An Optimal Distribution of Actuatorsin Active Beam Vibration – Some Aspects, Theoretical Considerations

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 & Przybylowicz, 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; Kozien, 2006; Przybylowicz, 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


Introduction
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 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.
• 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 & C. Wang, 2001) • dynamic effects of the glue (between actuators and structure) on the structure vibration (Pietrzakowski, 2004;Sheu et al., 2008). • 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 al., 2007), • appropriate actuators distribution on the structure (Bruant et al., 2010), • others, but they play a minor part. 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: • quadratic cost function of the measure error and the control energy (Bruant et al., 2001), • maximization of dissipation energy during the control (Yang, 2005), • spatial 2 H norm of the closed-loop transfer matrix from the disturbance to the distributed controlled output (Liu et al., 2006), • simultaneous simple H ∞ controller (Guney & Eskinat, 2007). 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: • maximization controllability/observability criterion using the gramian matrices (Bruant & Proslier, 2005;Jha & Inman, 2003), • modal controllability index based on singular value analysis of the control vector (Dhuri, & Seshu, 2006), • maximization of the control forces transmitted by the actuators to the structure (Q. Wang & C. Wang, 2001), • using the 2 H norm (Halim & Reza Moheimani, 2003;Qiu et al., 2007). 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 & 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.
Basing on the quasi-optimal distribution of the actuators, the protection beam vibration is achieved 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  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; 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.

Active beam vibration reduction with additional elements
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 ensured by at least one concentrated mass.

Uniform beam vibration with damping
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 classical beam theory, is applied here. It is simple and provides reasonable results for formulated problem. Let be the beam as depicted in Fig. 1. The Bernoulli-Euler equation governs transverse vibration (or bending or lateral vibration) of the beam has a following standard form (Kaliski, 1986;Pietrzakowski, 2004), where uu ( x , t ) = -beam deflection at the point x and the time t , f f(x,t) = -load force, hereafter the rest symbols are jointly explained. To solve Eq. (1) explicitly, four boundary conditions, at the ends of the beam, are needed. In general, boundary conditions represent displacement, slope, moment and shear respectively. Here, it is assumed that the beam is simple supported, then both displacement and the bending moment equal zero To solve over determined problem, one needs to know initial conditions. But here, the harmonic steady state plays a major part, so that the initial conditions are omitted.

Beam vibration with concentrated masses
To solve the intended problem, Eq. (1) must be rounded out. First of all, to obtain asymmetric modes and consistently asymmetric general vibration, a few concentrated masses are added to the beam (Low & Naguleswaran, 1998;Majkut, 2010;Naguleswaran, 1999 . For s n aktuators ( s n glue layers) and r n concentrated masses, Eq. (1) takes the form The Eq. (6) may be written down quite similar like Eq. (1), namely On the ground of the EJ , S ρ and r α form, Eq. (7) can not be understood in a classical manner. To solve it, some methods may be applied. One of them is presented in (Ercoli & Laura, 1987;Kasprzyk & Wiciak, 2007;Majkut, 2010); another attitude may be found in (C.N. Bapat & C. Bapat, 1987) and it is applied here. At the latter attitude, the beam is divided into some uniform elements. The division may not be coincidental. To clearly explain this problem, for simplicity consider a set beam-one actuator (and glue)-one concentrated mass, Fig. 5. The division is imposed out of the change of physical properties namely, properties of the actuators (and glue) and concentrated masses. So, the beam is divided into j j 1, 2,..., n 4 == elements. All elements may be considered separately and the solution to Eq. (7) can be expressed as where j u (x, t) is the solution on j -element and it is fulfilled the following equation To find j u (x, t) with the separation of variables method, the eigenvalues and eigenfunctions for each element are needed.

Eigenvalues and eigenfunctions problem
In this problem it is assumed that ( , ) 0 j fx t= and j 0 µ = , hence based on Eq. (10) one obtains where jj EJ and jj S ρ may be different on the separate elements, but here, as depicted in Fig. 5 The boundary conditions for the j −element consist of boundary conditions of the problem and coupling conditions between neighboring elements. The concentrated mass r m is www.intechopen.com considered in coupling conditions between third and forth elements and therefore it is omitted in Eq. (11). Let the solution be represented by a product of spatial and temporal functions Substituting (14) into (11) gives where the dispersion relationship is given by The Eq. (17) is very important and the solution to it is where Krylov functions are defined as, (Kaliski, 1986),   (22) it appears that 1 A0 = , 1 C0 = . In the same way, the rest of conditions given by Eqs. (23) − (26) lead to the set of algebraic equations and it may be written in the matrix form The matrix A is too large, to presented it in explicit form. Hence, its elements fall into blocks so that the matrix A can be written as

Orthogonality of modes
Orthogonality condition of the uniform beam modes may be found in (Kaliski, 1986;de Silva, 2000). First of all, based on twice integration by parts, one has ( ) The separate modes X (x) For standard boundary conditions, the right-hand-side equals zero. The procedure outlined above can be used to the problem presented in Fig. 6 Considering both boundary conditions of the problem and coupling conditions between neighboring elements, Eqs. (22) The Eq. (47) in particular case is used in deriving the solution to the forced vibration problem.

Forced vibrations with damping
A point departure for further consideration is Eq. (7); for j −element one has ( ) The solution to Eq. (48) Substituting Eqs. (49) and (50) The solution of the above equation is given by After some calculation, the constants j C ν are expressed by ( In the end, the problem of the forced j −element beam vibration with damping, excited with the force j f (x) is solved; in the harmonic steady state it is given by In current problem, two form of the forces have the practical meaning namely, the force with constant amplitude j00 f( x ) f = and the force acting at discrete point jai f (x ) . The former may be interpreted as the spread excitation forced, for example with plane acoustic wave, but the latter is the control force due to actuators, henceforth

Interaction between beam and actuators
It is assumed that the actuator is perfectly bonded to the beam surface. Exciting actuator, the interaction between actuator and the beam is appeared. The interaction process is explained in (Hansen & Snyder, 1997;Fuller at al, 1997) in detail and references cited therein. Assuming the spatially uniform actuator, it provides boundary induction solely in terms of the external line moment distributed along its edges (Burke & Hubbard, 1991;Sullivan et al., 1996). So, the bending moment in y −direction is given by the formula (Hansen & Snyder 1997), Fig. 8, where a C -constant depending on geometry and mechanical properties of the actuator and plate, 31 d − piezoelectric material strain constants, V -voltage in the direction of polarization.

Fig. 8. External line moments of the actuator
The problem is to determine of the a C , because it depends on the analysis method of the mutual interaction between beam-actuator (Hansen & Snyder 1997;Pietrzakowski, 2004). Let the static force coupling model is taken into account. If relatively thin actuator compared with beam thickness is assumed (so uniform normal stress distribution is accepted) and furthermore by ignoring the neutral axis displacement d , see

Beam vibration reduction through actuators
For the problem presented in Fig. 5, the total load of the beam, described by Eq. (56), is given by where the symbol ja f is replace by js f in order to express, in the future, the interaction sum of actuators and the glue on the beam. An expression in brackets is the sum of interacting forces actuator-beam. Hence, the integral j I ν , Eq. (54), for j f (x) expressed by above equation is given by The expression in square bracket constitutes the second-order central finite difference. Since the distance between nodes s  is constant, then the difference can be transformed into The j s (x ) ν κ is the curvature of the mode j X( x ) ν at the point s xx = (Brański & Szela, 2007;Brański & Szela, 2008). The sign of the j s (x ) ν κ is contractual namely, if the bending of the beam is directed upwards, the sign is positive and vice versa. Substituting Eq. (64) into Eq. (62), one obtains Next, substituting Eq. (65) into Eq. (52) through Eq. (53), the reduction vibration is obtained Note, that the amplitude jf A ν is the direct quantity which is liable to the reduction, in explicit form is At the same time, together with the vibration reduction amplitude jf A ν , the curvature is subjected to the reduction and based on Eq. (66) is Furthermore, the reduction of the jf A ν leads to the reduction of the shear force jf Q( x ) a n d bending moment jf M( x ) ( B r a ński et al., 2010; Kaliski, 1986;Kozień, 2006), As can be seen, the vibration reduction undergo on the following amplitudes: of the beam vibration jf X( x ) − Eq. (66), of the shear force jf Q( x ) − Eq. (69) and of the bending moment jf M( x ) − Eq. (70). Hereafter, the notion e.g. "shear force reduction" is used instead of "the reduction of the amplitude of the shear force", and so on.
The Eqs (66), (69) and (70) and may be written commonly Omitting for simplicity the index "f", for entire system beam-actuators one has jj jj It is appeared from Eq. (75) that the active vibration reduction depends on the following parameters:

Optimal actuators distribution problem
Before the optimization problem will be formulated, any coefficients of the vibration reduction should be defined.

Reduction and effectiveness coefficients
Let be the difference between any quantities of the beam vibration where ( The difference (x) ΔΨ is interpreted as the quantity of the vibration reduction and it is the first measure of this reduction namely, the quantity reduction coefficient. The second measure of the vibration reduction is defined as It is called as the reduction coefficient and it may be expressed in per cent. Note, that if the reduction coefficient equals one, the vibration reduction is total, R (x) 0 Ψ= . An effectiveness of the vibration reduction is defined as a quotient of some vibration reduction measure by an amount of the energy W provided to the system in order to excite actuators. Hence, thirst measure of the vibration reduction may be defined by so called the effectiveness coefficient The energy W provided to the system is translated into couples of forces, Fig. 9. Therefore, the energy W may be replaced by forces Rs s f4 f The Eqs. (77) -(82) define the appropriate factors of the vibration reduction at the point x . In many cases, it is convenient to calculate mean values of these coefficients at whole beam domain or at the beam sub-domains. First of them is the mean quantity reduction coefficient and it is defined by the formula Consequently, the mean reduction coefficient and the mean effectiveness coefficient are defined respectively mm m The coefficients defined above may constitute the base to formulate the optimization problem; hereafter the R( x ) Ψ is chosen.

Formulation of the optimization problem
In this chapter, one formulates the following problem: find the optimal actuators distribution { } s x which maximize of the reduction coefficient R( x ) Ψ ; hence R( x ) Ψ is assumed as an objective function. In this case the maximal value of R( x ) Ψ equals one and it www.intechopen.com means p-reduction; such instance is considered in Brański & Lipiński, 2011) and it seems that it is possible only in for separate mode. Let the energy provided to the actuators be constant and hence, the s f is always constant. Now, for clarity of the disquisition, rewrite the effectiveness coefficient in explicit form

Heuristic analysis of the optimization problem
Note, that the amplitude R A comprises the factor C 0 * ≠ , but it is constant and this is the factor 0R III Σ += which is changed. In practice, instead of the condition (87), the following condition of the reduction must be fulfilled For future considerations the sign of R I is very important. The vibrations are reduced with actuators, if the I  is positive, but must be fulfilled the following condition: R0 III0 Σ =+ ≥; R I0 = assures the total reduction. If this condition is not fulfilled, the actuators excite vibrations and thereby they are not accomplished owns role. Note, that the sign of 0 I is always negative, see Eq. (65). Then, the sign of I  must be positive and it depends on the signs both forces s f and curvatures s (x ) κ . From physical point of view, the sign of (x) κ is changed and as established above; it is positive, if the bending of the beam is directed upwards. Then, for many actuators one has  x . Hence, the distribution has a great significance; this problem was solved in (Brański & Szela, 2007;Brański & Szela, 2008;Szela, 2009). Interpreting Eq. (88) through Eq. (90) it is appear that the actuators ought www.intechopen.com to be bonded on the beam sub-domains in which the curvatures reach their extremum and consequently the highest and lowest values respectively, see Fig. 10. This is so called quasioptimal actuators distribution and it is described with Qs xx ≡ points, their number is Qs nn ≡ . κ in Eq. (90) have the same signs and all terms are positive. Furthermore, the actuators distribution described with Qs xx ≡ ensures the maximum of the reduction coefficient. The heuristic analysis described above was substantiated numerically for the separate beam and triangular modes and the details may be found in own papers.

Analytical analysis of the optimization problem
The aim of this section is to work out of the analytical method, which will describe such distribution of the actuators in order to assure the maximum of the reduction coefficient. It is expected that the analytical method will confirm the quasi-optimal distribution which has been found above with heuristic method. Therefore the assumptions are the same like in heuristic method, namely s n , s f and s  are settled.  (Fichtenholtz, 1999). Because 0 I is constant than a necessary condition for existing extreme value is κ and s f is quite the same as in heuristic analysis. Analytical analysis was applied for p-reduction and for the separate beam modes (Brański & Lipiński, 2011). As pointed out there, the analytical solution to the optimal actuators distribution problem confirms the results obtained with heuristic solution.

Conclusion
Deriving the shape of (x) κ , the influence both masses and stiffness of the actuators and glue on the shape of X(x) , and consistently on the shape of (x) κ , were omitted; if not, an adaptation method must be applied. But after determining shape of (x) κ , all these parameters were considered. As can be note, the actuators optimal distribution is attained assuming that the added energy to excite actuators is constant. It is translated into constant s f . Having the optimal distribution, the reduction coefficient may be improved by adding more energy or in order words, by increasing s f . This way, presented optimal method corresponds to that one presented in (Q. Wang & C. Wang, 2001), namely "maximization of the control forces transmitted by the actuators to the structure". Based on theoretical considerations, and numerical ones presented in own papers, the following conclusion may be formulated. 1. The optimization problem of the actuators distribution assuring the maximal active vibration reduction of the beam, measured with reduction coefficient, may be solved both heuristically and analytically. In analyzed problem, it turned out that both methods give the same results. 2. The following algorithm of analytical method may be worked out: to determine the value of the reduction coefficient, • to increase the value s f , through the energy increase which excites actuators, until the reduction coefficient will attain its maximum. It seems that proposed optimization method is very simple and may be useful in many technical problems of active vibration reduction. This work is a starting point for many computer simulations and experiments.