x + v tt v t +... D x = D x(d x ) D t = D t (D t ) According to Theorem.31 [8], pr () X[ρu tt {u xx + E ρ (u xx+ψv 3 xx +3u x u xx +ψu x v xx +ψu xx v x +v x v xx )}] F =0 =0, (8) pr () X[ρv tt {v xx + E ρ (ψu xx+ψ v 3 xx +u x v xx +u xx v x +3ψv x v xx +ψu x u xx )}] F =0 =0. (9) where the notation F =0 means that the second prolongation of X is applied to the solutions of Equations (4) and (5). This implies ργ tt {γ xx + E ρ 3 (γxx + ψη xx + γ x u xx + γ xx u x + γ x ψv xx + η xx ψu x + γ xx ψv x

6 64 W. Chatanin, A. Loutsiouk and S. Chucheepsakul +η x ψu xx + η x v xx + η xx v x )} =0, (10) ρη tt {η xx + E ρ 3 (y xη xx + ψγ xx + γ x v xx + η xx u x + γ xx v x + η x u xx +η x ψv xx +η xx ψv x + γ x ψu xx + γ xx ψu x + η x ψv xx + η xx ψv x )} =0. (11) Substituting expressions (7, 7 ) and (4) - (5) into Equations (10) and (11) and then equating the coefficients of all independent monomials to zero, we obtain the determining equations. After solving the determining equations, the functions α, β, γ, and η can be expressed as follows α(x, t, u, v) =k 1 x + k, (1) β(x, t, u, v) =k 1 t + k 3, (13) γ(x, t, u, v) =k 1 u + k 4 t + k 5, (14) where k 1,..., k 7 are arbitrary constants. η(x, t, u, v) =k 1 v + k 6 t + k 7. (15) Thus, the infinitesimal generators have the form X =(k 1 x+k ) x +(k 1t+k 3 ) t +(k 1u+k 4 t+k 5 ) u +(k 1v+k 6 t+k 7 ) v, (16) The infinitesimal generator contains seven arbitrary constants. Consequently, the Lie algebra derived from the governing equations is spanned by the following seven linearly independent generators: X 1 = x x + t t + u u + v v, X = x, X 3 = t, X 4 = t u, X 5 = u, X 6 = t v, X 7 = v. From the commutator table (Table 1.), we get by considering the adjoint representation that the following system of one- dimensional subalgebras spanned by the following vectors is optimal: X 1 + a 4 X 4 + a 6 X 6, (a 4 and a 6 are arbitrary constants) X + a 4 X 4 + a 6 X 6, (a 4 + a 6 > 0)

7 Equations of motion of inclined cables 65 Table 1: Commutator table X 1 X X 3 X 4 X 5 X 6 X 7 X 1 0 X X 3 0 X 5 0 X 7 X X X 3 X X 5 0 X 7 0 X X X 5 X X X X 7 X a X + a 5 X 5 + a 7 X 7, (a + a 5 + a 7 =1) a X + a 3 X 3, (a + a 3 =1) X + a 5 X 5 + a 6 X 6, (a 6 0) X 3 + a 4 X 4 + a 6 X 6. (a 4 and a 6 are arbitrary constants) 4 Invariant Solutions Given the infinitesimal generators X = α + β + γ + η of the one x t u v - parameter symmetry groups of Equations (4) and (5), the invariant solutions with respect to the one-parameter group generated by X can be found. Invariants I(x, t, u, v) are calculated by solving the characteristic system dx α = dt β = du γ = dv η. (17) After expressing independent variables as functions of invariants and substituting them into dependent variables, a system of ordinary differential equation will result. Some cases of generator X which reduce the system of partial differential equations to a system of ordinary differential equations will be considered here: Case1 For X = X + a 4 X 4 + a 6 X 6 = + a x 4t + a u 6t, v there are three independent invariants which are obtained by solving the equation XI = I x + a 4t I u + a 6t I v =0.

8 66 W. Chatanin, A. Loutsiouk and S. Chucheepsakul One of them is I 1 = t, while other invariants are found from the characteristic system dx 1 = du a 4 t = dv a 6 t. By integrating the equation dx = du, we get a 1 a 4 t 4tx + C 1 = u. This gives the invariant I = u a 4 tx. Likewise, we get I 3 = v a 6 tx. Let u a 4 tx = φ 1 (t) and v a 6 tx = φ (t). It follows that u = φ 1 (t)+a 4 tx, (18) v = φ (t)+a 6 tx. (19) Their derivatives are u x = a 4 t, u t = φ 1 + a 4 x, u xx =0, u tt = φ 1, v x = a 6 t, v t = φ + a 6x, v xx =0,v tt = φ. After substituting these derivatives into Equations (4) and (5), the system of partial differential equations reduces to the system of ordinary differential equations 1 =0, (0) =0. (1) The functions φ 1 = at + b and φ = ct + d are solutions to this system, where a, b, c and d are arbitrary constants. By substituting φ 1 and φ into Equation (18) and Equation (19), we get the exact solutions u = at + b + a 4 tx, () v = ct + d + a 6 tx. (3) Case For X = a X + a 3 X 3 = a x + a 3 t, the three independent invariants are I 1 = u, I = v and I 3 = a 3 x a t. Let u = φ 1 (z) and v = φ (z), where z = a 3 x a t. This yields the reduced system of ordinary differential equations a ρφ 1 a 3 {φ 1 + E ρ 3 (φ 1 + ψφ +3a 3φ 1 φ 1 + a 3ψφ 1 φ + a 3ψφ 1 φ + a 3φ φ )} =0, (4) a a 3{φ + E ρ 3 (ψφ 1 +ψ φ +a 3 φ 1φ +a 3 φ 1φ +3a 3 ψφ φ +a 3 ψφ 1φ 1)} =0. (5) The functions φ 1 = az + b and φ = cz + d are solutions to this system, where a, b, c and d are arbitrary constants. Hence the following are exact solutions to the initial system

9 Equations of motion of inclined cables 67 u = a(a 3 x a t)+b, (6) v = c(a 3 x a t)+d. (7) Case3 For X = a X + a 5 X 5 + a 7 X 7 = a + a x 5t + a u 7t, v the three independent invariants are I 1 = t, I = a 5 a x u and I 3 = a 7 a x v. Let a 5 a x u = φ 1 (t) and a 7 a x v = φ (t). This yields a system of ordinary differential equations 1 =0, (8) =0. (9) The functions φ 1 = at + b and φ = ct + d are solutions to this system, where a, b, c and d are arbitrary constants. We get the exact solutions u = a 5 x (at + b), a (30) v = a 7 x (ct + d). a (31) Case4 For X = X + a 5 X 5 + a 6 X 6 = + a x 5 + a u 6t, v the three independent invariants are I 1 = t, I = u a 5 x and I 3 = v a 6 tx. Let u a 5 x = φ 1 (t) and v a 6 tx = φ (t). It follows that u = φ 1 (t) +a 5 x, v = φ (t)+a 6 tx and we obtain a system of ordinary differential equations 1 =0, (3) =0. (33) The functions φ 1 = at + b and φ = ct + d are solutions to this system, where a, b, c and d are arbitrary constants. Thus, we get the exact solutions u = at + b + a 5 x, (34) v = ct + d + a 6 tx. (35) Case5 For X = X 3 + a 4 X 4 + a 6 X 6 = + a t 4t + a u 6t, v the three independent invariants are I 1 = x, I = u a 4 t and I 3 = v a 6 t. Let u a 4 t = φ 1 (x) and v a 6 t = φ (x). It follows that u = φ 1 (x)+ a 4 t, v = φ (x)+ a 6 t. This yields a system of ordinary differential equations a 4 ρ {φ 1 + E ρ 3 (φ 1 + ψφ +3φ 1φ 1 + ψφ 1φ + ψφ 1φ + φ φ )} =0, (36)

10 68 W. Chatanin, A. Loutsiouk and S. Chucheepsakul a 6 ρ {φ + E ρ 3 (ψφ 1 + ψ φ + φ 1 φ + φ 1 φ +3ψφ φ + ψφ 1 φ 1 )} =0. (37) Case6 For X = X 1 = x + t + u + v, x t u v the three independent invariants are I 1 = x,i t = u and I x 3 = v. Let u = φ x x 1(z) and v = φ x (z) where z = x. It follows that u = xφ t 1(z), v= xφ (z). This yields a system of ordinary differential equations ρ(z φ 1 + zφ 1)+{z φ 1 +zφ 1 + E ρ 3 (z φ 1 +zφ 1 + ψ(z φ +zφ ) +3(zφ 1+φ 1 )(z φ 1 +zφ 1)+ψ(zφ 1+φ 1 )(z φ +zφ )+ψ(z φ 1 +zφ 1)(zφ +φ ) +(zφ + φ )(z φ +zφ ))} =0, (38) ρ(z φ + zφ )+{z φ +zφ + E ρ 3 (ψ(z φ 1 +zφ 1)+ψ (z φ +zφ ) +(zφ 1 +φ 1 )(z φ +zφ )+(z φ 1 +zφ 1)(zφ +φ )+3ψ(zφ +φ )(z φ +zφ ) +ψ(zφ 1 + φ 1 )(z φ 1 +zφ 1))} =0. (39) 5 Conclusion In this paper invariant solutions of the equations of motion of the inclined unsagged cables are investigated. By using the infinitesimal generators, the system of partial differential equations can be reduced to a system of ordinary differential equations. In some cases it is possible to solve these systems analytically. ACKNOWLEDGEMENTS. The first author would like to express deep gratitude to Development and Promotion for Science and Technology talents project of Thailand (DPST) for the financial support. References [1] E. Ozkaya and M. Pakdemirli, Lie Group Theory and Analytical Solutions for the Axially Accelerating String Problem. J. Sound Vib, 30(000),

11 Equations of motion of inclined cables 69 [] G. Bluman and S. Kumei, Properties of the Wave Eqution, J. Math. Phys, 8 (1987), [3] G. Rega, N. Srinil, W. Lacarbonara and S. Chucheepsakul, Resonant Nonlinear Normal Modes of Inclined Sagged Cables, J. EUROMECH, 457 (004), [4] J.E. Peters and W.F. Ames, Group Properties of the Non-Linear Dynamic Equations of Elastic String, J. Non-Linear Mech, 5 (1990), [5] M. Torissi and A. Valenti, Group Properties and Invariant Solutions for Infinitesimal Transformations of a Non-Linear Wave Equation, J. Non- Linear Mech, 0 (1985), [6] N. Srinil, G. Rega and S. Chucheepsakul, Three-Dimensional Non-linear Coupling and Dynamic Tension in the Large - Amplitude Free Vibrations of Arbitrarily Sagged Cables, J. Sound Vib, 69 (004), [7] N. Srinil, S. Chucheepsakul and G. Rega, Internally Resonant Nonlinear Free Vibrations of Horrizontal/Inclined Sagged Cables, in proc. of IDECT 05, ASME (005). [8] P.J. Olver, Application of Lie Group to Differential Equations, Springer, New York, [9] R.F. Fung, Y.C. Wang and J.W. Wu, Group Properties and Group- Invariant Solutions for Infinitesimal Transformations of the Non-Linearly Traveling String, J. Int. J. Non-Linear Mech, 34 (1999), [10] W.F. Ames, R.J. Lohner and E. Adams, Group Properties of u tt = [f(u)u x ] x, J. Non-Linear Mech, 16 (1981), Received: February 10, 008