Dynamical analysis and controllers performance evaluation for single degree-of-freedom system

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VOLUME 13 , ISSUE 1 (Jan 2020) > List of articles

### Dynamical analysis and controllers performance evaluation for single degree-of-freedom system

Keywords : FLC, LPID, NPID, MSDS, SDOF, Mamdani system, Defuzzification

Citation Information : International Journal on Smart Sensing and Intelligent Systems. Volume 13, Issue 1, Pages 1-12, DOI: https://doi.org/10.21307/ijssis-2020-018

Received Date : 02-July-2020 / Published Online: 12-August-2020

### ARTICLE

#### ABSTRACT

In automation and mechatronics applications, mass spring damper system (MSDS) plays a significant role in ensuring model serviceability and safety. The system’s dynamic of this mechanical system is quite challenging to control. In this paper, the system is a single degree of freedom (SDOF) spring mass system. The issue of performance evaluation of three controllers, linear proportional integral derivate (LPID), nonlinear PID (NPID), and fuzzy logic controller (FLC), is presented. FLC demands that experience be built on information based on a rule. It has two inputs, one of which is the displacement and the other is the velocity, and force is calculated for forced damped vibration control for a single degree of freedom system. The method used for defuzzification is the center of gravity (COG). However, effective control of a system depends largely on the accuracy of the mathematical model that predicts its dynamics behavior. The mathematical model for the MSDS is based on a set of nonlinear second-order ordinary differential equations to simulate the dynamic accurately. The proposed control schemes are implemented with the aid of MATLAB and SIMULINK to investigate the system performance. The sequence of the three controllers in terms of performance is as follows: the first of them is FLC and then in some dominant cases this is NPID by comparison with the LPID and one without any controller cases. Also, FLC displays more effectiveness and efficiency than the system without this controller after a comparative of system analysis performance evaluation.

#### Nomenclature

$m$

mass

$k$

spring stiffness coefficient

$b$

viscous damping coefficient

$F$

force applied to the system externally

$x$

displacement

$ẋ$

velocity

$ẍ$

acceleration ( $a$ )

Z

zero

PM

positive medium

PS

positive small

NM

negative medium

NS

negative small

$xi$

point in the conclusion universe (i = 1, 2, 3,…)

$μc(xi)$

membership value of the result of the conclusion set

$γ$

design parameter

$ω$

natural frequency of the exciting force

Vibration is a big problem in many engineering systems. Active control for vibrations is very important topic with a lot of applications and fiscal benefits. An SDOF spring mass damper system represents many engineering structures. For example, in facing road bump, the vibration in a car can be modeled using the SDOF system. Also, the SDOF system is easy to analyze. It is easier to develop complicated systems after starting from a benchmark problem like SDOF problem. The comfort and maneuverability of a vehicle depend quite a lot on the vibration of the vehicle. Most of the vehicles implement a semi-active suspension system. Although they perform adequately, demand for smooth and cost-effective active-vibration controlled suspension is high (Sharma and Singh, 2014).

The momentous contributions of MSDS are notably reflected in the fields of automation and mechatronics engineering. It is extensively used as a compliant actuator in robot manipulator (Li and Yin, 2017), serves as a shock absorber in the vehicle suspension system (Akpakpavi, 2017; Allamraju, 2016; El-Nasser et al., 2015; Katal and Singh, 2012; Srinivasan et al., 2016), piezoelectric vibration energy harvester (Bahiah et al., 2017; Caliò et al., 2014; Kundu and Nemade, 2016), used as anti-vibration devices for sky-scraping building to improve serviceability and suitable for teaching dynamic modeling in control systems. In recent times, the deployment of the MSD system is significantly on high demand in health services to aid the control of rehabilitation robots in reinstating the incapacitated and aged people and in a hybrid vehicle suspension system to enhance passenger trip comfort and vehicle stability.

Furthermore, several control methods had been suggested and tested and new control schemes are still emerging: Vargas and Bruneau (2004) defined metallic dampers were to be structural fuses. Sánchez and Pugnaloni (2011) studied the response of an SDOF mechanical system composed of a primary mass and a particle damper.

Also, Shen et al. (2015) studied analytically the SDOF system under the control of three semi-active methods. The nonlinear model of the rotary axial eddy current damper and the energy dissipation was investigated by Liang et al. (2019). Bhutta et al. (2016) studied mathematical modeling of a double forced double MSDS.

Moreover, Orhorhoro et al. (2016) reported the research work carried on the MSD model in a phase variable form. Sharma and Singh (2014) controlled the vibrations of MSDS utilizing fuzzy logic.

However, vibration technology grew and took on a more interdisciplinary character. This was caused by more demanding performance criteria and specifications for the design of all types of machines and structures. So, considerable attention has been paid to research, development, and implementation of structural control devices. The essential aim of this project is to dynamical analysis for MSDS with SDOF and to compare performance evaluation of three different controllers including constructing a rule base that works effectively.

The rest of this paper is ordered as follows: methodology of mechanical system model with SDOF and controller design is given in the second section. The third section clarifies simulation results and discussion. Conclusion is given in the fourth section.

### Methodology of mechanical system model with SDOF and controller design

Many times, the SDOF framework can be used for modeling the machine. In addition, visualization and analysis are very simple. When designing a control approach, an SDOF problem is recommended, which is a benchmark problem (Sharma and Singh, 2014; Frankovský et al., 2011). The SDOF mechanical system is shown in Figure 1 with MSD. A harmonically variable force: $F(t)=F0sin(ωt)$F(t) = F0 sin (ωt) is used for the model excitation.

##### Figure 1:

SDOF mechanical system.

Therefore, trying to establish a mathematical model that depicts the dynamics of the MSDS and manipulates certain system parameters to restrict the force to have a desired response and performance of the system.

Dynamic motion equation that applies to the mechanical system (Fig. 1) in the form of a vector:

$(1)ma=Fr+Fd+F(t).$

Substituting the forces equation of the motion:

$(2)mẍ=−Fr−Fd+F(t),$
(2)where Fr is the reaction force in the spring; Fd the damping force; and F(t) the excitation force.

The reaction force in the spring is:

$(3)Fr=kx.$

The force of damping is as follows:

$(4)Fd=bẋ.$

The force of excitation is as follows:

$(5)F(t)=F0sin(ωt).$

After substitution of the relation (3), (4), (5) to Equation (2), we obtain the mechanical system acceleration with SDOF as follows:

$(6)ẍ=−bmẋ−kmx+1mF(t).$

As a consequence, we obtain the mechanical system’s kinematic values (displacement, velocity, and acceleration) in relation to time t (Table 1).

##### Table 1.

Values of system parameters.

Figure 2 shows the system of MSDS without any controller.

##### Figure 2:

MSDS without controller.

In addressing the control of position and velocity of MSDS, several control algorithms have been proposed and tested. Three different controllers are designed and a comparative analysis is performed on SDOF of MSDS for model derived through Newton’s law. These suggested controllers are designed independently from each other as follows.

### LPID controller

PID (proportional integral derivative) control is a control loop feedback mechanism that is mostly used in industrial control systems. PID control calculates an error value and applies a correction based on proportional, integral, and derivative terms (Samin et al., 2011; Nafea et al., 2018; Sartika et al., 2019).

For the same system in Figure 2, we insert PID controller before the plant. In continuous time, the ideal PID controller is given as:

$(7)uPID=kpe+ki∫edt+kdė.$

Ziegler and Nichols proposed rules for determining values of kp, ki, and kd based on Katsuhiko (2010) and Ang et al. (2005) and clarified in Table 2.

Values of gains.

### NPID controller

The nonlinear PID control law is proposed based on Al-Samarraie et al. (2016) by replacing the integral of the error function with the integral of the error saturation function, by adjusting the parameters of the saturation function:

$(8)uNPID=kpe+ki∫0tsatγedt+kdė,$
(8)where Satγ is the saturation function given by:
$(9)Satγ(e)=γ×sign(e).$

Moreover, the design parameter γ for the NPID controller, γ  = 1.5.

### FLC

Velocity and displacement of the SDOF system were taken as input to the fuzzy controller and the actuation force was output, whereas fuzzy logic-based control system was used in the nonlinear active vibration control system. Now, as shown in Figure 3 that clarifies the system of MSD with FLC, five membership function in the first input and five membership function in the second input so that (25 rules) was designed as FLC in addition to, five membership function in the output. These all by utilizing MATLAB fuzzy logic toolbox.

##### Figure 3:

MSDS with FLC.

FLC utilizes the Mamdani system which in the consequent part uses fuzzy sets. To make an FLC (Mamdani controller), the first step is to specify the use of variables of linguistic and also define the statement of the problem for this function, as shown in Figures 4-7.

Output variable.

##### Figure 4:

FIS (fuzzy inference system).

##### Figure 5:

1st input displacement (error) variable.

##### Figure 6:

2nd input velocity (change in error) variable.

Velocity and displacement of the sprung mass were fed as input to the fuzzy controller, which was based on 25 rules developed by heuristics. The controller was found to work satisfactorily in perturbed conditions also. Illustration of the rule base of MSDS control is as follows.

For discrete sets, center of gravity for singletons (COGS) is determined as follows (Alan, 2010):

$(10)uCOGS=∑iμc(xi)xi∑iμc(xi).$

## Simulation results and discussion

The simulation results are obtained in two ways: without a controller and with controller, i.e. FLC, LPID, and NPID. The three cases for the force are considered here as follows.

### Case 1

$F=0.5N.$F = 0.5N.

Figure 8 illustrates rule viewer, and Figures 8 and 9 clarify a fuzzy inference system (FIS) rule viewer and three-dimensional plot of the output surface viewer and simulation results controller for MSDS, respectively.

##### Figure 8:

FIS in rule viewer.

##### Figure 9:

Surface viewer for error vs. change of error.

Moreover, the displacement and velocity with three independent controllers and without controller are displayed in Figures 10 and 11, respectively.

##### Figure 10:

Displacement without controlled and with controlled.

##### Figure 11:

Velocity without controlled and with controlled.

Table 3 shows the response of SDOF system and percentage reduction in response output for case 1.

##### Table 3.

Response of SDOF system and percentage reduction in response output.

### Case 2

$F=F0×sin(wt),$

where $A$A is the amplitude, $α=2πT$ .

Take T  = 2πs, F = 0.5N, so that, α  = 1 rad/s, and F  = sin(t)N.

However, the displacement and velocity with three independent controllers and without controller are displayed in Figures 12 and 13, respectively.

##### Figure 12:

Displacement without controlled and with controlled.

##### Figure 13:

Velocity without controlled and with controlled.

Table 4 shows the response of SDOF system and percentage reduction in response output for case 2.

##### Table 4.

Response of SDOF system and percentage reduction in response output.

### Case 3

Figures 14 and 15 show its ability in response of SDOF system and percentage reduction in response output of case 3.

##### Figure 14:

Displacement without controlled and with controlled.

##### Figure 15:

Velocity without controlled and with controlled.

Table 5 shows the response of SDOF system and percentage reduction in response output for case 3.

##### Table 5.

Response of SDOF system and percentage reduction in response output.

Generally, Figure 16 shows the control forces with using all these three different controllers (FLC, LPID, and NPID).

##### Figure 16:

Control signal.

The performance of the proposed controllers is tested. The results prove the effectiveness of the FLC controller and its ability in response of SDOF system and percentage reduction in response output as clarified in chart of percentage reduction in response in Figure 17.

##### Figure 17:

Chart of percentage reduction in response.

Although many studies have reported on the use of MSDS in various applications and its important role besides the growth of vibration technology and its take on a more interdisciplinary character, little research has been reported on the effectiveness of this use. Consequently, the below table shows the comparison between our system and the system in (Ahmed, 2018) as follows.

## Conclusion

Particularly, the significant contributions of the MSDS are reflected in the fields of automation and mechatronics engineering. This paper extended to velocity and acceleration performance evaluation, moreover, the consideration of the nonlinearity. In addition, fuzzy membership functions are an important part of a fuzzy system usually defined by chartered experts. A comparative analysis of the proposed control algorithms is analytically examined. It is evident from the simulation result that the FLC is active and appropriate for refining the system’s response and suitable for hands-on controlling of the position and velocity of an MSDS. Simulation results showed that the FLC satisfied all control specifications except in some cases where the NPID showed superiority as clarified in Figure 3 that shows chart of percentage reduction in response. However, FLC response is more satisfactory in terms of robustness, response of system and percentage reduction in response output. Besides, the FLC displays more efficiency than the system without this controller after comparing the performance evaluation of the system analysis, i.e. a significant reduction occurs in the response output of displacement and velocity.

## References

1. Ahmed, A. 2018. Fuzzy logic control of classic SDOF spring mass dashpot system. International Journal of Advanced Research in Engineering and Technology, 9(6): 51–58.
2. Akpakpavi, M. 2017. Modeling and control of a car suspension system using P, PI, PID, GA-PID and auto-tuned PID controller in MATLAB/Simulink. Journal of Multidisciplinary Engineering Science Studies 3(3): 1506–1513.
3. Alan, H. 2010. Fuzzy system case study, intelligent systems: lectures. London: Universitat de Girona.
4. Allamraju, K. V. 2016. Dynamic analysis of an automobile suspension system. IPASJ International Journal of Mechanical Engineering 4(7): 15–21.
5. Al-Samarraie, S. A. , Midhat, B. F. and Gorial, I. I. 2016. Nonlinear integral control design for DC motor speed control with unknown and variable external torque. Journal of Engineering and Sustainable Development 20(4): 19–33.
6. Ang, K. H. , Chong, G. and Li, Y. 2005. PID control system analysis, design, and technology. IEEE Transactions on Control Systems Technology 13(4): 559–576.
7. Bahiah, N. , Noor, M. and Ahmad, M. R. 2017. Modeling the vibrational dynamics of piezoelectric actuator by system identification technique. 7(3): 1506–1512.
8. Bhutta, H. A. , Mansoor, H. and Tariq, M. 2016. Proceedings of the 3rd International Conference on Engineering & Emerging Technologies (ICEET), Lahore, PK: Superior University, April 7–8.
9. Caliò, R. , Rongala, U. , Camboni, D. , Milazzo, M. , Stefanini, C. , de Petris, G. and Oddo, C. 2014. Piezoelectric energy harvesting solutions. Sensors 14(12): 4755–4790.
10. El-Nasser, A. , Ahmed, S. , Ali, A. S. , Ghazaly, N. M. and Abd El-Jaber, G. T. 2015. PID controller of active suspension system for a quarter car model. International Journal of Advances in Engineering & Technology 8(6): 899–909.
11. Frankovský, P. , Delyová, I. and Hroncová, D. 2011. Modeling of a mechanical system with one degree of freedom of movement in the MATLAB program. pp. 71-73.
12. Katal, N. and Singh, S. K. 2012. Optimization of PID controller for quarter-car suspension system using genetic algorithm. International Journal of Advance Research in Computer Engineering & Technology 1(7): 30–32.
13. Katsuhiko, O. 2010. Modern Control Engineering 5th ed., Prentice Hall.
14. Kundu, S. and Nemade, H. B. 2016. Modeling and simulation of a piezoelectric vibration energy harvester. Procedia Engineering 144: 568–575.
15. Li, Z. and Yin, Z. 2017. Position tracking control of mass spring damper system with time-varying coefficients. 29th Chinese Control and Decision Conference (CCDC), IEEE, 4994–4998.
16. Liang, L. , Feng, Z. and Chen, Z. 2019. “Seismic control of SDOF systems with nonlinear eddy current dampers”, Applied Sciences, 9(16): 1–18.
17. Nafea, M. , Mohammad Ali, A. R. , Baliah, J. and Mohamed Ali, M. S. 2018. Metamodel-based optimization of a PID controller parameters for a coupled-tank system. TELKOMNIKA Telecommunication Computer Electronics and Control 16(4): 1590–1596.
18. Orhorhoro, E. K. , Onogbotsere, M. E. and Ikpe, A. E. 2016. Simulation of a mass spring damper model in phase variable. ELK Asia Pacific Journal of Mechanical Engineering Research 2(2): 1–16.
19. Samin, R. E. , Jie, L. M. and Zawawi, M. A. 2011. PID implementation of heating tank in mini automation plant using programmable logic controller (PLC). International Conference on Electrical, Control and Computer Engineering, pp. 515–519.
20. Sánchez, M. and Pugnaloni, L. A. 2011. Effective mass overshoot in single degree of freedom mechanical systems with a particle damper. Journal of Sound and Vibration 330(24): 5812–5819.
21. Sartika, E. M. , Sarjono, T. R. and Saputra, D. D. 2019. Prediction of PID control model on PLC. TELKOMNIKA 17(1): 529–536.
22. Sharma, M. and Singh, S. P. 2014. Fuzzy logic-based vibration control of a single degree of freedom system. 2014 Recent Advances in Engineering and Computational Sciences (RAECS), IEEE, pp. 1–6.
23. Shen, Y. , Fan, M. , Li, X. , Yang, S. and Xing, H. 2015. Dynamical analysis on single degree-of-freedom semi active control system by using fractional-order derivative. Mathematical Problems in Engineering 2015: 1–14.
24. Srinivasan, G. , Kumar, M. S. and Basha, A. M. J. 2016. Mathematical modeling and PID controller design using transfer function and root locus method for active suspension system. Middle-East Journal of Scientific Research 24(3): 622–627.
25. Vargas, R. and Bruneau, M. 2004. “Seismic response of single degree of freedom structural fuse systems”, 13th World Conference on Earthquake Engineering, Vancouver, August 1–6.

### FIGURES & TABLES

Figure 1:

SDOF mechanical system.

Figure 2:

MSDS without controller.

Figure 3:

MSDS with FLC.

Figure 4:

FIS (fuzzy inference system).

Figure 5:

1st input displacement (error) variable.

Figure 6:

2nd input velocity (change in error) variable.

Figure 7:

Output variable.

Figure 8:

FIS in rule viewer.

Figure 9:

Surface viewer for error vs. change of error.

Figure 10:

Displacement without controlled and with controlled.

Figure 11:

Velocity without controlled and with controlled.

Figure 12:

Displacement without controlled and with controlled.

Figure 13:

Velocity without controlled and with controlled.

Figure 14:

Displacement without controlled and with controlled.

Figure 15:

Velocity without controlled and with controlled.

Figure 16:

Control signal.

Figure 17:

Chart of percentage reduction in response.

2nd input velocity (change in error) variable.

Control signal.

### REFERENCES

1. Ahmed, A. 2018. Fuzzy logic control of classic SDOF spring mass dashpot system. International Journal of Advanced Research in Engineering and Technology, 9(6): 51–58.
2. Akpakpavi, M. 2017. Modeling and control of a car suspension system using P, PI, PID, GA-PID and auto-tuned PID controller in MATLAB/Simulink. Journal of Multidisciplinary Engineering Science Studies 3(3): 1506–1513.
3. Alan, H. 2010. Fuzzy system case study, intelligent systems: lectures. London: Universitat de Girona.
4. Allamraju, K. V. 2016. Dynamic analysis of an automobile suspension system. IPASJ International Journal of Mechanical Engineering 4(7): 15–21.
5. Al-Samarraie, S. A. , Midhat, B. F. and Gorial, I. I. 2016. Nonlinear integral control design for DC motor speed control with unknown and variable external torque. Journal of Engineering and Sustainable Development 20(4): 19–33.
6. Ang, K. H. , Chong, G. and Li, Y. 2005. PID control system analysis, design, and technology. IEEE Transactions on Control Systems Technology 13(4): 559–576.
7. Bahiah, N. , Noor, M. and Ahmad, M. R. 2017. Modeling the vibrational dynamics of piezoelectric actuator by system identification technique. 7(3): 1506–1512.
8. Bhutta, H. A. , Mansoor, H. and Tariq, M. 2016. Proceedings of the 3rd International Conference on Engineering & Emerging Technologies (ICEET), Lahore, PK: Superior University, April 7–8.
9. Caliò, R. , Rongala, U. , Camboni, D. , Milazzo, M. , Stefanini, C. , de Petris, G. and Oddo, C. 2014. Piezoelectric energy harvesting solutions. Sensors 14(12): 4755–4790.
10. El-Nasser, A. , Ahmed, S. , Ali, A. S. , Ghazaly, N. M. and Abd El-Jaber, G. T. 2015. PID controller of active suspension system for a quarter car model. International Journal of Advances in Engineering & Technology 8(6): 899–909.
11. Frankovský, P. , Delyová, I. and Hroncová, D. 2011. Modeling of a mechanical system with one degree of freedom of movement in the MATLAB program. pp. 71-73.
12. Katal, N. and Singh, S. K. 2012. Optimization of PID controller for quarter-car suspension system using genetic algorithm. International Journal of Advance Research in Computer Engineering & Technology 1(7): 30–32.
13. Katsuhiko, O. 2010. Modern Control Engineering 5th ed., Prentice Hall.
14. Kundu, S. and Nemade, H. B. 2016. Modeling and simulation of a piezoelectric vibration energy harvester. Procedia Engineering 144: 568–575.
15. Li, Z. and Yin, Z. 2017. Position tracking control of mass spring damper system with time-varying coefficients. 29th Chinese Control and Decision Conference (CCDC), IEEE, 4994–4998.
16. Liang, L. , Feng, Z. and Chen, Z. 2019. “Seismic control of SDOF systems with nonlinear eddy current dampers”, Applied Sciences, 9(16): 1–18.
17. Nafea, M. , Mohammad Ali, A. R. , Baliah, J. and Mohamed Ali, M. S. 2018. Metamodel-based optimization of a PID controller parameters for a coupled-tank system. TELKOMNIKA Telecommunication Computer Electronics and Control 16(4): 1590–1596.
18. Orhorhoro, E. K. , Onogbotsere, M. E. and Ikpe, A. E. 2016. Simulation of a mass spring damper model in phase variable. ELK Asia Pacific Journal of Mechanical Engineering Research 2(2): 1–16.
19. Samin, R. E. , Jie, L. M. and Zawawi, M. A. 2011. PID implementation of heating tank in mini automation plant using programmable logic controller (PLC). International Conference on Electrical, Control and Computer Engineering, pp. 515–519.
20. Sánchez, M. and Pugnaloni, L. A. 2011. Effective mass overshoot in single degree of freedom mechanical systems with a particle damper. Journal of Sound and Vibration 330(24): 5812–5819.
21. Sartika, E. M. , Sarjono, T. R. and Saputra, D. D. 2019. Prediction of PID control model on PLC. TELKOMNIKA 17(1): 529–536.
22. Sharma, M. and Singh, S. P. 2014. Fuzzy logic-based vibration control of a single degree of freedom system. 2014 Recent Advances in Engineering and Computational Sciences (RAECS), IEEE, pp. 1–6.
23. Shen, Y. , Fan, M. , Li, X. , Yang, S. and Xing, H. 2015. Dynamical analysis on single degree-of-freedom semi active control system by using fractional-order derivative. Mathematical Problems in Engineering 2015: 1–14.
24. Srinivasan, G. , Kumar, M. S. and Basha, A. M. J. 2016. Mathematical modeling and PID controller design using transfer function and root locus method for active suspension system. Middle-East Journal of Scientific Research 24(3): 622–627.
25. Vargas, R. and Bruneau, M. 2004. “Seismic response of single degree of freedom structural fuse systems”, 13th World Conference on Earthquake Engineering, Vancouver, August 1–6.