Abstract

The single degree-of-freedom (SDOF) system under the control of three semiactive methods is analytically studied in this paper, where a fractional-order derivative is used in the mathematical model. The three semiactive control methods are on-off control, limited relative displacement (LRD) control, and relative control, respectively. The averaging method is adopted to provide an analytical study on the performance of the three different control methods. Based on the comparison between the analytical solutions with the numerical ones, it could be proved that the analytical solutions are accurate enough. The effects of the fractional-order parameters on the control performance, especially the relative and absolute displacement transmissibility, are analyzed. The research results indicate that the steady-state amplitudes of the three semiactive systems with fractional-order derivative in the model could be significantly reduced and the control performance can be greatly improved.

1. Introduction

Machinery equipment will vibrate if it is excited, and more damage can be caused when it is under the condition of resonance. In order to reduce vibration, Frahm invented the original shock absorber based on classical vibration theory. Then Den Hartog propounded the design rule for damping dynamic shock absorber based on the fixed-points theory [1]. Subsequently, many improved shock absorbers were presented based on this one. At that stage, the control method should be named as passive control one. The damping force of passive vibration control system could not be adjusted instantaneously, so that it may be limited effective. With the development of science and technology, active control and semiactive control methods came into being. These two kinds of control methods performed noticeably well and they were adopted in many fields such as vehicle suspension and vibration absorption system [2–5].

Vehicle suspension could be simplified to a single degree-of-freedom (SDOF) semiactive control system under some assumptions. The study on semiactive control methods originated from Karnopp et al. who presented the on-off control method in 1974 [2]. After that, many scholars carried out research directing at semiactive suspension system, and the research efforts were focused on the control methods and/or design of control devices [6–8]. Some other scholars preferred analytical method than numerical one, because the analytical study may present more information about the adjustment or design of the controllable parameters. Shen et al. [9] applied averaging method to present analytical investigations for different control methods. Eslaminasab et al. [10] adopted averaging method to provide an analytical platform for analyzing the performance of relative control method. Shen et al. [11–13] studied the approximately analytical solutions and parameters optimization of four semiactive on-off dynamic vibration absorbers and researched a single degree-of-freedom semiactive oscillator with time delay.

In fact, it is rather hard to accurately describe the constitutive relations of most real materials by the classical integer-order model, especially for some viscoelastic and/or rheological materials. Therefore, some scholars started to research fractional-order derivative models. As far back as 1985, Bagley and Torvik [14–16] introduced fractional-order derivative into fluid mechanics and modelled the constitutive relation of the material viscoelasticity in many vibration mitigations. Rossikhin and Shitikova [17] proposed a method to analyze the free damped vibrations of a fractional-order oscillator. Tavazoei et al. [18] and Pinto and Tenreiro Machado [19] studied the fractional-order van der Pol oscillator and found multiple limit cycles existing in the system. Shen et al. [20–23] analytically studied the primary resonance of some nonlinear oscillators with different fractional-order derivatives by the averaging method and illustrated the effects of fractional-order parameters on dynamical response. Wahi and Chatterjee [24] studied an oscillator with special fractional-order derivative and time-delay by averaging method. Chen and Zhu [25], Wang and He [26], Huang and Jin [27], and Xu et al. [28] also investigated different fractional-order systems and presented some important results by analytical research.

In this paper, we introduce fractional-order derivative into semiactive SDOF systems, where the semiactive methods are on-off control, limited relative displacement (LRD) control, and relative control, respectively. The analytical solutions of the three semiactive systems are obtained by averaging method. Then we study the stability of those steady-state solutions by Lyapunov stability theory. Furthermore, the effects of fractional-order parameters on the steady-state amplitude and control performance are researched. At last the main conclusions are made.

2. Analytical Solutions of SDOF Semiactive System

The considered SDOF semiactive system is shown in Figure 1. If the fractional-order viscoelastic device is used, its mathematical model should bewhere is the external excitation to the controlled system with and as the excitation amplitude and excitation frequency. , , and are the system mass, system damping coefficient, and stiffness, respectively. is the fractional-order derivative of to with the fractional coefficient and the fractional-order . There are several definitions for fractional-order derivative, and they are equivalent under some conditions for a wide class of functions. In Caputo’s sense, the definition of order derivative of to iswhere is Gamma function that satisfies .

Figure 1 

The model of SDOF semiactive system by using fractional-order derivative.

Defining , (1) becomesUsing the following transformation of coordinatesequation (3) becomes

Supposing the solution of (5) iswhere , one could getFrom (5), (6), and (7), one could obtainwhere

The derivatives of the generalized amplitude and phase could be solved asThen, one could apply the standard averaging method to (10) in time interval Furthermore, (11) could be written aswhere

According to the averaging method, one could select the time terminal as if the integrand is periodic function or if the integrand is aperiodic one. Accordingly, one could obtain

In order to obtain the definite integral in (13c) and (14c), two important formulae, which have been deduced in [20, 21], are introducedBased on Caputo’s definition, (13c) becomesIntroducing and , (17) becomesIn fact, the second part in (18) would vanish according to (16a) and (16b). Defining the first part in (18) as and integrating it by parts, one could obtainBased on (16a) and (16b), one could find that the second part in (19) would vanish and the first part in (19) will beThat meansAfter the similar procedure, (14c) will beEquations (13b) and (14b) can be rewritten as The solutions of (23a) and (23b) depend on the specific semiactive control methods.

Accordingly, (12) becomeswhich could be rewritten aswhere and are defined as the equivalent damping coefficient and equivalent stiffness coefficient in semiactive systems, respectively. In the SDOF semiactive system, the fractional-order derivative serves as damping and stiffness at the same time. When , will be changed into spring force, whereas it will be almost the same as damping force if . When , will have little effect on control performance and the semiactive system will degrade into the traditional semiactive one.

3. Cases for Three Kinds of Semiactive Control Systems

3.1. Analytical Investigation for on-off Control System

The strategy of semiactive on-off control isOriginally, is selected as in [2], which may be inaccurate in real engineering. Here we select it as .

After transformation of coordinates, (27) becomesExpanding the critical conditionone could getwhereEquation (28) can be rewritten as

Using the averaging method, one could obtain the simplified forms of (13b) and (14b) aswhere

Combining (12), (15a), (15b), (21), (22), and (33), one could obtainObviously, the equivalent damping coefficient in this case could be written asFrom (36), it could be found that the equivalent damping coefficient mainly depends on the fractional-order parameters and , associated with the critical angle .

Now we study the steady-state solution, which is more important and meaningful in vibration engineering. Letting and , one could obtain the amplitude and phase of steady-state responsewhere

The transmissibility for relative displacement isThe absolute displacement of on-off control system iswhere

The displacement transmissibility is

Next, we study the stability of steady-state solution. Letting and and then substituting them into (35), one could getBased on (37), one could eliminate and obtain the characteristic determinantExpanding the determinant, one could obtain the characteristic equationSolving (45), one could obtain characteristic valuesObviously, is less than 0. Accordingly, all real parts of characteristic roots are negative, which means the steady-state solution of this kind of semiactive system is unconditionally stable.

3.2. Analytical Investigation for LRD Control System

The strategy of LRD control is where and is the space between the mass and excitation point. Here we select .

After transformation of coordinates, (47) becomesand the control strategy can be rewritten aswhere .

Using the averaging method, one could obtain the simplified forms of (13b) and (14b)where , .

Combining (12), (15a), (15b), (21), (22), and (50), one could obtainObviously, one could easily obtain the equivalent damping coefficientFrom (52), it could be found that the equivalent damping coefficient mainly depends on the fractional-order parameters and , associated with the critical angle .

Now we study the steady-state solution, which is more important and meaningful in vibration engineering. Letting and , one could obtainwhere and are the amplitude and phase of steady-state response, respectively.

The transmissibility for relative displacement isThe absolute displacement of LRD control system iswhere where