EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2703-2728 ISSN 1307-5543 – ejpam.com Published by New York Business Global On controlling, vibration performance and energy transfer of an offshore wind turbine tower system via PPF controller Ateq Alsaadi1,∗, Y. S. Hamed1 1Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia Abstract. In this paper, we focus on the control, energy transfer, and vibration performance under multi mixed excitations for the offshore wind turbine tower (OWTT) system. For reducing the controlled system oscillations, the positive position feedback (PPF) controller is applied. The energy transfers occur in the system of wind turbine by adding the PPF controller to the system equations. With the help of the phase plane approach, frequency response equations, and Poincare maps, the bifurcation and stability at worst resonance cases are sought and investigated. The vibration behaviors are studied numerically at different parameters values for the wind turbine system. Additionally, the response and numerical outcomes are examined, also, the approach of multiple scales is used to establish the approximate solutions of the wind turbine-controlled system. Besides that, MAPLE and MATLAB algorithms are used to implement the numerical results and compare analytical solutions with numerical behavior. The results also be compared to previous research that has been published. 2020 Mathematics Subject Classifications: 34A34, 34C15, 34C23, 34C25, 34D20, 34E13, 37N15, 70B05, 70K05, 70K40, 70K42, 70K50 Key Words and Phrases: Offshore wind turbine system, Energy transfer, Vibration control, Stability, positive position feedback control 1. Introduction Renewable energy has grown in importance as a topic of these investigations over the past few decades. On a worldwide scale, the demand for energy has been steadily rising. This increase motivates us to start looking at alternative energy production strategies. One of the most difficult issues our society is currently experiencing is the creation of a sustainable and renewable energy economy. The cost of wind energy has been the main motivator for the shift. Wind energy is projected to contribute significantly in terms of renewable energy in the coming decades to reach such an energy source. By about a third ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4929 Email addresses: ateq@tu.edu.sa (A. Alsaadi), yasersalah@ tu.edu.sa (Y. S. Hamed) https://www.ejpam.com 2703 © 2023 EJPAM All rights reserved. A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2704 since 2009, wind turbine prices have fallen. Production of wind energy is one of the most effective conservation projects [1, 2]. The Rayleigh’s energy approach and ANSYS FSI analysis are used to determine the dynamic effects of the blades on the tower as well as the effects of the earthquake forces, wave, and wind on wind turbine dynamic behavior [3]. Several parameters’ effects on the behavior of the (OWTT) system and the responses of the soil monopile tower system are the subject of this investigation [4]. The standard model of large rotating wind turbines active observer with effective control approach is studied [5]. A tuned mass damper and passive control approach are used to study the vibrations of the nacelle of offshore wind turbines and spar oscillations [6]. The effects of an active control method are investigated for a hybrid mass damper and a barge float- ing wind turbine type [7]. Mathematical analysis study for the wind turbines dynamics with time scale simulations and control was performed [8]. The wind turbines structural vibrations and its high oscitations was suppressed through active controller and has been proposed in [9]. With two different approaches, the environmental forces and seismic loads effect on the offshore wind tower behavior is performed [10]. The transfer the energy and reducing the high oscillations amplitudes of the wind turbine system using both PD and NPD controller are well studied. Furthermore, the Poincaré maps and averaging method were used to study the bifurcation analysis and stability at the worst resonance cases. Additionally, the wind turbine system’s frequency and force response curves were drawn before and after the addition of the control unit. In addition, the Numerical simulations were carried out and control parameters performances on the vibration magnitudes was performed applying with Maple and MATLAB algorithms. Obtained results demonstrate the efficiency of the NPD controller in reducing the nonlinear wind oscillations [11-12]. To eliminate the oscillator’s van der pol duffing with mixed excitations, the nonlinear integral positive position feedback (NIPPF), and integral resonant (IRC) controllers are applied. The multiple time scales approach and frequency response equations are used to perform the approximate solutions and stability analysis at the worst resonance cases [13]. The de- tailed of analysis is founded for some dynamical systems with different forces in the books [14-17]. The model of atomic force microscopy (AFM) was controlled utilizing the PD controller with time delayed and the PD control efficiency for putting down the nonlinear oscillations are performed to the obtained results [18]. The controlling of a rotating blade’s oscillation via nonlinear saturation control, the energy transfer and stability using tech- nique of Lyapunov’s linearization are investigated [19]. The PPF and PD signals control are merged to minimize the levels of AMBs system vibration involving constant stiffness and 16 poles. Also, Approximate solutions are extracted, and response curves are plotted before and after using PPF [20]. 2D and 3D Visualizations of the mass-damper-spring model was controlled with a servo controlled linear actuator (SCLA) with using signal generating (PPF) controller. In addition, they derived the system equations, phases and amplitudes using Krylov-Bogoliubov averaging perturbation method [21]. The approxi- mate solutions, stability, and control for the car model with the harmonic balance and averaging methods with nonlinear saturation controller (NSC) controller a cubic-position negative-velocity and a cubic-position negative-velocity feedback (CPNV) controllers [22- 23]. The mathematical methods are presented and explained for obtaining approximate A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2705 analytical solutions to differential and difference equations that cannot be solved exactly. The mathematical methods discussed in this book are known collectively as¬ asymptotic and perturbative analysis [24]. The stability analysis of even multidimensional strongly nonlinear systems described by PDEs are studied [25]. Tuned mass and liquid column dampers are used to reduce the excessive vibration responses of offshore wind turbines [26]. A tuned mass damper with passive control in a floating platform can reduce and im- prove the dynamic responses of the offshore wind turbine [27]. The high frequency tuned mass damper installed in the nacelle is examined using numerical methods to see whether it effectively reduces tower response while having little effect on platform response [28]. A new technique for minimizing the motion of a two-degree-of-freedom tuned mass damper is applied for offshore wind turbines on floating spar supports [29]. Using an electromagnetic shunt tuned mass damper, floating offshore wind turbine vibration can be reduced [30]. According to the articles described above, none of them discuss controlling the primary and internal resonance response via the positive position feedback (PPF) controller, which was the main motive of such a response. The current work suggests using this control ap- proach. In the present work, we focused on the control, energy transfer, and vibration performance under multi mixed excitations for the system of offshore wind turbine tower. For reducing the controlled system oscillations and energy transfers, the PPF controller be applied. With the help of the phase plane approach, frequency response equations, and Poincare maps, the bifurcation and stability at worst resonance cases are sought and in- vestigated. The vibration behaviors are studied numerically at different parameters values for the wind turbine system. 2. System modelling The parts of the OWTT system are the hub, blade, tower, and concentrated mass. This system is investigated in terms of some excitation forces as in Fig. 1(a). As in Figure 1(b), the structural model with multiple external forces, including the effects of wave FH , earthquake Feqk, and wind Faero forces. A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2706 (a) (b) Figure 1: (a) The wind turbine tower model, (b)The wind tower’s forces. The wind forces Faero acting on the Rotor Nacelle Assembly (RNA) and associate instrumentation can be considered as concentrated forces (hydrodynamic forces) which depend on the wind velocity. The wave force FH were utilized to compute the hydrody- namic wave forces acting on the wind tower, where the wave height H, wavelength L, and wave period T are the major wave parameters. The strong force acting on the turbine tower structure is the earthquake force due to the seismic excitation. The below equation governing the SDOF system’s equation of motion, which was obtained from [10]: mz̈(t) + ϵcż(t) + kz(t) + ϵF (t) = ϵG(t) (1a) Where z̈(t) is the acceleration, ż(t) is the velocity, z(t) is the coordinate vector, F (t) is the total force, G(t) is controller signal, m is the mass, c and k are the damping and stiffness coefficients, respectively. As proceeding in Ref [10], then the system’s equation of motion is derived and modified by the next ordinary differential equation: z̈ϵµ1ż + ω1 2z + ϵfeqk + ϵfa sinΩ1(t) = ϵfh cosΩ1(t)| cosΩ1(t)|. (1b) Where µ1 = c m , ω2 1 = k m , F (t) = 1 m(Feqk+Faero sinω1(t), G(t) = 1 m(FH cosω1(t)| cosω1(t)|). Applying the positive position feedback (PPF) controller donate by equation (2b) to the wind turbine tower system in equation (1b). In order to control the oscillation of the tower system, the control unit must acquire the feedback signal from the tower system, A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2707 the updated equation (1b) would be as follows: z̈ + ϵµ1z + ω2 1z + ϵfa sin(ω1(t)) = ϵfh cos(ω1(t))| cos(ω1(t))|+ ϵβ1y. (2a) ÿ + ϵµ2y + ω2 2y = ϵβ2z. (2b) Where µ1, µ2 are linear damping factors of the system and control unit, ϵ is small pertur- bation, fa, fh are excitation wind and wave forces, feqk = α cos(πt) is earthquake force, ω1,ω2 are natural frequencies tower system and control unit,ω1 is the excitation frequency and β1, β2 are control units. 3. Approximate analytical solution The goal of perturbation methods is to propose approximately analytical solutions to problems that do not have exact solutions depending on a small parameter ϵ where 0 ≪ ϵ ≪ 1. During the solution process, (unwanted) secular terms are eliminated using the new independent variables. The latter applies solvability conditions, which are constraints on the approximate solution. Approximate solutions of problem (2) are produced using the Multiple Scale Perturbation (MSP) approach [14-17], and the solutions have the following form: z(t, ϵ) = z0(T0, T1) + ϵz1(T0, T1) +O(ϵ2) (3a) y(t, ϵ) = y0(T0, T1) + ϵy1(T0, T1) +O(ϵ2) (3b) We provide the derivatives in the following format: d dt = { D0 + ϵD1, d2 dt2 = D2 0 + 2ϵD0D1. (3c) where T0 = t, T1 = ϵt and D0 = ∂ ∂T0, D1 = ∂ ∂T1 are the time scales and derivatives respectively and substituting equations (3a) (3b) through (3c) into equation (2), we get (D2 0z0 + ω2 1z0) + ϵ(D2 0z1 + ω2 1z1 + 2D0D1z0 + µ1D0z0 + α cos(πT0) + fa sin(Ω1T0) −fh cos(Ω1T0)| cos(Ω1T0)| − β1y0) + ϵ2(D2 0z2 + ω2 1z2 + 2D0D1z1 +D2 1z0 − fh cos(Ω1T0) | cos(Ω1T0)| − β1y0) + ϵ2(D2 0z2 + ω2 1z2 + 2D0D1z1 +D2 1z0 + µ1(D1z0 +D0z1)− β1y1) = 0. (4a) (D2 0y0 + ω2 2y0) + ϵ(D2 0y1 + ω2 2y1 + 2D0D1y0 + µ2D0y0 − β2z0) + ϵ2(D2 0y2+ ω2 2y2 + 2D0D1y1 +D2 1y0 + µ2(D1y0 +D0y1 − β1z1) = 0. (4b) A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2708 Equating the coefficients of ϵ in equation (4) leads: O(ϵ0): (D2 0 + ω2 1)z0 = 0. (5a) (D2 0 + ω2 2)y0 = 0. (5b) O(ϵ1): (D2 0+ω2 1)z1 = −2D0D1z0−µ1D0z0−α cos(πT0)−fa sin(ω1T0)+fh cos 2(ω1T0)+β1y0, cos(ω1T0) ≥ 0. (6a) or (D2 0+ω2 1)z1 = −2D0D1z0−µ1D0z0−α cos(πT0)−fa sin(ω1T0)−fh cos 2(ω1T0)+β1y0, cos(ω1T0) < 0. (6b) (D2 0 + ω2 2)y1 = −2D0D1y0 − µ2D0y0 + β2z0. (6c) Equation (5) has a general solution expressed as: z0 = A1(T1)e iω1T0Ā1(T1)e −iω1T0 . (7a) y0 = A2(T1)e iω2T0 + Ā2(T1)e −iω2T0 (7b) Substituting equation (7) into equation (6), the following are obtained: (D2 0 + ω2 1)z1 = [−iω1(2D1A1 + µ1A1)]e iω1T0 + [iω1(2D1Ā1 + µ1Ā1)]e −iω1T0 +β1A2e iω2T0 + β1Ā2e −iω2T0 + (ifa)/2(e iΩ1T0 − e−iΩ1T0) + fh 4 (e(2iΩ1T0+ e−2iΩ1T0 + 2)− α 2 eiπT0 − α 2 e−iπT0), cos(Ω1T0) ≥ 0. (8a) or (D2 0 + ω2 1)z1 = [−iω1(2D1A1 + µ1A1)]e iω1T0 + [iω1(2D1Ā1 + µ1Ā1)]e −iω1T0 +β1A2e iω2T0 + β1Ā2e −iω2T0 + (ifa)/2(e iΩ1T0 − e−iΩ1T0) + fh 4 (e(2iΩ1T0+ e−2iΩ1T0 + 2)− α 2 eiπT0 − α 2 e−iπT0), cos(Ω1T0) < 0. (8b) (D2 0+ω2 2)y1 = [−iω2(2D1A2+µ1A2)]e iω2T0+[iω2(2D1Ā2+µ1Ā2)]e −iω2T0+β1A1e iω2T0+β1Ā1e −iω1T0 . (8c) A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2709 After the secular terms e( ± iω1T0) and e( ± iω2T0) were eliminated from equations (8a) -(8c), the general solutions given as z1 = (A3e iω1T0 + Ā3e iω1T0) + β1 (ω2 1 − ω2 2) (A2e iω2T0 + Ā2e −iω2T0) + (ifa) 2 (ω2 1 − Ω2 1)(e iΩ1T0 − e−iΩ1T0) + fh 4 (ω2 1 − 4Ω2 1)(e 2iΩ1T0 + e−2iΩ1T0) + fh (2ω2 1) − α 2 (ωA2 1 − π2)(eiπT0 + e−iπT0). (9a) or z1 = (A3e iω1T0 + Ā3e iω1T0) + β1 (ω2 1 − ω2 2) (A2e iω2T0 + Ā2e −iω2T0) + (ifa) 2 (ω2 1 − Ω2 1)(e iΩ1T0 − e−iΩ1T0)− fh 4 (ω2 1 − 4Ω2 1)(e 2iΩ1T0 + e−2iΩ1T0) − fh (2ω2 1) − α 2 (ω2 1 − π2)(eiπT0 + e−iπT0). (9b) y1 = (A4e (iω2T0) + Ā4e −iω2T0) + β2 (ω2 2 − ω2 1) (A1e iω1T0 + Ā1e −iω1T0). (9c) The resonances were obtained from the approximations and are given as follows: a . Primary, super-harmonic and Internal resonance are: Ω1 ≊ ±ω1,Ω1 ≊ ±ω12 and ω1 ≊ ±ω2. b With any combination of the above resonance cases, simultaneous resonance is ob- tained. Analysis and studies are done on the solution’s stability in the primary with internal resonance caseΩ1 ≊ ω1, ω1 ≊ ω2 , Where the detuning σ1 and σ2 are given as: Ω1 = ω1 + ϵσ1, ω2 = ω1 + ϵσ2. (10) The first order approximation’s solvability conditions are given by inserting equation (10) into equation (8a) -(8b) and removing the secular then we get: −iω1(2D1A1 + µ1A1) + β1A2e iσ2T1 + (ifa) 2 eiσ1T1 = 0. (11a) A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2710 −iω2(2D1A2 + µ2A2) + β1A1e ( − iσ2T0) = 0. (11b) Let’s introduce the polar form as: A1 = 1 2 a1(T1)e iϕ1(T1), A2 = 1 2 a2(T1)e iϕ2(T1) (12) where ϕ1, ϕ2 are the motion’s phases and a1, a2 are the amplitudes steady state. By inserting equation (12) into equations (11a)–(11b) and equating the real and imaginary parts, the modulation of the phases and amplitudes is achieved, and the result is as follows. ȧ1 = −µ1a1 2 + (β1a2) 2ω1 sin θ2 + fa 2ω1 cos θ1 (13a) a1ϕ̇1 = −β1a2 2ω1 cos θ2 + fa 2ω1 sin θ1 (13b) ȧ2 = −µ2a2 2 − β2a1 2ω2 sin θ2 (14a) a2ϕ̇2 = −β2a1 2ω2 cos θ2 (14b) Where θ1 = σ1T1 − ϕ1, θ2 = σ2T1 − ϕ1 + ϕ2 (15) From equation (15) we have ϕ̇1 = σ1 − θ̇1, ϕ̇2 = σ2 − σ1 − θ̇2 + θ1 (16) Substituting equation (16) into equations (13a) -(14b), we obtained ȧ1 = −µ1a1 2 + β1a2) 2ω1 sin θ2 + fa (2ω1 cos θ1. (17a) a1θ̇1 = a1σ1 + β1a2 2ω1 cos θ2 − fa 2ω1 sin θ1 (17b) ȧ2 = −µ2a2 2 − β2a1 2ω2 sin θ2 (18a) a2θ̇2 = a2σ2 − β2a1 2ω2 cos θ2 + β1a 2 2 2ω1a1 cos θ2 − faa2 2ω1a1 sin θ1 (18b) It is too difficult to find the solutions of equations (17) and (18) analytically, but we can obtain the solutions numerically. To obtain the approximate periodic solutions, substitut- ing equations (7a), (7b) and (12) into equations (3a), (3b) and neglecting terms of o(ϵ), we have z = 1 2 a1e i(ω1T0+ϕ1) + 1 2 a1e −i(ω1T0+ϕ1) (19a) y = 1 2 a2e i(ω2T0+ϕ2)) + 1 2 a2e −i(ω2T0+ϕ2) (19b) A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2711 We can write the first approximation periodic solution in the form: z = a1 cos(Ω1T0 − θ1) (20a) y = a2 cos(Ω1T0 + θ2 − θ1) (20b) where a1, a2, θ1 and θ2 are the solutions of the equations (17) and (18) at the steady state. 4. Stability investigations To get vibration amplitudes of the steady state, we set ȧ1 = ȧ2 = θ̇1 = θ̇2 = 0, into equations (17a) -(18b), we get: µ1a1 2 = β1a2 2ω1 sin θ2 + fa 2ω1 cos θ1 (21a) a1σ1 = −β1a2 2ω1 cos θ2 + fa 2ω1 sin θ1 (21b) µ2a2 2 = −β2a1 2ω2 sin θ2 (22a) a2(σ2 − σ1) = β2a1 2ω2 cos θ2 . (22b) Eliminating θ1 and θ2 from equations (21) and (22) leads to: a1 2 = ω2 2 β2 2 (µ2 2 + 4(σ2 − σ1) 2)a2 2. (23) f2 a 4ω2 1 = ( µ1a1 2 + β1a 2 2µ2ω2 2β2a1ω1 )2 + ( σ1a1 + β1a2) 2ω2 β2a1ω1 (σ2 − σ1) )2 . (24) Equations (23) and (24) are the frequency-response equations for the system steady-state behavior at the practical case (a1, a2 ̸= 0). By computationally resolving the above alge- braic problem using the Newton-Raphson technique, we can get the steady state solution of equations (2a) and (2b). Using Lyapunov’s direct method [15], the solution stability is evaluated. Assume the following equation: an = an0 + an1 and θn = θn0 + θn1(n = 1, 2). (25) Where an0 and an1 are perturbative and steady-state amplitudes. Similarly, θn0 and θn1 are perturbative phases and steady-state phases. Inserting equation (25) into equations (17-18) and linearizing give us ȧ11 θ̇11 ȧ21 θ̇21 =J  a11 θ11 a21 θ21 =  r11 r12 r13 r14 r21 r22 r23 r24 r31 r41 r32 r42 r33 r43 r34 r44   a11 θ11 a21 θ21  A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2712 where J is the Jacobian matrix, and its constituent parts are listed in the appendix. J is defining equation of a fourth-degree equation in the following form. λ4+R1λ 3+R2λ 2+R3λ+R4= 0 (26) where Ri (i= 1,· · ·,4) are listed in the appendix. The roots of equation (26) are the eigenvalues of J and they determine whether the system solutions are stable or not. The system’s periodic solution is unstable if the real part of the eigenvalue is positive; if not, the system becomes stable. The Routh-Huriwitz criterion demonstrates that the roots of equation (26) had negative real parts, If and only if the following equation holds true: R1> 0, R1R2−R3> 0, R3 (R1R2−R3)−R2 1R4> 0, R4> 0 (27) 5. Discussion on the primary and internal resonance curves As it has been concluded in equations (23) and (24) that there two cases: 1 . (a1, a2 ̸= 0) where the controller is active. 2 . (a1, a2 = 0) where the controller is inactive. The system parameters are given by µ1 = 0.04, µ2 = 0.02, β1 = 5, β2 = 0.5, α = 9.73,Ω1 = 1.26, ω1 = 1.26, ω2 = 1.26, fa = 67.17, fh = 5.129. 5.1. Effect of some important different parameters on the wind tower system before control In this part, we looked at how various parameters affected the behavior of the wind tower system before control. The system behavior is shown in Figures 2(a, b) as a mono- tonically decreasing function of the damping coefficient µ1 and a monotonically increasing function in the excitation wind force fa. (a) (b) Figure 2: (a) Effect of linear damping µ1, (b)Effect of excitation wind force fa on wind tower system response. A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2713 5.2. Curves of force and frequency response for the system without con- trol The equilibrium behavior of the wind turbine tower amplitude a1 in terms of the forcing amplitude fa before control (β1 = 0, β2 = 0) is shown in Fig. 3. As the parameters µ1 and ω1 are increased, the stable nontrivial amplitudes decreased. In Fig. 4 (a, b, c), the equilibrium behavior of the amplitude a1 in terms of the frequency detuning σ1 is plotted before control (β1 = 0, β2 = 0). In this figure, increasing the parameters µ1 and ω1 can reduce the oscillations of the system, but the oscillations is increased with the increase the parameters fa. (a) (b) Figure 3: (a) Curves of force response for OWTT system before control (β1 = 0, β2 = 0) at different values of linear damping µ1, (b)Curves of force response for OWTT system before control (β1 = 0, β2 = 0) at different values of natural frequency ω1. A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2714 (a) (b) (c) Figure 4: (a) Curves of force response for OWTT system before control (β1 = 0, β2 = 0) at different values of linear damping µ1 , (b) Curves of force response for OWTT system before control (β1 = 0, β2 = 0) at different values of natural frequency ω1, (c) Curves of force response for OWTT system before control (β1 = 0, β2 = 0) at different values of wind amplitude force fa. 5.3. Curves of frequency response for the controlled system Using frequency response curves, we investigated the effects of the various factors on the controlled system’s stability zone. In Figs. (5-10), the equilibrium behavior of the (WTT)amplitudes a1, a2 in terms of the frequency detuning σ1 is plotted before control (β1 = 5, β2 = 0.5).According to Fig. 5, the controlled system behavior is a monotonically increasing function in the parameter fa. Figures (6–8) demonstrate the controlled system’s behavior as a monotonic decreasing function of the parameters µ1, µ2, ω1 andω2. We utilize these parameters to reduce the oscillations of the OWTT system based on these figures. Additionally, the saturation occurs, and the curves are shifted to the right for the system behavior with the change in values β1, β2, but the control behavior is decreased with increasing β1 and increased with increasing β2 as shown in Figs. (9, 10). A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2715 Figure 5: Curves demonstrating the controlled system’s frequency response after control (β1 = 5, β2 = 0.5) at different values of wind amplitude force fa. Figure 6: Curves demonstrating the controlled system’s frequency response after control (β1 = 5, β2 = 0.5) at different values of linear damping µ1. Figure 7: Curves demonstrating the controlled system’s frequency response after control (β1 = 5, β2 = 0.5) at different values of linear damping µ2. Figure 8: Curves demonstrating the controlled system’s frequency response after control (β1 = 5, β2 = 0.5) at different values of natural frequencies ω1, ω2. A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2716 Figure 9: Curves demonstrating the controlled system’s frequency response after control (β1 = 5, β2 = 0.5) at different values of control unit β1. Figure 10: Curves demonstrating the controlled system’s frequency response after control (β1 = 5, β2 = 0.5) at different values of control unit β2. 6. Discussion of analytical and numerical results Using the fourth order Rung-Kutta numerical approach, Figures 11 to 14 simulate the wind turbine tower vibrations both before and after control. In terms of time before control, Figure 11 shows the wind turbine tower’s nontrivial vibrations. Figure 11(a) depicts the system oscillations, which have an amplitude of about 43 centimeters, while Figure 11(b) depicts a steady-state system oscillation of last 20 seconds to demonstrate the reader the specifics of the solution waveform. Additionally, the controlled system’s oscillation amplitude can reach 0.6 centimeters, and the unit control’s effectiveness (Ea= the amplitude before control/amplitude after control) equal Ea=72 when (β1 = 5, β2 = 0.5, ω1 = 1.26) as shown in Fig. 12, this mean that the steady oscillations of the wind turbine had been reduced from their former state by about 98.5. Also, the oscillation amplitude of the controlled system can reach approximately 7 centimeters and Ea=6.2 when (β1 = 5, β2 = 0.05, ω1 = 1.26) as shown in Fig 13, and reach approximately 1 centimeters and Ea=43 when (β1 = 5, β2 = 0.5, ω1 = 2.01) as shown in Fig. 14. In According to Fig. 15, with the addition of a PPF controller, energy is moved from the system before control to the system after control and the performance of PPF control is more effective at the primary and internal resonance case Ω1 ≊ ω1 ≊ ω2 with small values of values ω1 and ω2. A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2717 (a) (b) Figure 11: The OWTT vibrations before control atβ1 = 0, β2 = 0, ω1 = 1.26 (a) all-round response, (b) last 20 seconds of the all-round response (a) (b) Figure 12: The OWTT vibrations after control at β1 = 5, β2 = 0.5, ω1 = 1.26 (a) all-round response, (b) last 20 seconds of the all-round response A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2718 (a) (b) Figure 13: The OWTT vibrations after control at β1 = 5, β2 = 0.05, ω1 = 1.26 (a) all- round response, (b) last 20 seconds of the all-round response (a) (b) Figure 14: The OWTT vibrations after control at β1 = 5, β2 = 0.5, ω1 = 2.01 (a) all-round response, (b) last 20 seconds of the all-round response A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2719 Figure 15: Energy flow between uncontrolled and controlled systems occurs at Ω1 = ω1 = ω2 = 1.26 6.1. Comparison of analytical and numerical simulation As illustrated in Figures 16 and 17, this subsection compares numerical simulation for the controlled system of equation (2) with perturbation solutions of equations (17) and (18) at various values of the controller values β1 and β2 at primary and internal resonance Ω1 = ω1 = ω2. While the blue line denotes numerical integration, the red line represents the perturbation solution. These graphs show good agreement between the analytical and numerical simulation results. Figs. 17(a) and 18(a) represent the system response after control, but Figs. 17(b) and 18(b) represent the controller response. A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2720 (a) (b) Figure 16: Comparison of the controlled system’s analytical and numerical simulations at β1 = 5, β2 = 0.5, ω1 = 1.26 (a) system response, (b) control response. A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2721 (a) (b) Figure 17: Comparison of the controlled system’s analytical and numerical simulations at β1 = 5, β2 = 0.05, ω1 = 1.26 (a) system response, (b) control response. 6.2. The Poincaré maps This section examines stability and uses Poincaré maps to plot bifurcation diagrams. These maps are used to transform the complex response in phase space to a discrete map in the lower dimensional space. Figures 18 and 19 analyzed the wind turbine behavior under the simultaneous primary and internal resonance case Ω1 ≊ ω1, ω1 ≊ ω2 using differ- ent values of control unit parameters β1, β2 with response and Poincaré map, respectively. Figure 18(a, b) demonstrates how the system begins with a chaotic reaction before stabi- lizing and exhibiting periodic motion on Poincare’s maps at β1 = 0, β2 = 0. The controlled system has a quasi-periodic motion at β1 = 5, β2 = 0.5, as depicted in Figure 19(a, b). A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2722 (a) (b) Figure 18: (a)Time responses of the system before control, (b) Poincare´ maps of the system before control at β1 = 0, β2 = 0, ω1 = 1.26, σ1 = 0, σ2 = 0, A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2723 (a) (b) (c) (d) Figure 19: (a) Time responses of the system after control (b) Poincare´ maps of the system after control (c) Time responses of the controller (d) Poincare´ maps of the controller at β1 = 5, β2 = 0.5, ω1 = 1.26, σ1 = 0, σ2 = 0. 6.3. Comparison with published works In comparison to previous studies, Dagli et al. [10] performed a dynamic vibration analysis for the single degree of freedom (SDOF) equation of motion using Rayleigh’s energy. Hamed et al. [12] used an NPD controller to suppress the OWTT system’s behavior with multiple external and parametric excitation forces. To analyze the response and stability, the averaging method is used. The steady oscillations of the wind turbine had been reduced from their former state by about 91 and the effectiveness of control is A. Alsaadi , Y. S. Hamed / Eur. J. Pure Appl. Math, 16 (4) (2023), 2703-2728 2724 Ea=12. With this work, the dynamics behavior and stability of the offshore wind turbine system with external excitation and applying PPF controller are studied. The steady oscillations of the wind turbine had been reduced from their former state by about 98.5 and the effectiveness of control is Ea=72. 7. Concluding Remarks This study proposed a PPF controller to mitigate the offshore wind turbine tower model’s nontrivial oscillations. Using the multiple time scale method, the equation of mo- tion has been approximately solved. Using the phase plane technique, frequency response equations, and Poincare maps, the stability at worst resonance cases will be checked and analyzed. The whole work results can be summarized as follows: 1 . Before control, the wind turbine caused strong vibrations and jumps due to the ex- istence of bifurcation points. after control, the wind turbine showed stable solutions without jumps due to the absence of bifurcation points. 2 . The wind turbine vibrations reached minimum levels in the range σ1 ∈ [−0.3, 0.3] especially at σ1=0. 3 . Wind turbines operates safely in the range σ1 ∈ [−0.65, 0.65] when σ1 = σ2. 4 . Before control, the amplitude is 43 and after applying PPF control the amplitude become 0.6 then the effectiveness of control is Ea=72. 5 . The control signal gain β1 and the feedback signal gain β2 can be used to change the minimum amplitudes bandwidth. 6 . The wind turbine vibration amplitudes are very sensitive to the wind force am- plitude of fa before and after control and it is reasonable to channel most of the vibration energy to the controller. 7 . The steady oscillations of the wind turbine had been reduced from their former state by about 98.5. 8 . The controller damping µ1 and µ2 and natural frequencies ω1 and ω2 are inversely proportional with the minimum vibratory level reached at σ1 = σ2. 9 . Energy is moved from the system before control to the system after adding control at varying values ω1, ω2 and Ω1. 10 . Saturation phenomenon exhibit with change values of the control parameters β1 and β2. 11 . On Poincare’s maps, periodic motion may be seen before control and a quasi- periodic motion appear control. REFERENCES 2725 Funding: This research did not receive any specific external funding other than the fund- ing by the Deanship of Scientific Research, Taif University, Saudi Arabia with Project Number (1-443-1). Acknowledgements This research was supported by the Deanship of Scientific Research, Taif University, Saudi Arabia with Project Number (1-443-1). Conflicts of Interest: The authors declare no conflict of interest. References [0] M. A. Silva, J. S. Arora, and R. M. Brasil, “Formulations for the optimal design of RC wind turbine towers,” in Proceedings of the International Conference on Engineering Optimization (EngOpt 2008), Rio de Janeiro, Brazil, June 2008. [0] W. 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Appendix The elements rij (i, j= 1,· · ·, 4) of the Jacobian matrix J given in Eq. 26 r11 = −µ1 2 , r12 = fa 2ω1 sinθ10, r13 = β1 2ω1 sinθ20 r14 = β1a20 2ω1 cosθ20 r21 = − β1a20 2ω1a210 cosθ20 + fa 2ω1a210 sinθ10, r22 = − fa 2ω1a10 cosθ10, r23 = β1 2ω1a10 cosθ20, r24 = − β1a20 2ω1a10 sinθ20 r31 = − β2 2ω2 sinθ20, r32 = 0, r33 = −µ2 2 r34 = −β2a10 2ω2 cosθ20 r41 = − β2 2ω2a20 cosθ20 − β1a20 2ω1a210 cosθ20 + fa 2ω1a210 sinθ10 , r42 = − fa 2ω1a10 cosθ10 r43 = β2a10 2ω2a220 cosθ20 + θ1 2θ1a10 cosθ20 , r44 = β2a10 2ω2a20 sinθ20 − β1a20 2ω1a10 sinθ20 APPENDIX 2728 The elements Ri (i= 1,· · ·, 4) of Eq. 27: R1= − ∑4 i=1 rii R2= 1 2! ∑4 i=1 ∑4 j=1 riirjj−rijrji R3= − 1 3! ∑4 i=1 ∑4 j=1 ∑4 k=1 riirjjrkk−3riirjkrkj+2rijrjkrki R4= 1 4! ∑4 i=1 ∑4 j=1 ∑4 k=1 ∑4 l=1 riirjjrkkrll−6riirjjrklrlk+8riirjkrklrlj+3rijrjirklrlk−6rijrjkrklrli