Utilizing Macro Fiber Composite to Control Rotating Blade Vibrations

This work applies an active control algorithm, using a macro fiber composite (MFC) to mitigate the unwanted vibrations of a rotating blade. The algorithm is a second-order oscillator, having the positive displacement signal of the blade for input and the suitable control force to actuate the blade for output. This oscillator is linearly coupled with the blade, having in mind that their natural frequencies must be in the vicinity of each other. The rotating blade is modeled by representing two vibrational directions that are linearly coupled. An asymptotic analysis is considered to understand the resulting nonlinear phenomena. Several responses are included to portray the dynamical behavior of the system under control. From the results, we observe the asymmetry between the blade’s vibrational directions. Moreover, a verification is included for comparing the analytical and numerical results.


Introduction
Rotating blades are important structures, being fundamental in many industrial fields like robotics, aerospace engineering, turbomachinery, and others. Due to their high rotating speeds, they may suffer from unwanted oscillations, causing disturbance and damage to, or even destruction of, the whole mechanism. Consequently, many researchers have focused their attention on the analysis and control of these vibrations to achieve maximum safety and optimum operation of such dynamical systems.
Yoo et al. [1] investigated, numerically, the effects of some dimensionless parameters on the vibration characteristics of a rotating blade and considered combinatory effects among these parameters. Lin and Chen [2] used a finite element analysis to study the stability problems of a rotating pre-twisted blade with a viscoelastic core constrained by a laminated face layer and subjected to a periodic axial load. They derived a system of equations of motion, governing the bending and extensional displacements through the Hamilton principle. Oh et al. [3] developed a structural beam model accounting for the fibrous composite material effects and the induced elastic couplings. Librescu et al. [4] considered the modeling of a spinning, thin-walled beam made of functionally graded materials, featuring a pre-twist and experiencing bending-bending coupled motion. Sinha [5] derived a complete set of coupled dynamic equations to analyze the effect of a Coulomb damper near the blade tip with a flexible blade mounted on a flexible rotating shaft. Hamdan and El-Sinawi [6] developed a slender, flexible arm dynamic model undergoing relatively large planar flexural deformations with a setting angle and a The motivation of this work was to merge MFC and one of the active control algorithms. The PPF active control algorithm is effective at suppressing mechanical vibrations. The role of MFC sensors is to supply the PPF controller with the feedback signal, and the role of the MFC actuators is to receive the control signal from the PPF controller. In this paper, we control the rotating blade vibrations theoretically by applying the PPF control algorithm through MFC sensors and actuators, as presented in Figure 1. The whole control process is pictured in Figure 1 to show how the MFC sensors acquired the feedback signal to be processed and conditioned, and then create a control signal to be supplied to the MFC actuators. The main difficulty was in finding a small-error approximate solution for the nonlinear model of the blade. Using the multiple scales perturbation technique, we obtained an approximate solution for the studied model in good agreement with the numerical simulations. Several response curves are included to portray the dynamical behavior of the blade under control. Moreover, a section of verification curves is included to compare the analytical and numerical results. The motivation of this work was to merge MFC and one of the active control algorithms. The PPF active control algorithm is effective at suppressing mechanical vibrations. The role of MFC sensors is to supply the PPF controller with the feedback signal, and the role of the MFC actuators is to receive the control signal from the PPF controller. In this paper, we control the rotating blade vibrations theoretically by applying the PPF control algorithm through MFC sensors and actuators, as presented in Figure 1. The whole control process is pictured in Figure 1 to show how the MFC sensors acquired the feedback signal to be processed and conditioned, and then create a control signal to be supplied to the MFC actuators. The main difficulty was in finding a small-error approximate solution for the nonlinear model of the blade. Using the multiple scales perturbation technique, we obtained an approximate solution for the studied model in good agreement with the numerical simulations. Several response curves are included to portray the dynamical behavior of the blade under control. Moreover, a section of verification curves is included to compare the analytical and numerical results.
The rest of the paper includes three sections. Section 2 presents the equations of motion of the controlled rotating blade model and the approximate solutions to these equations. Section 3 analyzes the different response curves of the blade vibrations before and after being controlled. Moreover, verification of the responses is also included. Section 4 summarizes the conclusions of the work.

Equations of Motion and Their Approximate Solutions
The partial differential equations governing the horizontal and vertical deflections-( ) and ( )-of the blade's cross-section were derived briefly in Appendix A and are detailed in [12,13]. The equations were discretized by a one-term Galerkin's procedure to have the ordinary differential equations governing the temporal horizontal ( ) and vertical ( ) displacements of the blade's cross-section. The extracted equations of motion can be written as follows: Applying the nonlinear PPF controller to Equation (1), the motion equations can be written as + Ω sin(Ω ) + (2a) The rest of the paper includes three sections. Section 2 presents the equations of motion of the controlled rotating blade model and the approximate solutions to these equations. Section 3 analyzes the different response curves of the blade vibrations before and after being controlled. Moreover, verification of the responses is also included. Section 4 summarizes the conclusions of the work.

Equations of Motion and Their Approximate Solutions
The partial differential equations governing the horizontal and vertical deflections-p(t) and q(t)-of the blade's cross-section were derived briefly in Appendix A and are detailed in [12,13]. The equations were discretized by a one-term Galerkin's procedure to have the ordinary differential equations governing the temporal horizontal p(t) and vertical q(t) displacements of the blade's cross-section. The extracted equations of motion can be written as follows: ..
Applying the nonlinear PPF controller to Equation (1), the motion equations can be written as .. ..
For O ε 0 : For O ε 1 : Based upon the theory of linear differential equations, the complex form solutions of Equation (6) are Preliminary numerical simulations of the controlled blade model revealed that the worst resonance occurred for cases where Ω ω and ω c ω. Therefore, the parameters σ 1 and σ 2 were imposed to express the detuning of the considered resonance case so that To avoid any unbounded solutions, and from the theory of linear differential equations, we can eliminate the secular terms in Equation (7) by equalizing the coefficients of e iωT 0 and e iω c T 0 to zero. Equations (8) and (9) are inserted into Equation (7). Then, using Equation (3), we can return every parameter to the real time t, which leads to −2iω The functions A n (n = 1, 2, 3) can be formulated in a polar form as A n = a n 2 e iδ n ⇒ . A n = . a n 2 e iδ n + i a n 2 .
δ n e iδ n (11) where a n and δ n are the amplitudes and phases of the blade vibrational directions and controller signal, respectively. Inserting Equation (11) into Equation (10), separating real and imaginary parts, and simplification gives us The steady-state form of Equation (12) can be reached by imposing the condition that . a n = . φ n = 0 (n = 1, 2, 3), but the resulting expressions cannot be solved explicitly, at which point we should resort to the Newton-Raphson numerical technique. A stability analysis needs to be done to examine whether the steady-state solutions are stable or not. Suppose that the amplitudes a n are composed of perturbation amplitudes a np added to the steady-state amplitudes a ns . Similarly, suppose that the phases φ n are composed of perturbation phases φ np added to the steady-state phases φ ns . This can be formulated as follows: a n = a np + a ns ⇒ . a n = . a np (13a) Inserting Equation (13) into Equation (12) and linearizing give us where J is the Jacobian matrix whose elements are given in Appendix B. The characteristic equation of J is a sixth degree equation in the following form: where γ i (i = 1, · · · , 6) are given in Appendix B. The roots λ of Equation (15) are the eigenvalues of J, and they determine whether the system solutions are stable or not. If the real parts of the eigenvalues λ are negative, then the steady state solution will be asymptotically stable; otherwise, it will be unstable.
To guarantee the stability of the solutions, the Routh-Hurwitz theorem is considered to derive the criteria for stable solutions: Moreover, if there is a sign change in a real eigenvalue, then a saddle-node bifurcation point will appear. This corresponds to the condition γ 6 = 0. Additionally, if there is a sign change in the real parts of a pair of complex conjugate eigenvalues, then a Hopf bifurcation point will appear. This corresponds to the following conditions:

Results and Discussion
In this section, we analyze the different responses of the blade vibrations and the control signal to variations of the excitation frequency and force amplitude. These responses are plotted using Equations (2) and (12), with the aid of the stability criteria in Equation (16).

Effects of the Parameter Variation on the Blade Steady-State Vibrational Amplitudes
Hereafter, we adopt the parameter values µ = 0.5, µ c = 0.005, Ω = ω = ω c , β 11 = −0.003, β 21 = −0.001, β 13 = β 22 = −0.82, β 14 = 0.55, β 24 = 0.5, β 5 = 0.9, β 16 = 6.55, f 0 = 7, f = 2, α = 0, c 1 = c 2 = 1000, and σ 1 = σ 2 = 0. The saddle-node bifurcation and Hopf bifurcation points are denoted by SN and H, respectively. These points were explored due to the criteria in Equation (17). Figure 2 shows how the blade's horizontal and vertical amplitude responded to the excitation frequency detuning σ 1 before control. It is clear that the blade was in a stable motion until it got into the range of 0 < σ 1 < 3, where jumps may happen due to the existence of bifurcation points. After control, the blade's horizontal and vertical amplitudes responded to the excitation frequency detuning σ 1 , as shown in Figure 3, achieving a stable motion all over the range of σ 1 . The bifurcation points disappeared, and no jumps were present. The blade is preferred to operate in the region of σ 1 ∈ [−4, 4], especially at the point σ 1 = 0, for which the blade is at its minimum vibratory level.           Figure 7 shows how the variation of the controller frequency detuning σ 2 affected the blade and the controller responses to the excitation frequency detuning σ 1 . We verified that the blade and the controller were at minimum vibratory values when −5 ≤ σ 1 = σ 2 ≤ +5. This can be achieved only by equalizing the controller natural frequency ω c and the blade excitation frequency Ω.  Figures 8 and 9 illustrate the hardening or softening effects that could be imposed on the blade and the controller responses to the excitation frequency detuning by varying the controller nonlinearity parameter . However, the bifurcation points, which led to the problem of jump phenomena, emerged again. Based on Figures 8 and 9, we verified that the value of should be within the range ∈ −3, 3 .     Figure 10 portrays the blade's horizontal and vertical responses to the excitation amplitude f before and after control at σ 1 = σ 2 = 0. The blade vibration amplitudes were very sensitive to small rises of the excitation amplitude before control. After control, the blade vibrations were saturated at a vibration level of almost zero, and most of the vibration energy was channeled from the blade sections to the controller. From Figures 2-10, we can notice the asymmetry between the blade's vibrational directions.

Time Responses
The blade's horizontal and vertical temporal displacements, before and after control, are represented in Figures 11 and 12, respectively. The simulation was conducted using MATLAB and the ODE45 package to integrate Equation (2) numerically. From the figures, we can see that the control algorithm was successful in suppressing the blade vibrations to almost a level of zero. This optimum level can be the best approach if the tuning condition = is guaranteed, as was discussed before.

Time Responses
The blade's horizontal and vertical temporal displacements, before and after control, are represented in Figures 11 and 12, respectively. The simulation was conducted using MATLAB and the ODE45 package to integrate Equation (2) numerically. From the figures, we can see that the control algorithm was successful in suppressing the blade vibrations to almost a level of zero. This optimum level can be the best approach if the tuning condition σ 1 = σ 2 is guaranteed, as was discussed before.

Validation Curves
This section shows the closeness between the analytical and numerical solutions in Figures 13-19 for the responses shown above. These results confirm the validity of the proposed modeling and control strategy.

Validation Curves
This section shows the closeness between the analytical and numerical solutions in Figures 13-19 for the responses shown above. These results confirm the validity of the proposed modeling and control strategy.

Validation Curves
This section shows the closeness between the analytical and numerical solutions in Figures 13-19 for the responses shown above. These results confirm the validity of the proposed modeling and control strategy.

Conclusions
This work addressed the control of the rotating blade vibrations via MFC sensors and actuators by applying the PPF algorithm. An asymptotic analysis extracted the equations governing the nonlinear dynamics of the controlled blade. Several responses were included to portray the dynamical behavior of the blade under control. Moreover, an extensive comparison was fulfilled to show the closeness between the analytical and numerical results.
We can summarize the results as follows: 1.
Before control, the blade suffered from severe vibrations and jumps due to the existence of bifurcation points. After control, the blade exhibited stable solutions without jumps due to the absence of bifurcation points; 2.
The minimum amplitude bandwidth could be adjusted via the control signal gain c 1 or the feedback signal gain c 2 ; 4.
The controller damping µ c was inversely proportional with the minimum vibratory level reached at σ 1 = σ 2 ; 6.
The controller nonlinearity parameter α should stay in the range of α ∈ (−3, 3), either for hardening or softening effects in the response curves without creating new jumps; 7.
The blade vibration amplitudes were very sensitive to small rises in the excitation amplitude f before control. However, after control, they became saturated at a level of almost zero thanks to channeling most of the vibration energy to the controller.
The considered thin-walled blade of length and thickness ℎ , shown in Figure A1, is connected to a rigid hub, which spun with an angular velocity Ω. A harmonic excitation = + cos(Ω ) may affect the blade's rotation. Due to the blade's flexural vibration, an angle appeared, and for twisting, another angle = / appeared. A bunch of assumptions should be imposed to continue the analysis: 1. There will be no deformation in the cross-section for the long-term operation; 2. The blade's thickness is very small, compared with its radius of gyration; 3. The blade can be considered an Euler-Bernoulli beam to neglect the shear force transversally; 4. In addition, we can neglect the elongation axially compared to the shown deflections. Figure A1. Cross-section plane of the studied blade.
According to axes rotation rules, the rotary axes and could be related to the stationary axes and . The extended Hamilton principle was used for deriving the equations of motion. Considering and are the kinetic and strain energies, respectively, is the virtual work due external forces, and is the variation operator, we have In [12,13], detailed substitutions and extractions have been fulfilled to get the normalized equations of motion as According to axes rotation rules, the rotary axes x p and y p could be related to the stationary axes x and y. The extended Hamilton principle was used for deriving the equations of motion. Considering K and U are the kinetic and strain energies, respectively, W is the virtual work due to external forces, and δ is the variation operator, we have t 0 (δK − δU + δW)dt = 0 (A1) In [12,13], detailed substitutions and extractions have been fulfilled to get the normalized equations of motion as ..
The single-term Galerkin procedure is applied to Equation (A2) to represent the vibrational modes as where p(t), and q(t) are the temporal deflections of the blade. The linear free undamped mode G(z) takes the form p + β 21 p + β 5 p 2 q + β 5 q 3 = 2 f 0 f β 24 q cos(Ωt) + f 2 β 24 q cos 2 (Ωt) where all the parameters are given in detail in [12,13].

Appendix B
The elements r ij (i, j = 1, · · · , 4) of the Jacobian matrix J given in Equation (14) are as follows: The elements of the stability criteria given in Equation (16) are as follows: r ii r jj − r ij r ji 6 j=1 6 k=1 r ii r jj r kk − 3r ii r jk r k j + 2r ij r jk r ki 6 j=1 6 k=1 6 l=1 r ii r jj r kk r ll − 6r ii r jj r kl r lk + 8r ii r jk r kl r lj + 3r ij r ji r kl r lk − 6r ij r jk r kl r li (r ii r jj r kk r ll r mm − 10r ii r jj r kk r lm r ml + 20r ii r jj r kl r lm r mk −20r ij r ji r kl r lm r mk + 15r ii r jk r k j r lm r ml − 30r ii r jk r kl r lm r mj + 24r ij r jk r kl r lm r mi ) (r ii r jj r kk r ll r mm r nn − 15r ii r jj r kk r ll r mn r nm +40r ii r jj r kk r lm r mn r nl − 90r ii r jj r kl r lm r mn r nk + 144r ii r jk r kl r lm r mn r nj +45r ii r jj r kl r lk r mn r nm − 15r ij r ji r kl r lk r mn r nm − 120r ij r jk r kl r lm r mn r ni +40r ij r jk r ki r lm r mn r nl + 90r ij r ji r kl r lm r mn r nk − 120r ii r jk r k j r lm r mn r nl )