# Optimal Design of Damped Structure with Inerter System Based on Modified Harmony Search Algorithm

Authors: Chao Pan; Xiao Han; Hao Wang; Dengke Yang; Li Zhang
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

An inerter system can amplify the deformation of its internal energy dissipation device, thereby improving the efficiency of energy dissipation and shock absorption. This is the so-called damping enhancement mechanism, one of the key mechanisms of the inerter system. Although the theoretical framework for damping enhancement of inerter systems has been established, the implementation of this principle for the design of an inerter system requires solving a complicated constrained optimization problem, which is not easy to be figured out using traditional approaches. To obtain valid design results through a lucid and robust method, it is proposed to optimize the damping parameters through a metaheuristic algorithm named harmony search algorithm in order to maximize the damping enhancement degree of the inerter system with the satisfaction of structural performance. First, the closed-form seismic response solutions of a single-degree-of-freedom (SDOF) structure with an inerter system are derived based on the theory of random vibration. Then, the mathematical expression of the constrained optimization problem is established. Due to the inefficiency of the original harmony search algorithm to solve the constrained optimization problem, the algorithm is modified by introducing a new harmony generating method and an adaptive strategy for parameter adjustment. The modified harmony search algorithm is compiled to solve the optimal design problem of the inerter system. The algorithm is verified by designing a structure with an inerter system. It is found that the number of iterations and time consumption until convergence required by the modified harmony search algorithm can be reduced by about 20%∼90% compared with the original algorithm, which confirms the effectiveness of the modified algorithm. The results of dynamic analyses show that the structure have achieved the preset performance demands under different cases and the damping enhancement characteristic of the inerter system is fully utilized.


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## Body

## 1. Introduction

The earthquake has become one of the main natural disasters that threaten the safety of human life and property because of its huge destructiveness. The destruction of engineering structures is the critical cause of earthquake disaster. Numerous studies have shown that structure control technology [1, 2] involving adding damping devices to the structure is an effective means to mitigate the structural response under earthquakes or strong winds or any other hazard that can cause vibration [3–5]. Inerter, a newly developed two-terminal acceleration-related element for structure control, can adjust the vibration characteristics of the structure flexibly and efficiently as its dynamic mass can reach thousands of times the physical mass [6, 7]. The realization mechanism of the inerter includes ball screw [8], rack-and-pinion [9], hydraulic [10], and electromagnetic mechanisms [11, 12]. The inerter system that is composed of an inerter, a spring, and an energy dissipation element (EDE; such as viscous, eddy current, magnetorheological dampers, etc.) has become an emerging shock absorption device with the excellent characteristic of damping enhancement. There are three typical types of inerter systems [13, 14]: series-layout inerter systems (SIS), series-parallel-layout type I inerter systems (SPIS-I, also known as TID), and series-parallel-layout type II inerter systems (SPIS-II, also known as TVMD), which can be used in different cases to meet various demands for structure control.The research team headed by Inoue and Ikago et al. from Tohoku University firstly proposed the inerter system (termed tuned viscous mass damper, TVMD for short) at the end of the last century. Ikago et al. [6] derived the formulae for estimating parameters of TVMD in a single-degree-of-freedom (SDOF) structure based on the fixed-point theory, which has become the classical design method for the inerter system. Subsequently, analogous to the tuned mass damper (TMD), Lazar et al. [15] proposed a tuned inerter damper (TID) and proved that the TIDs installed between adjacent floors of multistorey buildings can preferably amplify the damping effect compared with traditional TMDs. Afterward, many scholars studied the further characteristics of the TID. Among them, Gonzalez-Buelga et al. [16] studied the nonlinear characteristics of TID and conducted real-time hybrid simulation experiments. De Domenico et al. [17, 18] attached TID to the base isolation layer for optimized design. With the continuous deepening of research, Marian et al. [19] proposed a tuned mass damping inertial (TMDI) container, which is characterized by installing an inerter in the TMD system. They found that the TMDI is better than the classic TMD in reducing the displacement response. Pietrosanti et al. [20] optimized the SDOF structure by minimizing structural displacement and acceleration under white noise excitation and maximizing the ratio of TMDI energy consumption to total input energy. Park and Giarails et al. [21] pointed out that TMDI can reduce the additional mass and more effectively control the acceleration responses compared with traditional TMD. De Domenico and Ricciardi [22] applied TMDI to a base isolation structure to effectively reduce the displacement of the isolation layer, the bottom shear force, and the displacement angle of the upper structure. Jin et al. [23] comparatively studied the two connection forms of TMDI and analyzed the influence of the mass ratio of the tuned mass damper and the inertance-mass ratio on the performance of the system. Zhang et al. [24] proposed the strategy of using a distributed tuned inerter system to control the seismic response in multiple modes in the high-rise chimney, which achieved the lightweight effect. Faraj et al. [25] proposed a new method to adjust the inertial mass of the inerter by using impact energy management to absorb impact and store it in a special device. Javidialesaadi and Wierschem [26] proposed a new type of nonlinear energy absorber equipped with an inerter system for passive vibration control of structures. Zhao et al. [27, 28] proposed an equation from the perspective of energy, which produces an analysis method for evaluating the input power of an inerter-based structure and quantifying the energy dissipation effect of the inerter system. Pan and Zhang et al. [29–31] proposed principles and methods for the optimal design of inerter system considering structural performance demand and control cost simultaneously based on random vibration theory. Subsequently, Zhang and Pan et al. [32–34] discovered the principle of damping enhancement of the inerter system based on the analytical solution of the random vibration response, and the principle can provide a clear physical explanation of the mechanism of the inerter system.Damping enhancement, a mechanism of inerter system with clear physical meanings, has been adopted as the optimal design principle for a structure with an inerter system. However, traditional numerical optimization methods require complicated gradient expressions and appropriate initial guesses, which are not easy to obtain. Gradient metaheuristic algorithms provide an alternative way to perform optimization in engineering practice [35–39]. Although new algorithms emerge one after another nowadays, such as grey wolf optimizer [40], chaos game optimization [41], aquila optimizer [42], and atomic orbital search [43], the classical algorithms are still vital because they are simple, easy to implement, well-known, and widely accepted. The harmony search algorithm is just one of the widely used classical algorithms in the field of structural control. Bekda and Nigdeli [44] proposed using harmony search algorithm to find the best parameters for tuned mass dampers. Yazdi et al. [45] proposed an improved harmony search algorithm for designing tuned mass dampers, and it is found that the improved harmony search algorithm is more efficient than the gene algorithm and the particle swarm optimization algorithm. Nigdeli and Alhan [46] optimized the parameters of the seismic isolation system through the harmony search algorithm. Ocak et al. [47] optimized TLDs via metaheuristic methods, including the harmony search algorithm. Ulusoy et al. [48] used the harmony search algorithm to investigate the PID parameters of a multistorey structure incorporating active tendon control systems. These studies show the good performance of the harmony search algorithm and provide a theoretical basis for this paper.To perform the optimization of the inerter system, one of the novel structural control systems, more efficiently and robustly, a modified harmony search algorithm with better performance is proposed in this study. First, the closed-form stochastic seismic response solutions of an SDOF structure with an inerter system are derived based on the theory of random vibration under white-noise excitation, and then the mathematical description of the damping enhancement maximization principle of the inerter system is provided. To obtain the solutions more efficiently, a new harmony generation method is proposed as an improvement to the original harmony search algorithm. Besides, an adaptive adjustment strategy of the parameters in harmony search algorithm is adopted to improve the solving efficiency further. Then a computer program is developed to solve the nonlinear constraint optimization problem of the inerter system by the modified harmony search algorithm. Finally, the design cases are executed to prove the effectiveness of the proposed algorithm compared with the original algorithm and an existing harmony search algorithm. Meanwhile, the structural dynamic analysis of the design cases shows that the designed parameters have met the preset performance demands and the damping enhancement characteristics of the inerter system are fully utilized. Moreover, what needs to be pointed out is that the focus of the paper is the modification of the harmony search algorithm, so that only the simple SDOF structure is concerned in this paper. In fact, optimized parameters obtained from an SDOF structure can also be used in a multi-degree-of-freedom structure through certain mathematical transformations [31]. That is to say, the research works on an SDOF structure with an inerter system are meaningful.


## 2. Theoretical Basis of Structure with the Inerter System

### 2.1. Equation of Motion for SDOF Structure with the Inerter System

When an SDOF structure with an inerter system (as shown in Figure1) is subjected to a ground motion excitation (the acceleration of the ground motion is ag ), the equation of motion can be established as follows:(1)mu¨+cu˙+ku+FIS=−mag.Figure 1
Mechanical model of an SDOF system with the inerter system.The corresponding extra equations with respect toFIS are as follows:(2)FIS=kdu−uinminu¨in+cdu˙in=kdu−uin.The definitions of the other symbols can be found in Table1.Table 1
Notations.
NotationDefinitionptExternal excitationagAcceleration of ground motionmMass of the primary SDOF structurecDamping coefficient of the primary SDOF structurekStiffness of the primary SDOF structureuDisplacement response of the primary SDOF structureu˙Velocity response of the primary SDOF structureu¨Acceleration response of the primary SDOF structureFISThe control force of the inerter elementkdStiffness of the spring element in the inerter systemcdDamping coefficient of the EDE in the inerter systemminInertance of the inerteruinDisplacement response of the inerteru˙inVelocity response of the inerteru¨inAcceleration response of the inerterω0=k/mOriginal circular frequency of the primary SDOF structureζ=c/2mωInherent damping ratio of the primary SDOF structureμ=min/mInertance-mass ratio of the inerter systemk=kd/kStiffness ratio of the inerter systemξ=cd/2mωNominal damping ratio of the inerter systemULaplace transformation ofuHUiωThe frequency-domain transfer function ofuAgLaplace transformation ofagS0The amplitude of the input power spectral density of white-noise excitationσu2Stochastic mean square deformation response of the structureσu,d2Stochastic mean square deformation response of the EDE in the inerter systemσu,02Stochastic mean square displacement response of the original SDOF structureγStochastic response mitigation ratioαDamping deformation enhancement factor (DDEF)udDeformation of the EDE in the inerter systemγtTarget response mitigation ratio according to performance demandsωCircular frequency of harmonic excitationβ=ω/ω0Relative frequency of harmonic excitationHMHarmony memoryHMCRHarmony memory considering ratePARPitch adjusting rateHMSHarmony memory sizeTherefore, the integrated equation of motion for an SDOF system with the inerter system is(3)mu¨+cu˙+ku+kdu−uin=−magminu¨in+cdu˙in=kdu−uin.


### 2.2. Closed-Form Expressions of Stochastic Response

Both sides of equation (3) are divided by m to normalize as follows:(4)u¨+2ζω0u˙+ω02u+κω02u−uin=−agμu¨in+2ξω0u˙in=κω02u−uin.The definitions of key symbols in (4) are shown in Table 1. It can be found that μ,ξ,andκ are the three key parameters for the inerter system.Equation (4) can be transformed into the Laplace domain, and then the displacement frequency-domain transfer function of the primary SDOF structure can be obtained as(5)HUs=s2μ+2sξω0+κω02κ2ω04−s2μ+2sξω0+κω02⋅s2+2sζω0+1+κω02.According to the random vibration theory [49], under the white-noise excitation (power spectral density is S0), the mean square value of the structure displacement response is as follows:(6)σ2=∫−∞∞Hiω2S0dω.By substituting (5) into (6), the closed-form expression of the stochastic mean square value of the deformation response of the SDOF structure with inerter system can be obtained [28]:(7)σu2=πS02ω03⋅AB,(8)A=4ζ2κμξ+ξκ2+κ−2+μμ+μ2+4ξ2+ζκ2μ2+4κξ2+4μξ2,(9)B=4ζ3κμξ+κ2ξ2+ζ2κ2μ2+4μξ2+4κ1+μξ2+ζξμ2+κ21+μ2+4ξ2+2κ−μ+μ2+2ξ2.The closed-form expression of stochastic mean square deformation response of the EDE in the inerter system is(10)σu,d2=πS02ω03⋅κζ1+κμ2+κξ+4ζ2μξ+4ζξ2F,(11)F=4ζ3κμξ+κ2ξ2+ζ2κ2μ2+4μξ2+4κ1+μξ2+ζξμ2+κ21+μ2+4ξ2+2κ−μ+μ2+2ξ2.For a classic SDOF structure, the stochastic mean square deformation response can be expressed as follows:(12)σu,02=πS02ω03⋅1ζ.Based on the random vibration response expression, the stochastic response mitigation ratio of the structures with inerter systemγ can be defined as [29](13)γ=responseofstructurewithinertersystemresponseoforiginalstructure=σu2ζ,μ,ξ,κσu,02ζ.By substituting (7) and (12) into (13), the expression of the stochastic response mitigation ratio of the structure inerter system is as follows:(14)P=4ζ2κμξ+ξκ2+κ−2+μμ+μ2+4ξ2+ζκ2μ2+4κξ2+4μξ2Q=4ζ3κμξ+κ2ξ2+ζ2κ2μ2+4μξ2+4κ1+μξ2+ζξμ2+κ21+μ2+4ξ2+2κ−μ+μ2+2ξ2.


### 2.3. Optimization Problem of Structure with the Inerter System

When the inerter system resonates with the external excitation, the internal freedom of the inerter system has a phase difference with respect to the vibration of the primary structure [32], so that the displacement of the internal freedom of the inerter system will be greater than its deformation of the primary structure. Therefore, the inerter system can absorb and dissipate more input energy compared with a directly installed traditional EDE and can reduce the dynamic response of the structure more efficiently, as shown in Figure 2.Figure 2
Damping enhancement of inerter system.To measure the degree of damping enhancement, Zhang et al. [32] defined a dimensionless parameter damping deformation enhancement factor (DDEF) α as follows:(15)α=deformationofEDEoftheinertersystemdeformationoftheinertersystem.Considering (7) and (10), the specific expression of DDEF for an inerter system in an SDOF structure is as follows:(16)α=κζ1+κμ2+κξ+4ζ2μξ+4ζξ24ζ2κμξ+ζκ2μ2+4κξ2+4μξ2+ξκ2+κμ−2μ+μ2+4ξ2.To give full play to the damping enhancement characteristics of the inerter system, the key parameters of the inerter system should be optimized based on the principle of maximizing the damping enhancement degree of the inerter system with satisfaction of structural performance. The above optimization problem can be expressed as a constrained optimization problem:(17)maxfμ,ξ,κ=αμ,ξ,κs.t.γμ,ξ,κ=γt0≤μ≤10≤ξ≤10≤κ≤1.In (17), fμ,ξ,κ=αμ,ξ,κ is the objective function; γt is the target stochastic response mitigation ratio [29], according to the damping enhancement equation of the inerter system [32]:(18)α2=ζξ1γ2−1.Whenγ is constant, the DDEF of the inerter system is inversely proportional to its nominal damping ratio. In other words, when α reaches the maximum value, ξ would reach the minimum value. Therefore, the constrained optimization problem can be rewritten as follows:(19)minfμ,ξ,κ=ξs.t.γμ,ξ,κ=γt0≤μ≤10≤ξ≤10≤κ≤1.For the established constrained optimization problem, if the classical numerical calculation method is adopted, the calculation process is complicated due to the cumbersome mathematical expressions. In order to simplify the calculation, zero inherent damping is usually assumed for the primary SDOF structure, which is not accurate enough. As a metaheuristic algorithm, the harmony search algorithm has the advantages of simple principle, simple operation, easy implementation, and strong search diversity. By adopting the algorithm, it is not necessary to ignore the existence of the inherent damping ratio of the primary SDOF structure and is easy to adjust when the topological form of the inerter system or the structure is changed. Therefore, this paper chooses to modify the harmony search algorithm to optimize the key parameters of the structure with inerter system.


## 2.1. Equation of Motion for SDOF Structure with the Inerter System

When an SDOF structure with an inerter system (as shown in Figure1) is subjected to a ground motion excitation (the acceleration of the ground motion is ag ), the equation of motion can be established as follows:(1)mu¨+cu˙+ku+FIS=−mag.Figure 1
Mechanical model of an SDOF system with the inerter system.The corresponding extra equations with respect toFIS are as follows:(2)FIS=kdu−uinminu¨in+cdu˙in=kdu−uin.The definitions of the other symbols can be found in Table1.Table 1
Notations.
NotationDefinitionptExternal excitationagAcceleration of ground motionmMass of the primary SDOF structurecDamping coefficient of the primary SDOF structurekStiffness of the primary SDOF structureuDisplacement response of the primary SDOF structureu˙Velocity response of the primary SDOF structureu¨Acceleration response of the primary SDOF structureFISThe control force of the inerter elementkdStiffness of the spring element in the inerter systemcdDamping coefficient of the EDE in the inerter systemminInertance of the inerteruinDisplacement response of the inerteru˙inVelocity response of the inerteru¨inAcceleration response of the inerterω0=k/mOriginal circular frequency of the primary SDOF structureζ=c/2mωInherent damping ratio of the primary SDOF structureμ=min/mInertance-mass ratio of the inerter systemk=kd/kStiffness ratio of the inerter systemξ=cd/2mωNominal damping ratio of the inerter systemULaplace transformation ofuHUiωThe frequency-domain transfer function ofuAgLaplace transformation ofagS0The amplitude of the input power spectral density of white-noise excitationσu2Stochastic mean square deformation response of the structureσu,d2Stochastic mean square deformation response of the EDE in the inerter systemσu,02Stochastic mean square displacement response of the original SDOF structureγStochastic response mitigation ratioαDamping deformation enhancement factor (DDEF)udDeformation of the EDE in the inerter systemγtTarget response mitigation ratio according to performance demandsωCircular frequency of harmonic excitationβ=ω/ω0Relative frequency of harmonic excitationHMHarmony memoryHMCRHarmony memory considering ratePARPitch adjusting rateHMSHarmony memory sizeTherefore, the integrated equation of motion for an SDOF system with the inerter system is(3)mu¨+cu˙+ku+kdu−uin=−magminu¨in+cdu˙in=kdu−uin.


## 2.2. Closed-Form Expressions of Stochastic Response

Both sides of equation (3) are divided by m to normalize as follows:(4)u¨+2ζω0u˙+ω02u+κω02u−uin=−agμu¨in+2ξω0u˙in=κω02u−uin.The definitions of key symbols in (4) are shown in Table 1. It can be found that μ,ξ,andκ are the three key parameters for the inerter system.Equation (4) can be transformed into the Laplace domain, and then the displacement frequency-domain transfer function of the primary SDOF structure can be obtained as(5)HUs=s2μ+2sξω0+κω02κ2ω04−s2μ+2sξω0+κω02⋅s2+2sζω0+1+κω02.According to the random vibration theory [49], under the white-noise excitation (power spectral density is S0), the mean square value of the structure displacement response is as follows:(6)σ2=∫−∞∞Hiω2S0dω.By substituting (5) into (6), the closed-form expression of the stochastic mean square value of the deformation response of the SDOF structure with inerter system can be obtained [28]:(7)σu2=πS02ω03⋅AB,(8)A=4ζ2κμξ+ξκ2+κ−2+μμ+μ2+4ξ2+ζκ2μ2+4κξ2+4μξ2,(9)B=4ζ3κμξ+κ2ξ2+ζ2κ2μ2+4μξ2+4κ1+μξ2+ζξμ2+κ21+μ2+4ξ2+2κ−μ+μ2+2ξ2.The closed-form expression of stochastic mean square deformation response of the EDE in the inerter system is(10)σu,d2=πS02ω03⋅κζ1+κμ2+κξ+4ζ2μξ+4ζξ2F,(11)F=4ζ3κμξ+κ2ξ2+ζ2κ2μ2+4μξ2+4κ1+μξ2+ζξμ2+κ21+μ2+4ξ2+2κ−μ+μ2+2ξ2.For a classic SDOF structure, the stochastic mean square deformation response can be expressed as follows:(12)σu,02=πS02ω03⋅1ζ.Based on the random vibration response expression, the stochastic response mitigation ratio of the structures with inerter systemγ can be defined as [29](13)γ=responseofstructurewithinertersystemresponseoforiginalstructure=σu2ζ,μ,ξ,κσu,02ζ.By substituting (7) and (12) into (13), the expression of the stochastic response mitigation ratio of the structure inerter system is as follows:(14)P=4ζ2κμξ+ξκ2+κ−2+μμ+μ2+4ξ2+ζκ2μ2+4κξ2+4μξ2Q=4ζ3κμξ+κ2ξ2+ζ2κ2μ2+4μξ2+4κ1+μξ2+ζξμ2+κ21+μ2+4ξ2+2κ−μ+μ2+2ξ2.


## 2.3. Optimization Problem of Structure with the Inerter System

When the inerter system resonates with the external excitation, the internal freedom of the inerter system has a phase difference with respect to the vibration of the primary structure [32], so that the displacement of the internal freedom of the inerter system will be greater than its deformation of the primary structure. Therefore, the inerter system can absorb and dissipate more input energy compared with a directly installed traditional EDE and can reduce the dynamic response of the structure more efficiently, as shown in Figure 2.Figure 2
Damping enhancement of inerter system.To measure the degree of damping enhancement, Zhang et al. [32] defined a dimensionless parameter damping deformation enhancement factor (DDEF) α as follows:(15)α=deformationofEDEoftheinertersystemdeformationoftheinertersystem.Considering (7) and (10), the specific expression of DDEF for an inerter system in an SDOF structure is as follows:(16)α=κζ1+κμ2+κξ+4ζ2μξ+4ζξ24ζ2κμξ+ζκ2μ2+4κξ2+4μξ2+ξκ2+κμ−2μ+μ2+4ξ2.To give full play to the damping enhancement characteristics of the inerter system, the key parameters of the inerter system should be optimized based on the principle of maximizing the damping enhancement degree of the inerter system with satisfaction of structural performance. The above optimization problem can be expressed as a constrained optimization problem:(17)maxfμ,ξ,κ=αμ,ξ,κs.t.γμ,ξ,κ=γt0≤μ≤10≤ξ≤10≤κ≤1.In (17), fμ,ξ,κ=αμ,ξ,κ is the objective function; γt is the target stochastic response mitigation ratio [29], according to the damping enhancement equation of the inerter system [32]:(18)α2=ζξ1γ2−1.Whenγ is constant, the DDEF of the inerter system is inversely proportional to its nominal damping ratio. In other words, when α reaches the maximum value, ξ would reach the minimum value. Therefore, the constrained optimization problem can be rewritten as follows:(19)minfμ,ξ,κ=ξs.t.γμ,ξ,κ=γt0≤μ≤10≤ξ≤10≤κ≤1.For the established constrained optimization problem, if the classical numerical calculation method is adopted, the calculation process is complicated due to the cumbersome mathematical expressions. In order to simplify the calculation, zero inherent damping is usually assumed for the primary SDOF structure, which is not accurate enough. As a metaheuristic algorithm, the harmony search algorithm has the advantages of simple principle, simple operation, easy implementation, and strong search diversity. By adopting the algorithm, it is not necessary to ignore the existence of the inherent damping ratio of the primary SDOF structure and is easy to adjust when the topological form of the inerter system or the structure is changed. Therefore, this paper chooses to modify the harmony search algorithm to optimize the key parameters of the structure with inerter system.


## 3. Harmony Search Algorithm and Modification

### 3.1. Original Algorithm

The harmony search algorithm is a novel intelligent optimization algorithm proposed by Z. W. Geem et al. in 2001. The harmony search algorithm simulates the procedure of music creation. The musicians rely on their memory to understand the pitch of the instruments in the band. Repeated adjustments finally made the band play a beautiful harmony state. The analogy is shown in Table2 [50].Table 2
Comparison of optimization and music performance.
Analogy diagramAnalogy elementOptimization processRealization processBest stateGlobal bestExcellent harmonyBe evaluated by…Objective functionAesthetic evaluationEvaluate with…Design variable valuePitch of the instrumentProcess unitEvery iterationEvery practiceThe procedure of the original harmony search algorithm (Figure3) can be summarized as follows.Figure 3
Flowchart of harmony search algorithm.Step 1.
Establish the optimization problem (take solving the minimization problem as an example).(20)minfX⟶,where fX⟶ is the objective function to be optimized; X⟶=x1,x2,⋯,xi⋯,xn is the solution vector and xii=1,2,⋯,n is the decision variable that constitutes the solution vector; n is the dimension of solution space.Step 2.
InitializeHM. Randomly generate HMS harmonies, and put them into HM; the mathematical form of HM is as follows:(21)HM=X⟶1X⟶2⋮X⟶HMS=x11x12⋮x1HMSx21x22⋮x2HMS⋯⋯⋯⋯xn1xn2⋮xnHMS.Step 3.
Generate a new harmony. When a random number is greater than the retention probabilityHMCR, the new harmony is randomly generated in the feasible region. Otherwise, a new harmony is generated as follows:(22)x⟶newi=HMR,i,where R is a random number between 1 and HMS.
For every tone in the new harmony, if a randomly generated random number is less thanPAR, the tone is randomly generated near the original tone:(23)x⟶new=x⟶new+bw⋅2Rand⟶−0.5,where bw is bandwidth for pitch adjusting; Rand⟶ is a vector that contains randomly generated values between 0 and 1.Step 4.
Update theHM. If the new harmony vector is better than the worst vector in the HM, then the new vector replaces the worst harmony vector; otherwise, no operation is performed.Step 5.
Check whether the stop condition is reached; if the stop condition is reached, the algorithm terminates; otherwise, the algorithm repeats steps 3-4.


### 3.2. Modification of Harmony Search Algorithm

With the widespread application of the harmony search algorithm, scholars continue to develop some modifications over the original harmony search algorithm. For example, Omran and Mahdavi [51] improved the search mechanism of harmony and proposed a global harmony search. Fesangharya et al. [52] proposed a hybrid harmony search algorithm and so on.The original harmony search algorithm is not efficient for solving constrained optimization problems. Given this, a simple modification is made to replace the classic behavior of generating new harmony in this paper, that is, select two harmoniesx⟶i,x⟶j in the HM and generate a new harmony as follows:(24)x⟶new=x⟶i+2Rand⟶−0.5x⟶j−x⟶i.In (24), Rand⟶ is a random vector, and each value of it is between 0 and 1. The selection of the harmonies x⟶i,x⟶j is based on the individual fitness value fi and the cumulative probability pi, which are calculated as follows:(25)pi=∑k=1ifk/∑k=1nfk.If a generated random number is greater thanpi−1 and less than pi, the i-th harmony is selected.PAR and bw play an important role in searching for high-quality harmony. To carry out an effective search in the whole solution space and improve the efficiency of the algorithm, PAR and bw are set to change adaptively as follows [53]:(26)PAR=PARmin+PARmax−PARminNiter⋅t,(27)bw=bwmax⋅etNiterlogbwminbwmax.The damping enhancement optimization problem of the inerter system is a constrained optimization problem, and the modified harmony search algorithm is essentially an unconstrained search method. Therefore, the penalty function method is adopted to convert the optimization problem of the inerter system into an unconstrained optimization problem, so that the modified harmony search algorithm can be applied. After introducing the penalty function, the expression of the optimization problem of the inerter system is as follows:(28)minimizeμ,ξ,κ∈0,1fμ,ξ,κ=ξ+h⋅γμ,ξ,κ−γt,where h is the penalty factor that can eliminate individuals who do not meet the constraint. However, a proper penalty factor is not easy to be found to obtain a satisfactory solution. Given this, an adaptive penalty factor is adopted in this paper, and its expression is(29)h=h0⋅δts.In (29), δ and h0 are constants, and their value ranges are suggested to be δ∈1,1.1 and h0∈0.01,0.1; ts is the number of successful iterations, that is, the serial number of the iterative step when the new harmony is better than all the harmonies in the HM.Based on the above descriptions, the Python programming language is used to develop an object-oriented computer program termed Modified Harmony Search Algorithm for Optimization of Inerter System (MHSAOIS) to solve the parameter optimization problem of the inerter system.


### 3.3. Benchmark Test of the Modified Algorithm

Benchmark functions in [50, 54] are selected to make a preliminary test of the performance of the modified harmony search algorithm. The benchmark functions are listed in Table 3, where Dim denotes the dimension of the function and fmin denotes the analytical minimum value of the function.Table 3
The three benchmark functions.
Function categoryFunction nameFunctionDimRangefminUnimodal benchmark functionSphere functionf1x=∑i=1nxi230[−100, 100]0Generalized Rosenbrock’s functionf5x=∑i=1n−1100xi+1−xi22+xi−1230[−30, 30]0Multimodal benchmark functionGeneralized Rastrigin’s functionf9x=∑i=1nxi2−10cos2πxi+1030[−5.12, 5.12]0Constrained minimization problemminfx=x1−22+x2−12s.t.x1−2x2+1=0−x12/4−x22+1≥02[0, 1]1.3935Each benchmark problem is solved 100 times independently by the original, improved [45], and modified harmony search algorithms, and the average values of the benchmark functions are listed in Table 4. It can be found that the values of the benchmark functions solved by the modified and improved algorithms are obviously closer to the analytical solutions than that of the original algorithm. That is, the efficiency of the modified harmony search algorithm is testified.Table 4
The average solution of benchmark functions.
Function categoryFunctionAverage value of objective functionOriginal harmony search algorithmImproved harmony search algorithm [45]Modified harmony search algorithmUnimodal benchmark functionf15.0751970.0001540.000185f51.7968110.0000170.078591Multimodal benchmark functionf93.4409180.0116260.032503Constrained minimization problem1.3938971.3894281.393615


## 3.1. Original Algorithm

The harmony search algorithm is a novel intelligent optimization algorithm proposed by Z. W. Geem et al. in 2001. The harmony search algorithm simulates the procedure of music creation. The musicians rely on their memory to understand the pitch of the instruments in the band. Repeated adjustments finally made the band play a beautiful harmony state. The analogy is shown in Table2 [50].Table 2
Comparison of optimization and music performance.
Analogy diagramAnalogy elementOptimization processRealization processBest stateGlobal bestExcellent harmonyBe evaluated by…Objective functionAesthetic evaluationEvaluate with…Design variable valuePitch of the instrumentProcess unitEvery iterationEvery practiceThe procedure of the original harmony search algorithm (Figure3) can be summarized as follows.Figure 3
Flowchart of harmony search algorithm.Step 1.
Establish the optimization problem (take solving the minimization problem as an example).(20)minfX⟶,where fX⟶ is the objective function to be optimized; X⟶=x1,x2,⋯,xi⋯,xn is the solution vector and xii=1,2,⋯,n is the decision variable that constitutes the solution vector; n is the dimension of solution space.Step 2.
InitializeHM. Randomly generate HMS harmonies, and put them into HM; the mathematical form of HM is as follows:(21)HM=X⟶1X⟶2⋮X⟶HMS=x11x12⋮x1HMSx21x22⋮x2HMS⋯⋯⋯⋯xn1xn2⋮xnHMS.Step 3.
Generate a new harmony. When a random number is greater than the retention probabilityHMCR, the new harmony is randomly generated in the feasible region. Otherwise, a new harmony is generated as follows:(22)x⟶newi=HMR,i,where R is a random number between 1 and HMS.
For every tone in the new harmony, if a randomly generated random number is less thanPAR, the tone is randomly generated near the original tone:(23)x⟶new=x⟶new+bw⋅2Rand⟶−0.5,where bw is bandwidth for pitch adjusting; Rand⟶ is a vector that contains randomly generated values between 0 and 1.Step 4.
Update theHM. If the new harmony vector is better than the worst vector in the HM, then the new vector replaces the worst harmony vector; otherwise, no operation is performed.Step 5.
Check whether the stop condition is reached; if the stop condition is reached, the algorithm terminates; otherwise, the algorithm repeats steps 3-4.


## 3.2. Modification of Harmony Search Algorithm

With the widespread application of the harmony search algorithm, scholars continue to develop some modifications over the original harmony search algorithm. For example, Omran and Mahdavi [51] improved the search mechanism of harmony and proposed a global harmony search. Fesangharya et al. [52] proposed a hybrid harmony search algorithm and so on.The original harmony search algorithm is not efficient for solving constrained optimization problems. Given this, a simple modification is made to replace the classic behavior of generating new harmony in this paper, that is, select two harmoniesx⟶i,x⟶j in the HM and generate a new harmony as follows:(24)x⟶new=x⟶i+2Rand⟶−0.5x⟶j−x⟶i.In (24), Rand⟶ is a random vector, and each value of it is between 0 and 1. The selection of the harmonies x⟶i,x⟶j is based on the individual fitness value fi and the cumulative probability pi, which are calculated as follows:(25)pi=∑k=1ifk/∑k=1nfk.If a generated random number is greater thanpi−1 and less than pi, the i-th harmony is selected.PAR and bw play an important role in searching for high-quality harmony. To carry out an effective search in the whole solution space and improve the efficiency of the algorithm, PAR and bw are set to change adaptively as follows [53]:(26)PAR=PARmin+PARmax−PARminNiter⋅t,(27)bw=bwmax⋅etNiterlogbwminbwmax.The damping enhancement optimization problem of the inerter system is a constrained optimization problem, and the modified harmony search algorithm is essentially an unconstrained search method. Therefore, the penalty function method is adopted to convert the optimization problem of the inerter system into an unconstrained optimization problem, so that the modified harmony search algorithm can be applied. After introducing the penalty function, the expression of the optimization problem of the inerter system is as follows:(28)minimizeμ,ξ,κ∈0,1fμ,ξ,κ=ξ+h⋅γμ,ξ,κ−γt,where h is the penalty factor that can eliminate individuals who do not meet the constraint. However, a proper penalty factor is not easy to be found to obtain a satisfactory solution. Given this, an adaptive penalty factor is adopted in this paper, and its expression is(29)h=h0⋅δts.In (29), δ and h0 are constants, and their value ranges are suggested to be δ∈1,1.1 and h0∈0.01,0.1; ts is the number of successful iterations, that is, the serial number of the iterative step when the new harmony is better than all the harmonies in the HM.Based on the above descriptions, the Python programming language is used to develop an object-oriented computer program termed Modified Harmony Search Algorithm for Optimization of Inerter System (MHSAOIS) to solve the parameter optimization problem of the inerter system.


## 3.3. Benchmark Test of the Modified Algorithm

Benchmark functions in [50, 54] are selected to make a preliminary test of the performance of the modified harmony search algorithm. The benchmark functions are listed in Table 3, where Dim denotes the dimension of the function and fmin denotes the analytical minimum value of the function.Table 3
The three benchmark functions.
Function categoryFunction nameFunctionDimRangefminUnimodal benchmark functionSphere functionf1x=∑i=1nxi230[−100, 100]0Generalized Rosenbrock’s functionf5x=∑i=1n−1100xi+1−xi22+xi−1230[−30, 30]0Multimodal benchmark functionGeneralized Rastrigin’s functionf9x=∑i=1nxi2−10cos2πxi+1030[−5.12, 5.12]0Constrained minimization problemminfx=x1−22+x2−12s.t.x1−2x2+1=0−x12/4−x22+1≥02[0, 1]1.3935Each benchmark problem is solved 100 times independently by the original, improved [45], and modified harmony search algorithms, and the average values of the benchmark functions are listed in Table 4. It can be found that the values of the benchmark functions solved by the modified and improved algorithms are obviously closer to the analytical solutions than that of the original algorithm. That is, the efficiency of the modified harmony search algorithm is testified.Table 4
The average solution of benchmark functions.
Function categoryFunctionAverage value of objective functionOriginal harmony search algorithmImproved harmony search algorithm [45]Modified harmony search algorithmUnimodal benchmark functionf15.0751970.0001540.000185f51.7968110.0000170.078591Multimodal benchmark functionf93.4409180.0116260.032503Constrained minimization problem1.3938971.3894281.393615


## 4. Case Study

### 4.1. Optimal Design Cases

An SDOF structure is selected and its key parameters are as follows: massm=1900ton, stiffness k=280000kN/m, natural frequency f=1.93Hz, and period of free vibration T=1/f=0.52s. It is planned to equip an inerter system to control its seismic response. Considering the difference in inherent damping ratio ζ and target response mitigation ratio γt, 8 design cases are set in Table 5, and the MHSAOIS program is used to solve the optimal design of the key parameters of the inerter system. The parameters of the algorithm are set as HMS=50,HMCR=0.9,PARmin=0.3,PARmax=0.9,bwmin=0.00001,andbwmax=0.1.Table 5
Design optimization parameters of the structure with the inerter system based on the modified harmony search algorithm.
Optimized parametersCaseζγtμξκγα10.020.700.01130.00040.01130.69817.2523620.020.600.02820.00160.02890.59814.7364930.020.500.06800.00610.07270.50053.1318940.020.450.10580.01230.11760.45002.5316350.050.700.06550.00530.06920.70003.133360.050.600.14370.01880.16480.59992.175070.050.500.28320.05810.38480.50001.6067280.050.450.38000.09810.59490.45001.41687Note. The inherent damping ratio ζ and the target response mitigation ratio γt are the performance indices.After optimization, the designed parameters (μ,ξ,andκ) and performance indices (γ and α) are also listed in Table 5. From Table 5, the actual stochastic response mitigation ratio γ based on the modified harmony search algorithm is consistent with the target response mitigation ratio γt. The correctness and rationality of the optimized parameters will be verified later.


### 4.2. Verification of Optimization Objective

To verify the correctness of the obtained parameters, a large number of parameter analyses are carried out for obtaining the values of DDEFs of the inerter system as the parameters change. The results are shown in Figure4.In Figure4, the horizontal and vertical coordinates are the inertance-mass ratio μ and the stiffness ratio κ, respectively, and the height is the DDEF calculated through (16). It can be found that each contour line in the figure is closed and each contour has a single peak value, which is the maximum value of DDEF. The key parameters obtained from the modified harmony search algorithm are marked in the contours, and they are found to be consistent with the peaks of the contours, which verifies that the designed parameters realize the maximization of damping enhancement.Figure 4
Contours of the damping deformation enhancement factorα of an SDOF structure with the inerter system (note: the black square in the figure represents the parameters optimized by the modified harmony search algorithm). (a) Case 3. (b) Case 4. (c) Case 7. (d) Case 5.
(a)(b)(c)(d)


### 4.3. Efficiency Verification of Modified Harmony Search Algorithm

To investigate the performance and efficiency of the modified harmony search algorithm, the influences of the modified harmony generating strategy and the adaptive penalty weight strategy on the solution efficiency are investigated. Compared with the original harmony search algorithm (HCR=0.9,PAR=0.3,andbw=0.01) and the improved harmony search algorithm [45], the solution efficiency of the modified harmony generating strategy is studied. All design cases in Table 5 are solved 100 times by the original algorithm, the improved algorithm [45], and the modified algorithm, respectively. The maximum number of iterations of each algorithm is set as 100,000. Since the correctness of the optimized parameters in Table 5 is proved by Figure 4, the values of ξ are regarded as the criteria to judge if the solution is converged. That is, if the solved value of ξ is closed (relative error is less than 10-2) to the corresponding value in Table 5, the solution is regarded as converged. The average converged number of iterations and iteration time required for each case are listed in Table 6 (since the improved harmony search algorithm [45] did not converge in all solutions, the results are not listed). It can be seen from Table 6 that the number of iterations and consumed time required by the modified harmony search algorithm are reduced by more than 50% on average compared with the original harmony search algorithm. Among the design cases, the most obvious solution efficiency enhancement occurs in case 3, and the corresponding number of iterations for convergence and consumed time are reduced by about 90%. This means that the modified algorithm has a stronger solving ability. The variation trends of the nominal damping ratio ξ of the inerter system and the value of the constraint function of each case as the number of iteration increases are shown in Figures 5–12. It can be seen from Figures 5–12 that all the designed cases are converged under the modified harmony search algorithm, but there are nonconvergent cases under the original harmony search algorithm, such as case 1, case 2, case 3, and case 5. As for the improved harmony search algorithm [45], although its performance in the benchmark test (Section 3.3) is the best for unconstrained problems, it failed to converge in all cases here. The variation trends of these nonconvergent cases show that the original harmony search algorithm and the improved harmony search algorithm [45] are easy to fall into a local optimal solution and no longer change during the iteration process. In addition, although the values of the objective functions for case 4, case 6, case 7, and case 8 for both the original algorithm and the improved algorithm [45] are less than that of the modified algorithm, it can be seen that the values of the constraint functions are about 3 orders of magnitude smaller, which means that the constraint conditions are more satisfied. These results prove the advantages of the modified harmony search algorithm over the original one.Table 6
Average convergent iterations of modified and original harmony search algorithms.
CaseOriginal harmony search algorithmModified harmony search algorithmRate of increase (%)Number of iterations at the convergenceNumber of the objective function evaluationsConsumed Time (s)Number of iterations at the convergenceNumber of the objective function evaluationsConsumed time (s)Number of iterations at the convergenceConsumed time1>100000301701>7.2129927924323.04>70.07>57.862>100000304551>7.2813700432511.36>86.30>81.833>100000306651>7.327922259170.81>92.08>88.914521001636013.8119260623311.9063.0350.005>100000304701>7.649097293920.86>90.90>88.74627397893422.0116789558681.6438.7218.477394311303442.8620565682461.9947.8530.398425821448973.1222371750142.2847.4626.80Note. The symbol “>” indicates that, during the 100 solutions of the case, there are 1 or more nonconvergent solutions. The computer for calculation is a common office computer with an Intel Core i7-6700 CPU and an 8 GB RAM. Since the improved harmony search algorithm [45] did not converge in all solutions, the results are not listed here.Figure 5
The variation trends of key indices during the optimization process for case 1. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 6
The variation trends of key indices during the optimization process for case 2. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 7
The variation trends of key indices during the optimization process for case 3. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 8
The variation trends of key indices during the optimization process for case 4. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 9
The variation trends of key indices during the optimization process for case 5. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 10
The variation trends of key indices during the optimization process for case 6. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 11
The variation trends of key indices during the optimization process for case 7. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 12
The variation trends of key indices during the optimization process for case 8. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Design case 3 is selected to conduct detailed statistical analysis. The problem is solved 100 times independently by the original and modified harmony search algorithms. The value of nominal damping ratio (i.e., the value of the objective function) for design case 3 in Table5 is used as the criterion for judging convergence, and the tolerances of the relative errors are set as 20%, 10%, 5%, and 2%, respectively, to demonstrate the convergency of the original and modified algorithms. The results are shown in Figure 13 and Table 7. It can be found that the average number of iterations for convergence of the modified algorithm is significantly smaller than that of the original algorithms. As the tolerance decreases, the convergence probability of the original algorithm becomes lower and lower, while the modified algorithm can easily converge after fewer iterations. This proves that the modified harmony search algorithm has a more powerful ability for solving optimization problems.Figure 13
The convergence efficiency of original and modified harmony search algorithms. (a) Relative converge tolerance 20%, (b) relative converge tolerance 10%, (c) relative converge tolerance 5%, and (d) relative converge tolerance 2%.
(a)(b)(c)(d)Table 7
Mean and standard deviation of the count of iterations at convergence for different algorithms.
Relative converge tolerance (%)Modified harmony search algorithmOriginal harmony search algorithmMeanStandard deviationMeanStandard deviation203748.881362.16>48727.18>31498.65104924.593695.10>72731.49>32507.0455494.833323.34>82034.17>26944.2026151.284454.48>93258.13>18715.29Note. The symbol “>” indicates that, during the 100 solutions, there are 1 or more nonconvergent solutions.The next step is to verify the specific advantages of adopting an adaptive penalty weight strategy (equation (28)). The fixed penalty weight strategy and different forms of adaptive penalty weight functions are tested, and the results are shown in Table 8. It can be seen that the fixed penalty weight strategy requires more iterations than the adaptive penalty weight strategy. Furthermore, there are nonconvergent cases under the fixed penalty weight strategy. Meanwhile, it can be seen that the adopted expression of adaptive penalty weight h (h=h0⋅δts, i.e., (28)) can give the best solution effect compared with other forms of penalty weight functions because h=h0⋅δts can guarantee the convergency but the other penalty weight functions cannot. Therefore, the currently adopted adaptive penalty weight strategy is more powerful in solving the optimization problems of inerter systems.Table 8
Approximate number of iterations for convergency under different penalty weight strategies.
CaseFixed penalty weight strategyAdaptive penalty weight strategyh=0.01h=0.1h=1h=10h=h0⋅δtsh=h0t100h=h0+t/100012100052000NN30000100000N2N2500055000N14000100000N3N1200034000N800080000N4NN32500N1900045000N5NNN13000350004600085006NN90001800031000900090007NN13500440001700023000N8N1100047000N900055000NNote. N in the table means that the solution does not converge after 100000 iterations.


## 4.1. Optimal Design Cases

An SDOF structure is selected and its key parameters are as follows: massm=1900ton, stiffness k=280000kN/m, natural frequency f=1.93Hz, and period of free vibration T=1/f=0.52s. It is planned to equip an inerter system to control its seismic response. Considering the difference in inherent damping ratio ζ and target response mitigation ratio γt, 8 design cases are set in Table 5, and the MHSAOIS program is used to solve the optimal design of the key parameters of the inerter system. The parameters of the algorithm are set as HMS=50,HMCR=0.9,PARmin=0.3,PARmax=0.9,bwmin=0.00001,andbwmax=0.1.Table 5
Design optimization parameters of the structure with the inerter system based on the modified harmony search algorithm.
Optimized parametersCaseζγtμξκγα10.020.700.01130.00040.01130.69817.2523620.020.600.02820.00160.02890.59814.7364930.020.500.06800.00610.07270.50053.1318940.020.450.10580.01230.11760.45002.5316350.050.700.06550.00530.06920.70003.133360.050.600.14370.01880.16480.59992.175070.050.500.28320.05810.38480.50001.6067280.050.450.38000.09810.59490.45001.41687Note. The inherent damping ratio ζ and the target response mitigation ratio γt are the performance indices.After optimization, the designed parameters (μ,ξ,andκ) and performance indices (γ and α) are also listed in Table 5. From Table 5, the actual stochastic response mitigation ratio γ based on the modified harmony search algorithm is consistent with the target response mitigation ratio γt. The correctness and rationality of the optimized parameters will be verified later.


## 4.2. Verification of Optimization Objective

To verify the correctness of the obtained parameters, a large number of parameter analyses are carried out for obtaining the values of DDEFs of the inerter system as the parameters change. The results are shown in Figure4.In Figure4, the horizontal and vertical coordinates are the inertance-mass ratio μ and the stiffness ratio κ, respectively, and the height is the DDEF calculated through (16). It can be found that each contour line in the figure is closed and each contour has a single peak value, which is the maximum value of DDEF. The key parameters obtained from the modified harmony search algorithm are marked in the contours, and they are found to be consistent with the peaks of the contours, which verifies that the designed parameters realize the maximization of damping enhancement.Figure 4
Contours of the damping deformation enhancement factorα of an SDOF structure with the inerter system (note: the black square in the figure represents the parameters optimized by the modified harmony search algorithm). (a) Case 3. (b) Case 4. (c) Case 7. (d) Case 5.
(a)(b)(c)(d)


## 4.3. Efficiency Verification of Modified Harmony Search Algorithm

To investigate the performance and efficiency of the modified harmony search algorithm, the influences of the modified harmony generating strategy and the adaptive penalty weight strategy on the solution efficiency are investigated. Compared with the original harmony search algorithm (HCR=0.9,PAR=0.3,andbw=0.01) and the improved harmony search algorithm [45], the solution efficiency of the modified harmony generating strategy is studied. All design cases in Table 5 are solved 100 times by the original algorithm, the improved algorithm [45], and the modified algorithm, respectively. The maximum number of iterations of each algorithm is set as 100,000. Since the correctness of the optimized parameters in Table 5 is proved by Figure 4, the values of ξ are regarded as the criteria to judge if the solution is converged. That is, if the solved value of ξ is closed (relative error is less than 10-2) to the corresponding value in Table 5, the solution is regarded as converged. The average converged number of iterations and iteration time required for each case are listed in Table 6 (since the improved harmony search algorithm [45] did not converge in all solutions, the results are not listed). It can be seen from Table 6 that the number of iterations and consumed time required by the modified harmony search algorithm are reduced by more than 50% on average compared with the original harmony search algorithm. Among the design cases, the most obvious solution efficiency enhancement occurs in case 3, and the corresponding number of iterations for convergence and consumed time are reduced by about 90%. This means that the modified algorithm has a stronger solving ability. The variation trends of the nominal damping ratio ξ of the inerter system and the value of the constraint function of each case as the number of iteration increases are shown in Figures 5–12. It can be seen from Figures 5–12 that all the designed cases are converged under the modified harmony search algorithm, but there are nonconvergent cases under the original harmony search algorithm, such as case 1, case 2, case 3, and case 5. As for the improved harmony search algorithm [45], although its performance in the benchmark test (Section 3.3) is the best for unconstrained problems, it failed to converge in all cases here. The variation trends of these nonconvergent cases show that the original harmony search algorithm and the improved harmony search algorithm [45] are easy to fall into a local optimal solution and no longer change during the iteration process. In addition, although the values of the objective functions for case 4, case 6, case 7, and case 8 for both the original algorithm and the improved algorithm [45] are less than that of the modified algorithm, it can be seen that the values of the constraint functions are about 3 orders of magnitude smaller, which means that the constraint conditions are more satisfied. These results prove the advantages of the modified harmony search algorithm over the original one.Table 6
Average convergent iterations of modified and original harmony search algorithms.
CaseOriginal harmony search algorithmModified harmony search algorithmRate of increase (%)Number of iterations at the convergenceNumber of the objective function evaluationsConsumed Time (s)Number of iterations at the convergenceNumber of the objective function evaluationsConsumed time (s)Number of iterations at the convergenceConsumed time1>100000301701>7.2129927924323.04>70.07>57.862>100000304551>7.2813700432511.36>86.30>81.833>100000306651>7.327922259170.81>92.08>88.914521001636013.8119260623311.9063.0350.005>100000304701>7.649097293920.86>90.90>88.74627397893422.0116789558681.6438.7218.477394311303442.8620565682461.9947.8530.398425821448973.1222371750142.2847.4626.80Note. The symbol “>” indicates that, during the 100 solutions of the case, there are 1 or more nonconvergent solutions. The computer for calculation is a common office computer with an Intel Core i7-6700 CPU and an 8 GB RAM. Since the improved harmony search algorithm [45] did not converge in all solutions, the results are not listed here.Figure 5
The variation trends of key indices during the optimization process for case 1. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 6
The variation trends of key indices during the optimization process for case 2. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 7
The variation trends of key indices during the optimization process for case 3. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 8
The variation trends of key indices during the optimization process for case 4. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 9
The variation trends of key indices during the optimization process for case 5. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 10
The variation trends of key indices during the optimization process for case 6. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 11
The variation trends of key indices during the optimization process for case 7. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Figure 12
The variation trends of key indices during the optimization process for case 8. (a) The nominal damping ratioξ. (b) The constraint function γμ,ξ,κ−γt.
(a)(b)Design case 3 is selected to conduct detailed statistical analysis. The problem is solved 100 times independently by the original and modified harmony search algorithms. The value of nominal damping ratio (i.e., the value of the objective function) for design case 3 in Table5 is used as the criterion for judging convergence, and the tolerances of the relative errors are set as 20%, 10%, 5%, and 2%, respectively, to demonstrate the convergency of the original and modified algorithms. The results are shown in Figure 13 and Table 7. It can be found that the average number of iterations for convergence of the modified algorithm is significantly smaller than that of the original algorithms. As the tolerance decreases, the convergence probability of the original algorithm becomes lower and lower, while the modified algorithm can easily converge after fewer iterations. This proves that the modified harmony search algorithm has a more powerful ability for solving optimization problems.Figure 13
The convergence efficiency of original and modified harmony search algorithms. (a) Relative converge tolerance 20%, (b) relative converge tolerance 10%, (c) relative converge tolerance 5%, and (d) relative converge tolerance 2%.
(a)(b)(c)(d)Table 7
Mean and standard deviation of the count of iterations at convergence for different algorithms.
Relative converge tolerance (%)Modified harmony search algorithmOriginal harmony search algorithmMeanStandard deviationMeanStandard deviation203748.881362.16>48727.18>31498.65104924.593695.10>72731.49>32507.0455494.833323.34>82034.17>26944.2026151.284454.48>93258.13>18715.29Note. The symbol “>” indicates that, during the 100 solutions, there are 1 or more nonconvergent solutions.The next step is to verify the specific advantages of adopting an adaptive penalty weight strategy (equation (28)). The fixed penalty weight strategy and different forms of adaptive penalty weight functions are tested, and the results are shown in Table 8. It can be seen that the fixed penalty weight strategy requires more iterations than the adaptive penalty weight strategy. Furthermore, there are nonconvergent cases under the fixed penalty weight strategy. Meanwhile, it can be seen that the adopted expression of adaptive penalty weight h (h=h0⋅δts, i.e., (28)) can give the best solution effect compared with other forms of penalty weight functions because h=h0⋅δts can guarantee the convergency but the other penalty weight functions cannot. Therefore, the currently adopted adaptive penalty weight strategy is more powerful in solving the optimization problems of inerter systems.Table 8
Approximate number of iterations for convergency under different penalty weight strategies.
CaseFixed penalty weight strategyAdaptive penalty weight strategyh=0.01h=0.1h=1h=10h=h0⋅δtsh=h0t100h=h0+t/100012100052000NN30000100000N2N2500055000N14000100000N3N1200034000N800080000N4NN32500N1900045000N5NNN13000350004600085006NN90001800031000900090007NN13500440001700023000N8N1100047000N900055000NNote. N in the table means that the solution does not converge after 100000 iterations.


## 5. Structural Responses in the Frequency Domain and Time Domain

Substituting the parameters in Table5 into the analytical expression of the displacement response frequency response function ((5)), the frequency-domain displacement transfer function curves are shown in Figure 14. It can be seen that the parameters obtained can effectively control the seismic response of the SDOF structure.Figure 14
Frequency-domain displacement transfer function curves of SDOF structure with the inerter system.To visually demonstrate the damping enhancement effect, the design parameters of case 2 are chosen as example, and the structure with inerter system and the original structure are analyzed under dynamic excitation. Seismic excitations with different spectral characteristics (white noise excitation, artificial seismic excitation, and natural seismic excitation) are considered, and the displacement responses are shown in Figure15. It can be found that the displacement response of the structure with the inerter system is smaller than that of the original structure under different seismic excitations, and the root-mean-square response mitigation ratio is similar to the target one (γt=0.6), which proves the effectiveness of the proposed method and the rationality of design parameters.Figure 15
Displacement response curves of structures with inerter systems under different seismic excitations. (a) White noise excitation. (b) Artificial seismic excitation. (c) Natural seismic excitation.
(a)(b)(c)In addition, a comparison between the inerter system and the traditional viscous damper is also conducted. An inerter system and a viscous damper (they have the same viscous damping coefficient) are installed in the structure simultaneously. The responses of dampers are shown in Figure16. The comparison of the hysteresis curves shows that the deformation and damping force of the damper in the inerter system is greater than that of the traditional viscous damper. As for the enclosed area of the hysteresis curve, that is, the dissipated energy, the damper in the inerter system is also significantly larger than the traditional viscous damper.Figure 16
Hysteresis curves of a viscous damper within an inerter system and traditional viscous damper (the damping coefficients are identical) under seismic excitation.


## 6. Conclusion

In this paper, a modified harmony search algorithm is proposed to design the inerter system for the maximization of the damping enhancement effect. It is verified that the algorithm is effective and easy to implement, and the following conclusions are drawn:(1)
The key parameters of the inerter system can be effectively obtained according to the target demands by the modified harmony search algorithm. No assumptions or engineering experiences are needed during the optimal design.(2)
The design problem of the maximum damping enhancement of the inerter system can be described as a mathematical optimization problem and the constraint condition can be replaced by a penalty function.(3)
With respect to the original harmony search algorithm, the convergence performance of the modified harmony search algorithm is significantly improved. The number of iterations at the convergence of the modified harmony algorithm can be reduced by 38%∼92%, and the consumed time can be reduced by 18% ∼ 89%.(4)
This paper only discusses the single-degree-of-freedom structure with the inerter system. The optimization of the multi-degree-of-freedom structure with inerter systems by the modified harmony search algorithm will be further investigated in detail in the future.



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