Performance Comparison of Different Vibration Control Strategies

Abstract

Structural engineers are always struggling with the unpredictable complexities of the extreme vibration due to natural or human induced loads. It is not possible to eliminate the vibration from structures entirely what so ever. Therefore, in order to keep the structures safe and healthy, the vibration needs to be mitigated and that can be done by adopting passive, active or semi-active type control systems. Those aforementioned technologies come with a price, hence, it is not so straightforward to decide which technology should be adopted. The passive vibration systems (e.g. tuned mass damper, base-isolator) can be found both in many old and new structures due to their feasibility and simplicity. On the other hand, many modern structures are adopting active and semi-active control systems (e.g. dampers) as an alternative to achieve better control on structure. Both the active and semi-active control systems are more expensive than passive control systems but they offer better control on structures. In order to understand the discussed issue, herein, a comparative study has been performed to evaluated their performances. A detail comparison among passive, active and semi-active control alternatives have been conducted. The outcome shows that the passive control systems can be suitable where deformations are and the other alternatives would be beneficial where large deformations are expected.

1 Introduction

Vibration induced problems in the area of structural engineering is quite old and well-known issue but still require further attention due to the complexity of the matter. There are many vibration reduction technologies are available in the real-life applications [1,2,3,4,5,6,7,8,9]. However, even among those alternatives there is none neither flawless nor foolproof. As a result, the scope is still open for further development and update of vibration control tools and technologies. Among available vibration mitigation strategies, they broadly can be separated into three main categories, such as; (i) passive [8], (ii) active [10], and (iii) semi-active [11]. The working principles of those technologies are quite different from one to other. However, regardless the working mechanism the main goal remains same that is the vibration reduction. Based on their mechanism and efficacy, the feasibility is a point that needs to be taken care of [12]. In this regard, the passive strategy might be the cheapest solution in comparison to its alternatives. Additionally, the aforementioned systems do not require any external energy supply into it during the operation. Hence, the application of passive system is quite easy to adopt it. Nevertheless, the performance in terms of vibration reduction of the passive systems might not be the best choice in contrast to active or semi-active systems.

The active and semi-active systems require the energy/power supply during operation. Therefore, it might be a serious problem when an extreme event happens such as earthquake or tsunami. However, in case of aforementioned situation it might an option have a system that is operable during an extreme event without external power supply. For the early mentioned reason, the semi-active or hybrid type systems might be beneficial as they can be operated with and without external power supply. In the real-life applications almost, all of those technologies can be found for various purpose. The passive systems have been adopted in many structures for vibration mitigation such as; offshore structure [13], tall buildings [14], bridge [15]. Additionally, many research can be found that are conducted to update the performance by developing new devices such as a hybrid type passive system for multi-storied building [16], a novel translational tuned mass damper [17], elastomeric bearing [15].

The active and semi-active systems have employed in many structures, for instance, adaptive semi-active control for suspension systems [11], energy-based active control for tensegrity structures [18]. A validation of novel semi-active control scheme has been performed in [19], and reported that system identification and vibration control possible via the use of the unscented Kalman filter. Further, an energy-based control for swinging pendulums have been reported in [20], simultaneous control and monitoring via adaptive control in [21], intelligent control [4].

It is clear from the above discussion that to understand which vibration mechanism works better for what application is quite complicated. Hence, this study has focused to investigate the inter-comparison to understand main three categories of vibration strategies. To achieve the goal of the study, a 3-toried dynamical system is considered and all three control strategies have been implemented. The outcome of the study has been accommodated in the later section of this paper. The rest of the paper contains problems statement, results and discussion and finally, a summary of the study.

2 Problem Statement

The simplified formulation for the modeling of any dynamical problems are widely adopted instead of solving individual equation of motion. Therefore, in this study, the simulations are performed by adopting simplified formulation of structural dynamics so-called the lumped-mass model. In order to solve the dynamical system using the concept of the lumped-mass model, the equation of motion needs to be formulated. The early mentioned equation contains all the necessary information of the dynamical systems. Herein, the studied 3-degree-of-freedom (DOF) system has been described as,

$${M}_{3\times 3}\left\{\begin{array}{c}{\ddot{x}}_{1}\\ {\ddot{x}}_{2}\\ {\ddot{x}}_{3}\end{array}\right\}(t)+{C}_{3\times 3}\left\{\begin{array}{c}{\dot{x}}_{1}\\ {\dot{x}}_{2}\\ {\dot{x}}_{3}\end{array}\right\}(t)+{K}_{3\times 3}\left\{\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right\}(t)=\beta p(t)$$

(1)

where the mass matrix is given by ${M}_{3\times 3}=\left[\begin{array}{ccc}{m}_{1}& 0& 0\\ 0& {m}_{2}& 0\\ 0& 0& {m}_{3}\end{array}\right],$ while the damping matrix is ${C}_{3\times 3}=\left[\begin{array}{ccc}{c}_{1}+{c}_{2}& -{c}_{2}& 0\\ -{c}_{2}& {c}_{2}+{c}_{3}& -{c}_{3}\\ 0& -{c}_{3}& {c}_{3}\end{array}\right],$ and the stiffness matrix is ${K}_{3\times 3}=\left[\begin{array}{ccc}{k}_{1}+{k}_{2}& -{k}_{2}& 0\\ -{k}_{2}& {k}_{2}+{k}_{3}& -{k}_{3}\\ 0& -{k}_{3}& {k}_{3}\end{array}\right]$, $t$ is the time vector, $x$ is the displacement, $\dot{x}$ means the velocity, $\ddot{x}$ is the acceleration, $\beta$ is the control vector that control the location of the applied load, $p$ is the external loads.

Later, the equation motion has been transformed into a more compact formulation known as the state space formulation. The state space formulation consists of two main equations (i) system/process equation, and (ii) observation/measurement equation. Briefly, those two equations of state space formulation are given by,

$$\dot{X}\left({\text{t}}\right)=AX\left(t\right)+BU\left(t\right) \,{\text{and}}\, Y(t)=CX(t)+DU(t)$$

(2)

where $A$ is the system matrix that contains mass, stiffness and damping information of the system, $B$ is input control matrix, $C$ is the output matrix, $D$ is the feedthrough matrix, $Y$ reprents the output vector, $X$ is the state vector.

The comparison of the different control strategies have been performed by considering a three storied structure. The performances are evaluated for both uncontrolled and controlled structures. For the controlled structures the control systems are assumed be laced at the ground floor. The passive control system is given by the simplest form that is given by Eq. (3). It can be found that in the early mentioned equation that only a stiffness component is needed to represent a passive control device. Usually, the passive system is assumed to provide a linear control force that is controlled by the define stiffness of the device.

$${f}_{pa}={k}_{pa}x$$

(3)

In order estimate the control force for the active and semi-active control systems the control forces are defined as shown via Eqs. (5) and (6), respectively. It needs to be mentioned that depending on the choice of the control algorithm (e.g. full-state feedback or partial feedback) all floors displacement and velocity might be essential to calculate the desired control force. Herein is both the active and semi-active full state feedback type control law has been adopted namely the linear-quadratic-regulator (LQR). In other words, the adopted control algorithm required all floors displacements and velocities. The LQR control algorithm is widely used in the real-life applications and the detail can be found in many existing literatures. Briefly, the main equation that minimizes the cost function ($J$) and thereby optimized the controller performance. The cost function ($J$) is given as,

$$J=\sum\nolimits_{0}^{N}\left({X}^{T}QX+{U}^{T}RU\right)and U=\left[-{k}_{ac}^{d}-{k}_{ac}^{v}\right]X$$

(4)

$${f}_{ac}=\left[-{k}_{ac}^{d}-{k}_{ac}^{v}\right]\left[\begin{array}{c}x\\ \dot{x}\end{array}\right]$$

(5)

$${f}_{sac}=\left[-{k}_{sac}^{d}-{k}_{sac}^{v}\right]\left[\begin{array}{c}x\\ \dot{x}\end{array}\right]$$

(6)

where the ${f}_{ac}$ represents the active control force, ${f}_{sac}$ is the semi-active control force, $x$ and $\dot{x}$ are the displacement and velocity of the system. It is not necessary that ones have to use full-state feedback type control algorithm, depending on the individual problem type he/she may select the partial state feedback type algorithms. Typically, both the active and semi-active control systems are nonlinear and quite complex to model. However, it is assumed that the semi-active control strategies may ignore unwanted active components (see Fig. 6c), if that’s not the case then it would behave as active control system. Hence the employed control algorithm needs to be efficient enough to estimate the representative behavior correctly.

3 Results and Discussion

This study investigates the performances of the vibration reduction strategies. To do so, a 3-DOFs is considered and three approaches (i) passive, (ii) active, (iii) semi-active, have been implemented. The simulations are performed for 10 seconds with a sampling frequency of 1000 Hz. The floor masses are assumed to be around 72 Kg while the floor stiffnesses are considered to be around 10000 kN/m. The damping matrix is estimated based on the mass and stiffness properties.

As a first step, all of the floor’s displacement have been evaluated and the uncontrolled time-history response has been compared with controlled time-history data. The first, second, and third floors displacement time-histories have been depicted in Fig. 1, 2 and 3, respectively. The uncontrolled displacement data is given by the red line, while the passive controlled case by the black line, magenta line indicates the active controlled case and the green line represents the semi-active controlled case. It can be found in those aforementioned figures that all of the controlled strategies can be beneficial than without having any control. However, it is also noticeable that both the active and semi-active cases may perform better than passive system.

It is necessary to mention that the performance of any control strategy may vary significantly depending on how well-tuned or optimized. The tuning process may play a crucial role in case of active and semi-active control than passive. It is due the complex phenomenon of the active and semi-active control schemes. In other words, tuning process for passive control systems is quite straightforward as there are only two parameters e.g. mass ratio and stiffness.

Fig. 1.

The comparison of 1^st floor uncontrolled and controlled displacements.

Fig. 2.

The comparison of 2^nd floor uncontrolled and controlled displacements.

Fig. 3.

The comparison of 3^rd floor uncontrolled and controlled displacements.

In addition to displacements time-histories, the 3^rd floor’s velocities (see Fig. 4) and accelerations (see Fig. 5) are also compared. In those figures, it is also visible that the implemented control strategies worked quite effectively.

Fig. 4.

The comparison of 3^rd floor uncontrolled and controlled velocities.

Fig. 5.

The comparison of 3^rd floor uncontrolled and controlled accelerations.

In order to understand that the applied control strategies work, a confirmation of control force versus displacement plot of each control approach has been provided in Fig. 6. The early mentioned figure confirms that the passive (see Fig. 6a), active (see Fig. 6b), and semi-active control strategies (see Fig. 6c) work quite well.

Fig. 6.

Overview and confirmation of the implemented control strategies.

To be more precise, ones may easily see that the left sub-figure shows a clear elastic force that was applied by the passive system. Whereas, the middle and the right sub-figures show a complex hysteretic behavior that is what usually expected from any smart control strategy. However, there is a little difference between the middle and the right sub figures, on right sub-figure (semi-active) case note the x-axis because that confirms that this strategy is not same as active control case.

4 Conclusions

This work investigates the performances of different vibration control strategies numerically. To do this end, a 3-DOF system has been adopted and three control concepts have been implemented. The performance of any vibration scheme would come with some drawbacks hence serious assessment is essential by the designers. For instance, herein, it has been observed that the passive mitigation technology may not be suitable for extreme excitations. While the active or semi-active scheme might be beneficial but that would cost more than the passive system. However, considering all limitations (e.g. energy consumption) active and semi-active technologies would be better where extreme level of vibration is expected due to their inherent mechanism that helps to reduce extreme vibration. The authors believe that further study is necessary to enhance the performance as well as the control strategies need to be adopted in more complex structural systems.