Abstract
A parameter optimal design method for a tank with an inerter system is proposed in this study based on the requirements of tank vibration control to improve the effectiveness and efficiency of vibration control. Moreover, a response indicator and a cost control indicator are selected based on the control targets for liquid storage tanks for simultaneously minimizing the dynamic response and controlling costs. These indicators are reformulated through a random vibration analysis under virtual excitation. The problem is then transformed from a multiobjective optimization problem to a single-objective nonlinear problem using the -constraint method, which is consistent with the demand-based method. White noise excitation can be used to design the tank with the inerter system under seismic excitation to simplify the calculation. Subsequently, a MATLAB-based calculation program is compiled, and several optimization cases are examined under different excitation conditions. The effectiveness of the demand-based method is proven through a time history analysis. The results show that specific vibration control requirements can be met at the lowest cost with a simultaneous reduction in base shears and overturning base moments.
1. Introduction
As a type of significant facility for public safety, liquid storage tanks play an important role in water supply systems, chemical industries, nuclear plants, and other fields. Storage tanks are designed in a wide range of capacity sizes ranging from small to large [1]. The vulnerability of tanks during earthquakes has been demonstrated in previous studies [2–6]. Based on the observations of several other researchers [7–10], the failure modes of tank damage can be divided into several types, such as buckling of tank walls, failure of link facilities, spillover of stored liquid, and uplifting of anchorage systems. Strengthening, isolation, and hybrid methods have been promoted by several researchers to ensure that storage tanks are not damaged during earthquakes [11–18].
The conventional technique of strengthening tanks to safeguard them against damage caused by earthquakes is to increase the size of different members to resist forces to a higher extent. Different from the strengthening methods, tanks are isolated from the period of the dominant component of ground motion, and the stiffness of the structures is adjusted using the isolation technique, which employs isolators between the base and the foundation of the tanks to increase their fundamental natural period. Shaking table tests have also been conducted to determine the effectiveness of the isolation method and demonstrate the reduction in the hydrodynamic pressure of these tanks and their seismic response [19, 20]. However, this type of method can lead to an increase in the sloshing response [8, 21]. The setting of obstacles, such as baffles, has been widely used to hinder liquid sloshing inside tanks and control structural damping. With baffles [22], the damping ratio of the sloshing mode increases while the liquid sloshing response decreases. The seismic energy is absorbed, and the resonant frequency of the sloshing mode increases because of the obstacles set in oscillating liquid. For the method involving baffles, there are some studies showing that baffles may cause an increase in the value of the base shear, and the ability of baffles in reducing sloshing effects in liquid storage tanks with lower aspect ratios is also under question [23]. Additionally, the provision for setting baffles is not available for some tanks with specific uses.
A hybrid control method that uses a viscous inerter system (VIS) and an isolator was proposed to control the earthquake response of storage tanks and simultaneously adjust the structure stiffness, damping ratio, and mass [21]. As a type of inerter system, the enhancement effect of mass and damping can be produced [8]. Therefore, the mass, damping ratio, and stiffness of the structure can be adjusted according to the structural requirement. The seismic response of tanks, such as their hydrodynamic pressure and sloshing response, can be efficiently reduced using the hybrid control method. A typical VIS is divided into two parts, namely, an inerter element and a damping element. The VIS used in this study is a type of parallel system, which consists of an inerter enhancement element and a viscosity element in parallel. The inerter element is a two-terminal mass element, which is different from the existing mass elements. Relative acceleration is allowed and constrained in the two-terminal inerter element model. The VIS can significantly dissipate seismic energy through the viscous element with the help of the inerter element. Luo et al. [21] proved the validity of the hybrid control method using theory and numerical calculation. The sloshing heights of the liquid, base shears, and overturning base moments of the tanks are decreased by applying the inerter system and an isolator. However, the methods of setting the design parameters of the inerter system, which achieves effective vibration control at the minimum cost under a specific controlling target, have not yet been proposed. The structural response of the tank is a complex problem under multiparameter settings, and structural control capability is continually adjusted as the parameters change.
Selecting the objective function of the design is critical in optimizing the design of the storage tank with the inerter system and determining the algorithm for calculating the function under multiple parameter changes. The fixed-point method proposed by Den Hartog [24] is applied to optimize the design of a tuned mass damper and can be extended for use in the VIS. Ikago et al. [8] adopted the fixed-point method for a tuned viscous mass damper. The disadvantage of the fixed method is that it neglects the inherent damping of the primary structure, and a limitation is introduced because an empirically equivalent mass ratio is used. Pan et al. [25] developed a demand-based optimal design for a single-degree-of-freedom structure considering the inherent damping ratio of the primary structure.
The dynamic response and force of the inerter system are reformulated according to the theory of random vibration to develop a suitable algorithm considering the storage tank’s performance demand and synthetically controlling expense. Several variable parameters are changed in the investigation to optimize the selected objective indicators. Moreover, the optimization of the tank with the inerter system was converted to a multiobjective problem with the objective of synchronously minimizing the dynamic response and controlling cost. Considering two objective indicators, the -constraint approach is applied to convert the optimization problem to a demand-based (performance-based) design method with a demand level based on the limit values of the response of the sloshing height and the displacement of the isolation layer. A calculation program was compiled using the proposed demand-based design method to calculate the parameter optimization design. Finally, several optimization cases were considered under different excitation conditions, and a time history analysis was performed to exemplify the validity and accuracy of the proposed optimization method.
2. Theoretical Analysis of the Storage Tank Controlled with the Inerter System
2.1. Simplified Theoretical Model of the Storage Tank
Several theoretical models for analyzing the seismic response of storage tanks have been proposed to simplify the tank model for theoretical analysis and numerical calculation. Housner [27] proposed a type of lumped-mass model for theoretical analysis by analogizing the theoretical calculation model of shear force. The hypotheses were that the wall of the storage tank is rigid, the fluid displacements are small, fluid is incompressible, and the liquid in the storage tank can be divided into two components (i.e., convective and impulsive). Under the force of an earthquake, the impulsive component synchronously moves with the tank wall, whereas the convective component exhibits a sloshing motion. The researchers performed a shaking table experiment and proposed a more accurate consideration of higher-order vibration modes to verify the correctness of this model. Referring to a previously proposed model [28], a more accurate calculation model is selected in this study considering higher-sloshing modes to evaluate the seismic response of storage tanks.
Considering the first three sloshing mode components, the convective component in this model is divided into these three components. Its sloshing motion can be expressed as a linear combination of these modes. A rigid cylindrical tank is regarded as a four-lumped-mass model. Figure 1 shows the diagram of a tank controlled with the VIS and isolators. represents the diameter of the rigid tank, while represents the height of the liquid in the tank. , , , and denote the mass, stiffness, damping coefficient, and height of the th sloshing mode, respectively. and denote the liquid density and the tank radius, respectively. , , , and can be calculated by referring to the study of Luo et al. [21].
2.2. Simplified Model of Rubber Bearing
A rubber bearing (RB), which is set in the hybrid device, consists of a spring and a dashpot connected in parallel. The output force, , provided by the RB can be expressed as follows:where denotes the stiffness of the spring, while represents the damping coefficient of the dashpot. and denote the relative base displacement and relative base velocity, respectively.
2.3. Simplified Model of the VIS
The VIS used in this study is a type of inerter system comprising the following two parts: inerter element and viscous damping element.
Figure 2 shows the inerter enhancement element, which is a two-terminal element in which relative displacement is allowed and constrained. Different from the existing mass model, the two-terminal element allows for relative acceleration within itself. More energy, which is efficiently dissipated by the viscous element, can be dissipated with the help of the amplified equivalent mass [8]. Therefore, according to the principle of Newtonian mechanics, the output force can be expressed as the vector product of the mass and the relative acceleration of the element.
As mentioned earlier, the inerter element is connected with the viscous damping element in parallel. The output force can be expressed using the following equation:where and denote the equivalent damping coefficient and the equivalent amplified mass of the VIS, respectively. and denote the velocity and the acceleration of the VIS, respectively.
2.4. Analytical Model
The equation of motion for the tank controlled with the VIS and the isolators under the ground motion excitation, , can be written as follows:where , , and are the mass, stiffness, and damping matrices of the tank, respectively; and are the additional mass and damping matrices provided by the VIS, respectively; , , and are the vectors of displacement, velocity, and acceleration relative to the ground, respectively; and is the influence coefficient vector. The matrices and vectors used in (3) are expressed as follows:
A solution for the stochastic vibration response was obtained herein, based on the complex mode and pseudoexcitation method [29]. The equation of motion (i.e., (3)) is transformed into the equation of state of space as follows:
The state space equation can be expressed as follows, considering the case of intense harmonic motion :
Equation (5) can be expressed as follows to evaluate its eigenvalues and eigenvectors:
The eigenvalue matrix and the eigenvector matrix are obtained by solving (7) as follows:where denotes the complex eigenvalue matrix and denotes the complex eigenvector matrix.
Considering and substituting this into (5), the equation is diagonalized and expressed as follows:where the parameters used in (9) are explained as follows:where , represents the ground input excitation power spectral density, and denotes the frequency of the external harmonic excitation. Differential equation (9) can be solved as follows:
According to the expression in (11), the response of the structure under virtual harmonic vibration excitation can be expressed as follows:where denotes the number of degrees of freedom.
The response of the power spectral density matrix can be expressed as follows:where , , and denote the conjugates of the matrices of displacement, velocity, and acceleration relative to the ground, respectively. , , and denote the transposes of the matrices of displacement, velocity, and acceleration relative to the ground, respectively. The output force of the VIS power spectral density function can be determined as follows:
3. Optimization of the Storage Tank Controlled with the Inerter System
3.1. Parametric Study of the Storage Tanks Controlled with the Inerter System
A parametric study was conducted to determine the responses by changing parameters , , and in the viable definition domain and investigate the reduction in the seismic response. The three parameters are defined as follows:where is the aspect ratio of the tank (i.e., the ratio of the height of the tank to its radius); is the additional damping ratio, which is the ratio of the damping coefficients of the VIS and RB; and denotes the ratio of the equivalent amplified masses of the VIS and the impulsive components. Two dimensionless response variation ratios (i.e., the sloshing height and displacement of the isolation layer ratio, , and the output force ratio, ) were defined as follows to make the results of the study universally applicable:
is the sloshing height and the displacement of the isolation layer ratio (i.e., the ratio of the sloshing height and the displacement of the tank with the inerter system and the uncontrolled tank) of the tank. is the output force ratio, which is the ratio of the output force of the tank to the inerter system and the uncontrolled tank (caused by inherent damping). and are the root mean square (RMS) values of the output force and the sloshing height responses, respectively, combined with the displacement of the isolation layer of the uncontrolled tank. and are the RMS values of the output force and the sloshing height responses, respectively, combined with the displacement of the isolation layer of the tank controlled with the VIS.
Considering the control of the sloshing height of the liquid in the tank, the sloshing height response, , refers to the sum of the sloshing heights of the first three modes of liquid vibration. As mentioned above, the sloshing motion of the convective component can be expressed as a linear superposition of the natural modes of the liquid vibration. The sloshing height, , can then be determined using the following equation:where denotes the sloshing height in the th mode; denotes the values for which the first derivative of the Bessel function of the first kind and first order is zero; and , , and are 1.841, 5.331, and 8.536, respectively. Therefore, in (16) can be determined as follows:where and denote the power spectral density of the acceleration response and the ground motion of the th mode, respectively. is the displacement of the power spectral density of the impulsive components. Substituting (14) into (16) and (19), these two objective indicators can be accurately calculated. A toolbox for integration in MATLAB is used for overcoming the considerable difficulties in performing the integration in (16) and (17). Indicators and can be obtained as the sum of the corresponding index values at different time points by adopting the iterative calculation method and defining the numerical calculation steps.
A three-dimensional (3D) space can be established to demonstrate the changes in the indicator values and enhance the visual representation of the calculation results considering the three undetermined parameters, , , and . The parameters in the 3D space are set as the orthogonal coordinate axes. Each point in the parametric space can be assigned a specified color based on the corresponding value of the response of the controlled tank to clearly visualize the results of the responses. Slices can be made at specific parametric planes to investigate the variation in the response with respect to the changes in the undetermined parameters (, , and ). The target benchmark is set as the storage tank for which the values of , , and are 0.5, 0.5, and 1.5, respectively.
A set of array-specific slices at constant , , and are used in the 3D contour map to observe and obtain an intuitive prediction of the variation in the objective function over the three parameters. The corresponding trends of indicators and that changed with the three parameters are also compared. Figure 3 shows the following adopted parametric ranges:
(a) Slices at the constant amplified mass ratio values (, 0.3, 0.5, 0.7, 0.9)
(b) Slices at the constant aspect ratio values (, 1.0, 1.5, 2.0, 2.5, 3.0)
(c) Slices at the constant damping ratio values (, 0.3, 0.5, 0.7, 0.9)
The ratio of the sloshing height to the displacement of the isolation layer, , of the viscous liquid was selected as the objective indicator in the 3D space considering the seismic response of the tank controlled using the hybrid method and the liquid stability. The slices shown in Figure 3 denote that only two parameters can be varied in each contour plot. Obtaining the influence of parameter variation on the seismic response of the controlled tank is possible by analyzing the change in the objective indicator for different slices.
The analysis of the global contour color implies that the response of the sloshing height of the liquid in the tank varies with the three variables. The minimum value of is observed at a fixed point or a boundary of the contour plot, denoting that the structural control objectives will be achieved under the conditions of maximum output force and mass force if the height-to-diameter ratio of the tank, , damping ratio, , and mass parameters of the tank are adjusted freely during the seismic tank design. In actual design, the tank geometry is always preset based on specific requirements and manufacturing capacity. Therefore, the optimal design based on the fixed aspect ratio is more valuable in practical view.(1)To analyze the slices at several selected aspect ratios, it can be considered that, under the conditions of a specific height-to-diameter ratio, there is no closed loop in the variation of parameters and . This implies that, without a preset parameter variation range, the indicator index of the controlled tank will be minimized when the damping ratio and the amplified mass ratio reach their maximum values. The minimum value of described above will be at the expense of high control cost of the VIS.(2)It can be clearly observed in the slices at several selected damping ratios that, for a specific damping ratio, the value of indicator changes with the two parameters. As the aspect ratio of the tank decreases, the tank with a low aspect ratio exhibits a better controlling effect of the sloshing height and the displacement of the isolation layer under the same damping ratio. The comparison of different slices shows that the variation in has a negligible effect on the numerical value of the control target. Therefore, the efficiency of optimizing is not satisfactory under the condition of a preset specific damping ratio. As shown in these slices, the optimization of the tank geometry is extremely efficient and operable.(3)The same trend as that for the slices at several selected damping ratios is observed in the analysis of the slices at several selected amplified mass ratios. A comparison of the slices at different amplified ratios shows that the value of is extremely sensitive to the changes in the aspect ratio. The value of for tanks with a low aspect ratio does not significantly change with the damping ratio. Conversely, the value becomes more sensitive to the changes in the damping ratio as the aspect ratio of the tank increases.
3.2. Optimization of the Hybrid Method Design
Preliminary conclusions on the control optimization can be obtained in the analysis of the 3D contour plot through a dynamic analysis of the controlled tank. According to the actual production capacity and device design requirements, the fluctuation in the sloshing height of the liquid in the storage tank will reach the minimum value at a high control cost within the range of the three variable parameters (, , and ). In the actual optimization design of the VIS, certain control requirements are necessary to lower the control costs [25]. As mentioned above, and are selected as the objective indicators. reflects the cost of structural control to a certain extent.
Furthermore, the target function for the optimization of the tank controlled with the VIS and isolations can be expressed as follows, considering the two indicators:where and denote the target functions for the sloshing height and the output force of the VIS. denotes the vector of the three selected parameters. is the range of the definition field. and are determined by selecting and , respectively. The target function (21) can be rewritten as follows, based on the optimization objectives:
In function (22), , , and represent the lower bounds of the ranges of values, while , , and represent the upper bounds of the ranges of values.
The -constraint method of the control theory is adopted to determine the optimal solution of the objective function. As discussed earlier, this optimization problem is a multiobjective problem (MOP) with two optimization goals. A few simplifications are made according to the control theory to solve this problem effectively. An objective function of the MOP is selected as the objective function of the SOP to transform the problem into an equivalent optimization problem with a single-objective function (i.e., a single-objective problem (SOP)). The other objective function is treated as an extra constraint condition for the SOP by setting bound values. Similarly, in performance-based structural design theory, the performance levels are always expressed as response limit values. Target function (22) can be modified and expressed as follows using this method:where denotes the upper limit of , which can be determined by design or functional requirements.
During the process of controlling the tank, a few supplementary constraints are required to be set to increase the controlling force provided by the inerter enhancement element and ensure that the inerter enhancement element of the VIS is in a dominant position in the seismic response during control. In other words, this constraint condition ensures that the damper is a type of inerter system that will not degenerate into a viscous damper. The constraint condition can be determined as follows based on this discussion:
Using this constraint condition, the inequality for factor ensures that the horizontal force provided by the rotating mass element dominates the VIS. In addition, this constraint satisfies the requirements of the type of inerter system used in the actual design optimization of the damping device.
A numerical calculation is conducted to confirm the existence of a unique optimization result. Based on the requirement of performance and its geometry design, the hypothesis in this benchmark model is that the aspect ratio is 1.5. For variation in only two parameters, the problem can be demonstrated in a plane (Figure 4). A series of response control demands and the constraint optimization problem are clearly demonstrated in the mixed contour plots of this plane. In addition, in each control demand, the lowest point on the contour of the damping ratio along any solid line represents a unique optimization result. Figure 4 shows a color-filled contour that represents the values of and solid lines in the plane that represent the values of . For every solid line, a unique lowest point in the color-filled contour of the output force represents a unique solution to the constraint optimization problem. In addition, the contour plot shows that the lower level of has less influence on the control target, and the mass coefficient is more sensitive to this variation. This finding illustrates the effectiveness of larger masses for sloshing height control.
(a) Combined contour plot of and at
(b) Combined contour plot of and at
The target benchmark is selected as the storage tank with , , and as 0.5, 0.5, and 1.5, respectively, to clearly illustrate the control principle of the VIS. Figure 5 shows the response density curves of the tank with the VIS and an uncontrolled tank under simple harmonic vibration excitation. The figure illustrates that peaks are observed in the output force of the VIS and the dynamic response of the sloshing height when the external excitation frequency is close to that of the tank. With the VIS attached to the tank, the effect of sloshing height control is significantly enhanced, accompanied by a substantial increase in the output force, which implies lower control targets at the expense of higher control costs. In other words, the objective of the optimal design of the tank is to find the balance of the control target and the control cost.
(a) Power spectral density of the sloshing height of the tank with the VIS and the uncontrolled tank
(b) PSD of output force of the tank with the VIS and the uncontrolled tank
3.3. Analysis of the Base Shears and the Overturning Base Moments
The base shears and the overturning base moments are considered and calculated according to the following expressions to complete the analysis of the dynamic response of the tank:
The power spectral density of the base shears and the overturning base moments can be expressed as follows, based on (25):
The power spectral density of the overturning base moments and the base shears in the standard model can be calculated using (26). Figure 6 shows the results.
(a) PSD of the base shears of the tank with the VIS and the uncontrolled tank
(b) PSD of the overturning base moments of the tank with the VIS and the uncontrolled tank
and are defined as the ratios of the base shears and the overturning base moments, respectively, to simplify the curves of the power spectral density and intuitively compare the benchmark and uncontrolled tanks. Figure 7 depicts the curves of and , implying that the base shears and the overturning base moments of the tank with the VIS simultaneously decrease during the short-period external excitation and increase during the long-period excitation. With the increase in the external excitation period, the dominant period can be close to the first period of the sloshing mode, which is an essential and larger modal participation factor of the base shears and the overturning base moments.
4. Design Example
Based on the abovementioned discussion and the establishment of the tank calculation model under the control of the VIS, MATLAB is adopted to write the optimization program for the demand-based calculation. In the optimization process, the target index is optimized using the nonlinear constrained optimization algorithm in a MATLAB toolbox. This algorithm is compiled to obtain a constrained minimum of a scalar function of several variables starting from an initial estimate. This is typically referred to as the constrained nonlinear optimization or nonlinear programming.
Based on the discussion of Luo et al. [21], the tank controlled using the hybrid method can be effectively controlled during earthquakes. The actual tank vibration control is frequently based on the preset tank geometry. Therefore, the tank with an aspect ratio of 1.5 is selected as the benchmark. Table 1 lists the design conditions of the tanks at different sloshing height ratios. Parameter , which is set as 1.5, is included in the optimization calculation to ensure the dominance of the mass block in the VIS.
Table 2 lists the optimization results of each parameter and illustrates that the parameter optimization design under white noise excitation is similar to the other types of noises. In addition, it shows that white noise can be used to simplify the optimization design.
A series of time history dynamic analyses were conducted based on the optimization results for group WN-1 to verify the controlling effect of the optimally designed VIS-controlled tank. The target-based baseline correction method can correct the inconsistent vibration acceleration signals [30, 31]. Figure 8 shows the selected earthquake waves. Figures 9–12 depict the representative time history response curves, which illustrate that the sloshing height response of every tank is effectively controlled by the VIS. Figure 13 presents the corresponding maximum sloshing heights. Meanwhile, Figures 14–17 illustrate that the base shears and the overturning base moments of the tank with the VIS are effectively reduced under the excitation of the El Centro, Kobe, and Coalinga earthquakes. In the case of the Chi-Chi earthquake, the base shears and the overturning base moments are observed to increase in the long-period earthquake. Figures 18–21 illustrate that the output force is higher in the early stage of earthquake excitation and decreases as the earthquake continues. Through parameter optimization, the output force can be effectively reduced under the premise of setting the vibration control target. In addition, the dynamic analysis results confirm the validity and the efficiency of the demand-based design optimization method of tank control.
5. Conclusions
This study proposes an optimal design method based on the tank vibration performance and applies this to the optimization of tank control with an inerter system. This design method provides an effective solution for the parameter optimization design of the VIS and overcomes the problems in existing design methods. Based on the theory of stochastic vibration, the optimization program is compiled based on selected optimization indicators. The effectiveness of the method is also proven through a time history analysis. The following conclusions can be drawn from the study:(1)The optimal design of VIS control should achieve a balance between response control and provisioning costs. The demand-based design approach proposed in this study can minimize costs and ensure that the response does not exceed a given limit.(2)White noise excitation can be used to design the tank with the VIS under seismic excitation and simplify the calculation.(3)The VIS is an effective structural control device for reducing the sloshing height responses of a tank system under dynamic excitation, such as white noise and earthquake excitations.(4)The parameter optimization design of the storage tank is based on the nonlinear optimization of a specific control objective in the feasible region of each parameter. Typically, the optimum design parameters for the tank are a combination of the upper limit of parameter and the other parameters. The storage tank with the VIS can be effectively controlled at a low cost to a considerable extent under the premise of small values.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the Shanghai Pujiang Program under Grant no. 17PJ1409200, the National Natural Science Foundation of China under Grants nos. 51308418 and 51478356, and the Collaborative Research Project under International Joint Research Laboratory of Earthquake Engineering under Grant no. ILEE-IJRP-P1-P3-2016.