Abstract
This work presents an adaptive model predictive control (MPC) strategy to suppress the vibration in a laminated composite beam. The control method incorporates a system identification algorithm to estimate the system parameters online, which provides a precise simulation of system dynamics. A fixed-free cantilever composite beam equipped with piezoelectric actuators was used to evaluate the efficacy of the control method. The sensors and actuators are securely bonded to the upper and lower surfaces at arbitrary locations along the beam’s length. A unified mechanical displacement field is applied to all layers, while displacements are considered independently for each layer. The beam is composed of eight layers of material, each with a thickness of 0.2 mm and orientations specified as (90°/0°/90°/0°). To achieve the best performance, the parameters of the MPC were adjusted numerically. The numerical analysis revealed that placing the actuator near the clamped end at the fixed end resulted in superior control outcomes, with a settling time of approximately 1.8 s. Conversely, the longest settling time occurred when the actuator was positioned at the free end, taking around 4 s. This model could potentially be expanded to address vibration in more intricate beams exhibiting nonlinear characteristics. The deflection readings measured at the end of the beam have been utilized as feedback control signals for predicting future behavior over a predetermined control horizon. The subsequent cost function is minimized through a quadratic equation to determine the sequence of optimal yet constrained control inputs. The suggested active vibration control system is then implemented and assessed numerically to examine the effectiveness of the control method.
1 Introduction
Although the beam as a structure itself is simple in nature, the dynamic characteristics of fixed-free cantilever beams can emulate the behavior observed in real-life structures, such as aircraft wings and helicopter rotors. On the other hand, lightweight composite structures play a crucial role in various industries, especially in aerospace structure applications. Composite structures offer numerous advantages for aircraft and spacecraft, primarily due to their flexibility and adaptability [1,2]. The rising use of composite materials instead of traditional ones in structural tasks is due to their strong strength-to-weight ratios and additional benefits like fatigue resistance and low friction coefficients. Typically, orthotropic materials involve rigid, continuous, one-directional reinforcing fibers within a softer matrix including different laminating technique. The superiority of these fiber-reinforced materials for structural uses has led to extensive research in the mechanics of such materials [3]. The Honeycomb geometry is the most common type of smart composite beams [4]. However, simulating the vibration of smart composite beams requires rigorous mathematical modeling depending on the nature of the structure, for instance, simulating laminated glass responses to extreme loading scenarios in protective structure design [5], externally bonded spent catalyst based ferrocement laminates [6], and the simulation can be extended further for more complicated 2D problem such as laminated composite plate [7]. Researchers have focused on developing approximate solutions for certain weak nonlinear problems, such as the mean technique and the method of small parameters, to address concerns regarding how generally orthotropic beams respond to free vibrations. One effective approach proven to handle such nonlinear problems is the trial solution method [8], and more details can be found in the study by Zuo [9]. However, these lightweight designs are prone to excessive vibrations, including phenomena like flutter and fatigue failure. However, there has not been an extensive study on the control and dynamic characteristics of laminated composite beams when compared to plates and shells [10,11].
The dynamics of composite beams usually involve inherent nonlinearities that emerge from material layers, nature of their physical properties, which is considered a challenge in the analysis of the beam dynamics when the nonlinear analysis is considered. In general, the behavior of the linear system is predictable, unlike that of the nonlinear system. To overcome the challenges associated with the nonlinear analysis of composite beams, the researchers developed advanced numerical methods that were able to handle the strong system nonlinearities and simulate it accurately, and eventually these methods were validated experimentally.
In most cases, vibration in beams is undesirable; however, controlling the vibration in composite beams required accurate simulation and dynamics analysis in order to design a robust control system. A simple control system can handle the linear behavior of the beam; however, a more advanced control methodology is crucial to address the system’s nonlinearities and reject discontinuities effectively. In fact, simulating the nonlinear dynamics of composite beams is computationally expensive and requires advanced coding and programming efforts. Some adaptive control strategies, such as model predictive controller (MPC), are proven to show superior performance when integrated into system with strong nonlinearities where a system identification algorithm is included to identify the system characteristics momentarily during the operation regardless of the strength of the system nonlinearity and undesired disturbances.
Active control methodologies are the most effective among the other control methods, namely, the passive ones [12]. The interest in active vibration control methods is motivated by their cost efficiency, which is technically supported by the rapid advancements in electronic technologies. Various control techniques have been conducted for the vibration suppression of flexible structures. For instance, sliding-mode controller was implemented to control the vibration of flexible beam by imposing force through piezoceramic sensors and actuators [13]. Another study on active control method was presented by Abreu et al. [14] who implemented a linear quadratic Gaussian controller along with piezoelectric sensors and actuators to effectively control the vibrations of a flexible beam. Different classical strategies, such as pole allocation and optimal control, can be implemented when an accurate mathematical model is developed. However, the majority of real applications involve inherent nonlinearities; hence, classical methods may not be effective. For instance, some models do not consider the properties of individual elements, such as the voltage amplifier and piezoelectric actuator; therefore, the control force achieved is assumed to be identical to the one applied by the actuator [15].
An active control strategy was reported by Chen et al. [16] to suppress the vibration in high-speed flexible structures using complex mode method and nonlinear control scheme. Linear and nonlinear control algorithms were utilized to control a single L-beam by using a MFC actuator [17]. The focus of the authors was on saturation control, and they outlined the advantages and disadvantages of implementing these methods of control. Another study introduced the application of the Positive position feedback (PPF) controller to control the vibration in a flexible manipulator [18], where various vibration modes were considered in the control strategy by utilizing a linear mathematical simplified model of the plant. Experimental investigations showed that the PPF algorithm demonstrated effective performance. A similar methodology was conducted by Warminski et al. [15], where the efficacy of applying the PPF algorithm to a solar panel model was assessed. The utilization of the block-inverse technique enabled the control of more modes than the actual number of applied actuators and sensors. The design of a nonlinear controller, serving as a vibration absorber for a linear model of a cantilever beam considering saturation phenomena, was investigated by Oueini et al. [19], where the analytical multiple scale method was implemented to demonstrate the impact of loop gains and controller damping. For systems with strong nonlinearity, the Model Predictive Controller (MPC) strategy can be effectively implemented in different means to eliminate the undesired vibration and reject the disturbance [20]. Model Predictive Control (MPC), also referred to as receding-horizon control, is commonly utilized for its superior tracking performance and adept handling of system constraints, especially in settings prone to periodic disturbances. Nevertheless, the practical implementation of its implicit versions encounters real-time challenges. Generally, an MPC algorithm relies on a linear quadratic cost function to define controller performance. The system variables are momentary updated via an online system identification algorithm to provide an accurate estimation of the system characteristics.
This work presents a nonlinear adaptive control strategy to suppress the vibration in composite beams. The control method utilizes feedback-measured signals received by an adaptive model-predictive controller (MPC), which operates along with a system identification algorithm to update the system parameters (characteristics) momentarily. The effectiveness of the methodology is tested on clamped-free ends of laminated beams, where the numerical model of the beam dynamics is obtained by using the finite element method (FEM) incorporated with modal analysis. The tip deflection at the free end is measured and fed back to the MPC, where its Generalized predictive control (GPC) block minimizes the cost function for an optimal set of upcoming control actions. Utilizing nonlinear MPC effectively allows future possible studies to address beam vibration problems with more complicated nonlinearities such as vibration in composite plates.
2 Methodology
2.1 Mathematical model
This section presents the mathematical dynamics model of a composite beam where a pair of piezoelectric actuators and sensors are located at a predetermined location on both sides along the beam length. It is assumed that the plane stress case holds since both elastic and piezoelectric layers are considered to be thin. The displacement field for all layers are considered unified, whereas the electrical displacements are assumed to be independent for each layer. The numerical model incorporates the dynamics of a composite beam of length (L) clamped at one end, as shown in Figure 1. The figure illustrates the coordinate system specific to a laminated composite beam. u(x, z, t) and w(x, z, t) are the displacements in the x and z directions, respectively. The orientation of the layers with respect to the x-axis is represented by (
Schematic diagram of the composite beam and piezoelectric patch.
By utilizing the third-order shear deformation theory, the beam is discretized into a number of finite segments using the FEM. This formulation results in a coupled finite element model incorporating both mechanical degrees of freedom (displacements) and electrical degrees of freedom (potentials of piezoelectric patches) [21].
where
and
The equations of motions described in equation (1) can be simplified by utilizing modal analysis technique to consider only first modes and can be rewritten as [21]
where
2.2 Design of the MPC
The basic idea behind developing an MPC method is to utilize models to predict future outcomes (Np) by taking different actions into account over a specific period of time (Nu). In order to enhance the system’s response, it also considers previous acts. The idea is to ensure that next steps maintain the system very close to the target path by understanding what was done previously. Using a system model to improve controller performance gives MPC multiple advantages over other control strategies approaches, particularly when handling complicated dynamics. Furthermore, by updating its understanding in real-time throughout the process, it can adjust to changes in the behavior of the system. By applying modifications to the system model as it proceeds, the nonlinearity of the system can be handled online as shown in Figure 2.
Feedback control loop of adaptive nonlinear MPC.
In this study, a GPC is implemented to the linearized form of a composite beam. The beam model is represented by the autoregressive with extra input (ARX) model, which is described as follows [12]:
where u(n) and y(n) denote the input (voltage) and output (displacement) of the control system, respectively. The backward shift operator q (−1) signifies a one-step delay in discrete time. The system disturbances e(k) represents the white noise (negligible in this work) as follows:
where
where
and calculated by using:
where
where
The model parameters were calculated and updated online, through system identification algorithm, which significantly enhanced the model estimation when compared to fixed model parameters run. Subsequently, the ARX model estimation was utilized to optimize the system’s control action through the implementation of the GPC algorithm. This algorithm is designed to minimize the cost function J as follows [22]:
where
2.3 Simulation procedure
The study was designed with a focus on simulation to assess the effectiveness of an MPC in suppressing vibrations in a composite beam. The design phase involved a finite element analysis (FEA) model of the beam that was implemented to simulate its dynamic behavior under various conditions. The MPC was designed to predict the future states of the beam based on its model and to compute control actions that minimize vibrations. The MPC adjusts the inputs of actuators attached to the beam to achieve the required level of vibration suppression. Model predictive controllers (MPCs) are commonly used in process control. They work by comparing what we want a process to do (the setpoint) with what it is actually doing (the measured variable). Then, they adjust the process to minimize the difference between these two values. The MPC has parts that help it understand the system better, which can change how the system works temporarily. When we give the beam a push and it starts vibrating, a sensor called a piezoelectric transducer (PZT) feels this movement and notifies the MPC controller. The MPC then acts to reduce the vibration by adjusting things based on a cost-benefit analysis. MPCs are flexible and can handle tricky systems with delays, changes, and outside disturbances. The MATLAB simulation package is used to develop a code for the MPC algorithm and the FEA. The output responses from the code were plotted and demonstrated to illustrated the effectiveness of the control method.
3 Numerical results
The proposed control strategy was tested by suppressing the vibration of a composite beam. The beam was excited by an initial displacement at the free tip such that the beam undergoes free vibration. In this scenario, we examine a cantilever-symmetric laminated beam with a length of 0.6 m and a width of 0.022 m. The beam comprises eight layers of material, each with a thickness of 0.2 mm and orientations specified as (90°/0°/90°/0°). The piezoelectric actuator and sensor are constructed from PZT, featuring thicknesses of 0.15 mm and lengths of 40 mm. The sensor is located at the tip of the beam, and the actuator location is to be assigned for each test. The permissible electric field for piezoceramic materials is within the range of 400–1,100 V/mm. The actuator thickness is 0.15 mm, and the maximum allowable voltage is set at 180 volts. The material properties of the graphite–epoxy layer and PZT are listed in Table 1.
Properties of the composite beam and PZT patch [23]
| Part | Properties |
|---|---|
| Composite beam | E x = 40.59 GPa, E y = 13.96 GPa, E z = 13.96 GPa |
| μ xy = 0.22, μ yz = 0.11, μ xz = 0.11 | |
| G xy = 3.1 GPa, G yz = 1.55 GPa, G xz = 3.1 GPa | |
| ρ = 1,830 kg/m3 | |
| PZT | E = 139 GPa, μ = 0.3, ρ = 7,350 kg/m3 |
The system identification block must first be tested to determine its capacity for accurate system state estimation before the numerical algorithm blocks of the MPC can be used. In order to identify the system, the response to a pseudorandom generated tip displacement input signal that varies over a random period of time for 15 s was simulated, as shown in Figure 3. The system’s control action was then optimized using the predicted ARX model by applying the GPC algorithm, which minimizes the cost function, J, described in equation (12). We can now move forward to test the controller since it is evident that the system identification process was able to estimate the dynamics of the system for each input that was applied.
Normalized output from system identification algorithm block for simulated system to pseudorandom generated inputs.
Similar to the PID controller, the MPC controller also requires tuning. The weighting parameter (
Normalized responses for the composite beam with and without controller during MPC tuning process. (a) Nu = 5 and (b) Nu = 10.
Figure 5 illustrates the system responses to the active control system for three actuator locations, namely, fixed end, midpoint of the beam, and free end. The controlled response showed that installing the actuator at the fixed end (near the clamped end) results in better control action where the settling time is around 1.8 s. Whereas the longest settling time was observed when the actuator was located at the free end (around 4 s). This is attributed to the fact that controlling vibrations at points with lower amplitudes is a path to effective vibration control. Also, by applying control forces to the fixed end, the transmission of control inputs throughout the entire beam becomes more efficient. However, the performance of the MPC controller remains effective in suppressing vibration within a relatively short time frame, as evidenced by its comparison to conventional control methods. Moreover, the effectiveness of Model Predictive Control (MPC) rises from its predictive abilities, capability to handle constraints, and rejecting undesired disturbances. Also, incorporating system identification algorithms into the MPC control loop enhances its suitability for nonlinear systems.
Normalized tip displacement with and without control, when the actuator is placed at (a) the fixed end, (b) the midpoint of the beam, and (c) at the free end of the beam.
4 Conclusion
This work proposes an active control methodology to suppress the vibration in a composite cantilever beam. A Model Predictive Control (MPC) strategy was implemented, and the primary goal of the control is to restore the structure from a perturbed state to its initial state with minimal control effort. In order to improve the performance and robustness of the MPC controller, this study proposed the use of system identification algorithm to estimate system states online to approximate the characteristics of the dynamic model. The control action was imposed by piezoelectric actuator located at arbitrary locations along the beam length. By utilizing the third-order shear deformation theory, the beam is discretized into a number of finite segments using the FEM. The beam was excited by applying an initial displacement at the free tip such that the beam undergoes free vibration. In this scenario, we examined a cantilever-symmetric laminated beam with a length of 0.6 m and a width of 0.022 m. The beam comprises eight layers of material, each with a thickness of 0.2 mm and orientations specified as (90°/0°/90°/0°). The parameters of the MPC controller were tuned numerically to obtain the optimal performance. The numerical results showed that installing the actuator at the fixed end (near the clamped end) results in better control results where the settling time is around 1.8 s. Whereas the longest settling time was observed when the actuator was located at the free end (around 4 s). The proposed model can be extended to address vibration of more complex beams with nonlinear characteristics. The complexities arising from the interactions among different material layers and the intricate load-bearing mechanisms make nonlinearities in composite beams particularly challenging and requires effective control system. However, although the MPC is powerful enough to handle nonlinear systems and provide fast and stable control system, it is considered computationally expensive and complicated to code. The method can be implemented effectively in more complicated highly nonlinear problems such as large deformation in laminated plates.
-
Conflict of interest: Authors state no conflict of interest.
-
Data availability statement: Most datasets generated and analyzed in this study are comprised in this submitted manuscript. The other datasets are available on reasonable request from the corresponding author with the attached information.
References
[1] Nuhoğlu K, Aktaş E, Tanoğlu M, Martin S, İplikçi H, Barisik M, et al. Analysis of adhesively bonded joints of laser surface treated composite primary components of aircraft structures. Int J Adhes Adhes. 2023;126:103456. 10.1016/j.ijadhadh.2023.103456.Search in Google Scholar
[2] Duc DM, Hung TQ, Tu TM. Analytical and mesh-free approaches to dynamic analysis and active control of smart FGP-GPLRC beam. Structures. 2023;56:105020. 10.1016/j.istruc.2023.105020.Search in Google Scholar
[3] He C-H, El-Dib YO. A heuristic review on the homotopy perturbation method for non-conservative oscillators. J Low Freq Noise Vib Active Control. Dec. 2021;41(2):572–603. 10.1177/14613484211059264.Search in Google Scholar
[4] Safaei B, Onyibo EC, Goren M, Kotrasova K, Yang Z, Arman S, et al. Free vibration investigation on RVE of proposed honeycomb sandwich beam and material selection optimization. Facta Univ Ser Mech Eng. Apr. 2023;21:31–50. 10.22190/FUME220806042S.Search in Google Scholar
[5] Hejazi MA, Sari A. A comparative study on the optimal modeling of laminated glass. Civ Eng J. Nov. 2023;9(11):2896–913. 10.28991/CEJ-2023-09-11-018.Search in Google Scholar
[6] Balamuralikrishnan R, Al-Mawaali ASH, Al-Yaarubi MYY, Al-Mukhaini BB, Kaleem A. Seismic upgradation of RC beams strengthened with externally bonded spent catalyst based ferrocement laminates. HighTech Innov J. Mar. 2023;4(1):189–209. 10.28991/HIJ-2023-04-01-013.Search in Google Scholar
[7] Mohammed WH, Shambina S, Ammash HK. Reducing effects of initial imperfection by investment in the orthotropic characteristics of laminated composite plate. Civ Eng J. Jul. 2023;9(7):1579–96. 10.28991/CEJ-2023-09-07-03.Search in Google Scholar
[8] He C-H, Liu C. A modified frequency–amplitude formulation for fractal vibration systems. Fractals. Dec. 2021;30(3):2250046. 10.1142/S0218348X22500463.Search in Google Scholar
[9] Zuo Y. A gecko-like fractal receptor of a three-dimensional printing technology: a fractal oscillator. J Math Chem. Mar. 2021;59:1–10. 10.1007/s10910-021-01212-y.Search in Google Scholar
[10] Feng L, Zhong H, Hao Z, Wu D. Free vibration analysis of laminated composite beams using differential quadrature method. Tsinghua Sci Technol. 2002;7(6):567–73.Search in Google Scholar
[11] Pankaj S, Kant, Jawalkar CS. Comparative modal analysis of cantilever beam made of biocomposites using finite element analysis. Mater Today Proc. 2022;49:2330–4. 10.1016/j.matpr.2021.09.351.Search in Google Scholar
[12] Hameury C, Ferrari G, Buabdulla A, Silva TM, Balasubramanian P, Franchini G, et al. Multiple-input multiple-output active vibration control of a composite sandwich beam by fractional order positive position feedback. Mech Syst Signal Process. 2023;200:110633. 10.1016/j.ymssp.2023.110633.Search in Google Scholar
[13] Ma Y, Lou X, Wu W. Sliding mode vibration control of Euler–Bernoulli beam with unknown bounded disturbances. Computers Electr Eng. 2021;96:107504. 10.1016/j.compeleceng.2021.107504.Search in Google Scholar
[14] Abreu G, Ribeiro J, Steffen Valder J. Experiments on optimal vibration control of a flexible beam containing piezoelectric sensors and actuators. Shock Vib. Jan. 2003;10:283–300. 10.1155/2003/594083.Search in Google Scholar
[15] Warminski J, Cartmell MP, Mitura A, Bochenski M. Active vibration control of a nonlinear beam with self- and external excitations. Shock Vib. 2013;20:792795. 10.3233/SAV-130821.Search in Google Scholar
[16] Chen J, Qu W, Ye C, Zhao Z, Wang H. Nonlinear vibration and dynamic stability of dielectric sandwich micro-beams. Int J Mech Sci. 2023;262:108738. 10.1016/j.ijmecsci.2023.108738.Search in Google Scholar
[17] Warminski J, Bochenski M, Jarzyna W, Filipek P, Augustyniak M. Active suppression of nonlinear composite beam vibrations by selected control algorithms. Commun Nonlinear Sci Numer Simul. 2011;16(5):2237–48. 10.1016/j.cnsns.2010.04.055.Search in Google Scholar
[18] Shan J, Liu H-T, Sun D. Slewing and vibration control of a single-link flexible manipulator by positive position feedback (PPF). Mechatronics. 2005;15(4):487–503. 10.1016/j.mechatronics.2004.10.003.Search in Google Scholar
[19] Oueini SS, Nayfeh AH, Pratt JR. A nonlinear vibration absorber for flexible structures. Nonlinear Dyn. 1998;15(3):259–82. 10.1023/A:1008250524547.Search in Google Scholar
[20] Alsahlani A, Ozalp N. Experimental performance of a nonlinear control strategy to regulate temperature of a high-temperature solar reactor. J Sol Energy Eng. May 2023;145(5):051011. 10.1115/1.4062483.Search in Google Scholar
[21] Zabihollah A, Sedagahti R, Ganesan R. Active vibration suppression of smart laminated beams using layerwise theory and an optimal control strategy. Smart Mater Struct. 2007;16(6):2190. 10.1088/0964-1726/16/6/022.Search in Google Scholar
[22] Al Sahlani A, Ozalp N. An overview of process temperature control in solar thermochemical energy storage systems. 2023.10.1615/AnnualRevHeatTransfer.2023047757Search in Google Scholar
[23] Yashavantha Kumar GA, Sathish Kumar KM. Free vibration analysis of smart composite beam. Mater Today: Proc. 2017;4:2487–91. 10.1016/j.matpr.2017.02.101.Search in Google Scholar
© 2024 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
