Vibration control of smart cantilever beam using finite element method

Vibration control of smart cantilever beam using finite element method

Alexandria Engineering Journal (2019) xxx, xxx–xxx H O S T E D BY Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej ...

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Alexandria Engineering Journal (2019) xxx, xxx–xxx

H O S T E D BY

Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

Vibration control of smart cantilever beam using finite element method Mark A. Kamel a, Khalil Ibrahim b, Abo El-Makarem Ahmed b a b

LEONI Wiring Systems Egypt Company, Egypt Department of Mechanical Engineering, Faculty of Engineering, Assiut University, Assiut, Egypt

Received 20 July 2018; revised 22 December 2018; accepted 12 May 2019

KEYWORDS Dynamic model; Finite element; Modal analysis; Harmonic analysis; PZT; PID; PD; Fuzzy; PID-AT; STFC

Abstract This work aims at developing a new dynamic model for the vibration control using the finite element analysis using comparing the effects of different controllers on system performance. The finite element method (FEM) is used to derive the flexible beam model. Frequency analysis (modal and harmonic) of the model is performed using ANSYS software in 4 different cases. In each of the 4 cases, the state space model of the beam was extracted in workspace of MATLAB based on result of its frequency analysis done in ANSYS. A PID controller was designed based on the model obtained from the first case in a MATLAB based environment and validated on the other 3 cases. To achieve better system performance, three more intelligent controller systems were considered: PD like Fuzzy, PID-AT (PID auto-tuning) and STFC (self-tuning fuzzy controller). In comparison with PID, PD like Fuzzy and PID-AT controllers, the system under effect of STFC shows the least overshoot, least rise time and least settling time. The three studied intelligent controller types based on fuzzy logic (PD like Fuzzy, PID-AT and STFC) achieved much improved system performance with the STFC showing the best system performance. Ó 2019 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Vibrations occur in many mechanical and structural systems. If uncontrolled, vibration can lead to catastrophic situations. The proposed techniques to minimize the structural vibrations, in general, consist of two main categories; passive control systems and active control systems. Active control methods can be used to eliminate undesired vibrations in engineering structures. Using piezoelectric smart structures for the active E-mail addresses: [email protected] (K. Ibrahim), [email protected] (A. El-Makarem Ahmed) Peer review under responsibility of Faculty of Engineering, Alexandria University.

vibration control has great potential in engineering applications. In the actively controlled system, control forces are generated using an external energy source and applied to the structure through actuators according to a prescribed control algorithm. Modeling and response analysis of dynamic systems by using ANSYS and MATLAB was made by Khot and Yelve [1]. A state model of a cantilever beam was generated in MATLAB based upon the result of modal analysis of its finite element model through the finite elemnt software ANSYS. An active vibration control of a cantilever beam by using a PID based output feedback controller Khot et al. [2]. It was found that the frequency responses of the full and reduced models are very similar. Extraction of system model through finite element method and simulation analysis of active control of vibrations

https://doi.org/10.1016/j.aej.2019.05.009 1110-0168 Ó 2019 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: M.A. Kamel et al., Vibration control of smart cantilever beam using finite element method, Alexandria Eng. J. (2019), https://doi. org/10.1016/j.aej.2019.05.009

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was made by Khot et al. [3]. To demonstrate the methodology a case of cantilever beam was considered. Finite element analysis of piezoelectric cantilever beam was studied by More [4]. His work aimed at modeling of vibration in piezoelectricbased devices using MATLAB and ANSYS software. Static and dynamic analysis of piezoelectric cantilever beam was performed inside ANSYS. Li and Li [5], applied ANSYS APDL in the design of a piezoelectric transducer (metal plate plus PZT ceramic ring). Development of dynamic model for control of vibrations in flexible beam was studied by Adel et al. [6], which aimed at acquiring a new dynamic model for vibration control of a composite carbon cantilever beam. Aleksandar et al. [7], developed an experimental setup of an active vibration control system. Moreover, the Comparison between the different control algorithms are developed. Junqiang et al. [8], simulation and experimental work are developed of the flexible arm by using optimal multi-poles placement control. Also, the comparing results among the simulation and experimental work are very closed. A hybrid control strategy of a smart flexible manipulator is presented [9,10]. The simulation and experiments results are demonstrated. However, the settling time of the setup descends approximately 55%. The objective of this work is to develop a new dynamic model for the vibration control using the finite element analysis through ANSYS software and applying both classical and intelligent control algorithms on the developmental dynamic model and comparing their effects on system performance using SIMULINK software. 2. Model analysis of flexible beam based on FEA ANSYS is a general-purpose finite-element modeling package for numerically solving a wide variety of mechanical problems. These problems include static/dynamic, structural analysis, heat transfer, and fluid problems. The basic steps involved in any finite element analysis consist of the following: preprocessing phase, solution phase and post-processing phase. FEM was used due to its high mathematical capability in describing system performance plus the availability of a variety of software applying the FEM [6]. For the present analysis, a composite carbon cantilever beam is studied as shown in Fig. 1. The beam dimensions are similar to those used by Nasser et al. [11]. The beam geometry is created in ANSYS. The mechanical properties of composite carbon cantilever beam are shown in Table 1. Element type BEAM3 element is used for the analysis. The beam is meshed and divided into 10

Fig. 1

Table 1 Mechanical properties of composite carbon cantilever beam. Young’s modulus Poisson’s ratio Density Area moment of inertia

Y = 41.5 GPa m = 0.042 q = 1480 kg/m3 I = 1.83  1011 m4

elements and is now ready for dynamic analysis as shown in Fig. 2. (i) Free vibration modal analysis The method of Block Lanczos (used for large symmetric eigenvalue problems. This method uses the sparse matrix solver) is used as shown in to find the eigenvalues and the eigenvectors normalized with respect to the beam mass. Eigenvectors relative to the Y-component degrees of freedom are used in the vibration analysis and three mode shapes were extracted. (ii) Free vibration of composite beam with piezoelectric element Now a free vibration modal analysis is performed on the beam in which SOLID45 (solid structure unit 8-nodes brick) is used for composite carbon beam and SOLID5 (3-D coupled-field solid) is used for piezoelectric element. The piezoelectric element selected is PZT-4 similar to the one used by Fuxu Li and Guangji Li [5]. The configuration of the model is shown in Fig. 3. The structure of this type has been studied many in research papers (Manning et al. [12], Gaudenzi et al. [13], Bruant et al. [14], Singh et al. [15], Xu and Koko [16] and Weldegiorgis et al. [17]). The dimensional structure of the cantilever beam and the distances of a second case studies are shown in Table 2. A macro is written using APDL. The macro starts with the definition of the variables for the dimensions of the structure. Then the three-dimensional material properties are assigned. The part of the macro where the material properties are assigned is given below. Material 1 (PZT-4) is the piezoelectric material and Material 2 is the composite carbon beam piezoelectric material. The macro is continued to create nodes and finite elements. The beam with the attached PZT element is shown in Fig. 4.

Composite carbon beam.

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Vibration control of smart cantilever beam

3

Fig. 2

Fig. 3

Table 2

Structure configuration of flexible beam.

Dimensions and distances for case 2.

Parameter

Dimensions of PZT element (mm)

l b h d

72 100 0.61 12

Fig. 4

Beam meshing.

The boundary conditions of the beam are defined for the nodes at x = 0. The degrees of freedom, VOLT, are coupled for the nodes at the top and bottom surfaces of the piezoelectric element by the ANSYS APDL command prompt. The bottom surface of the PZT is glued to the upper surface of the beam. Modal analysis is carried out to determine structure performance.

Beam and attached PZT element configuration.

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M.A. Kamel et al. (iii) PZT beam subjected to tip loading of 2.5 N

The data in case 2 remain unchanged, but this time the beam is subjected this time to a tip loading of magnitude 2.5 N as shown in Fig. 5. For this case, a harmonic analysis is performed using the full solution method. Frequency range is set from 0 to 100 Hz with 100 substeps and stepped b.c. Then the Y component of displacement (UY) is plotted against frequency showing the peak frequencies of the uncontrolled system. To get a better view of the response the log scale of UY is viewed.

is plotted against frequency showing the peak frequencies. It is necessary to summarize the previous results. Table 3 shows Case 1 vs. Case 2 first three mode shapes. For harmonic analysis of the 3rd and 4th cases, it is shown in Fig. 7. Table 4 summarizes frequency values obtained in all four cases. From the previous data, it could be shown the effect of the PZT actuator on the beam. The PZT actuator has a stabilizing effect on the beam at the third mode which is maintained even when the beam is exposed to tip loading conditions which normally should increase system instability (notice the increasing frequency in first and second modes).

(iv) PZT beam subjected to tip loading of 5 N The data in case 3 remain unchanged, but this time the beam is subjected to a tip loading of magnitude 5 N as shown in Fig. 6. As in the previous case, a harmonic analysis is performed using the full solution method. Frequency range is set from 0 to 100 Hz with 100 substeps and stepped b.c. Then the log scale of the Y component of uncontrolled displacement (UY)

Fig. 5

Fig. 6

3. Design of control algorithms for vibration control of flexible beam (i) PID controller The required end position angle was found to be approximately 20 degrees (0.35 rad) as deduced from the tip deflection by FE analysis of the beam. For the current study PID controller is designed using the Simulink Design Optimization

Case 3 tip load of 2.5 N.

Case 4 tip load of 5 N.

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Vibration control of smart cantilever beam Table 3 Mode No.

5

Case 1 vs. Case 2 for the first three mode shapes. Case 1 Free vibration beam

Case 2 Free vibration PZT beam

1

2

3

(SDO) tool in MATLAB. The signal constraint block GUI is updated during optimization so that the optimization progress can be displayed (Ibrahim et al. [18]). Optimization iterations are performed on varying ranges of Kp, Ki, Kd and N (filter coefficient) parameters. The system response to a position step input based on optimized parameters is simulated. From the signal constraint optimization, the PID controller parameters were found to be as follows: Kp = 3.5, Ki = 1, Kd = 0.025 and N = 100. The system response of the model to a position step input of 0.35 rad is simulated in each of the four studied cases. The variations of the full model system response with time are plotted in Fig. 8 for step input position 10 degrees (0.17 rad). In order to validate the efficiency of the introduced controller, a different input for example a sine wave pattern with amplitude of 0.35 rad (desired reference) and a frequency of 1 rad/sec. The controlled response is simulated and plotted in MATLAB command window as shown in Fig. 9 (target input is shown by a blue curve while the actual output is shown by a red line) As seen from the previous figure the designed controller shows a good effect on different types of inputs (for instance the controlled system output tracks almost exactly the desired input sine wave). Thus the designed controller would be suitable for testing in the next cases. (ii) PD like Fuzzy controller In this type of controller, the signal constraint (signal design optimization) method discussed earlier is used to tune

Kp (the proportional gain), Kd (the derivative gain) and gain K. Initially, Kp is set to 1, Kd is set to 0.001 and K is set to 1. The fuzzy controller consists of two inputs (error and error change and one output. The fuzzy system has three membership functions for each input and three membership functions for the output. The rule base consists of 9 rules. The desired position is set at 0.35 rad (=20 deg). The optimization is based on the reduced state space model of the first case (free vibration beam). The block diagram of the system is shown in Fig. 10. Through SDO, it was found that the optimized values for the design parameters. Kp = 4.5, Kd = 0.005 and K = 0.699. The simulation was run based on the optimized values of the design parameters. The controlled system response is plotted for each of the four studied flexible beam cases. (iii) (PID-AT) Controller with Fuzzy Self-Tuning Because it is relatively simple in nature, this strategy has been studied in various researches. This strategy consists of optimizing the tuning rules of Ziegler–Nichols by means of a single parameter@, then using an online fuzzy-based tuning to adapt this single parameter. Under this strategy, the three PID parameters could be written as (Nasser et al. [19]): kp ¼ 1:2@ ðtÞkcr Ki ¼ 0:75

1 Pcr 1 þ @ ðt Þ

Kd ¼ 0:25Ki

ð1Þ ð2Þ ð3Þ

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Fig. 7

Table 4

Harmonic analysis for the 3rd and 4th cases.

Frequency values obtained in all four cases.

Case

1st Mode (Modal) 1st Peak (Harmonic) Frequency in Hz

2nd Mode (Modal) 2nd Peak (Harmonic) Frequency in Hz

3rd Mode (Modal) 3rd Peak (Harmonic) Frequency in Hz

1 2 3 4

5.16884 6.31855 6.45 6.55

32.3933 34.4918 35.5 36.5

90.721 85.7442 86.5 87.5

where kcr and Pcr are the critical gain and critical period, respectively. The value of @ ðtÞ is deduced by the following equation:  @ ðtÞ þ chðtÞð1  @ ðtÞÞ for @ ðtÞ > 0:5 @ ðt þ 1Þ ¼ ð4Þ @ ðtÞ þ chðtÞ@ ðtÞ for @ ðtÞ  0:5 where hðtÞ is the output of the fuzzy optimizing system and c is a constant that has to be chosen in the range [0.2–0.6]. The

fuzzy system has three membership functions for each of the two inputs (e and_e) and three membership functions for the output. The rule-base consists of 9 rules. The preliminary value of @ ðtÞ is chosen to be 0.5, which matches with the Ziegler–Nichols tuning rules. With respect to the methodology, the tuning of the scaling value of the fuzzy model and of the parameter c is left to the researcher. The control structure of beam model with auto-tuned PID controller with

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Vibration control of smart cantilever beam

Fig. 8

Fig. 9

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Variations of the controlled system response to an input position of 0.17 rad.

Response of a sine wave input of amplitude 0.35 rad and a frequency of 1 rad/sec.

Fig. 10

PD like Fuzzy Controller of smart cantilever beam.

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Fuzzy self-optimization of a single parameter is given in Fig. 11. Based on the above equations, an m-code tuning algorithm is designed to self-tune the parameter aðtÞ and accordingly the values of Kp, Ki and Kd. The value of c is set at 0.4. The desired position is set at 0.35 rad (20 deg). After running the m-code, the system results obtained are simulated through a built model based on the reduced state space model of the first case in Simulink. The end values of Kp, Ki and Kd were found to be 0.1560, 0.1040 and 0.0897 respectively. The system response is simulated for each of the four studied flexible beam case.

Let E be defined by: E ¼ e2 þ l_e2

ð5Þ

e ¼ SP  yðkÞ

ð6Þ

where E Cost function l Error change weight sp Set point (desired position) yðkÞ System output (actual position) e Error

(iv) Self-tuning fuzzy control (STFC) Learning rule is: In this type of controller, the back-propagation (BP) algorithm is used to get the updating laws of the scaling factors S1 , S2 and S3 as shown in Fig. 12. The aim is to reduce a cost function E, so that training pattern k is proportional to the square of the difference between the set point ðsp Þ and the system output yðkÞ (angle error) and the square of the angle error change.

Fig. 11

Fig. 12

Si ðk þ 1Þ ¼ Si ðkÞ  gi

@E þ ai DSi ðkÞ @Si

ð7Þ

where Si adjusted Scaling factor (SF) i = 1, 2, 3 gi the learning rate

PID-AT controller with Fuzzy self-tuning of smart cantilever beam.

Self-tuning scaling factors, fuzzy controller of the smart cantilever beam.

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Vibration control of smart cantilever beam

9

ai the momentum parameter ai ; 0 < ai < 1

able system response is achieved. The end values of scaling factors S1 , S2 and S3 are found to be as follows: 0.0311, 3.5222 and 0.3516 respectively.

DSi ðkÞ ¼ Si ðkÞ  Si ðk  1Þ

ð8Þ

the scaling factors are updating according to the following equations: S3 ðk þ 1Þ ¼ S3 ðkÞ þ g3 d4 ðkÞY3 ðkÞ þ a3 DS3 ðkÞ

ð9Þ

S1 ðk þ 1Þ ¼ S1 ðkÞ þ g1 d21 ðkÞeðkÞ þ a1 DS1 ðkÞ

ð10Þ

S2 ðk þ 1Þ ¼ S2 ðkÞ þ g2 d22 ðkÞDeðkÞ þ a2 DS2 ðkÞ

ð11Þ

The updating rules of the scaling factors S1 , S2 and S3 are integrated into an m-code tuning algorithm. The set point is set at the desired angle of 0.35 rad (20 deg). The constants g, a and l are set at 0.001, 0.04 and 0.5 respectively. The m-code is run and with each run the system response is simulated and observed through a block diagram designed based on the reduced state space of the first case in Simulink until an accept-

Fig. 13

4. Simulation results and discussion The system response is simulated in each of the four studied flexible beam cases. The system response in the first case of each of the four studied controller types is compared in Fig. 13(a)–(d). The average of controlled system response parameters to a position step input of 0.35 rad in four studied cases is shown in Table 5 and Fig. 14(a)–(c). The updating rules of the scaling factors S1 , S2 and S3 are integrated into an m-code tuning algorithm. The set point is set at the desired angle of 0.35 rad (20 deg). The constants g, a and l are set at 0.001, 0.04 and 0.5 respectively. The m-code is run and with each run the system response is simulated and observed through a block diagram designed based on the reduced state space of the first case in Simulink until an acceptable system response is achieved. The end values of scaling factors S1 , S2 and S3 are found to be as follows: 0.0311, 3.5222 and 0.3516 respectively.

Comparison between different control algorithms.

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Table 5

The average of controlled system response parameters to a position step input of 0.35 rad in four studied cases.

Controller type

MP %

Tr, sec

Ts, sec

ess

PID PD-like Fuzzy PID-AT STFC

52.775 15.522 8.333 1.487

0.885 0.405 0.299 0.241

3.210 1.361 0.945 0.610

zero zero zero zero

Fig. 14

Comparison of controlled system response parameters to a position step input of 0.35 rad in four studied cases.

5. Conclusions Finite element analysis using ANSYS software was shown to be a reliable and effective method when modeling dynamic structures. Four different cases by using ANSYS are achieved. Full state space model of the smart cantilever beam is built in MATLAB. Simulation result using ANSYS/MATLAB is obtained. The three studied, intelligent controller types based on fuzzy logic (PD like Fuzzy, PID-AT and STFC) achieved much improved system performance with the STFC showing the best system performance. Exploring more advanced options in ANSYS, using different types of beam material and dimensions, in addition to experimental results based on the derived model and the designed controller types all inspire future work. References [1] S. Khot, N.P. Yelve, Modeling and response analysis of dynamic systems by using ANSYSÓ and MATLABÓ, J. Vib. Control 17 (2011) 953–958. [2] S. Khot, N.P. Yelve, R. Iyer, Extraction of system model form finite element model and simulation study of active vibration control, Adv. Vib. Eng. 11 (2012) 259–280. [3] S. Khot, N.P. Yelve, R. Tomar, S. Desai, S. Vittal, Active vibration control of cantilever beam by using PID based output feedback controller, J. Vib. Control 18 (2012) 366–372.

[4] N.N. More, Finite element analysis of piezoelectric cantilever, Int. J. Innov. Eng. Technol. (IJIET) 2 (2013) 100–105. [5] F. Li, G. Li, Application of ANSYS APDL in the Design of Piezoelectric Transducer, 2015. [6] M. Adel, K. Ibrahim, A.R. Gad, A.E.M. Khalil, Development of dynamic model for vibration control of flexible beam, J. Eng. Sci. 44 (5) (2016) 555–565. [7] Aleksandar M. Simonovic´, Miroslav M. Jovanovic´, Nebojsˇ a S. Lukic´, Nemanja D. Zoric´, Slobodan N. Stupar, Slobodan S. Ilic´, Experimental studies on active vibration control of smart plate using a modified PID controller with optimal orientation of piezoelectric actuator, J. Vib. Control (2014) 1–13. [8] Junqiang Lou, Jiangjiang Liao, Yanding Wei, Yiling Yang, Guoping Li, Experimental identification and vibration control of a piezoelectric flexible manipulator using optimal multi-poles placement control, Appl. Sci. 7–309 (March 2017) 1–20. [9] Khalil Ibrahim, Abdel Badie Sharkawy, A hybrid PID control scheme for flexible joint manipulators and a comparison with sliding mode control, Ain Shams Eng. J. 9 (2018) 3451–3457. [10] J.Q. Lou, Y.D. Wei, Y.L. Yang, F.R. Xie, Hybrid PD and effective multi-mode positive position feedback control for slewing and vibration suppression of a smart flexible manipulator, Smart Mater. Struct. (2015). [11] H. Nasser, E.-H. Kiefer-Kamal, H. Hu, S. Belouettar, E. Barkanov, Active vibration damping of composite structures using a nonlinear fuzzy controller, Compos. Struct. 94 (2012) 1385–1390. [12] W. Manning, A.R. Plummer, M. Levesley, Vibration control of a flexible beam with integrated actuators and sensors, Smart Mater. Struct. 9 (2000) 932.

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Vibration control of smart cantilever beam [13] P. Gaudenzi, R. Carbonaro, E. Benzi, Control of beam vibrations by means of piezoelectric devices: theory and experiments, Compos. Struct. 50 (2000) 373–379. [14] I. Bruant, G. Coffignal, F. Lene, M. Verge, Active control of beam structures with piezoelectric actuators and sensors: modeling and simulation, Smart Mater. Struct. 10 (2001) 404. [15] S. Singh, H.S. Pruthi, V. Agarwal, Efficient modal control strategies for active control of vibrations, J. Sound Vib. 262 (2003) 563–575. [16] S. Xu, T. Koko, Finite element analysis and design of actively controlled piezoelectric smart structures, Finite Elem. Anal. Des. 40 (2004) 241–262.

11 [17] R. Weldegiorgis, P. Krishna, K.V. Gangadharan, Vibration control of smart cantilever beam using strain rate feedback, Elsevier, Proc. Mater. Sci. 5 (2014) 113–122. [18] K. Ibrahim, Ahmed Ramadan, Mohamed Fanni, Yo Kobayashi, Ahmed A. Aboismail, Masakatsu G. Fujie, Screw theory based-design and tracking control of an endoscopic parallel manipulator for laparoscopic surgery, 2013 IEEE International Conference on Robotics and Automation, ICRA2013, Germany, Karlsruhe, May 6–10, 2013. [19] G.A. Nasser, A.B. Sharkawy, M.-E.S. Soliman, An auto-tuning method for the scaling factors of fuzzy logic controllers with application to siso mechanical system, Int. J. Mater., Mech. Manuf. 3 (2015) 49–55.

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