Discontinuity, Nonlinearity, and Complexity 6(3) (2017) 329–391

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Bifurcation Trees of Period-1 Motions to Chaos of a Nonlinear Cable Galloping Bo Yu1 , Albert C. J. Luo2† 1

Department of Mechanical and Industrial Engineering, University of Wisconsin-Platteville, Platteville, WI 53818, USA 2 Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA Submission Info Communicated by V. Afraimovich Received 6 October 2016 Accepted 23 January 2017 Available online 1 October 2017 Keywords Nonlinear cable galloping Period-m motions Hopf bifurcation Saddle-node bifurcation Bifurcation trees

Abstract In this paper, period-m motions on the bifurcation trees of peiod-1 to chaos for nonlinear cable galloping are studied analytically, and the analytical solutions of the period-m motions in the form of the finite Fourier series are obtained through the generalized harmonic balance method, and the corresponding stability and bifurcation analyses of the period-m motions in the galloping system of nonlinear cable are carried out. The bifurcation trees of period-m motions to chaos are presented through harmonic frequency-amplitudes. Numerical illustrations of trajectories and amplitude spectra are given for periodic motions in nonlinear cables. From such analytical solutions of periodic motions to chaos, galloping phenomenon in flow-induced vibration can be further understood. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The galloping vibration of power transmission lines has been discussed since the early 1930s. In the early stage, den Hartog [1] developed a single-degree-of-freedom (SDOF) system for the galloping vibration of cables, and such a model was further investigated by Parkinson [2] and Blevins [3]. In 1956, Edwards and Madeyski [4] observed the torsional effects on galloping motions in power transmission lines in field observations. For a better description of galloping motions induced by fluid flow. In 1974, Blevins and Iwan [5] presented the two-degree-of-freedom (2-DOF) model to study the galloping phenomenon (also see, Blevins [3, 6]). In 1981, Nigol and Buchan [7] discussed the torsional effects on conductor galloping, and Richardson [8] investigated the galloping dynamics of lightly iced transmission power line through the two-degree-of-freedom oscillator. Based on such an idea, in 1988, Richardson [9] discussed the bluff body aerodynamics. Compared to the SDOF system, the 2-DOF model considers the torsional effects. The twisting motion also plays an important role on the initiation of galloping vibrations. In 1990, Desai et al [10] used the two-degree-of-freedom oscillator to investigate the galloping instability. In 1993, Yu et al [11] developed a three degree-of-freedom oscillator to † Corresponding

author. Email address: [email protected], [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2017.09.007

330

Bo Yu, Albert C. J. Luo / Discontinuity, Nonlinearity, and Complexity 6(3) (2017) 329–391

investigate the galloping behavior in the plunge, twist and horizontal directions (long-wind direction). From this 3-DOF model, the explicit expressions for the periodic and quasi-periodic solutions of galloping were obtained from the perturbation analysis. It is assumed that the power transmission lines are deformed linearly with wind load. The conductors are modeled as linear oscillators. However, the nonlinearities of structures (materials and geometries) are the important factors for galloping motions. For a better understanding of cable galloping, the analytical dynamics of the power transmission cables under both aerodynamic loads and external forces will be discussed in this paper. The external forces are in the form of sinusoidal waves. Based on this model, different kinds of periodic vibrations can be obtained analytically for specific parameters. The modelling of fluid force acting on the structures is very difficult because of the irregular cross sections and vortices. It is impossible to obtain a true model for the fluid forces on different bluff structures. If the oscillation of a structure is small enough, the aerodynamic force may be modeled as a linear function of angle attack (e.g., airfoils). However, in most of cases, the aerodynamic forces are nonlinear and coupled with the structures. Especially for the aerodynamic loads on the transmission lines, the fluid flow is separated by the structural cross section, and the fluid force is a nonlinear function of angle of attack. Generally, the fluid models are determined by nonlinear curve fitting to the experimental data measured from the wind tunnel test. The aerodynamic force on the bluff structures can be written as a polynomial. In 1959, Slate [12] used a polynomial of order as high as 25 to model the nonlinear aerodynamic force. However, such a model requires heavy-duty computations for determining galloping instability. For all the previous models of transmission cables, dynamic responses of the transmission power lines are considered only under the aerodynamic forces. From the aforementioned mechanical models for fluid-induced structural galloping instability, in the previous research, numerical simulations and perturbation methods were employed to obtain dynamical responses of structural galloping. However, the transmission lines are slightly damped. The steady-state periodic motions of the transmission lines are obtained by the conventional time-marching techniques. Even for a single degree-of-freedom model, the possible steady-state galloping vibration cannot be easy to obtain. Because the aero-dynamical forces models are nonlinear, the galloping motions cannot be obtained. Such a galloping motion is a stable limit cycle of a system of linear oscillators under the nonlinear aerodynamic forces. Thus, one tried to use analytical approaches to find the steady-state solutions of galloping motions. In 1989, Parkinson [2] used the harmonic balance method to determine such steady-state solutions of periodic galloping motions. In 1990, Desai et al [10] used the Krylov-Bogoliubov method to determine the limit cycle of stable galloping motions of iced transmission power lines. Other researchers used the multiple scale method (e.g., Nayfeh [13]) to find the analytical solutions of cable galloping motions. In 1974, Blevins and Iwan [5] estimated maximum galloping amplitude to determine the strength of galloping motion. Such results cannot explain and predict galloping dynamics very well. The analytical solutions of periodic motions of galloping motions were obtained only when the ratio of any two linear natural frequencies is close to a ratio of two positive integers (e.g., Blevins and Iwan [5]; Desai et al [10]). Until now, one cannot find an appropriate method to find the limit cycle of the fluid-induced galloping motion. The limit cycle is an isolated periodic motion in nonlinear dynamical systems. To accurately determine periodic motions in nonlinear dynamical systems, in 2012, Luo [14] systematically developed a generalized harmonic balance method. The comprehensive description of such a method is presented in Luo [15] for periodic and quasiperiodic motions in nonlinear dynamics systems. Through such a method, the bifurcation trees of periodic motions to chaos can be determined. This method provides a finiteharmonic-term transformation with different time scales to obtain an autonomous nonlinear system of coefficients in the Fourier series form with finite harmonics. In 2012, Luo and Huang [16] used the generalized harmonic balance method with finite terms for the analytical solutions of period-1 motions of the Duffing oscillator with a twin-well potential. Luo and Huang [17] developed analytical solutions of period-m motions in such a Duffing oscillator through a generalized harmonic balance method. The analytical bifurcation trees of periodic motions to chaos in the Duffing oscillator were obtained (also see, Luo and Huang [18, 19]). Such analytical bifurcation trees give all stable and unstable periodic motions. For a better understanding of nonlinear behaviors in nonlinear dynamical systems, the analytical solutions for the bifurcation trees from period-1 motion to chaos

Bo Yu, Albert C. J. Luo / Discontinuity, Nonlinearity, and Complexity 6(3) (2017) 329–391

331

in a periodically forced oscillator with quadratic nonlinearity were presented in Luo and Yu [20–22], and periodm motions in the periodically forced, van der Pol equation was presented in Luo and Laken [23]. The analytical solutions for the van der pol oscillator can be used to verify the conclusions in Cartwright and Littlewood [24] and Levinson [25]. In 2014, Luo and Laken [26] discussed the bifurcation trees of periodic motions to chaos in the van der Pol-Duffing oscillator. In fact, the fluid-induced structural vibration possesses the similar behaviors of the van der Pol-Duffing oscillator. To study the galloping instability of power transmission lines, a two-degree-of freedom nonlinear oscillator will be employed, and the analytical solutions of periodic motions in the two-degree-of-freedom nonlinear system will be developed, and the nonlinearity and frequency-amplitude characteristics of periodic motions of cable galloping will be determined. As before, one used perturbation method for the periodic motions of the two-degree-of-freedom nonlinear systems, and the nonlinear modal shapes were employed. In 2014, Luo and Huang [27,28] used a two-degree-freedom nonlinear oscillator to discuss periodic motions to chaos of a nonlinear Jeffcott rotor through the generalized harmonic balance method, and the analytical solutions of stable and unstable periodic motions were obtained. In 2015, Luo and Yu [29, 30] used the generalized harmonic method to study the analytical solutions of period-1 motions in the two-degree-of-freedom nonlinear oscillators and the corresponding bifurcation trees, and the traditional nonlinear modes in many degree-of-freedom nonlinear systems cannot be observed. In 2016, Yu and Luo [31] used a two-degree-of-freedom nonlinear oscillator to investigate the analytical galloping dynamics of linear cables. The nonlinearity in the two-degree-of-freedom oscillator was considered only from aero-dynamic forces caused by the uniform airflow. The galloping motions of iced linear cables possess the same mechanism of periodic motions in the van der Pol nonlinear oscillator. To further investigate galloping motions of an iced power transmission line, the geometrical nonlinearity of cable will be considered. The mechanism of periodic motions should be similar to the van der Pol-Duffing oscillator. In this paper, the analytical solutions of periodic motions for nonlinear cable galloping will be investigated through a two-degree-of-freedom nonlinear oscillator, and the generalized harmonic method with the finite Fourier series will be used to determine analytical solutions of periodic motions. The corresponding stability and bifurcation analyses of the periodic motions in the galloping system of nonlinear cables will be completedly. The harmonic frequency-amplitude characteristics of periodic motions to chaos will be presented. Numerical illustrations of trajectories and amplitude spectra will be presented for galloping motions in nonlinear iced cables.

2 Mechanical model 2.1

Nonlinear cable

Consider a tightly stretched cable of length l subject to a transverse distributed force f (x,t) per unit length and an external distributed torque m(x,t) per unit length, as shown in Fig.1. ϕ (x,t) denotes the angle between the tension N(x,t) and horizontal axis. T (x,t) is the twisting moment. The transverse and torsional displacements are w(x,t) and Θ(x,t), respectively. The distributed forces and moments on the cable include damping forces, external distributed forces and aero-dynamical forces, expressed by f (x,t) = fy (x,t) − cy w, ˙ ˙ m(x,t) = mΘ (x,t) − cΘ Θ,

(1)

where cy and cΘ are the damping coefficients in the transverse and torsional directions, respectively. Using the infinitesimal cable element, Newton’s second law gives the equations of motion as

∂ (N sin ϕ ) + fy − cy w˙ = ρ Aw, ¨ ∂x ∂T ˙ = I0 Θ ¨ + mΘ − cΘ Θ ∂x

(2)

332

Bo Yu, Albert C. J. Luo / Discontinuity, Nonlinearity, and Complexity 6(3) (2017) 329–391

where ρ is the mass per unit length and I0 is the mass polar moment of inertia of the cable per unit length. ˙ = ∂ Θ/∂ t. If the displacement w(x,t) and twisting angle Θ(x,t) are small, the following w˙ = ∂ w/∂ t and Θ approximations are used. w,x ≈ w,x , sin ϕ = p (1 + u,x )2 + (w,x )2 q 1 (3) N = N0 + EA( (1 + u,x )2 + (w,x )2 − 1) ≈ N0 + EAw2,x , 2 T = GJΘ,x , where E and G are Young’s modulus and shear modulus, and J is moments of polar inertia. w,x = ∂ w/∂ x and Θ,x = ∂ Θ/∂ x. The longitudinal displacement u possesses u,x

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Bifurcation Trees of Period-1 Motions to Chaos of a Nonlinear Cable Galloping Bo Yu1 , Albert C. J. Luo2† 1

Department of Mechanical and Industrial Engineering, University of Wisconsin-Platteville, Platteville, WI 53818, USA 2 Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA Submission Info Communicated by V. Afraimovich Received 6 October 2016 Accepted 23 January 2017 Available online 1 October 2017 Keywords Nonlinear cable galloping Period-m motions Hopf bifurcation Saddle-node bifurcation Bifurcation trees

Abstract In this paper, period-m motions on the bifurcation trees of peiod-1 to chaos for nonlinear cable galloping are studied analytically, and the analytical solutions of the period-m motions in the form of the finite Fourier series are obtained through the generalized harmonic balance method, and the corresponding stability and bifurcation analyses of the period-m motions in the galloping system of nonlinear cable are carried out. The bifurcation trees of period-m motions to chaos are presented through harmonic frequency-amplitudes. Numerical illustrations of trajectories and amplitude spectra are given for periodic motions in nonlinear cables. From such analytical solutions of periodic motions to chaos, galloping phenomenon in flow-induced vibration can be further understood. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The galloping vibration of power transmission lines has been discussed since the early 1930s. In the early stage, den Hartog [1] developed a single-degree-of-freedom (SDOF) system for the galloping vibration of cables, and such a model was further investigated by Parkinson [2] and Blevins [3]. In 1956, Edwards and Madeyski [4] observed the torsional effects on galloping motions in power transmission lines in field observations. For a better description of galloping motions induced by fluid flow. In 1974, Blevins and Iwan [5] presented the two-degree-of-freedom (2-DOF) model to study the galloping phenomenon (also see, Blevins [3, 6]). In 1981, Nigol and Buchan [7] discussed the torsional effects on conductor galloping, and Richardson [8] investigated the galloping dynamics of lightly iced transmission power line through the two-degree-of-freedom oscillator. Based on such an idea, in 1988, Richardson [9] discussed the bluff body aerodynamics. Compared to the SDOF system, the 2-DOF model considers the torsional effects. The twisting motion also plays an important role on the initiation of galloping vibrations. In 1990, Desai et al [10] used the two-degree-of-freedom oscillator to investigate the galloping instability. In 1993, Yu et al [11] developed a three degree-of-freedom oscillator to † Corresponding

author. Email address: [email protected], [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2017.09.007

330

Bo Yu, Albert C. J. Luo / Discontinuity, Nonlinearity, and Complexity 6(3) (2017) 329–391

investigate the galloping behavior in the plunge, twist and horizontal directions (long-wind direction). From this 3-DOF model, the explicit expressions for the periodic and quasi-periodic solutions of galloping were obtained from the perturbation analysis. It is assumed that the power transmission lines are deformed linearly with wind load. The conductors are modeled as linear oscillators. However, the nonlinearities of structures (materials and geometries) are the important factors for galloping motions. For a better understanding of cable galloping, the analytical dynamics of the power transmission cables under both aerodynamic loads and external forces will be discussed in this paper. The external forces are in the form of sinusoidal waves. Based on this model, different kinds of periodic vibrations can be obtained analytically for specific parameters. The modelling of fluid force acting on the structures is very difficult because of the irregular cross sections and vortices. It is impossible to obtain a true model for the fluid forces on different bluff structures. If the oscillation of a structure is small enough, the aerodynamic force may be modeled as a linear function of angle attack (e.g., airfoils). However, in most of cases, the aerodynamic forces are nonlinear and coupled with the structures. Especially for the aerodynamic loads on the transmission lines, the fluid flow is separated by the structural cross section, and the fluid force is a nonlinear function of angle of attack. Generally, the fluid models are determined by nonlinear curve fitting to the experimental data measured from the wind tunnel test. The aerodynamic force on the bluff structures can be written as a polynomial. In 1959, Slate [12] used a polynomial of order as high as 25 to model the nonlinear aerodynamic force. However, such a model requires heavy-duty computations for determining galloping instability. For all the previous models of transmission cables, dynamic responses of the transmission power lines are considered only under the aerodynamic forces. From the aforementioned mechanical models for fluid-induced structural galloping instability, in the previous research, numerical simulations and perturbation methods were employed to obtain dynamical responses of structural galloping. However, the transmission lines are slightly damped. The steady-state periodic motions of the transmission lines are obtained by the conventional time-marching techniques. Even for a single degree-of-freedom model, the possible steady-state galloping vibration cannot be easy to obtain. Because the aero-dynamical forces models are nonlinear, the galloping motions cannot be obtained. Such a galloping motion is a stable limit cycle of a system of linear oscillators under the nonlinear aerodynamic forces. Thus, one tried to use analytical approaches to find the steady-state solutions of galloping motions. In 1989, Parkinson [2] used the harmonic balance method to determine such steady-state solutions of periodic galloping motions. In 1990, Desai et al [10] used the Krylov-Bogoliubov method to determine the limit cycle of stable galloping motions of iced transmission power lines. Other researchers used the multiple scale method (e.g., Nayfeh [13]) to find the analytical solutions of cable galloping motions. In 1974, Blevins and Iwan [5] estimated maximum galloping amplitude to determine the strength of galloping motion. Such results cannot explain and predict galloping dynamics very well. The analytical solutions of periodic motions of galloping motions were obtained only when the ratio of any two linear natural frequencies is close to a ratio of two positive integers (e.g., Blevins and Iwan [5]; Desai et al [10]). Until now, one cannot find an appropriate method to find the limit cycle of the fluid-induced galloping motion. The limit cycle is an isolated periodic motion in nonlinear dynamical systems. To accurately determine periodic motions in nonlinear dynamical systems, in 2012, Luo [14] systematically developed a generalized harmonic balance method. The comprehensive description of such a method is presented in Luo [15] for periodic and quasiperiodic motions in nonlinear dynamics systems. Through such a method, the bifurcation trees of periodic motions to chaos can be determined. This method provides a finiteharmonic-term transformation with different time scales to obtain an autonomous nonlinear system of coefficients in the Fourier series form with finite harmonics. In 2012, Luo and Huang [16] used the generalized harmonic balance method with finite terms for the analytical solutions of period-1 motions of the Duffing oscillator with a twin-well potential. Luo and Huang [17] developed analytical solutions of period-m motions in such a Duffing oscillator through a generalized harmonic balance method. The analytical bifurcation trees of periodic motions to chaos in the Duffing oscillator were obtained (also see, Luo and Huang [18, 19]). Such analytical bifurcation trees give all stable and unstable periodic motions. For a better understanding of nonlinear behaviors in nonlinear dynamical systems, the analytical solutions for the bifurcation trees from period-1 motion to chaos

Bo Yu, Albert C. J. Luo / Discontinuity, Nonlinearity, and Complexity 6(3) (2017) 329–391

331

in a periodically forced oscillator with quadratic nonlinearity were presented in Luo and Yu [20–22], and periodm motions in the periodically forced, van der Pol equation was presented in Luo and Laken [23]. The analytical solutions for the van der pol oscillator can be used to verify the conclusions in Cartwright and Littlewood [24] and Levinson [25]. In 2014, Luo and Laken [26] discussed the bifurcation trees of periodic motions to chaos in the van der Pol-Duffing oscillator. In fact, the fluid-induced structural vibration possesses the similar behaviors of the van der Pol-Duffing oscillator. To study the galloping instability of power transmission lines, a two-degree-of freedom nonlinear oscillator will be employed, and the analytical solutions of periodic motions in the two-degree-of-freedom nonlinear system will be developed, and the nonlinearity and frequency-amplitude characteristics of periodic motions of cable galloping will be determined. As before, one used perturbation method for the periodic motions of the two-degree-of-freedom nonlinear systems, and the nonlinear modal shapes were employed. In 2014, Luo and Huang [27,28] used a two-degree-freedom nonlinear oscillator to discuss periodic motions to chaos of a nonlinear Jeffcott rotor through the generalized harmonic balance method, and the analytical solutions of stable and unstable periodic motions were obtained. In 2015, Luo and Yu [29, 30] used the generalized harmonic method to study the analytical solutions of period-1 motions in the two-degree-of-freedom nonlinear oscillators and the corresponding bifurcation trees, and the traditional nonlinear modes in many degree-of-freedom nonlinear systems cannot be observed. In 2016, Yu and Luo [31] used a two-degree-of-freedom nonlinear oscillator to investigate the analytical galloping dynamics of linear cables. The nonlinearity in the two-degree-of-freedom oscillator was considered only from aero-dynamic forces caused by the uniform airflow. The galloping motions of iced linear cables possess the same mechanism of periodic motions in the van der Pol nonlinear oscillator. To further investigate galloping motions of an iced power transmission line, the geometrical nonlinearity of cable will be considered. The mechanism of periodic motions should be similar to the van der Pol-Duffing oscillator. In this paper, the analytical solutions of periodic motions for nonlinear cable galloping will be investigated through a two-degree-of-freedom nonlinear oscillator, and the generalized harmonic method with the finite Fourier series will be used to determine analytical solutions of periodic motions. The corresponding stability and bifurcation analyses of the periodic motions in the galloping system of nonlinear cables will be completedly. The harmonic frequency-amplitude characteristics of periodic motions to chaos will be presented. Numerical illustrations of trajectories and amplitude spectra will be presented for galloping motions in nonlinear iced cables.

2 Mechanical model 2.1

Nonlinear cable

Consider a tightly stretched cable of length l subject to a transverse distributed force f (x,t) per unit length and an external distributed torque m(x,t) per unit length, as shown in Fig.1. ϕ (x,t) denotes the angle between the tension N(x,t) and horizontal axis. T (x,t) is the twisting moment. The transverse and torsional displacements are w(x,t) and Θ(x,t), respectively. The distributed forces and moments on the cable include damping forces, external distributed forces and aero-dynamical forces, expressed by f (x,t) = fy (x,t) − cy w, ˙ ˙ m(x,t) = mΘ (x,t) − cΘ Θ,

(1)

where cy and cΘ are the damping coefficients in the transverse and torsional directions, respectively. Using the infinitesimal cable element, Newton’s second law gives the equations of motion as

∂ (N sin ϕ ) + fy − cy w˙ = ρ Aw, ¨ ∂x ∂T ˙ = I0 Θ ¨ + mΘ − cΘ Θ ∂x

(2)

332

Bo Yu, Albert C. J. Luo / Discontinuity, Nonlinearity, and Complexity 6(3) (2017) 329–391

where ρ is the mass per unit length and I0 is the mass polar moment of inertia of the cable per unit length. ˙ = ∂ Θ/∂ t. If the displacement w(x,t) and twisting angle Θ(x,t) are small, the following w˙ = ∂ w/∂ t and Θ approximations are used. w,x ≈ w,x , sin ϕ = p (1 + u,x )2 + (w,x )2 q 1 (3) N = N0 + EA( (1 + u,x )2 + (w,x )2 − 1) ≈ N0 + EAw2,x , 2 T = GJΘ,x , where E and G are Young’s modulus and shear modulus, and J is moments of polar inertia. w,x = ∂ w/∂ x and Θ,x = ∂ Θ/∂ x. The longitudinal displacement u possesses u,x