Vibration control of laminated composite cantilever beam operating in elevated thermal environments using fuzzy logic controller

Abstract

In the present study, vibration control of laminated composite cantilever beam operating in the elevated thermal environment is achieved using combined experimental and numerical techniques. The impact hammer test is performed on the glass-epoxy cantilever beam at different temperatures. Experimentally recorded impact hammer force signals and piezoelectric accelerometer time-domain signals are processed through a system identification toolbox in MATLAB to obtain transfer functions of the plant models. A robust fuzzy logic controller is developed to accomplish the effective vibration control of a cantilever composite beam operating at different temperatures. The fuzzy logic controller with two inputs and one output is designed using the 20 if-then rules. The results are presented in both frequency and time domain, keeping the vibration attenuation of the fundamental frequency as the point of interest. The results indicate the proposed fuzzy logic control strategy can attenuate the vibrations of a cantilever composite beam for a wide temperature range.

Keywords

Vibration control effect of temperature fuzzy logic controller free vibration test laminated composite beam

Introduction

In dynamic engineering structures, the vibrations are inherent in nature and are not always desirable.¹ In the fields of automobile, aviation, civil, medicine, etc., low-frequency vibrations are a significant concern to engineers.² Mitigation of the impact of the undesirable vibrations is the most important task in high-end engineering applications as they may adversely affect the operation of the structure. The passive vibration control technique is one of the most effective tools for vibration attenuation. The passive vibration control techniques involve the usage of dynamic absorbers, or dampers, or neutralizers which are more significant in reducing high-frequency vibrations.³ For controlling the vibrations of lower frequencies, active vibration control (AVC) techniques are more predominantly used.^4,5 The development of piezoelectric materials motivated the implementation of the AVC strategy in a wide range of applications. Usage of these lightweight, fast response piezo materials embedded on the test structures as sensors and actuators can make the system smart by direct output feedback.^6–8

Composite structures are widely used in aerospace, naval, and high-performance civil engineering applications due to their lightness, high specific modulus, rigidity, and corrosion resistance.⁹ However, this flexibility may cause structural instability and allows undesirably large vibration amplitude, which influences the performance of the material, and leads to catastrophic failures.¹⁰ The abrupt thermal variations also cause thermal shocks, which may result in undesirable vibrations in the structure.¹¹ The vibrating structures operating in ambient conditions may behave differently on exposure to the elevated thermal environment. Hence, it is important to note that the different vibration control strategies are to be adopted for the AVC of the oscillating structures operating in different thermal environments.

The composite structures integrated with the sensor and actuator pair which can sense and mitigate the vibrations are known as smart structures.^6,12 Piezoelectric patches are predominantly used as the sensor and actuating elements to achieve the AVC of the system. The sensor reads the parameters like force, strain, and acceleration and the antivibration signals can be provided through the actuator. Sharma et al.¹¹ experimentally investigated the influence of temperature on the AVC of space antenna reflectors. The non-linear fuzzy logic controller was developed for vibration control of titanium cantilever beam operating in an elevated thermal environment. Gupta et al.¹³ achieved AVC of galvanized iron plate operating in an elevated thermal environment. The vibrating cantilever plate was controlled using a negative velocity control algorithm. Both Sharma et al.^11,14 and Gupta et al.¹³ used a specific thermal profile during the experimentation. The specimen was brought down to room temperature before achieving the next elevated temperature value. The specimen was allowed to be at room temperature for a minimum of 30 min, and vibration readings were obtained at elevated temperatures by allowing the specimen to cure for a minimum of 20 min. Riessom et al.¹⁵ used the system identification toolbox of the MATLAB simulation module to predict the dynamics of the beam in the form of a mathematical model. The strain rate feedback control algorithm was used to achieve up to 50% of vibration attenuation of the cantilever beam.

Kallannavar et al.^16–19 numerically investigated the influence of temperature on the vibration characteristics of laminated composite, hybrid composite, and sandwich plates. It was reported that considering the variation in the material properties with an increase in temperature yields more accurate results. Rath et al.⁹ performed an experimental investigation to understand the effect of the hygrothermal environment on the natural frequency of the system. It was noted that the system’s stiffness and natural frequency considerably reduce with an increase in temperature and moisture values. Sai Ram et al.²⁰ numerically investigated the effect of temperature and moisture on the free vibration characteristics of the laminated composite plate. It was noted that with a uniform increase in temperature and moisture values, the reduction in fundamental frequency was not linear. Rezaei et al.²¹ experimentally investigated the influence of temperature on the mechanical properties and failure modes of glass fiber-reinforced polymer composites and sandwich structure with polyethylene terephthalate foam core. The tensile test results indicate that with an increase in temperature, the elastic modulus and the ultimate strength of the material slightly reduced. Extensive softening of the resin was observed above 75°C (near to glass transition temperature T_g= 83°C), which caused a substantial reduction in shear strength and shear modulus of the laminate. Colakoglu²² investigated the effect of temperature on natural frequency and damping factor on kevlar and polyethylene laminate beams for −15°C to 60°C, between which the T_g doesn’t exist and found a decrease in natural frequency and increase in damping factor with increasing temperature.

Many researchers have achieved the AVC of laminated composite beams, plates, shells, etc., using different control algorithms such as proportional-integral-derivative (PID) controller, linear quadratic regulator (LQR) controller, and fuzzy logic controller. Sharma et al.^11,14,23 achieved AVC using a non-linear robust fuzzy logic controller to attenuate the beam structures operating elevated thermal environments. Both experimental and numerical methods were used to investigate the influence of temperature on control strategy and the damping parameters. Parameswaran et al.²⁴ attained up to 50% of vibration reduction of the beam structure using a PID control algorithm. The model-free sliding mode robust controller was developed in order to reduce the computational time and to reduce the demand for higher memory/hardware resources. It was also observed that the natural frequency of the system is higher than the open-loop model. This is due to the change in system dynamics upon application of the control force.^24,25 Sharma et al.^26,27 designed a fuzzy logic-based modal controller for AVC of a cantilevered beam using piezo patches as sensors and actuators. The results indicate that the controller is more effective for the fundamental mode compared to the higher modes of vibration. Takawa et al.²⁸ experimentally and numerically studied the mitigation of the intense non-linearities in electro rheological fluid actuator by neuro-fuzzy logic based on modern control theory.

From an extensive literature survey, it is observed that the pieces of literature pertaining to AVC of structures using numerical and analytical methods are in abundance. However, the experimental studies to understand the performance of AVC of laminated structures in elevated thermal environments are limited. In this study, a combined experimental and numerical approach is proposed to achieve the AVC of a glass-epoxy composite cantilever beam operating in a wide temperature range. An attempt has been made to develop a robust fuzzy logic controller which can attenuate the vibrations of laminated composite cantilever beam structures operating in different thermal environments.

Experimentation

The glass-epoxy laminated composite beams were fabricated by hand layup method keeping the weight fraction as 50:50. The epoxy resin was prepared by blending the araldite LY-556 resin and HY-951 hardener in the ratio of 10:1. The plastic sheet (OHP sheet) with mold releasing agent (petroleum jelly) was roofed on the metal base plate having a smooth surface. The blended resin mixture is then applied to the surface of the sheet as a gel coat. The gel coat helps in keeping the surface smooth and also protects the fibers from exposure to the environment. Then the 280 GSM bidirectional glass fiber mats were laid one above the other to achieve the desired thickness. A sufficient amount of resin mixture was applied before placing the next layer of the glass fiber mat. Care was taken to make sure that the glass fiber mats were completely wet. Metal rollers were used to remove any entrapped air bubbles in the layup. The fabricated laminates were then cured at room temperature (28°C) under the pressure of 50 bars for 24 h in a pressure-controlled hydraulic press. After proper curing, the laminates were freed from the plastic sheets (release films). The laminates were then cut to 2.5 cm × 25 cm dimensions, as shown in Figure 1. The average thickness of the prepared laminates was measured to be 0.25 cm.

Figure 1.

Fabrication of composite beam.

Thermal treatment of composite beam

The laminated composite beam samples were thermally aged in the thermal chamber which is capable of preserving the desired temperature with an accuracy of ±1°C. The experiments were conducted for temperatures ranging from 25°C to 100°C. Initially, the temperature in the chamber was reduced to 25°C from room temperature (28.5–30°C). The composite beam is allowed to stay at 25°C for 30 min. Then the temperature of the thermal chamber was raised to the required elevated temperatures and maintained for a minimum of 20 min before taking the reading. The climatic chamber was slightly opened to perform the impact hammer test. The care was taken such that the temperature in the thermal chamber was not dropped more than 1°C. The temperature of the thermal chamber was dropped down to the reference temperature every time after performing the free vibration test at an elevated temperature. The specimen was maintained in reference temperature for at least 30 min before performing the next experiment .^11,13,14 Figure 2 shows the temperature profile followed for conducting the test at different temperatures, that is, 25°C, 50°C, 75°C, and 100°C temperatures.

Figure 2.

Temperature profile followed for thermal chamber.

The desired thermal profile was fed into the thermal chamber control unit through a computer program, as shown in Figure 3. The thermal profile of 410 min was maintained in the thermal chamber with the maximum temperature deviation of ±0.5°C.

Figure 3.

Thermal chamber and its control unit.

Free vibration test

The composite beam specimen was fitted firmly to the fixture in a cantilever position. The beam specimen was excited using an impact hammer (PCB model: 086C03), and the resulting vibration signals were picked up by the high-temperature triaxial-accelerometer (PCB model: 339A31/NC). The accelerometer was attached to the tip of the specimen using flex bond instant adhesive, which can operate in high-temperature environments. The impact hammer and accelerometer are connected to an analog input module (NI-DAQ 9234) which was mounted in the USB chassis (NI-USB-9162), as shown in Figure 4. The input force and output acceleration (z-direction) time-domain signals were captured with a sampling frequency of 25.6 kHz.

Figure 4.

Pictorial representation of (a) cantilever composite beam inside the thermal chamber, (b) NI-DAQ 9234 with NI-USB-9162 chassis, and (c) computer with LabVIEW Program for signal processing.

Robust fuzzy logic controller

Every physical system can be expressed in terms of mathematic expressions known as the plant model. The plant models are also referred as transfer functions which relates the input and output response of the system. On exposure to the thermal environment, the bonding between the fiber and matrix of composite materials gets weak.²⁹ This may cause a change in material properties and may lead to variation in material behavior at elevated thermal environments. Hence, the plant model keeps constantly changing for structures operating in elevated thermal environments.

Developing a robust control strategy for structures with dynamically changing material properties is a challenging task. A classical PID controller needs different gains to control the constantly changing plant model. A PID controller with constant gain cannot suffice the robustness parameter. Hence, the controller with minimum dependency on the plant model is most suitable for the current application. The fuzzy logic controller is one such kind of controller that doesn’t depend entirely on the plant model. The fuzzy logic controller works on the set of rules given manually with experiences and is also capable of dealing with the non-linearities in the system.

For composite beam operating in the elevated thermal environment, the robust fuzzy logic controller is designed as two inputs and one output system using 20 if-then rules. The sensor (accelerometer) voltage and rate of change of sensor voltage are considered as inputs, and control actuator voltage is considered as the output of the controller. The widely used and accepted MAMDANI inference engine is used for the study, which is easy to understand, and the rule base of the engine is easily interpretable. MAMDANI systems are well suited for a system with predefined rules available as both input and output of the inference engine are linguistic variables, unlike Takagi-Sugeno-Kang inference engine, where the inputs are linguistic variables and output is linear equation.³⁰

The controller is modeled using 9 Gaussian and 2 trapezoidal membership functions, as shown in Figure 5. Generally, Gaussian and triangular membership functions were found to be giving better and close results when compared to other membership functions. The advantage of using Gaussian over triangular membership function is its smoothness, concise notation, and non-zero values at all points.³¹ Besides ideal cases, the vibration of a system will never be zero; hence, the Gaussian membership function is most suited for the current study. The two trapezoidal membership functions are selected to deal with the extreme ±5 V range.¹¹ The mean (μ) and standard deviation (σ) of Gaussian functions used are listed in Table 1. The 20 if-then rules were written using “AND” logic between the two inputs sensor voltage (V) and rate change of sensor voltage (V′). The fuzzy rules in the if-then form relating the input and output parameters are usually derived from human reasoning and experiences. The rules implemented in the current study are listed in Table 2.

Figure 5.

Membership functions of (a) sensor inputs, (b) rate change of sensor inputs, and (c) actuator output.

Table 1.

Mean (μ) and standard deviation (σ) values for Gaussian functions.

Gaussian	Sensor		Rate change of sensor		Actuator
Functions	μ	σ	μ	σ	μ	σ
-d	−0.85	0.2123	−0.85	0.2123	−0.85	0.2123
-c	−0.75	0.2123	−0.75	0.2123	−0.75	0.2123
-b	−0.5	0.2123	−0.65	0.2123	−0.5	0.2123
-a	−0.25	0.2123	−0.4	0.2123	−0.25	0.2123
ze	0	0.2123	0	0.3	0	0.2123
a	0.25	0.2123	0.4	0.2123	0.25	0.2123
b	0.5	0.2123	0.65	0.2123	0.5	0.2123
c	0.75	0.2123	0.75	0.2123	0.75	0.2123
d	0.85	0.2123	0.85	0.2123	0.85	0.2123

Table 2.

Rules for inference engine in fuzzy logic controller.

V′	V
V′	-e	-d	-c	-b	-a	ze	a	b	c	d	e
-e						ze
-d					a		-a
-c				b				-b
-b			c						-c
-a		d								-d
ze	e										-e
a		d								-d
b			c						-c
c				b				-b
d					a		-a
e						ze

Simulink modeling

The experimentally obtained impact hammer signals and accelerometer response were considered as input and output, respectively, to generate the transfer functions. While developing the control algorithm, a discrete impulse signal was given as input to the identified plant model with a low pass filter. The active control technique requires the production of an anti-phase signal to cancel out the original signal, and it requires some processing time of the controller to produce the signal. At higher frequencies, the amplitude signal varies at a faster rate, making it difficult for the controller to produce the canceling signal without any delays. The delayed signals may add up to the original signal instead of canceling the signals and result in instability of the system. Hence, it is well established that active vibration control is widely used to control low-frequency vibrations, whereas passive techniques are used to control higher frequency vibrations.^13,32,33 So a low pass filter was introduced to eliminate frequencies higher than 50 Hz. The output of the transfer function, that is, sensor signal (V), and rate change of sensor signal (V′) were passed through the fuzzy logic controller, as shown in Figure 6.

Figure 6.

Simulink Block diagram of AVC using fuzzy logic controller.

In the fuzzification process, these two numerical values are converted to linguistic variables based on the membership functions. This fuzzified value then enters into the inference engine. In the inference engine, the fuzzy output is determined in the form of a linguistic variable using the if-then rules given in the rule base between the two inputs. In the defuzzification process, this linguistic variable is converted to numerical value using the centroid method. This output of the fuzzy logic (actuator voltage) is fed back as an anti-vibrating signal to the input of the transfer function. The block diagram of the fuzzy logic controller is presented in Figure 7.

Figure 7.

Block diagram of fuzzy logic controller.

Results and discussion

The results obtained from free vibration tests are presented in both the time and frequency domain for the first fundamental vibration mode as shown in Figure 8. The lower vibration modes are associated with the lower energies and can easily be excited. Hence, it is very important to control the lower vibration modes immediately.³⁴ The vibration modes existing up to 50 Hz are targeted for the control. First, the experimental vibration data were collected for different temperatures, and the modal parameters were evaluated through LabVIEW spectral analysis module. The impact hammer and accelerometer signals recorded through experiments for various temperatures are then used to obtain the transfer functions of composite beams at different temperatures using the MATLAB System Identification Toolbox. The simulation results and the experimental results are compared in order to check the accuracy of the developed transfer function model. From Table 3, it is evident that the simulation results are in good agreement with the experimental results and the identified plant models are acceptable to continue the analysis of AVC of the composite beam for different temperatures.

Figure 8.

Frequency response plot of the glass-epoxy composite beam operating in room temperature (a) power density plot and (b) dB plot.

Table 3.

First mode experimental and simulation natural frequencies laminated composite beam.

Temperature (°C)	First mode (Hz) [experimental]	First mode (Hz) [simulation]	Difference (%)
25	14.98	14.860	0.8011
50	14.12	14.240	0.8498
75	13.58	13.620	0.2945
100	13	13.001	0.0076

The open-loop and closed-loop responses in both frequency and time domains are obtained for all the temperatures considered. The open-loop response corresponds to the material damping characteristics only, and the closed-loop response corresponds to the active force generated by the control voltage analogous to the fuzzy logic controller outputs. Figure 9 demonstrates the dynamic response of the laminated composite beam operating at 25°C. The percentage reduction up to 52.41% is observed in the peak amplitude (dB) value. Figure 9(a) suggests the control strategy adopted is quick and needs less than 1 s to attenuate the amplitude of vibrations.

Figure 9.

Active vibration control of composite beam using the fuzzy logic controller at 25°C (a) time-domain, (b) frequency domain (power density plot), and (c) frequency domain (dB plot).

Similarly, Figures 10 to 12 demonstrate the vibration response of laminated composite beam operating at 50°C, 75°C, and 100°C temperatures, respectively. From the results, it is evident that the time required for vibration attenuation considerably reduces with an increase in temperature. It implicates that the response time of the control strategy increases with an increase in temperature. The percentage reduction of 53.04%, 61.67%, and 59.85% is observed in the peak amplitude values for beams operating in 50°C, 75°C, and 100°C temperatures, respectively. The results indicate that with the increase in temperature values, the vibration attenuation characteristics of the control algorithm increase. The detailed observations are quantified in Table 4.

Figure 10.

Active vibration control of composite beam using the fuzzy logic controller at 50°C (a) time-domain, (b) frequency domain (power density plot), and (c) frequency domain (dB plot).

Figure 11.

Active vibration control of composite beam using the fuzzy logic controller at 75°C (a) time-domain, (b) frequency domain (power density plot), and (c) frequency domain (dB plot).

Figure 12.

Active vibration control of composite beam using the fuzzy logic controller at 100°C (a) time-domain, (b) frequency domain (power density plot), and (c) frequency domain (dB plot).

Table 4.

Reduction in the amplitude of vibrations at different temperatures.

Temp, °C	First mode (Hz)		Peakamplitude(dB)		Decrease (AVC)	Damping ratio( $ξ$ )
Temp, °C	Open	Closed	Open	Closed	(dB)	Open	Closed
25	14.86	16.09	−15.78	−24.02	8.24	0.0062	0.0127
50	14.24	15.48	−15.95	−24.41	8.46	0.0065	0.0132
75	13.62	14.86	−16.07	−25.98	9.91	0.0078	0.0139
100	13.001	13.62	−16.54	−26.44	9.90	0.0064	0.0128

The effect of temperature on the natural frequency of laminated composite beam is plotted in Figure 13. It can be observed that with an increase in temperature, the natural frequency of the system considerably reduces. This may be due to the reduction in stiffness of the laminate upon exposure to an elevated thermal environment.^9,16,20,35 It can also be noted that the vibration control phenomenon is also affecting the natural frequency of the beam. It is clear that the natural frequency in closed-loop mode (i.e., when control action is applied) is higher than that of open-loop mode. This may be due to the change in system dynamics when control force is applied.^24,25

Figure 13.

Influence of temperature on the open and closed-loop fundamental frequency of the laminated composite beam.

Figure 14 illustrates the influence of temperature on the peak amplitude of vibration. From the results, it is evident that the peak amplitude (dB) at a resonant frequency slightly reduces with an increase in temperature magnitude for the open-loop model. Whereas for the closed-loop model, the peak amplitude value reduces considerably with an increase in temperature. This development may be due to the increase in efficiency of the control algorithm with the increase in temperature. Similar observations were made by many researchers on AVC of isotropic materials at elevated thermal environments.^11,14 The effect of temperature on the damping ratio of the composite beam is plotted in Figure 15. The damping ratio ( $ξ$ ) is computed using the decay in the amplitude of time-domain signals. Using the method of logarithmic decrement ( $δ$ ), the damping ratio can be calculated as

ξ = \frac{δ}{\sqrt{4 π^{2} + δ^{2}}}, in which δ = \frac{1}{n} \ln (\frac{x_{n}}{x_{n + 1}})

Figure 14.

Influence of temperature on the peak amplitude (dB) of the laminated composite beam.

Figure 15.

Influence of temperature on the damping ratio of the laminated composite beam.

Where x_n and x_n+1 are amplitudes of two consecutive troughs or crests of the time-domain signal. From the results, it can be observed that as temperature increases, the damping ratio increases to 75°C. On exposure to elevated thermal environments, the material properties of the fiber-reinforced laminated composite structures deteriorate. The phenomenon is mainly due to the debonding between the constituent fiber and matrix of the composite material as shown in Figure 16. Several researchers have captured the debonding phenomenon of fiber-reinforced composite materials using scanning electron microscope images.^21,36 The mechanical and thermal properties of the fiber and matrix are different, and they act differently when exposed to elevated thermal environments. The debonding of fiber and matrix causes the reduction in stiffness of the structure. This, in turn, causes a reduction in the natural frequency of the system and enhancement in the damping ratio. Similar observations were made by many researchers.^11,13,33 It can also be observed that the damping ratio value reduced at 100°C. This may be due to the softening of the epoxy resin matrix of glass-epoxy composite laminate upon crossing its glass transition temperature.³⁷ Thus, the energy absorbing capability of laminated composites becomes poor due to the feeble molecular chains of the epoxy resin at this stage.

Figure 16.

Schematic representation fiber-matrix debonding at elevated thermal environments.

The influence of the temperature on the amplitude reduction of vibration is presented in Figure 17. From the results, it is apparent that with an increase in temperature values, the vibration attenuation characteristics also increase.^11,14 This trend can be observed till the laminated composite beam crosses its glass transition temperature.

Figure 17.

Influence of temperature on the amplitude reduction (dB) of the laminated composite beam.

Conclusion

Experimental and numerical methods are used to understand the influence of temperature on the AVC behavior of the laminated composite cantilever beam. The results are presented in both the time and frequency domain for a wide temperature range considered (25°C–100°C). Robust fuzzy logic control is successfully developed to attenuate the vibrating cantilever beam operating in an elevated thermal environment using 20 if-then rules. The simulation results are compared with the experimental results for the sake of validations of developed plant models. The influence of temperature on natural frequency, damping ratio, and peak amplitudes are presented for both open- and closed-loop models. The results indicate that the efficiency of the proposed control strategy increases with an increase in temperature values. The damping ratio and the magnitude of amplitude reduction decrease above glass transition temperature as the energy absorbing capability is getting decreased due to feeble molecular chains. In general, it can be concluded that the designed fuzzy logic controller can effectively attenuate the vibrations of laminated cantilever composite beams in different thermal environments.

Footnotes

Acknowledgements

The authors thank the Science and Engineering Research Board (DST-SERB, Project No. EEQ/2017/000744), Govt. of India, and Department of Mechanical Engineering, National Institute of Technology Karnataka Surathkal for supporting with all essential endowments and accommodations.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Data availability statement

All data generated or analyzed during this study are included in the article.

ORCID iD

Subhaschandra Kattimani

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. Study of mechanical and dynamic mechanical behavior of halloysite nanotube-reinforced multiscale syntactic foam. J Appl Polym Sci 2021; 138: 11–16.