**An Optimal Distribution of Actuators in Active Beam Vibration – Some Aspects, Theoretical Considerations**

Adam Brański *Rzeszow University of Technology Poland* 

## **1. Introduction**

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The reduction of the effects of mechanical vibration fall into the of vibration isolation, design for vibration or vibration control (de Silva, 2000). The vibration control is subdivided into two group: passive control and active one. The core of the vibration control is to detect the level of vibration in a system and to counteract the effects of the vibration, so it needs two devices.

Hence, the passive devices do not require external power for their operation. Hence, passive control is relatively simple, reliable and economical. But it has limitations namely, the control force depends entirely on the natural dynamics and it may not be adjust on line. Furthermore, in a passive device, there is no supply of power from an external source. It leads to the incomplete control, particularly in complex and high-order systems.

The shortcomings of passive control can be overcome using an active one. In this case, the system response is directly sensed on line and on that basis, the specific control actions are applied to any locations of the system. But the active control needs external power, namely to apply control forces to vibrating system through actuators and to measure vibration response using sensors.

Two different types of actuators can be applied (Shimon et al., 2005). The first, inertial actuators, make up a piezoelectric material to vibrate large masses. Their vibrations are used to counteract the vibrations of the structure (Jiang et al., 2000). The advantages and disadvantages are enumerated in above reference.

The second type of actuators is a layer of smart or intelligent materials. The sensors also belong to these materials; together they are well−known as piezoelectric elements (Tylikowski & Przybyłowicz, 2004). It was shown that these elements can offer excellent potential for an active vibration reduction of the structure vibrating with low frequencies (Croker, 2007; Fuller at al, 1997; Hansen & Snyder, 1997; Kozień, 2006; Przybyłowicz, 2002; Wiciak, 2008). As a general, piezoelectric elements are glued to the host structure. It makes the advantage, namely their incorporating into the structure is that the actuating mechanism becomes part of the structure. Both sensors and actuators are relatively light, compared to the structure, and can be made in arbitrary shape. The disadvantage is that they once bonded and they cannnot be used again. In recent years the measure of the vibration with the sensors are replaced by touch less measures. For this reason, hereafter in research the sensors are omitted and only second type actuators will be considered. Nowadays actuators

An Optimal Distribution of Actuators in

beam vibration were considered.

too.

Active Beam Vibration – Some Aspects, Theoretical Considerations 399

Basing on the quasi-optimal distribution of the actuators, the protection beam vibration is achieved (Brański et al., 2010; Brański & Lipiński, 2011). In this case always the separate modes were considered. The problem was solved based on heuristic reasons and was confirmed analytically. In the latest own research, the results presented in (Brański et al.,

In this chapter, the above attitude to the optimal actuators distribution is continued and extended. First at all, the optimal problem is formulated. For this purpose, the optimization criterion is defined. It is assumed that a measure of the vibration reduction is a reduction coefficient (Szela, 2009; Brański & Szela 2010) and here it becomes the objective function. This attitude is quite similar to the maximization of the control forces transmitted by the

Dynamics effects of the glue and actuators are also considered. Furthermore, the solution of active vibrations reduction is derived for general solution, not only for separate modes. Since analytical solution was attained with separation of variables method, first of all the modes of the problem are derived. Next, the orthogonality condition of the modes is derived

The simple supported beam is chosen as the research object. The study of beams is very important in a variety of practical cases, noteworthy, the vibration analysis of structures like bridges, tall buildings, and so on. Loosing a bit on generality, it is considerably easier to realize the aim of the paper. It is assumed that the beam is excited with evenly spread and harmonic force. The material inner damping coefficients of all elements of the research system are taken into account. It seems that all main factors having the influence on the

To solve the problem analytically, a few simplifications are made. Namely, the energy provided to the system is in the form of voltage applied to the surface of the actuators. Assuming that the charge is homogenously distributed, as a result of piezoelectric effect, the actuators interact with the beam with moments for couple of forces homogenously distributed along the actuators' edges. Next, these moments are replaced with the couple of

All problems were considered only theoretically; no calculations are run. It seems that presented considerations will be the base to many numerical simulations and experiments. To the author's knowledge, the theoretical description of the optimal actuators distribution

In this problem, the additional elements make the concentrated masses and actuators and all constitute the mechanical set beam-actuators-masses. Adding actuators (and the glue at the same time) is the technical necessity but they introduce to the mechanical set the additional dynamics effects namely, local stiffness and concentrated masses. As far as concentrated masses are concerned, adding them is substantiated as follows. The proposed optimal distribution of the actuators needs asymmetrical beam vibrations and these ones may be

There are four theories (models) for the transversely vibrating uniform beam (Han et al., 1999): Euler-Bernoulli, Rayleigh, shear and Timoshenko. The first of them, called the

on even simple structure like the beam, up to now have not been brought up.

**2. Active beam vibration reduction with additional elements** 

2010) were substantiated analytically (Brański & Lipiński, 2011).

actuators to the structure (Q. Wang & C. Wang, 2001).

forces and finally, they are counteracted the vibrations.

ensured by at least one concentrated mass.

**2.1 Uniform beam vibration with damping** 

are used to very original structures for example to the satellite boom (Moshrefi-Torbati et al., 2006) or to sun plate (Qiu et al., 2007).

To make the reduction more effective, many problems should be solved.


Reviewing the literature, it appears that the actuators distribution play a major part. Now, a question arises about an optimal distribution of actuators. In the recent year, a great number of papers has been published on this subject. It is obvious that there are a lot of optimization techniques; an excellent survey is given in (Bruant et al., 2010). Two main approaches are distinguished to this problem.

First of them is the coupling of the optimization of actuators/sensors locations and controller parameters. In this case the following criterions are taken into account for the optimization:


As can be seen, the optimization criterions are dependent on the choice of controllers. Therefore, the optimal location obtained using one controller may not be a suitable choice for another one.

At the latter approach, the optimal location is obtained independently of the controller definition. In this case, the following criterions are used:


In the quoted references, it was not provided the actuators distribution in explicite; only the general rules (criterions) were formulated. However, this problem was partially solved; it was proved in (Brański & Szela, 2007; Brański & Szela, 2008; Szela, 2009; Brański & Szela 2010; Brański & Lipiński, 2011) that the most effective actuators distribution was on the structure sub-domains with the largest curvatures; such distribution was called quasioptimal one. As the research object, a right-angled triangle plate with clamped-free-free boundary conditions was taken into account. The quasi-optimal distribution was deduced based on the heuristic reasons and the conclusions were confirmed only numerically. Furthermore, the problem was solved merely for the separate modes.

are used to very original structures for example to the satellite boom (Moshrefi-Torbati et al.,

• dynamic effects (mass loading and stiffness) of the actuators on the structure vibration (Charette et al., 1998; Gosiewski & Koszewnik, 2007; Hernandes et al., 2000; Q. Wang &

• dynamic effects of the glue (between actuators and structure) on the structure vibration

Reviewing the literature, it appears that the actuators distribution play a major part. Now, a question arises about an optimal distribution of actuators. In the recent year, a great number of papers has been published on this subject. It is obvious that there are a lot of optimization techniques; an excellent survey is given in (Bruant et al., 2010). Two main approaches are

First of them is the coupling of the optimization of actuators/sensors locations and controller parameters. In this case the following criterions are taken into account for the optimization: • quadratic cost function of the measure error and the control energy (Bruant et al., 2001),

• spatial H2 norm of the closed-loop transfer matrix from the disturbance to the

As can be seen, the optimization criterions are dependent on the choice of controllers. Therefore, the optimal location obtained using one controller may not be a suitable choice

At the latter approach, the optimal location is obtained independently of the controller

• maximization controllability/observability criterion using the gramian matrices (Bruant

• modal controllability index based on singular value analysis of the control vector

• maximization of the control forces transmitted by the actuators to the structure (Q.

In the quoted references, it was not provided the actuators distribution in explicite; only the general rules (criterions) were formulated. However, this problem was partially solved; it was proved in (Brański & Szela, 2007; Brański & Szela, 2008; Szela, 2009; Brański & Szela 2010; Brański & Lipiński, 2011) that the most effective actuators distribution was on the structure sub-domains with the largest curvatures; such distribution was called quasioptimal one. As the research object, a right-angled triangle plate with clamped-free-free boundary conditions was taken into account. The quasi-optimal distribution was deduced based on the heuristic reasons and the conclusions were confirmed only numerically.

• actuators' geometric-technical features (Frecker, 2003; Hong et al., 2007; Wang, 2007), • orientation of the actuators on the structure (Bruant et al., 2010; Ip & Tse, 2001; Qiu et

To make the reduction more effective, many problems should be solved.

• appropriate actuators distribution on the structure (Bruant et al., 2010),

• maximization of dissipation energy during the control (Yang, 2005),

• using the H2 norm (Halim & Reza Moheimani, 2003; Qiu et al., 2007).

Furthermore, the problem was solved merely for the separate modes.

• simultaneous simple H∞ controller (Guney & Eskinat, 2007).

distributed controlled output (Liu et al., 2006),

definition. In this case, the following criterions are used:

& Proslier, 2005; Jha & Inman, 2003),

(Dhuri, & Seshu, 2006),

Wang & C. Wang, 2001),

2006) or to sun plate (Qiu et al., 2007).

• others, but they play a minor part.

distinguished to this problem.

(Pietrzakowski, 2004; Sheu et al., 2008).

C. Wang, 2001)

al., 2007),

for another one.

Basing on the quasi-optimal distribution of the actuators, the protection beam vibration is achieved (Brański et al., 2010; Brański & Lipiński, 2011). In this case always the separate modes were considered. The problem was solved based on heuristic reasons and was confirmed analytically. In the latest own research, the results presented in (Brański et al., 2010) were substantiated analytically (Brański & Lipiński, 2011).

In this chapter, the above attitude to the optimal actuators distribution is continued and extended. First at all, the optimal problem is formulated. For this purpose, the optimization criterion is defined. It is assumed that a measure of the vibration reduction is a reduction coefficient (Szela, 2009; Brański & Szela 2010) and here it becomes the objective function. This attitude is quite similar to the maximization of the control forces transmitted by the actuators to the structure (Q. Wang & C. Wang, 2001).

Dynamics effects of the glue and actuators are also considered. Furthermore, the solution of active vibrations reduction is derived for general solution, not only for separate modes. Since analytical solution was attained with separation of variables method, first of all the modes of the problem are derived. Next, the orthogonality condition of the modes is derived too.

The simple supported beam is chosen as the research object. The study of beams is very important in a variety of practical cases, noteworthy, the vibration analysis of structures like bridges, tall buildings, and so on. Loosing a bit on generality, it is considerably easier to realize the aim of the paper. It is assumed that the beam is excited with evenly spread and harmonic force. The material inner damping coefficients of all elements of the research system are taken into account. It seems that all main factors having the influence on the beam vibration were considered.

To solve the problem analytically, a few simplifications are made. Namely, the energy provided to the system is in the form of voltage applied to the surface of the actuators. Assuming that the charge is homogenously distributed, as a result of piezoelectric effect, the actuators interact with the beam with moments for couple of forces homogenously distributed along the actuators' edges. Next, these moments are replaced with the couple of forces and finally, they are counteracted the vibrations.

All problems were considered only theoretically; no calculations are run. It seems that presented considerations will be the base to many numerical simulations and experiments. To the author's knowledge, the theoretical description of the optimal actuators distribution on even simple structure like the beam, up to now have not been brought up.
