Parameter optimization of an inerter-based isolator for passive vibration control of Michelangelo’s Rondanini Pietà

Mechanical Systems and Signal Processing 98 (2018) 667–683

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Parameter optimization of an inerter-based isolator for passive vibration control of Michelangelo’s Rondanini Pietà A. Siami ⇑, H.R. Karimi, A. Cigada, E. Zappa, E. Sabbioni Department of Mechanical Engineering, Politecnico di Milano, 20156 Milan, Italy

a r t i c l e

i n f o

Article history: Received 25 March 2017 Received in revised form 20 May 2017 Accepted 22 May 2017

Keywords: Inerter Passive vibration control Parameter optimization Performance index Firefly algorithm Tuned mass-damper-inerter

a b s t r a c t Preserving cultural heritage against earthquake and ambient vibrations can be an attractive topic in the field of vibration control. This paper proposes a passive vibration isolator methodology based on inerters for improving the performance of the isolation system of the famous statue of Michelangelo Buonarroti Pietà Rondanini. More specifically, a fivedegree-of-freedom (5DOF) model of the statue and the anti-seismic and anti-vibration base is presented and experimentally validated. The parameters of this model are tuned according to the experimental tests performed on the assembly of the isolator and the structure. Then, the developed model is used to investigate the impact of actuation devices such as tuned mass-damper (TMD) and tuned mass-damper-inerter (TMDI) in vibration reduction of the structure. The effect of implementation of TMDI on the 5DOF model is shown based on physical limitations of the system parameters. Simulation results are provided to illustrate effectiveness of the passive element of TMDI in reduction of the vibration transmitted to the statue in vertical direction. Moreover, the optimal design parameters of the passive system such as frequency and damping coefficient will be calculated using two different performance indexes. The obtained optimal parameters have been evaluated by using two different optimization algorithms: the sequential quadratic programming method and the Firefly algorithm. The results prove significant reduction in the transmitted vibration to the structure in the presence of the proposed tuned TMDI, without imposing a large amount of mass or modification to the structure of the isolator. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Protection of cultural heritage and famous statues against environmental vibration and earthquake are receiving more and more interest in recent years. Although there are so much researches in the context of passive, active and semiactive vibration control for buildings and suspension systems, these have not been extensively applied to historical objects so far. This field is challenging since it presents specific physical constrains for designing isolators for the object under consideration. In addition, preserving these structures for long time is a fundamental problem. Although aforementioned issues are of great importance, to the authors’ best knowledge, there is not so much research in this field, which is one of the motivations for this work.

⇑ Corresponding author. E-mail addresses: [email protected] (A. Siami), [email protected] (H.R. Karimi), [email protected] (A. Cigada), emanuele.zappa@polimi. it (E. Zappa), [email protected] (E. Sabbioni). http://dx.doi.org/10.1016/j.ymssp.2017.05.030 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

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In this paper, a relatively new concept in the field of vibration absorbers has been used. In addition, the methodology for finding the optimum values of design parameters of the proposed passive vibration control method has been expressed. In fact, inerter is a mechanical element with two terminals which can apply force at its two ends proportional to their relative acceleration. The inerter element has been introduced by Smith [1] in 2002. In [2] he presented mechanical mechanisms for inerters and discussed about different aspect of these mechanical elements. The inerter first used by McLaren Mercedes for suspension system in Formula one race car and after that interest to this element has been increased. Afterwards, application of this element in vibration control of different mechanical systems in terms of passive, semi-active and active control methods has been reported in open literature. Chen et al. [3] compared different combinations passive, semi-active and active inerters and dampers for a vehicle suspension. Based on their results, semi-active inerter and semi-active dampers resulted in the best performance, although they have not suggested a physical model for semi-active inerter. In Chen et al. [4] the influence of inerter on the natural frequencies of vibration system is discussed. The impact of this device on natural frequencies of single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) system has been investigated. Chen et al. [5] have recently presented application of the inerter in semi-active suspension systems. They used a semi-active full car model which equipped with six semi-active suspension struts. The results showed a significant improvement in the suspension system performance by using inerters. In addition, the effect of location of the device on natural frequencies of a six-degree-offreedom system has been investigated. Application of inerters in passive isolation systems with various configurations has been extensively researched in literature, see for instance Scheibe [6], Wang ([7,8]), Brzeski [9], Hu [10] and Shen [11]. In the above-mentioned references, results of different setups and applications of vibration isolation system combined with the inerter have been investigated. Li et al. [12] introduced adaptive inerter for increasing the suspension performance of a quarter-car model. An adjustable model of inerter is proposed in parallel with spring and damper in their model and the impact of using constant inertertance is compared with the situation that there is the adaptive device. Inerter, as an element with two-terminal mechanical device in order to add apparent mass to TMD, has been introduced by Marian at el. [13]. Salvi at el. in [14,15] studied the optimal design parameters including frequency ratio and damping factors in SDOF by using analytical and numerical methods and considering various types of excitation. Tubino at el. [16] designed an appropriate TMD to mitigate the pedestrian induced vibration on a bridge. In [17] mass-uncertain rolling pendulum TMDs have been used in order to reduce the vibration of buildings induced by the seismic excitation and H1 synthesis has been used to find design parameters. In another research, Liu et al. [18] proposed a new method for passive vibration isolation in the low frequency range by combination of lever-type isolator and X-shape supporting structure. In the sequel, Arfiadi et al. [19] studied optimum properties and placement of TMDs for vibration control in buildings. Genetic algorithm (GA) is utilized to optimize the properties of TMDs. Another research in order to design the optimum control of vibration for tall buildings based on the GA has been done by Shayeghi et al. [20]. Furthermore, Mohebbi et al. [21] obtained the optimal parameters of multiple tuned mass dampers (MTMD) using GA method for a structure subjected to the seismic excitation. Morga et al. [22] investigated the reduction of vibration induced by wind load using TMD in a slender structure. Moreover, the tuning frequency is optimized based on displacement and acceleration of the top end of the structure. In another research, Si et al. in [23] proposed TMDs for passive vibration control in a spar-type floating wind turbine. The authors represented the optimum values for the parameters of the TMD by considering different performance indexes and using different optimization methods. Fallah Jahromi et al. [24] proposed a semi-active vibration control design based on Magnetorheological (MR) dampers for suspension systems. More specifically, a Linear Quadratic Regulator strategy with two weighting matrices is presented to design controller in order to minimize the energy of the system. The weighting matrices have been found by combination of Sequential Quadratic Programing (SQP) method and the GA. In fact, the GA is used to choose the initial input near the global optimum point in the SQP algorithm. In the situation that the SQP methods have been used in optimization problems it is not possible to guarantee that the results are the global optimum points. In fact, in these methods the results are dependent to initial points. A gradient-based method has been used by Li et al. [25] for optimizing distributed multiple tuned mass dampers (MTMD). Their method can be utilized in the situation that there are uncertainties in parameters of the structure and TMDs. They obtained the optimal non-uniformly distributed MTMD with some limitations on the frequency spacing and damping. Zuo and Nayfeh [26] considered multi-degree-of-freedom (MDOF) systems with MDOF models for absorbers and proposed an approach based on the descent sub-gradient method to optimize the parameters of tuned mass dampers. The results have been evaluated experimentally. Groco et al. [27] studied robust optimum design of TMDs installed on a MDOF system under the stochastic seismic input disturbances by assuming uncertainties for the structural and seismic model parameters. Different performance indexes are defined for the system and the efficiency of the designed TMDs is investigated. On the other hand, one of the nature inspired methods for solving optimization problems which has been received interests recently, is Firefly algorithm (FA). This method has been developed by Yang [28] in 2008. The mentioned metaheuristic approach has been used for solving different optimization problems in various fields, for designing passive or semi-active vibration control systems ([29,30]) or finding the optimal values in design process of structural engineering ([31–32]) or solving optimization problem in different field of engineering ([33–37]). Furthermore, in the field of semi-active and active vibration control, there are some interesting results with the target of vibration reduction in buildings and large-scale structures that they can be found in [38–41]. The concept of these works can be extended for the structures such as tall and slender statues.

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Due to proportionality of the produced force of the inerter element to the acceleration of mass, this element can be added easily to the equation of motion of the physical system and its constant appears in the mass matrix [42]. Therefore, the using of inerter in isolators has two advantages; the mass ratio in the isolation system can be increased and in addition, there is no need to mount additional considerable mass on the object which needs to be isolated such as traditional TMDs. Encouraged by the issues mentioned above, the main contribution of this work is to develop TMDIs for passive vibration control of a specific type of vibration isolation system and tuning the optimal parameters for the TMDI. Using the inerter as a passive element in this kind of structure is completely novel according to the authors’ knowledge and by considering the limitations that there are in preserving cultural heritage and valuable statues, it can be very efficient for designing vibration isolator for these kinds of structures and objects. For this purpose, first a 5DOF model of the original system will be presented. This model has been validated by using the results of experimental test. Then, a TMDI will be added to the developed model and the equations of motion for the entire system are presented. The results of numerical solution will be provided to illustrate the effectiveness of the proposed passive vibration control method for this specific structure. The next step is to find optimum values of the design variables by using definition of two different cost functions or performances indexes. The optimum parameters will be calculated by two different methods; a sequential quadratic programming (SQP) technique and Firefly algorithm (FA). Then, the optimization results of these two methods will be compared in order to verify the optimal results. The rest of this paper is organized as follows. In Section 2, the structure of the isolation system of the statue is presented. In Section 3, the equations of motion of the original system are provided and the MDOF model is presented. In addition, the model of the isolator equipped with TMDI is expressed in this section. Section 4 includes investigation about the effect of using TMDI on the performance of the isolation system. Various indexes are used to show the efficiency of the TMDI. Section 5, is about introducing the performance indexes which are used to obtain the optimum design parameters of the MDOF model with the TMDI. Furthermore, two optimization methods are presented; SQP and FA that they are used in this research. In Section 6, the numerical results for optimal design values of the MDOF model are discussed. The comparison has been done for two different performance indexes used as the target functions for the optimization problem. The other part of this section is comparing the results of two different optimization algorithms (SQP and FA). Conclusions are presented in Section 7. Notation: The superscript ‘T’ denotes matrix transposition, Rn shows the n-dimensional Euclidean space. diagf  g means x are adopted to first a block diagonal matrix. minx ðf ðxÞÞ indicates the minimum value of f ðxÞ respect to x. In addition, x_ and € and second derivative of vector x respect to time. j:j denotes the norm of a vector. 2. The isolation system of the statue Pietà Rondanini statue has recently been moved to a new museum placed in the city centre of Milan. According to the new location of this famous statue and the significant level of vibration due to transportation systems, a new isolation system has been designed for the statue in order to protect it against vibration in two orthogonal directions, vertical and horizontal axes. The development and the testing of this isolation system were developed in previous works and are not the scope of this paper. A description of the previously developed base is given here but for a more complete information, please refer to Cigada et al. [43]. In horizontal direction, a sliding part has been added to the isolation system for protecting the structure against earthquake (optimized for frequency range lower than 10 Hz). A series of rubber bearings have been utilized to filter the vibration transmitted to statue and due to the transportation system in both horizontal and vertical directions. This type of vibration is mainly in the frequency range between 20 to 100 Hz. The schematic figure of the statue and its isolator is presented in Fig. 1. For ambient vibration control, 13 rubber bearings have been used in the isolation system. The distribution of these elements between upper and lower plates of the isolator is shown in Fig. 1. The values of rubber bearing parameters are reported in Table 1 and they were provided by the manufacturer. To guarantee the proper mechanical filtering of the isolation system, a compromise solution has been chosen by selecting rubber bearings allowing to get a natural frequency in the vertical (i.e. perpendicularly to the ground) and horizontal directions of approximately 8 Hz and 3 Hz, respectively. In order to build a multi-degree-of-freedom (MDOF) model, the results of experimental tests carried out with 1:1 marble copy of the statue were used. Fig. 2 shows the isolation system, the statue and the setup of experimental test. In order to find the frequency response function of the system, a sweep sine input was used in vertical and horizontal directions and the responses (accelerations) were measured for different points by using accelerometers. Position of sensors is shown in Fig. 2. 3. The equations of motion 3.1. The original system In this section, the equations of motion of multi-degree-of-freedom (MDOF) models are presented. The isolation system has been modelled using 5DOF model which is presented in Fig. 3. In fact, three DOFs for upper part (two translation and one rotation) and 2DOFs for lower plate (two translation) have been considered. This model can predict the dynamic behaviour of the system according to the results of the experiments. As presented in this figure, the 13 rubber bearings based on their

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Fig. 1. (a) Schematic plan of the statue and the isolating system, (b) Position of rubber bearings between upper and lower bases (Cigada et al. [43]).

Table 1 Main characteristics of the isolating system. Design mass

[kg]

2800–3100

Stiffness of restoring belts Vertical stiffness of rubber bearings (13) Horizontal stiffness of rubber bearings (13)

[N/mm] [N/mm] [N/mm]

1.26 568 128

Fig. 2. Experimental setup, position of accelerometers and reference system.

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Fig. 3. 5DOF model of the isolator system.

setup between two plates have been modelled with five spring-dampers. One group consists of three springs (k1 , k2 ; k5 ) and three dampers (c1 ; c2 ; c5 ) that they present the properties of three different columns of rubber bearings. And the other group includes elements named 3 and 4 (k3 ; c3 and k4 ; c4 ) in Fig. 1 that they contain information of the two rubber bearings in parallel. This relation is illustrated in Fig. 1. It should be noted this configuration (using five spring-damper elements) has been chosen in order to have same properties for rotational DOF in the model and real structure. In addition, connection between lower plate and ground is expressed with two spring-damper elements in different directions, represented by kgx ; cgx and kgy ; cgy . For the structure presented in Fig. 3, the dynamic equation of motion can be written as:

€ þ C 0 X_ þ K 0 X ¼ F g0 M0 X

ð1Þ

where; X ¼ ½ xu xl yu yl h T and M0 , C 0 and K 0 are mass, stiffness and damping matrices respectively. F g0 is the force applied to the structure due to horizontal ground motion. In this model, xu , yu and h are translation of mass centre of the upper part in x and y directions and rotation of upper part around the mass centre point respectively. xl , yl , xg and yg are the displacement of the lower plate in x and y directions and disturbances in horizontal and vertical directions, respectively. The matrices and vectors are presented in Appendix A. The result of frequency response function (FRF) between the ground plate and the statue in vertical and horizontal directions obtained from the 5DOF model and experimental test have been plotted in Fig. 4. In fact, the parameters of the model have been updated by comparing the FRFs obtained from the 5DOF model with the experimental ones. The FRFs presented here have been achieved between a sensor on the base connected to the shaker (point AT in Fig. 2) and the sensor on the head of statue (A6 in Fig. 2). In the next step, we want to reduce the peak of this FRF in Y-direction by using TMDI elements. It should be noted that the objective of this research is to reduce the vibration transmitted to the statue around the first mode in vertical direction. 3.2. The isolating system with TMDIs In this part, first the inerter and its mechanism are introduced in order to add it to the model of the isolator, then the equation of motion of the system is derived. In Fig. 5 the schematic figure of an inerter and its counterpart in electrical circuit has been shown. The force in the terminal is:

€1  u €2 Þ F ¼ bðu

ð2Þ

€ 1 and u € 2 are the acceleration of terminals 1 and 2, respectively. In the where b is the inertance with mass units (kg) and u electrical counterpart, i expresses the current and v 1 and v 2 are the voltage of points 1 and 2, respectively. It must be noticed that the actual mass of the inerter is very low compared to its apparent mass. Fig. 6 shows the mechanism of an inerter, which consists of pinions, gears, rack and flywheel. In this figure, r i and r pi are the radius of gears and pinions, respectively, rf is the radius of flywheel and r pf indicates the radius of flywheel pinion. In addition, mf is the mass of the flywheel. In fact, linear movement of the plunger drives the gears and flywheel. The inertance of this system can be expressed as:

b ¼ mf

r 2f r2pf

n Y r2

i r2 i¼1 pi

! ð3Þ

where b is the inertance of the system (kg) and n is the number of gears. This kind of mechanism produces a large apparent mass in comparison with real mass of device. Here, considering the required inertance for the isolation system which it will be used in next steps, one stage gear and pinion (n ¼ 1) can be sufficient.

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Fig. 4. Amplitude and phase of the FRFs between the seismic base (point AT) and the acceleration of points 6 (see Fig. 2). In (a) Y-direction and (b) Xdirection.

Fig. 5. An inerter and its electrical counterpart.

In the sequel, we want to add two ineters to our model in order to use the mass amplification effect of the added elements and reach more vibration reduction in the isolating system. In Fig. 7 the system with two TMDs in Y-direction (md1 and md2 ) and two inerters (b1 and b2 ) placed between the mass of TMD and the lower plate has been presented. The equations of motion of the isolator system with two TMDs in Y-direction combined with two inerters is presented as follows:

€ aug þ C X_ aug þ KX aug ¼ F g MX

ð4Þ

 T where the displacement vector is presented as X aug ¼ X T yd1 yd2 , where yd1 and yd2 are the displacement of two TMDs installed in Y-direction and X is a vector including the 5DOF system in (1). M, C and K are the mass, stiffness and damping of the 7DOF system, respectively. Matrices and vectors of the system have been expressed in Appendix B. It should be noted that the effect of the inerters appears in the mass matrix of the system. Now, based on the good correlation between our original MDOF model and experimental test results, we can verify the effect of TMDI on the isolation system and reach to an effective and practical passive method for improving the performance of the vibration control system.

Fig. 6. Schematic figure of an inerter with rack and pinion mechanism.

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Fig. 7. 7DOF model of the isolator system with two TMDs in Y-direction combined with two inerters.

4. Effect of the TMDIs on the response of the system In this section, according to the developed model in (4), we will investigate the impact of inerter on our system. In literature, various configurations have been used as inerter-based vibration suppression methods. Here, we use TMD combined with inerter (TMDI). Considering the amount of inertances reported for commercial types of inerters, this element has been added to the isolation system. In Table 2 the values considered for the parameters have been presented. It is an assumption that the mass and damping of inerters are negligible in comparison with the mass and damping of other parts. It should be noted the optimized value which is presented in this section, achieved by using closed-form relation for single-degree-of-freedom TMD represented by Den Hartog [44]. This value named here as fDH means the optimal value obtained for SDOF using the closed form solution presented by Den Hartog. Therefore, the optimized values for f and xd for the MDOF model will be calculated in the following sections. Using the parameters tabulated in Table 1, the comparison between the result of the system without passive vibration control, with TMD and with TMDI have been plotted in Fig. 8. Based on the results presented in Fig. 8, application of TMD can assist to reduce the amplitude up to 50 percent. If we increase the damping of TMD more than fDH , because of higher force between TMD and the object structure, the effectiveness of TMD will be reduced. Therefore, by using TMD it is not possible to mitigate the amplitude of vibration more than specific amount that achieved by optimum fDH . When the TMDI has been used, without adding a high amount of mass to the system, by using the inertance of the TMDI the level of vibration has been reached to the 30 percent of the case without control and even less. For better comparison of the response of the isolator system, three different indexes have been used as follows:

J acc

J sws

J rhd

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 T 2 € dt ¼ y T 0 u

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 T ¼ ðyu  yl Þ2 dt T 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 T ¼ ðyl  yg Þ2 dt T 0

ð5Þ

ð6Þ

ð7Þ

€u , yl and yg are the vertical acceleration of the upper plate, the displacement of the lower where T is the simulation time. y plate and ground plate, respectively. In addition, it is possible to combine all these three functions (5)–(7) as the root mean square and reach to the total performance J rms :

J rms

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 T ¼ q y€2 þ q2 ðyu  yl Þ2 þ q3 ðyl  yg Þ2 dt T 0 1 u

ð8Þ

where q1 ; q2 and q3 are weighting coefficients which are determined by designer and they have been used in the performance index. The other index that it is used here, is maximum value of FRF function that can be defined as:

J Gmax ¼ maxðjHi:j ðxÞjÞ

ð9Þ

where Hi:j ðxÞ is the frequency response function of the MDOF system for j input and i output. These indexes assist us to have better insight about the impact of the TMDI parameters on the performance of the isolation system. Here the FRF between the ground plate and upper part in y-direction has been considered. Therefore, in Table 3 the values of different indexes considering various cases have been presented. These values include the case without any controlling element, with TMDs and using TMDIs. From the results, using the TMDI elements provides this possibility to reach considerably less amplitude in comparison with the case of using just TMD. This fact becomes clear by considering the amount of

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A. Siami et al. / Mechanical Systems and Signal Processing 98 (2018) 667–683 Table 2 The values considered for the parameters of the TMD and the TMDI. Parameters

Value

mTMD1 , mTMD2

20 kg 53.3 rad/s 52.5 rad/s 0.087 mTMD1 x2d ½N=m

x0 xd

fDH kTMD1

mTMD2 x2d ½N=m 2xd mTMD1 fDH ½Ns=m 2xd mTMD2 fDH ½Ns=m 240 kg

kTMD1 cTMD1 cTMD1 b1 , b2

Fig. 8. Comparison of the FRF between the Ground plate and upper plate; without passive control, using TMD and TMDI with various damping ratios calculated using numerical solution.

Table 3 The result of numerical solution of the isolation system; representation of the indexes for different cases. Index 2

Jacc (m=s ) Jsws (mm) Jrhd (mm  106 ) Jrms JGmax

Case 1; uncontrolled (without TMD)

With TMD; f ¼ fDH

With TMD; f ¼ 2fDH

With TMDI; f ¼ fDH

With TMDI; f ¼ 4fDH

4.26 0.01 0.29

3.59 0.0049 0.29

3.71 0.006 0.29

3.67 0.002 0.3

3.14 0.0015 0.31

2.06 12.67

1.89 6.64

1.93 8.30

1.91 4.22

1.81 3.56

J Gmax for different cases. The values of J acc in Table 3 show the average acceleration transmitted to the statue will be reduced by adding TMDI. Furthermore, using TMDI provides the possibility to increase damping and obtain completely better results in comparing with TMD cases. In addition, the result obtained for J rms proves the effectiveness of using TMDI with high damping in comparison with the TMD case. The transfer functions correspond to the presented cases in Table 3 have been illustrated in Fig. 8. This figure shows by increasing the damping in the cases of using TMDI, the lower peaks which are introduced to the system by adding the passive elements can be changed. In order to find the lowest level of the peaks the optimum values of damping coefficient and tuning frequency should be found. 5. Optimization of tuning parameters In this part the formulation and methods which were used for finding the optimal values of the TMDIs’ parameters have been presented. First, different objective functions used for this problem will be presented and then two optimization methods which have been applied for obtaining the optimal parameters will be discussed. 5.1. Defining different objective functions The performance of the TMDIs depends on the tuning frequency of the TMDs and their damping factors. For a singledegree-of-freedom system, the optimized value for tuning frequency (t) and damping factor (f) have been obtained using

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closed-form solutions in [44] but these relations cannot be extended for MDOF systems. Therefore, here numerical solution has been used to find the optimized values of parameters. On purpose, we need to define a cost function for the optimization problem. Here, two different cost functions have been used. In order to have an insight to finding optimal parameters for MDOF system [13], we can calculate the modal mass for each frequency as:

mi ¼ uTi Mui

ð10Þ

where M is the mass matrix of the system without TMD and ui represents the eigenvector of the chosen frequency. This modal mass can be used if we want to define dimensionless masses in our problem. In our model, the FRF between the upper €g the ground acceleration can be expressed as: plate in vertical direction and y

HðxÞ ¼

Y u ð xÞ Y€ g ðxÞ

ð11Þ

€ g are spectrum of the displacement of the where HðxÞ is the frequency response for the chosen input and output, Y u ðxÞ and Y upper part and acceleration of the ground plate, respectively. Here, the design variables are tuning frequency and damping factor which are defined as follows:

t¼

xTMD x1

f1 ¼

ð12Þ

cd1 cd2 ;f ¼ 2xTMD ðmd1 þ b1 Þ 2 2xTMD ðmd2 þ b2 Þ

ð13Þ

In fact, it is assumed that the properties of two TMDIs used for the system which have been shown in Fig. 7 are the same, therefore we have:

f1 ¼ f2 ¼ f

ð14Þ

Considering these design variables, we want to define a cost function using the response of the system in a general case. Therefore, the mean square value of the response of the system to a random input can be written as

Z

y2 ¼

1

0

jHðxÞj2 Sxx ðxÞdx

ð15Þ

where Sxx ðxÞ is one-sided power spectrum of the stationary random input signal (acceleration) applied to the structure from the ground plate in vertical direction and HðxÞ is defined according to the Eq. (11). It should be mentioned here, the integral in Eq. (15) has been calculated for specific frequency range of input. In the other word, a limited frequency band force (disturbance) has been considered as the input for Eq. (15). Based on the mentioned relations, the dimensionless cost function that here named performance index is defined as

PI1 ¼

y2TMDI y20

ð16Þ

where y2TMDI and y20 are the mean square values of the displacement of the object that we want to reduce its vibration in the presence of TMDI and without it, respectively. Suppose the design variables t and f, and for the given values of b1 and b2 , we want to minimize the cost function. In fact, the result of this minimization of the performance index provides us the optimized value of the design variables (topt , fopt ). The second performance index has been defined in order to reduce the peak of FRF according to the design variables (t; f). €g ; the ground acceleration is Therefore, the maximum value of the FRF between the upper plate in vertical direction and y used as the second performance index and defined as follows:

PI2 ¼

max ðjHðxÞjÞ

x2ð0;xmax 

ð17Þ

where HðxÞ is the FRF between ground plate and upper part in y-direction. xmax is the maximum frequency which is considered for finding the maximum amplitude of the FRF. Minimizing this performance index for the design parameters enforces the FRF to have minimum amount in the maximum amplitude for a chosen frequency range, while minimizing PI1 provides the minimum value for the response summation to the random input. The relation between the proposed PIs and the system behaviour is explained in details. 5.2. Sequential Quadratic Programming (SQP) optimization method Sequential Quadratic Programming (SQP) method is one of the most powerful techniques for the numerical solution of constrained nonlinear optimization problems and engineering problems [45,46]. In other words, SQP is an iterative method for finding the solution of nonlinear optimization problems. This method can be described as follows:

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minf ðxÞ; x 2 Rn

ð18Þ

x

subject to cðxÞ 6 0

where f : Rn ! R is the objective function and c : Rn ! Rm describes equalities and inequalities constraints. In fact, SQP solves the nonlinear optimization problem for a given xk using Quadratic Programing (QP) sub-problem and after solving this sub-problem a new iterate xkþ1 has been constructed. This construction is performed in a way that the sequence converges to a local minimum of the nonlinear optimization problem. In order to apply this method on the performance indexes, a MATLAB built-in function has been used here. The objective function is the introduced performances indexes (PI1 or PI2 ) and the constraint of the optimization problem will be 0 < f < 1 and 0:5 < t < 1:4. 5.3. Firefly optimization algorithm The firefly algorithm (FA) was introduced first by Yang [35] in 2008 and it is based on the behaviour of fireflies and their flashing pattern to attract each other. This algorithm can be summarized in the following steps [47–49]: (1) A firefly can attract other fireflies without considering their sex. In other word, fireflies are unisex (2) For each two flashing fireflies, the brighter one will attract the less bright one towards itself. If there is no brighter one than a particular firefly, it will move randomly. (3) The brightness of a firefly can be determined by the objective function. The attractiveness of each firefly is proportional to its light intensity directly. The variation of attractiveness b can be defined as:

b ¼ b0 ecr

2

ð19Þ

where r is the distance between two adjacent fireflies, b0 is the attractiveness at r ¼ 0, and c is a constant named light absorption coefficient and it can control the speed of FA convergence. The movement of a firefly i towards position of a brighter (higher attractive) one is determined as follows:

xkþ1 ¼ xki þ b0 e i

cr2ij

ðxkj  xki Þ þ ak ki

ð20Þ

where ak is randomization parameter and  is a vector of random number drawn from Gaussian distribution at time k. r ij is distance between fireflies i and j. In each algorithm loop, the distances of each pair of fireflies are updated. For applying the algorithm, the population of fireflies N G and number of iteration N I should be specified. It should be noted that ak control the diversity of solutions and it can vary with the iteration counter k. Generally, for most application it can be chosen as pffiffiffi ak 2 ½0; 1. If L is introduced as the average scale of the problem of interest, c can be chosen as 1= L [46–49]. It should be k i

considered that the order of computation cost of the FA is OðN 2G N I Þ. Therefore, using the higher number of generation, although improve the results of algorithm, it can be limited by the computation cost. The FA can be summarized as follows. Define the number of fireflies (N G ) Define the number of iterations (N I ) Define the objective function f ðxÞ Generate initial population of fireflies xi Determine the light intensity Ii (Ii ¼ 1=f ðxi Þ) Determine the light absorption coefficient c while k < N I for i ¼ 1 : N I for j ¼ 1 : N I if (Ii > Ij ) Move firefly j towards firefly i Calculate new attractiveness via Eq. (19) Update the light intensity end if end for end for end while

Find the best solution by considering the values of f i Based on the explanation, this algorithm has been applied to the optimization problem. The result will be presented in the next section. It should be noted in this application, the objective function which is mentioned in the FA will be different

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677

performance indexes (PI1 or PI2 ) that they have been introduced previously. Here, we consider N G ¼ 15 and N I ¼ 40. The other parameters of the FA have been chosen in the range that explained earlier. In addition, it is assumed; :01 < f < 0:99 and 0:5 < t < 1:4. Therefore, the average scale will be L ¼ 0:9. 6. Results of numerical solution for the optimization 6.1. Optimized parameters In this section, the optimum design parameters are presented for the isolation system equipped with two TMDIs illustrated in Fig. 7 and the properties provided earlier. The results of the first performance index (PI1 ) and the second one (PI2 ) will be presented for a given mass of the TMD and different applicable inertance values. The results will provide a complete information about optimum tuning frequency and damping ratio for the isolating system considering different possible ineters and various cost functions. It should be mentioned that the frequency range considered for calculating the performance indexes is ½0; 100rad=s which is a frequency band around the first eigenfrequency of the system. At first step, the problem has been solved using the SQP method in MATLAB. In fact, the function that should be minimized is the performance index (here it can be PI1 or PI2 ) according to Eq. (18) and the constraints are the acceptable ranges for the damping factors and the tuning frequencies. Suppose two cost functions (or performance indexes) presented in Eq. (16) and (17), and the two design parameters t; f, for the given value of TMD mass (Table 2) and different given values for interances (between 100 and 1200 kg), the results of optimization problem are presented under the constraint 0 < f < 1. In Fig. 9 the optimal damping factor for various values of inertance has been plotted for both PIs. For PI1 the slope of the curve is sharper in the range of higher inertances. In fact, this cost function covers broad band frequency range and for attaining minimum of the cost function the higher damping will be needed. For instance, if we use an inerter with 1000kg inertance the required damping factor will be more than 0.7 for PI1 and around 0.45 for PI2 .Fig. 10 represents the performance indexes versus normalized frequency for various values of inertance for PI1 and PI2 . Each curve in figure (a) or (b) represents a TMDI with the specific constant and with optimal damping frequency obtained for various normalized frequency. The results for various inertance values between 100 and 1000 kg in the limited range t can be observed in these two figures. The black dashed line indicates the optimal tuning frequency obtained from the optimization solution. For PI1 , the rate of changing of tuning frequency is not so high and starts from near to 1 and decreases to less than 0.85 for a TMDI with a 1000kg inerter. The intersections of black dashed line with curves of different values of inertance are the minimum points of PIs. Fig. 10(b) shows the range of variation of optimum tuning frequency for attaining minimum performance index (PI2 ) is higher than the case of PI1 . It starts from less than 0.95 for inertance equal to 100kg and reach to around 0.63 for the TMDI with 1000kg inerter. Therefore, the PI2 will be more sensitive to the tuning frequency in comparison of the case that the cost function is PI1 . The presented curves assist us to choose a proper inerter according to the level of reduction and available values of inertance. In Fig. 11 the FRFs of the system for different values of inertance and using optimum values of normalized tuning frequency and damping factor (topt ; fopt ) have been presented for both performance indexes. The solid lines indicate the results calculated for the PI1 and the dashed lines expressed the result of the PI2 . This plot proves the significant effectiveness of optimal tuned TMDI. The black dashed lines show the case that we have not TMDI in our systems. Although using the PI2 as a cost function of optimization provides a flatter response without significant peaks, it seems using PI1 presents more reduction in wider frequency band. For instance, by choosing inerter with inertance equal to 500 kg we will reach to max-

Fig. 9. Optimum damping factor (f) vs. different values of inertance calculated for two different performance indexes (PI1 and PI2 ).

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Fig. 10. The index versus normalized frequency for different inertance values and the optimum normalized tuning frequency (dashed black curve) for (a) PI1 , (b) PI2 .

Fig. 11. The FRF in presence of TMDI with different inerter with optimum tuning frequency and damping factor (topt , fopt ), comparing the results of PI1 and PI2 .

imum peak lower than 3 in magnitude of the FRF if we choose PI1 as cost function and this value will be less if the PI2 has been chosen. It should be noted that the results obtained for the second performance index have better performance in all chosen frequency range. In Fig. 12 the 3D plots of the result of Fig. 11 for different values of inertance have been shown. According to this figure by increasing the inertance value the rate of reduction of the amplitude of the FRF will be reduced. These figures can give more insight about the impact of choosing higher inerters on the response of the system. As it is clear from these results, the FRFs do not have significant peaks around the resonance area and have approximately flat response to the ground plate excitation. Table 4 presents the various indexes calculated based on the optimal results obtained for the case which its properties presented in Table 2. The results of this table prove using the optimal value achieved for both performance indexes (PI1 and PI2 ) can increase the effectiveness of the TMDI considerably.

6.2. Comparing the results of SQP and FA methods In this part the optimum value for the problem will be obtained using FA. This algorithm can be applied for the optimization problem according to the steps of FA algorithm and Eq. (15). The design variables and performance indexes versus various inertances obtained using FA have been plotted in Fig. 13. In addition, the results obtained using the SQP method have been plotted simultaneously. Although there are some small deviations in the topt and fopt calculated from the FA and SQP, totally the results of the FA are similar to the results of the previous method. The percentage of error for the minimum performance indexes obtained by the SQP and FA are tabulated in Table 5. It is shown in this table that the amount of errors for two different optimization methods in various values of inertance are small. It should be noted by increasing the number of generation (N G ) the results can be better, but as it mentioned earlier the computation time will increase. Therefore, consid-

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Fig. 12. 3D plot of the FRF between the ground plate and upper in presence of TMDI with different inerter with optimum tuning frequency and damping factor(topt , fopt ) obtained for: (a) PI1 , (b) PI2 .

Table 4 The indexes obtained for the optimal values of design parameters (topt , fopt ) for b ¼ 240kg considering two different performances indexes (PI1 and PI2 ). Index

Uncontrolled

Optimum results for PI1

Optimum results for PI2

3.17

3.18

1:41  106

1:51  106

3:1  106 1.78 3.34

3:1  106 1.78 2.73

1 2

J acc (m=s ) J sws (mm)

4.26 0.01

3

J rhd (mm  106 ) J rms J Gmax

0.29

4 5

2

2.06 12.67

Fig. 13. Optimum normalized tuning frequency (t), Optimum damping factor (f) and Minimum performance indeces in different values of inertance obtaibed from the FA and SQP.

ering the computational effort, the SQP is faster than the FA. The specification of the computer which is used for this work as follows: Core i7 processor and 12 GB memory. In this case, computational time for finding the optimization problems with N G ¼ 15 and N I ¼ 30 can reach to 2 h for the firefly algorithm depending on the performance indexes. This time for the SQP method is less than 3 min.

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Table 5 The results of FA and SQP method for the different inertance values. Inertance (kg)

1 2 3 4 5 6 7 8 9 10

FA

100 150 200 300 400 500 600 800 1000 1200

SQP

Error (%)

PI1min

PI2min

PI1min

PI2min

PI1min

PI2min

0.346 0.299 0.265 0.220 0.190 0.168 0.152 0.129 0.114 0.104

3.640 3.210 2.889 2.493 2.241 2.070 1.930 1.746 1.623 1.534

0.346 0.298 0.265 0.220 0.190 0.168 0.152 0.129 0.114 0.104

3.630 3.180 2.880 2.483 2.232 2.056 1.925 1.742 1.620 1.532

0.00 0.33 0.00 0.00 0.05 0 0 0 0 0

0.27 0.94 0.31 0.40 0.40 0.68 0.26 0.23 0.18 0.13

7. Conclusion In this paper, an effective passive vibration control method using TMDI was proposed for the isolation system of a famous statue with the goal of further reduce the vibration level reaching the statue. The numerical results illustrated that using TMDs do not reduce the level of vibration more than the specific amount which depends on the mass of TMDs. Introducing TMDIs to this isolation system just will add small amount of mass to the upper part of the isolator but it can provide the possibility for achieving a significant improvement on the performance of the vibration isolation in vertical direction. In this case, to have higher effective mass, we can increase the damping of the system in order to dissipate more energy. It should be considered that the TMDI has not significant impact on the filtering effect of rubbers in higher frequencies. The better performance of TMDI has been proved by representing frequency response function as well as by examining different index functions. The main issue for designing TMDs, is the optimal amount of the parameters which can have a significant effect on the performance of the isolation system. In this case, considering a 7DOF system the traditional tuning methods with closed-form solution cannot be used. Therefore, using two different performance indexes and applying the SQP methodology, the optimal parameters which are the tuning frequency and the damping factor have been calculated for different values of inertances. In addition, Firefly algorithm which is a nature inspired method for solving optimization method has been used here in order to compare the optimum parameters with those obtained by the SQP method. The numerical results illustrated that both methods reach to similar values. Based on the proposed results, it will be possible to choose an appropriate inerter and corresponding proper tuning frequency and damping factor to reach the maximum performance of the TMDI. Our future work is to implement the proposed method on a laboratory scaled model of the isolator and to evaluate the results of the experiment. Acknowledgments The authors gratefully acknowledge Dr. Claudio Salsi and Dr. Giovanna Mori (Direction of Castello Sforzesco in Milan, Italy) and Dr. Elisabetta Giani (Istituto Superiore per la Conservazione ed il Restauro, Italian Ministry of Cultural Heritage) for the support given to the project on the protection of Pietà Rondanini against seismic and ambient vibrations. Appendix A Matrixes of the 5DOF model of the original isolator

M0 ¼ diagð½mt ; ml ; mt ; ml ; It Þ 2

cx

6 cx 6 6 0 C0 ¼ 6 6 6 0 4

0

0

cx r

ðcx þ cgx Þ 0

0 c yt

0 cyt

cx r ðc2y l2 þ c4y l4  c1y l1  c3y l3 Þ

cyt

ðcyt þ cgy Þ

ðc1y l1 þ c3y l3  c2y l2  c4y l4 Þ

0

2

2

cx r

kx 6 kx 6 6 0 K0 ¼ 6 6 6 0 4

kx

0

0

kx r

ðkx þ kgx Þ

0

0

kx r

kx r

2

2

3

0

kyt

kyt

ðk2y l2 þ k4y l4  k1y l1  k3y l3 Þ

0

kyt

ðkyt þ kgy Þ

ðk1y l1 þ k3y l3  k2y l2  k4y l4 Þ

kx r

7 7 7 7 7 7 5

ðc2y l2 þ c4y l4  c1y l1  c3y l3 Þ ðc1y l1 þ c3y l3  c2y l2  c4y l4 Þ ðcx r2 þ c1y l1 þ c2y l2 þ c3y l3 þ c4y l4 Þ

cx r 2

3

cx

2

2

2

7 7 7 7 7 7 5 2

ðk2y l2 þ k4y l4  k1y l1  k3y l3 Þ ðk1y l1 þ k3y l3  k2y l2  k4y l4 Þ ðkx r 2 þ k1y l1 þ k2y l2 þ k3y l3 þ k4y l4 Þ

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2

F g0

3

0

7 6 _ 6 cgx xg þ kgx xg 7 7 6 7 6 ¼6 0 7 7 6 _ 4 cgy yg þ kgy yg 5 0

wherekyt ¼ k1y þ k2y þ k3y þ k4y þ k5y , cyt ¼ c1y þ c2y þ c3y þ c4y þ c5y Appendix B The elements of stiffness and damping matrices in presence of TMDI (7DOF model):M ¼ M d þ M b where

2

mt 6 0 6 6 0 6 Md ¼ 6 6 0 6 0 6 4 0 0 2

kx 6 k 6 x 6 6 0 6 K¼6 6 0 6 k r 6 x 6 4 0 0 2

cx 6 cx 6 6 0 6 6 C¼6 0 6 6 cx r 6 4 0 0

0 ml 0 0 0 0 0

0 0 mt 0 0 0 0

0 0 0 ml 0 0 0

0 0 0 0 It 0 0

0 0 0 0 0 md 0

3 2 0 0 0 0 0 60 0 0 0 7 0 7 6 60 0 0 0 7 0 7 6 6 0 0 0 b1 þ b2 and M 0 7 ¼ b 7 6 60 0 0 0 7 0 7 6 40 0 0 0 5 b1 md 0 0 0 b2

kx 0 0 kx r ðkx þ kgx Þ 0 0 kx r kyt K 35 0 kyt þ ðK d1 þ K d1 Þ 0 kyt ðkyt þ kgy Þ K 45 kx r K 53 K 54 K 55 0 K d1 0 K d1 ld1 0 K d2 0 K d2 ld2 cx 0 0 cx r ðcx þ cgx Þ 0 0 cx r 0 cyt þ ðC d1 þ C d2 Þ cyt C 35 0 cyt ðcyt þ cgy Þ C 45 cx r C 53 C 54 C 55 0 cd1 0 C d1 ld1 0 cd21 0 C d2 ld2

0 0 K d1 0 K d1 ld1 kd1 0 0 0 C d1 0 C d1 ld1 C d1 0

0 0 0 0 0 0 0 b1 0 0 0 b1 0 0

3 0 0 7 7 0 7 7 b2 7 7 0 7 7 0 5 b2

3 0 7 0 7 7 K d2 7 7 7 0 7 K d2 ld2 7 7 7 5 0 K d2 3 0 0 7 7 C d2 7 7 7 0 7 7 C d2 ld2 7 7 0 5 C d2

where

kyt ¼ k1y þ k2y þ k3y þ k4y þ k5y ; cyt ¼ c1y þ c2y þ c3y þ c4y þ c5y C 35 ¼ C 53 ¼ ðc2y l2 þ c4y l4  c1y l1  c3y l3 Þ þ ðC d2 ld2  C d1 ld1 Þ; C 45 ¼ C 54 ¼ ðc1y l1 þ c3y l3  c2y l2  c4y l4 Þ 2

2

2

2

2

2

C 55 ¼ ðcx r 2 þ c1y l1 þ c2y l2 þ c3y l3 þ c4y l4 Þ þ ðC d1 ld1 þ C d2 ld2 Þ k35 ¼ k53 ¼ ðk2y l2 þ k4y l4  k1y l1  k3y l3 Þ þ ðK d2 ld2  K d1 ld1 Þ K 45 ¼ K 54 ¼ ðk1y l1 þ k3y l3  k2y l2  k4y l4 Þ 2

2

2

2

2

2

K 55 ¼ ðkx r 2 þ k1y l1 þ k2y l2 þ k3y l3 þ k4y l4 Þ þ ðK d1 ld1 þ K d2 ld2 Þ and the F g term can be expressed as:

2

6 c x_ 6 gx g 6 6 6 6 F g ¼ 6 cgy y_ g 6 6 6 6 4

0

3

þ kgx xg 7 7 7 7 0 7 7 þ kgy yg 7 7 7 0 7 7 0 5 0

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