Optimal Design of Hanging Truss Having SMA Wires (On ......In (Zhang et al., 2017b), the vibration isolation and attenuation capabilities of the hanging truss are improved on the basis

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Optimal Design of Hanging Truss Having SMA Wires (On Sectional Area of SMA Wire)

Journal: Transactions of the Canadian Society for Mechanical Engineering

Manuscript ID TCSME-2018-0150.R1

Manuscript Type: Article

Date Submitted by the Author: 06-Jan-2019

Complete List of Authors: Zhang, Xuan; Beijing Computing Center, Beijing Key Laboratory of Cloud Computing Key Technology and ApplicationHanahara, Kazuyuki; Iwate University, Department of Systems Innovation EngineeringTada, Yukio; Kobe University, Graduate School of System InformaticsPei, ZhiyongLi, ZheSun, Pengjie

Keywords: hanging truss, multi-objective optimization, shape memory alloy wire, vibration attenuation, vibration isolation

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

https://mc06.manuscriptcentral.com/tcsme-pubs

Transactions of the Canadian Society for Mechanical Engineering

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1

Optimal Design of Hanging Truss Having SMA Wires

(On Sectional Area of SMA Wire)

Xuan Zhang1,2, Kazuyuki Hanahara3, Yukio Tada4, Zhiyong Pei1,2, Zhe Li1,2 & Pengjie Sun1,2

1Beijing Key Laboratory of Cloud Computing Key Technology and Application, Beijing， China

2Beijing Computing Center, Beijing Academy of Science and Technology， Beijing, China

3Department of Systems Innovation Engineering, Faculty of Science and Engineering, Iwate

University, Iwate, Japan

4Department of Biomedical Engineering, Faculty of Life and Medical Sciences, Doshisha

University, Kyoto, Japan

Correspondence: Pengjie Sun, Beijing Key Laboratory of Cloud Computing Key Technology and

Application, Beijing， China. Beijing Computing Center, Beijing Academy of Science and

Technology, Beijing， China. E-mail: [email protected]. Tel: 010-59341888. Fax:

010-59341888

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Abstract

In this study, we deal with a dynamic problem of a kind of column-type hanging truss structural system having pseudo-elastic shape memory alloy (SMA) bracing wires. In the case that the sectional area values of the bracing SMA wire members are small enough to be negligible, it is close to the situation that there are no braces. In the case that the sectional area values of the bracing SMA wire members are large enough not to be negligible, the vibration amplitude of peripheral end apparatus is suppressed from the deformation point of view. In addition, energy attenuation efficiency is improved with the larger sectional area values due to the hysteretic characteristic of SMA. The small sectional area values of the bracing SMA wire members near the support ceiling or peripheral end are beneficial to vibration transmission reduction. These indicate that the positions of placement and their sectional area values of SMA wire members are both significant from the viewpoint of suppression of influence of the support ceiling vibration of the hanging truss. In this study, we obtain the optimal sectional area values of the SMA wire bracing members for the objectives of vibration isolation and attenuation. We discuss influences of different vibration conditions on the optimal solutions.

Keywords: hanging truss, multi-objective optimization, shape memory alloy wire, vibration attenuation, vibration isolation

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1. Introduction

As a kind of functional material, SMA material has been researched extensively in recent years. Especially, a large number of studies on vibration reduction by means of pseudo-elastic SMA materials have been performed. In the study of (Mekki et al., 2010), the behaviors of passive vibration control of cable-stayed bridges have been discussed. A piecewise linear model of pseudo-elastic SMA is used in the dynamic analysis of the cable-stayed bridges. In the research of (Choi et al., 2005), a new isolation device having SMA wires is put forward. The SMA wires are incorporated in an elastomeric bearing, which is able to solve the problem of residual deformation due to the pseudo-elastic SMA material. The coupled thermo-mechanical behaviors of a vibration isolator made of SMA material are demonstrated (Jose et al., 2017). Isolation performances are discussed in both of the cases of isothermal and non-isothermal conditions.

A large number of novel optimization approach have been put forward recently (Jalili & Hosseinzadeh, 2018), (Hosseinzadeh et al., 2015), (Jalili & Kashan, 2017). With the development of optimization approach regarding on evolutionary computation, there are a large number of researches on the construction of algorithms for multi-objective optimization problems and on the optimal designs of engineering applications (Madeira et al., 2005), (Fallah & Zamiri, 2013), (Choi et al., 2017), (Dlugosz & Burczynski, 2017), (Coello & Christiansen, 2000). In addition, the evolutionary algorithms for structural optimization are able to be modified in accordance with the practical engineering or academic problems. For instance, in the study of (Nonami et al., 2017), an evolutionary computation approach is modified for the purpose of saving time.

Before performing an optimization calculation, its objectives have to be concretely specified. Vibration isolation and attenuation effects are significant properties for the engineering applications of structural systems. A number of researches on optimization have been conducted to enhance the capabilities of vibration reduction. For example, the performance of vibration reduction of a structure subjected to earthquake loading is improved in (Salajegheh et al., 2007) by using artificial intelligence and wavelet transforms. The optimal distributions of piezoelectric active structural elements in smart structures are obtained to suppress vibration (Yang & Sedaghati, 2005). In that research, a heuristic-based simulated annealing is utilized to search the near-optimal solutions at an acceptable time cost. In the study of (Khot, 1988), an optimal design algorithm is developed for better active vibration control effect. Optimization on structural system brings us not only material saving but also improvement of the corresponding mechanical properties of the structures. In our current study, we consider both of the vibration isolation and attenuation effects of a hanging truss structural system by attaining the optimal distributions of sectional area values of bracing SMA wires.

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Column truss structure is unstable without bracing members. Under the hanging configuration, a `truss' system of this type is virtually not a structure but a mechanism, while it is stable due to the effect of the gravitational force. The system has the ability to isolate an apparatus installed at its peripheral end from the influence of vibration of the support ceiling. In the case that the sectional area values of the bracing wire members are negligibly small, it is virtually a situation that there are no braces. Vibration transmission from the support ceiling to the peripheral end does not occur significantly. However, the deformations of the truss structure are large due to the small stiffness of the truss structure in such cases. In the case that the sectional area values of the bracing wire members of the hanging truss are large enough not to be insignificant, it is stable from the deformation point of view. Efficiency of vibration attenuation is improved. However, vibration energy transmits significantly from the support ceiling to the peripheral end. Therefore, a trade-off exists between the values of the sectional area of the bracing wire members and the vibration reduction performance of the hanging truss structure. Vibration transmission is reduced significantly in case of small sectional area values of the bracing wire members in the truss units near the support ceiling and the peripheral end apparatus (Liu et al., 2005). Therefore, the optimization of distribution of the sectional area values of the bracing wire members is necessary for the improvement of vibration reduction performance of the hanging truss structural system. In relatively high temperature conditions, SMA material has a pseudo-elastic effect, which demonstrates a hysteretic loop. Structure having pseudo-elastic SMA material is expected to possess the capability of energy attenuation.

In (Zhang et al., 2017b), the vibration isolation and attenuation capabilities of the hanging truss are improved on the basis of placement optimization on the SMA wire members. However, the corresponding vibration reduction performance of the optimal trusses are not as effective as expected. It is supposed that not only the placement but also the sectional area values of the bracing SMA wire members are important factors for the vibration suppression effect. Appropriately small diameters of several bracing SMA wire members enable the hanging truss to demonstrate the characteristic of pendulum more significantly. Vibration transmission from the support ceiling to the peripheral end apparatus is reduced. Several bracing SMA wire members with large diameters in the hanging truss improves the efficiency of energy attenuation. In addition, the deformations of the hanging truss are suppressed to some extent. Therefore, by conducting optimization on sectional area values of bracing SMA wire members, the vibration reduction capability of the optimal solutions is improved. Vibration suppression performance of these optimal solutions is more effective than the optimal solutions obtained only by conducting placement optimization.

In this research, the influences of the sectional area values of the bracing SMA wire members on the dynamic behaviors of the hanging truss are discussed. Optimization problems on the sectional area values of the bracing wire members are dealt with from the vibration isolation and attenuation points of view. An optimization algorithm is constructed on the basis of non-dominated sorting genetic algorithm (NSGA). A Pareto solution in terms of the objective functions is shown. Relationships

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between the distributions of the sectional area values of the bracing SMA wire members and the effects of vibration reduction of the hanging truss structural system are discussed.

The structure of this paper is organized as follows. Section 2 describes the dynamic model of the hanging truss structure having SMA wire members. In section 3, a typical dynamic behavior of a hanging truss having SMA wire members is demonstrated. In section 4, discussions of the influence by the sectional area values of SMA wire members on the dynamic behaviors of hanging truss structural system are conducted. The optimization method is introduced in section 5 in order to search the optimal solutions of the section area values of SMA wire members from the vibration isolation and attenuation points of view. One of the optimization examples is shown. Influences of vibration frequency of the support ceiling on the optimal solutions are also taken into account. Section 6 concludes the main outcomes of this paper.

2. Dynamic Model

The hanging truss dealt with in this study is demonstrated as in Figure 1(a). The number of truss

member is denoted as . Elastic deformation of the th truss member is expressed N i ),...,2,1( Ni

as:

)1(SMAdtransformePhase

austeniteorRigid

t

iii

ii r

rR

where is the total deformation, is the natural length and is the phase transformation ir i ti

strain, respectively.

Kinetic energy, elastic strain energy, potential energy and the work done by the support ceiling are

expressed as , , and , respectively. In these

UMUT

Q21 RKR R

TU21

UGTP 1 Df 0TP

formulations, is the mass matrix, is the elastic deformation vector and M R

is the diagonal matrix that contains the stiffness informations of all ]/,...,/[diag 111 NNN AEAE RK

the truss members (Hanahara et al., 2016). Parameters and are the Young's modulus and the iE iA

sectional area value. Vector is the gravitational force vector and is the force vector exerted by G f

the support ceiling. The initial positional vector of the truss nodes is denoted as and the current 0P

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positional vector of the truss nodes is obtained as . In this formulation, is DDPUPP 000 U

the absolute displacement vector; is the displacement vector due to the motion of the support 0D

ceiling and is the deformation vector of the truss nodes due to the deformations of the truss D

members. By utilizing the basic formulation of virtual work , the dynamic equation tPPUQ d21

of motion is derived as:

)2(0 GqJDMKDDM

T

In the above dynamic equation of motion, is the stiffness matrix. Matrix JKJK RT TT )/( PLJ

is the coordinate transformation matrix. Vectors and are the current member length vector L q

and the nonlinear force vector corresponds to the characteristic of pseudo-elastic SMA wire members.

3. Typical Dynamic Behavior of Hanging Truss

In Figure 1(a), the larger black rectangle at the top represents the support ceiling and the smaller one at the bottom represents the peripheral end apparatus. The motion trajectory of the support ceiling is a sinusoidal wave shown as the broken line in Figure 1(b), the frequency and amplitude of which are 5Hz and 0.03m, respectively. The diameters of all the bracing SMA wire members are 1mm. Numbers assigned to the lines in the figure serve as the identifiers of the bracing SMA wire members. The specification of the hanging truss and material parameters for the SMA are listed as in Tables 1 and 2. Figures 1(b) and 1(c) are the displacement and acceleration behaviors.

The truss in Figure 1(a) is viewed as the reference truss that is to be compared with various truss designs. In order to demonstrate a characteristic behavior of the truss dealt with in the study from the viewpoint of natural frequencies, the time histories of the initial ten natural frequencies of the reference truss are demonstrated in Figure 2. Method for obtaining the time history of natural frequencies is demonstrated in (Zhang et al., 2017a). As can be seen in this result, owing to zero initial strain of all the bracing SMA wire members, at the initial time period of numerical integration, all the ten natural frequencies change significantly due to the the drastic change in the stiffness of the truss structural system. In the steady state of the time period of vibration of the support ceiling, the natural frequencies vary periodically. It should be noted that in case of both of the bracing SMA wire members in the same truss unit are in taut or slack states, natural frequencies become significantly large or small suddenly due to the drastic change in the stiffness matrix of the truss structural system at those moments. In the steady state after the vibration of support ceiling ceased, natural frequencies are

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approximately invariable due to the elastic area of the pseudo-elastic SMA and small deformations of the truss system itself.

4. Influences of Sectional Area of SMA Wires on the Dynamic Behaviors

Several dynamic behaviors of hanging truss structural system are discussed. Influences by sectional area values of the bracing SMA wire members on the dynamic behaviors are demonstrated. The adequacy of the numerical calculation process was confirmed from the energy conservation point of view.

The sectional area values of all the bracing SMA wire members in Figure 3(a) is 0. Figure 3(b) is the time history of displacement of the peripheral end apparatus. The deformation of the truss is large; however, from the acceleration point of view shown in Figure 3(c), the vibration isolation effect to the peripheral end apparatus is confirmed. This demonstrates the effect of vibration isolation analogous to the characteristic of pendulum. In the time period of 0s-4s, the RMS value of the acceleration of peripheral end apparatus is 1.60m/s2 and the RMS value of acceleration of support ceiling is 20.94m/s2. In the time period after the cease of the vibration of the support ceiling, the RMS value is 2.84m/s2.

Figure 4(a) is a configuration of hanging truss structure having SMA wires. The diameters of the SMA wire members are assumed to be 0.2mm. Figure 4(b) is the corresponding time history of displacement of peripheral end apparatus. Owing to the extremely small diameters of the bracing SMA wire members, the effect of pendulum characteristic is obvious. Behaviors in Figures 4(b) and 3(b) are similar. Figure 4(c) is the time history of acceleration of the peripheral end apparatus. Vibration isolation and attenuation effects of the truss in Figure 4(a) are more significant than the truss in Figure 3(a). In the time period of 0s-4s, the RMS value in Figure 4(c) is 1.23m/s2 and in the time period of 4s-8s, the RMS value is 1.17m/s2. This is due to the effect of mechanical behavior of the bracing SMA wire members with small diameters. The vibration of the support ceiling dose not transmit to the peripheral end significantly and the exerted energy from the support ceiling is attenuated by a non-insignificant degree, due to the hysteretic loop of pseudo-elasticity of the SMA wire members.

In Figure 5, the diameters of the bracing SMA wire members are 0.5mm. As the diameters of the bracing SMA wire members in Figure 5(a) are larger than that of the values in Figure 4(a), the amplitude of displacement in Figure 5(b) is more insignificant. In Figure 5(c), obvious vibration isolation and attenuation effects are seen. The RMS value in the time period of 0s-4s is 1.93m/s2 and in the time period of 4s-8s is 1.68m/s2. The vibration attenuation effect seen in Figure 5(b) is clearly significant than that in Figure 4(b). This indicates that with appropriate diameters of bracing SMA wire members, vibration attenuation capability of the hanging truss is enhanced.

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The reference truss already shown in Figure 1(a) has the bracing SMA wire members of 1mm diameter. The displacement and acceleration of the dynamic simulation are shown in Figures 1(b) and 1(c). It should be noted that the coordinate scale in Figures 1(b) and 6(b) are different from Figures 3(b), 4(b) and 5(b). The amplitude of displacement is significantly smaller than the above results; however, the amplitude of acceleration is larger than the above results. This is due to the significant vibration transmission from the support ceiling to the peripheral end by the thick bracing SMA wire members. In Figure 1(c), the RMS value of acceleration in the time period of 0s-4s is 7.50m/s2 and in the time period of 4s-8s is 4.45m/s2.

In Figure 6, the diameters of the bracing SMA wire members are 2mm. The bracing SMA wires of truss in Figure 6(a) are thicker than other trusses in Figures 3(a), 4(a), 5(a) and 1(a); accordingly, the amplitude of displacement is the most insignificant. Vibration is hardly isolated from the acceleration point of view. This is considered as the vibration transmission from the support ceiling to the peripheral end by the thick SMA wire members. Most of the transmitted energy are not attenuated due to the elastic relation in the pure austenite phase of the SMA wire members in the stage of residual vibration. The RMS value in the time periods of 0s-4s and 4s-8s are 26.69m/s2 and 15.22m/s2, respectively.

With the increasing of diameters of bracing SMA wire members, the deformation of the peripheral end becomes more insignificant and the vibration frequency of displacement becomes higher. In the case of small sectional area values of bracing SMA wire members, the effect of pendulum is obvious and vibration transmission does not occur significantly. However, in the case of no bracing SMA wire members as shown in Figure 3(c), the performances of vibration isolation and vibration attenuation are not the most effective. Optimization on the sectional area values of the bracing SMA wire members is of utmost importance for reducing vibration.

5. Conducted Optimal Design

5.1 Formulation of Optimal Design Problem

Large sectional area values of bracing SMA wire members are beneficial to vibration attenuation; however, vibration transmission occurs significantly. Small sectional area values of bracing SMA wire members are beneficial to reduction of vibration transmission. A trade-off exists between the effects of vibration isolation and attenuation in accordance with the cross-sectional area values of SMA bracing wires. The distribution of the values of sectional area of bracing SMA wire members plays a significant role in the performance of vibration isolation and attenuation. Performing optimization of sectional area values of the bracing wire members is able to improve the capability of vibration reduction.

Formulation of the optimization problem is as follows:

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maxmin

21

toSubject

)3(,...,,...,,torespectwith),(Minimize

DDDDDDD

VWFF

i

Ni

where the maximum and the minimum diameters are expressed as and . The objective of maxD minD

this model is to obtain the minimum Pareto ranking value in the space. Vibration isolation F WV

evaluation function is the RMS value of acceleration in the time period from to ; vibration W 0t 1t

attenuation evaluation function is the RMS value of acceleration in the time period from to :V 1t 2t

)2/1(2

12

)2/1(2

01

d1

)4(d1

2

1

1

0

tatt

V

tatt

W

tt

tt p

tt

tt p

In the above formulations, is the initial time, is the time of cease of support ceiling vibration 0t 1t

and is the final time. In the following examples, we assume , and . These 2t s00 t s41 t s82 t

evaluations are based on the horizontal acceleration of the peripheral end apparatus at time t, which is

denoted as . pa

The continuous design variable of SMA wire diameter is discretized by utilizing 16 bit binary number. The maximum diameter and the minimum diameter are expressed as 216-1 and 0, respectively. The

other values of diameter are attained on the basis of interpolations of and . Each of the maxD minD

chromosome is denoted as a 20×16 bit string.

The NSGA-based optimization algorithm (Deb et al., 2002) used in this research is expressed as follows:

step 1: Prepare individuals as the parents in generation .popN 1s

step 2: Obtain the values of and .W Vstep 3:

(a) : Determine the rank values in generation . 1s s

(b) : Select individuals from generations and as the parents of generation . 1s popN 1s s s

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step 4: Perform crossover and mutation operators to produce the offspring in generation .sstep 5: Repeat from step 2 to step 4, with .1 ss

The selection probability of the th individual (Murata et al., 1996) in step 4 is:j

)5()}1({

)1(

max

max

FFFF

pj

jj

where is the current Pareto ranking of the th individual and is the total number of Pareto jF j maxF

front layers. In terms of the selected two individuals, crossover operation is performed on the th SMA iwire in the first individual and the th SMA wire in the second individual.i

Pareto ranking describes the relationship of predominance among all of the values in the solution space.

The algorithm for determining the current Pareto ranking of the th individual is as follows:jF j

step 1: Set .1rstep 2: Find all the non-dominated designs. They are referred to as the th Pareto front. rstep 3: Eliminate the th Pareto front. rstep 4: Repeat from step 2 until all the designs are eliminated, with .1 rr

The relationship of predominance among all the solutions in the current iteration is attained rthrough the following algorithm:

step 1: Determine the rank values of all the individuals in terms of and , respectively. W Vstep 2: Set individual number .1kstep 3: Search the individuals with higher rank values than in terms of and , respectively.k W Vstep 4: For individual : k

(a) if & , then is dominated by .Wk

Wj FF V

kVj FF k j

(b) else is non-dominant. kstep 5: Repeat from step 3 to 4 until all the individuals are examined, with .1 kk

In the above algorithm symbol denotes the rank value merely corresponds to the objective WF

function and symbol denotes the rank value merely corresponds to the objective function . W VF V

In the step 3, the relationship of ranking corresponds to the separate objective function between the

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two solutions and is defined as: in the case that , the relationship is expressed as k j kj WW

; in the case that , the relationship is expressed as . Wk

Wj FF kj VV V

kVj FF

In the step 4, the dominant/non-dominant search process is as follows. Solutions with higher rank

values than a solution from the viewpoint of are arranged into one set and the solutions k W WS

with higher rank values than the solution from the viewpoint of are arranged into another set k V

. In the case that there is some solution in the set is the same as some solution in the set VS WS VS

(we denote this solution as just as the above algorithm) in the solution space of the Pareto front , j

the solution is dominated by the solution . In the case that there are no common solutions k j

between the sets and , the solution is a non-dominant solution. In a particular case that all WS VS k

the other individuals have lower rank values than in terms of and respectively, individual k W V is considered as the only solution in the current layer of the Pareto front solutions. k

5.2 Optimization Example

The optimization is conducted on the hanging truss shown in Figure 7(a), that consists of 20 SMA wire

members. Number of individual is 40 and number of evolution is 200. Probabilities of popN evoN

crossover operator and mutation operator are 0.8 and 0.01, respectively. The maximum and minimum

values of SMA diameter are and . Pareto solutions are demonstrated as in mm2max D mm0min D

Figure 7(b). There are 14 individuals in the Pareto front. From the left to the right, the solutions are

denoted as . Several intermediate Pareto fronts in the convergence history are demonstrated 141 CC

in Figure 8. In this result, we are able to see that the optimal solutions have a tendency of approaching the origin of coordinates with an increasing of iteration number. The optimal solutions in Figure 8(f) are the same as the optimal solutions in Figure 7(b). The difference between these two figures is the coordinate scale.

Figure 9 demonstrates the patterns of distributions of thickness of SMA wires of several optimal solutions. It should be noted in the figure that the thickness of SMA wires are exaggerated. From this result, we can see the similar tendency of distributions of sectional area values of SMA wires among

these optimal solutions. In the optimal solutions, the values of , , , , and 11D 15D 16D 17D 18D 20D

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are tremendously small. Especially, and are approximately equal to 0. The effect of 16D 20D

pendulum is obvious in case of small diameters of the bracing SMA wire members in the hanging truss

structural system. The values of , , , and are relatively larger than the other 4D 5D 10D 12D 13D

bracing wire members. Deformations are suppressed because of these bracing SMA wire members with large sectional area values. Vibration energy dissipation efficiency is also improved in such cases.

Sectional area values of bracing SMA wire members in the truss units near the support ceiling or peripheral end apparatus are negligibly small. Distribution tendency of small sectional area values of these bracing SMA wire members is analogous to the ordinary structures having vibration isolation capability. Mechanical connections between the support and the structure for vibration isolation should be flexible. Insignificant vibration transmission contributes to suppression of residual vibration.

The distributions of sectional area values of the optimal solutions obtained in Figure 9 demonstrate the same tendency with the optimal solutions in the reference of (Zhang et al., 2017b). In the optimal solutions of this reference, there are few SMA wire members placed at the truss units near the support ceiling and the peripheral end apparatus. Most of the SMA wire members concentrate in the middle units of the hanging truss. As shown in Figure 9, the sectional area values of the bracing SMA wire members in the truss units near the support ceiling as well as the peripheral end apparatus are negligibly small. The sectional area values of several bracing SMA wire members are large in the middle truss units. Vibration transmissions from the support ceiling to the truss and from the truss to the peripheral end are reduced; the transmitted energy is attenuated by the SMA bracing wire members with larger sectional area values placed in the middle truss units.

Vibration suppression capability of the optimized truss obtained in this research is more significant than the optimized truss attained in (Zhang et al., 2017b). The optimization problem conducted in this reference is only on the placement of the bracing SMA wire members. However, the sectional area values of bracing SMA wire members is also important for the vibration reduction performance. Several bracing SMA wire members with large diameters in the hanging truss improves the efficiency of vibration energy attenuation. In addition, the deformations of the hanging truss are suppressed to some extent due to these SMA wires. Therefore, by conducting optimization on sectional area values of bracing SMA wire members, the capability of vibration reduction of the optimal solutions is more significant than that of the solutions in the configuration optimization problem.

Dynamic characteristics of the optimal solutions and are obtained as in Figures 10(a) and 1C 14C

10(b). The Values of vibration isolation evaluation function and vibration attenuation evaluation function are listed as in Table 3 for comparison. It should be noted that the coordinate scale of ordinate in Figures 10(a) and 10(b) are different from Figures 1(c), 3(c), 4(c), 5(c) and 6(c). The values of the

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objective functions are , , and . 2s/m73.01

CV 2s/1.17m41

CV 2s/1.28m1

CW 2s/1.19m41

CW

These values of objective function are tremendously smaller than the result in Figure 3(c), which is Vcorresponding to the truss having no bracing wires. In the time period of the vibration of support ceiling, the isolation effect of the peripheral end apparatuses are shown; in addition, during the time period after the vibration of the support ceiling ceased, residual vibrations are small. Results of Figures 10(a) and 10(b) demonstrate that by performing optimization calculations on the sectional area values of the bracing SMA wire members, the improvement of capabilities of both vibration isolation and attenuation of the hanging truss is achieved.

Figure 11 is the time history of the initial ten natural frequencies of truss . We can see in this result 1C

that the initial 4 natural frequencies are significantly smaller than the vibration frequency of the support ceiling. The vibration of the hanging truss dose not occur significantly in such cases because the initial several natural frequencies are far away from the vibration frequency of the environment. In addition, the higher modes (in this example, the ninth and the tenth vibration modes) are quite far away from the vibration frequency of the support ceiling. The higher modes are not excited in such cases. However, the distribution tendency of natural frequencies in Figure 2, which is the result of a truss before performing optimization is not the same case. Therefore, by performing optimization on sectional area values of bracing SMA wire members, the differences between the frequency of vibration of the support ceiling and the initial several natural frequencies of the hanging truss become significantly large. Simultaneously, higher modes are not excited.

As shown in Figures 9(a) and 9(k), diameter value distributions of these two trusses are approximately

the same except for the values of , and . The corresponding diameters of truss are 2D 4D 15D 14C

slightly larger than the corresponding values of truss . Owing to these larger diameters of bracing 1C

SMA wire members, at the initial time period of vibration of support ceiling, vibration transmission of

truss is more immediate than truss . This phenomenon is shown in Figures 10(c) and 10(d) in 14C 1C

the time period of . Acceleration in Figure 10(d) changes more promptly than the s3.00.05s

acceleration changes in Figure 10(c). In addition, it is interesting that the residual vibration of truss 1C

is smaller than that of truss , while truss has thicker SMA wires. This is considered to be due 14C 14C

to that the difference in vibration isolation effect between trusses and is more significant in 1C 14C

this case than the difference in vibration attenuation effect between trusses and .14C 1C

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5.3 Deviation of the Optimization Method

Owing to the stochastic nature of the NSGA, the statistical results of this algorithm ought to be discussed. The distance between the Pareto solution and the origin of coordinates is utilized to

evaluate the results. The optimization process is conducted times. The average distance between aN

the optimal solutions and the origin of coordinates in the th Pareto front is defined as:c

)6()(11

2/122

cN

jcjcj

cc VW

Nx

where the subscript is the solution number in the th Pareto front, parameter is the total j c cN

number of optimal solutions in the th Pareto front. The deviation among average distances of c aN

Pareto fronts is:

)7()(1

12/1

1

2

aN

cc

a

xxN

where parameter is the average value of the average distances.

aN

cca xNx

1)/1( aN

The optimization process has been conducted times and the results are demonstrated in 5aN

Figure 12. The average distances of the Pareto fronts are , , , aN 50.11 x 2.262 x 2.223 x

, , respectively. The average value of the average distances is . On the 33.14 x 50.15 x aN 76.1

x

basis of formulation (7), the deviation is obtained as . Due to the relationships of 0.44

xx2

and , the degree of dispersion of the average distances should not be ignored. We regard

xx3

the second and the third Pareto fronts as one group due to that not only they are exceedingly close to each other but also the corresponding average distances are approximately the same. Similarly, we regard the first, the fourth and the fifth Pareto fronts as another group. However, these two groups are apparently different from each other. It is considered that the attained solutions in Figure 12 are local optimal solutions. The solutions in the first, the fourth and the fifth Pareto fronts are better than the solutions in the second and the third Pareto fronts.

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The differences between these two groups are due to the distributions of the sectional area values of the optimal designs. The number of large sectional area values of the SMA wire members of the optimal designs in the first group are obviously more than that of the number in the second group. The unattenuated energy from the support ceiling of local optimal solutions located in the first group are more than that of the unattenuated energy in the second group. Therefore, the behaviors of the optimal designs in the second group are better than the behaviors of the optimal designs in the first group. In addition, the optimization process should be performed several times in order to attain the optimal solutions more satisfactorily.

It should also be noted that the distributions of the sectional area values of the bracing SMA wire members of the optimal designs in the five Pareto fronts are significantly different, even though they are in the same group. It is considered that there are several types of distributions of the sectional area values of SMA wire bracing members that enable the truss structure to avoid resonance with the environmental vibration. Simultaneously, the vibration isolation and attenuation capabilities of the trusses with those kinds of distributions meet the expectations.

5.4 Optimization Results with Different Vibration Frequencies

We show the optimization results with different vibration frequencies (from to ) of the Hz2 7Hzsupport ceiling in Figure 13. In all of these optimal solutions, the sectional area values of the bracing SMA wire members in the truss units near the support ceiling as well as the peripheral end apparatus are almost negligibly small. On the contrary, the diameters of several bracing SMA wire members in the middle truss units are large. It should be noted that in case of high vibration frequency condition ( ), Hz7the sectional area values of the bracing SMA wire members are tremendously larger than the cases from to . The reason is that in such high frequency condition, divergence is apt to occur in Hz2 Hz6the case that the diameters of the bracing SMA wire members are not thick enough. The diameters of bracing SMA wire members near the support ceiling in the hanging truss from Figures 13(g) to 13(j) are larger than that of the values from Figures 13(a) to 13(f). This is also due to the high vibration frequency of the support ceiling. In case of the same vibration frequency of the support ceiling, the optimal trusses have approximately the same distributions of the sectional area values of the bracing SMA wire members. However, distribution of the sectional area values are tremendously different in case of different vibration frequencies of the support ceiling.

6. Conclusions

In this paper, optimization problem has been performed on a hanging truss structural system under an ideal vibration conditions. The vibration amplitude and frequency is determined in advance. The topology of the hanging truss is also invariable. Under such circumstances, influences of the sectional area values of the bracing SMA wire members on the dynamic behaviors of hanging truss have been

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discussed. The optimization of the sectional area values of the bracing wire members has been performed from the vibration isolation and attenuation points of view. An NSGA-based algorithm for the optimization problem has been constructed. The results demonstrate that the sectional area values of the bracing SMA wire members near the support ceiling as well as the peripheral end apparatus are negligibly small. Vibration transmissions from the support ceiling to the truss and from the truss to the peripheral end apparatus are reduced. Sectional area values of the bracing SMA wire members in the middle of the truss are relatively larger. The transmitted energy are absorbed by these bracing SMA wire members with larger sectional area values. A Pareto solution has been shown. The dynamic behaviors of the optimized truss structure demonstrate excellent vibration isolation and attenuation capabilities. For instance, vibration isolation capability of the truss in Figure 9(a) is 5.86 times, 1.25 times, 0.96 times, 1.51 times, 20.85 times of the trusses in Figures 1(a), 3(a), 4(a), 5(a), 6(a), respectively. The vibration attenuation capability of the truss in Figure 9(a) is 6.10 times, 3.89 times, 1.60 times, 2.30 times, 20.85 times of the trusses in Figures 1(a), 3(a), 4(a), 5(a), 6(a), respectively. The optimal solutions in case of different vibration frequencies of support ceiling show the same tendency of the distribution of the sectional area values of bracing SMA wire members. References

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Hanahara, K., Zhang, X., & Tada, Y. (2016). Dynamic simulation of adaptive truss consisting of various types of truss members. Mechanical Engineering Research, 6(1), 75-87. https://doi.org/10.5539/mer.v6n1p75

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Zhang, X., Hanahara, K., & Tada, Y. (2017b). Optimal design of hanging truss having SMA wires (From vibration isolation and attenuation viewpoints). Mechanical Engineering Research, 7(2), 53-63. https://doi.org/10.5539/mer.v7n2p53

Tables

Table 1: Specification of truss

Member diameter (mm) rigid 10mmRa

SMA 1mmsmaa

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Density (kg/m3) rigid 7860kg/m3R

Table 2: SMA characteristics

Maximum phase transformation strainYoung’s modulus of austenite phaseYoung’s modulus of martensite phase

0.05 70GPaAE 30GPa ME

Table 3: Values of objective functions

Figure W V Figure W V3(c)5(c)6(c)10(b)

1.6m/s2 2.84m/s2

1.93m/s2 1.68m/s2

26.69m/s2 15.22m/s2

1.19m/s2 1.17m/s2

4(c)1(c)10(a)

1.23m/s2 1.17m/s2

7.5m/s2 4.45m/s2

1.28m/s2 0.73m/s2

List of tables

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Table 1: Specification of trussTable 2: SMA characteristicsTable 3: Values of objective functions

List of figures

Figure 1: Dynamic behavior of the hanging truss in the case that the diameters of SMA wire are 1mmFigure 2: Time histories of the initial ten natural frequenciesFigure 3: Dynamic behavior of the hanging truss having no bracing wire membersFigure 4: Dynamic behavior of the hanging truss in the case that the diameters of SMA wires are 0.2mmFigure 5: Dynamic behavior of the hanging truss in the case that the diameters of SMA wires are 0.5mmFigure 6: Dynamic behavior of the hanging truss in the case that the diameters of SMA wires are 2mmFigure 7: Truss configuration and a Pareto frontFigure 8: Convergence history Pareto front in an optimization processFigure 9: Optimized solutions in the case that frequency of vibration is 5HzFigure 10: Dynamic behaviors of two optimized truss structures

Figure 11: Time histories of the initial ten natural frequencies of truss 1C

Figure 12: Final Pareto fronts fro analyzing deviation of optimization methodFigure 13: Optimized solutions in the case that frequencies of vibration are from 2Hz to 7Hz

Acknowledgments

Funding: BJAST young talent project (BGS201906) and Beijing Academy of Science and Technology Innovation Team Plan (IG201802C1, IG201602C2).

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Draft(a) Configuration (b) Displacement behavior (c) Acceleration behavior

Figure 1: Dynamic behavior of the hanging truss in the case that the diameters of SMA wires are 1mm

Figure 2: Time histories of the initial ten natural frequencies

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Figure 3: Dynamic behavior of the hanging truss having no bracing wire members

(a) Configuration (b) Displacement behavior (c) Acceleration behavior

Figure 4: Dynamic behavior of the hanging truss in the case that the diameters of SMA wires are 0.2mm

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Figure 5: Dynamic behavior of the hanging truss in the case that the diameters of SMA wires are 0.5mm

(a) Configuration (b) Displacement behavior (c) Acceleration behavior

Figure 6: Dynamic behavior of the hanging truss in the case that the diameters of SMA wires are 2mm

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(a ) Truss configuration (b) Pareto front

Figure 7: Truss configuration and a Pareto front

(a) 1st iteration (b) 10th iteration (c) 25th iteration

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(d)100th iteration (e) 150th iteration (f) 200th iteration

Figure 8: Convergence history of Pareto front in an optimization process

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)1C 2C 3C 4C 5C 6C 8C 01C 21C 31C 41C

Figure 9: Optimized solutions in the case that frequency of vibration is 5Hz

(a) Acceleration behavior of truss (b) Acceleration behavior of truss 1C 41C

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Draft(c)Acceleration of in the time period of 0.05s-0.3s (d)Acceleration of in the time period of 0.05s-0.3s 1C 41C

Figure 10: Dynamic behaviors of two optimized truss structures

Figure 11: Time histories of the initial ten natural frequencies of truss 1C

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Draft Figure 12: Final Pareto fronts for analyzing deviation of optimization method

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)1C 01C 1C 61C 1C 24C 1C 22C 1C 35C

(2Hz) (2Hz) (3Hz) (3Hz) (4Hz) (4Hz) (6Hz) (6Hz) (7Hz) (7Hz)

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Optimal Design of Hanging Truss Having SMA Wires (On ......In (Zhang et al., 2017b), the vibration isolation and attenuation capabilities of the hanging truss are improved on the basis