Analysis of Dynamic Characteristics of Attached High Rise Risers

Abstract

The exhaust chimney of third-generation nuclear power units is a typical attached high-rise riser structure. In this paper, the simplified mechanical model and dynamic model of China’s third-generation nuclear power Hualong-1 VNA system, including multiple nonlinear factors, are established for the first time. The DTM (differential transformation method) was first applied to solve the natural vibration characteristics of a multi-point constrained variable cross-section riser structure, and the effects of variable cross-section, variable mass, variable axial force, and different elastic constraint parameters on the natural vibration characteristics of the system were studied. The dynamic behavior of the VNA system under the combined action of internal flow velocity, vortex excitation, and foundation excitation was studied. The results show that the outer diameter function of the VNA system pipeline should be designed as a quadratic function or a near quadratic multi-segment constant value function. The “limiting” effect of constraining large stiffness can force low-order vibration modes with high constraint stiffness to jump to high-order vibration modes with low constraint stiffness. The elastic constraint arrangement scheme with near center symmetry can make the system vibration mode present a half stable and half-curved form. A new optimization design scheme has been proposed regarding the layout and stiffness parameters of the VNA system guide bracket. This can enable the VNA system pipeline to avoid severe oscillations near the response extreme values caused by multiple frequency excitations of seismic loads under design and accident conditions and ensure the service life of the equipment.

1. Introduction

With the maturity of China’s third-generation nuclear power technology, all nuclear power units and their supporting equipment need to meet the characteristics of advanced, economic, mature, and reliable third-generation nuclear power technology. As the exhaust system of nuclear power units, the ventilation of the nuclear auxiliary (VNA) system is close to the outer wall of the nuclear island, with a height of more than 50 m. It is a typical attached high-rise structure. The VNA system is in a dynamic environment with multiple constraints and excitations during use. Due to the installation of relatively heavy platforms and instrument equipment on its upper part, it is prone to dynamic deformation under the external excitations coming from the foundation load of the factory building and nuclear island shell, strong wind, and airflow. That may cause collisions between the pipe body and the support, the pipe body and the nuclear island shell, seriously affecting the operational efficiency and structural safety of the VNA system, even posing significant safety hazards to the entire nuclear power unit. At the same time, the pressure of the internal exhaust airflow on the pipe wall causes dynamic changes in the system stiffness. Therefore, studying and analyzing the dynamic characteristics of the VNA system helps to accurately avoid the resonance range of the structure under design and accident conditions and improve the system safety factor.

Research on the VNA system is barely reported domestically or internationally. But there are rich research results regarding the similar structures of flow pipelines. The flow pipeline is a typical multi-physical field coupling system due to the interaction between the structure and the fluid. Ashley and Haviland [1] were the first to study the natural vibration characteristics of oil pipelines in oil field projects spanning the Arabian Peninsula. Long [2] used experimental methods for the first time to analyze and study the natural frequency of vibration in flow-conveying pipelines. He observed that the natural frequency of the flow pipeline decreases with the increase inin internal flow velocity. Benjamin [3,4] first studied the dynamics of a cantilever flow-conveying pipeline system with one end fixed and one end free and found that the instability form of the cantilever pipeline is flutter instability, which is different from the instability form of the simply supported flow-conveying pipeline at both ends. Gregory and Païdoussis [5] conducted both theoretical analysis and experimental research on cantilever flow pipelines, confirming Benjamin’s research conclusion that the instability mode of cantilever pipelines is flutter instability. Païdousis and Issid [6] uniformly considered factors such as gravity, axial tension at both ends, external pressure of the structure, and internal flow loss on the flow pipeline and established a relatively complete linear dynamic equation of the system. Païdoussis [7] provided a comprehensive discussion on the linear vibration behavior and characteristics of fluid conveying pipeline systems. Olson and Jamison [8] first used the generalized finite element method to study the vibration problem of cantilever flow pipelines. Seiranyan [9] used the perturbation method to study the energy conversion between various vibration modes of twisted chain flow pipelines. The cantilever flow pipeline model studied by Xu and Huang [10] is different from previous models, but changes the shape of the free end outlet to the nozzle shape. They explored the stability issues of such systems. Marzani et al. [11] found that the Winkler-type elastic foundation coefficient can increase the critical internal velocity value, thereby increasing the stability performance of flow-conveying pipelines. Ryu et al. [12] determined the order in which the eigenvalues of each order cross the imaginary axis by studying the changes in different parameter characteristics of cantilever flow pipelines, thereby determining the modal order corresponding to system instability. Sinha et al. [13] conducted dynamic analysis on a type of open cantilever flow-conveying pipeline system and found that the additional fluid mass on the surface of the structure is a factor that cannot be ignored in the dynamic analysis of the system. Firouz Abadi et al. [14] conducted research and analysis on the dynamic problems of a class of inclined nozzle flow-conveying pipelines with both bending and torsion. Yu et al. [15] proposed two cantilever flow-conveying pipeline models with geometric periodicity and material periodicity, respectively. In addition, some scholars have attempted to extend the theory of planar pipeline models to three-dimensional flow-conveying pipeline systems, providing fundamental research methods and vibration characteristics of three-dimensional flow-conveying pipeline system dynamics [16,17,18]. Thurman and Mote [19], based on Hamilton’s principle, used local substitution of the vector field on the low dimensional manifold, and Semler et al. [20] adopted the energy method and the Newton method to derive three different forms of nonlinear motion equations for a simply supported pipeline. Holmes [21,22] used the second-order modal expansion Galerkin discretization method to study the instability forms of flow-conveying pipelines with boundary conditions supported at both ends. Modarres Sadeghi and Païdoussis [23] found in their study of flow pipelines with boundary conditions supported at both ends that when the fluid inside the pipeline is a steady flow, the pipeline will remain in a non-zero equilibrium position after buckling and instability. Namachchivaya and Tien [24] obtained the average equation of the system by using perturbation theory according to Holmes’ nonlinear dynamic equation of flow-conveying pipeline. When Wang [25] studied the nonlinear dynamic response of flow pipelines with boundary conditions supported at both ends, the existence of configuration constraints induced rich nonlinear dynamic phenomena such as period doubling bifurcation, almost periodic response, and chaotic motion in the system.

From existing literature and research results, it can be seen that most studies on vortex-induced dynamics in flow-conveying pipelines mainly focus on the constraint conditions of supports at both ends, and cantilever pipeline studies also mainly focus on horizontally placed flow-conveying pipelines. As far as it goes, little research is conducted on the vortex-induced vibration (VIV) of vertically placed cantilever pipelines and flow-conveying pipelines with multiple point constraints, which are particularly important for guiding the dynamic behavior research of the Hualong-1 VNA system. Because the structure of VNA pipe is slender elastic, it can generally be simplified as a beam model. According to the different functions in engineering fields, the pipeline’s structure properties, such as section, effective tension, and mass per unit length, may change along the length direction. That will make the governing equation a fourth-order partial differential equation with variable coefficients, which is difficult to obtain its analytical solution in theory [26,27]. In order to solve such problems, domestic and foreign scholars have successively adopted the WKB method [28], the Fourier transform analysis method [29], the dynamic stiffness method [30], the finite element method [31], the variable transformation method [32], and other numerical calculation methods to get the natural frequencies and vibration modes of pipelines under their respective engineering conditions. Khalid H. Almitani et al. [33] used the Laplace transform and its inversion to solve the nonlinear bending problems, buckling loads, and post-buckling configurations of complete and incomplete bionic spiral composite beams with linear rotation angles. Joon Kyu Lee [34] solved the differential control equations and boundary conditions of the transverse free vibration of rectangular flat bars by the Runge-Kutta scheme and the determinant search algorithm. The proposed method is considered a feasible tool for studying the free vibration of rectangular flat bars. Mashhour A. Alazwari et al. [35] used the DQM method to discretize the variable coefficient differential equation of motion of composite laminates under a variable axial load. The effects of axial load type, orthogonal ratio, slenderness ratio, fiber direction, and boundary conditions on the natural frequency of composite beams were analyzed. For now, though, there are still few studies on the natural frequencies and vibration modes of multi-point constrained risers such as the Hualong-1 VNA system [36,37], and some of the assumptions of numerical algorithms are not applicable to the boundary conditions of the Hualong-1 VNA system. It cannot be simply copied and applied to the multi-point-constrained state of the VNA pipeline.

The main contribution of this work is to reveal the natural vibration characteristics and nonlinear parametric dynamic behavior of the Hualong-1 VNA system. The dynamic model of the Hualong-1 VNA system with multiple nonlinear factors is established for the first time, and the DTM is used to solve the natural vibration characteristics of the VNA system for the first time, which verifies the effectiveness of the algorithm and observes the phenomenon of the reordering of the order of vibration modes. Finally, using nonlinear vibration theory, the dynamic effects of external flow parameters, internal flow parameters, and basic excitation parameters on the VNA system were explored, providing theoretical guidance for system design.

2. Natural Vibration Characteristics Analysis

2.1. Theoretical Model

The VNA system mainly includes horizontal and vertical pipes, steel structure platforms, fixed supports, guide supports, flow meters, lightning protection equipment, and other components. The specific structural model is shown in Figure 1a. Vertical pipes and horizontal pipes are decoupled by hoses connecting them.

The specific layout is shown in Figure 1b. The fixed support and guide support make the vertical section of the VNA system a multi-point constrained riser model. The fixed support restricts all degrees of freedom at the bottom of the riser, and the guide support only releases degrees of freedom of translation and rotation along the axis of the riser.

For the reason that there are many components in the VNA system and the contact relationship between each component is complex, it is necessary to simplify the structural model: (1) It is assumed that the steel structure platform and various instruments and equipment are located at the top of the riser. (2) The fixed support is simplified by the fixed constraint located at the bottom of the riser. (3) The guide bracket is simplified into two, which are located at different positions on the pipe body. The simplified physical model is shown in Figure 2.

Based on Euler-Bernoulli longitudinal and transverse bending beam theory [38,39], the free vibration control equation of riser is as follows:

$$\begin{array}{l}\frac{{\partial }^2}{\partial x^2}\left(E\left(x\right)I\left(x\right)\frac{{\partial }^{2}w(x,t)}{\partial x^2}\right)-\frac{\partial }{\partial x}\left(T\left(x\right)\frac{\partial w(x,t)}{\partial x}\right)+m\left(x\right)\frac{{\partial }^{2}w(x,t)}{\partial t^2}\\ +K_{1}w(x,t)\delta (x-l_a)+K_{2}w(x,t)\delta (x-l_b)=0\end{array}$$

(1)

where $E\left(x\right)I\left(x\right)$ is the bending stiffness, $T\left(x\right)$ is the effective axial force of the riser, $m\left(x\right)$ is the effective mass per unit length of the riser, $K_1$ is the stiffness of the low-position guide bracket, the distance from the bottom is $l_a$, $K_2$ is the stiffness of the high- position guide bracket, and the distance from the bottom is $l_b$. $T(x)=-m_{t}g-{\int }_x^{L}m(x)g\mathrm{d}x$, $m_t$ is the top concentrated mass, and the riser length is L.

Use the method of separating variables to solve Equation (1), make

$$w(x,t)=\varphi (x)\eta (t)$$

(2)

Substitute it into the Equation (1),

$$\begin{array}{l}\frac{EI}{m}\frac{1}{\varphi (x)}\frac{d^4\varphi (x)}{d{x}^4}-g\frac{1}{\varphi (x)}\frac{d\varphi (x)}{dx}+\frac{m_{t}g+mg(L-x)}{m}\frac{1}{\varphi (x)}\frac{d^2\varphi (x)}{d{x}^2}\\ +\frac{K_1\delta (x-l_a)+K_2\delta (x-l_b)}{m}=-\frac{1}{\eta (t)}\frac{d^2\eta (t)}{d{t}^2}\end{array}$$

(3)

Owing to the variable coefficient x on the left side of the Equation (4) and the spring constraints, the vibration mode function cannot be expressed as an elementary function, and the analytical solution cannot be obtained. It can only be solved by numerical methods. The dimensionless variables are defined as follows:

$$\begin{array}{l}\overline{x}=\frac{x}{L}, \overline{\varphi }(\overline{x})=\frac{\varphi \left(x\right)}{L}, s(\overline{x})=\frac{E\left(x\right)I\left(x\right)}{E\left(x_0\right)I\left(x_0\right)}, n(\overline{x})=\frac{T\left(x\right)L^2}{E\left(x_0\right)I\left(x_0\right)}, \\ q(\overline{x})=\frac{m\left(x\right)}{m\left(x_0\right)}, {\lambda }^4=\frac{m\left(x_0\right){\omega }^2{L}^4}{E\left(x_0\right)I\left(x_0\right)}\end{array}$$

the control equation becomes

$$\frac{d^2}{d{\overline{x}}^2}\left(s(\overline{x})\frac{d^2\overline{\varphi }(x)}{d{\overline{x}}^2}\right)-\frac{d}{d\overline{x}}\left(n(\overline{x})\frac{d\overline{\varphi }(x)}{d\overline{x}}\right)-q(\overline{x}){\lambda }^4\overline{\varphi }(x)=0$$

(4)

Due to the elastic constraints in the beam, the vibration mode of the cantilever beam will be significantly affected in cases of large stiffness. In order to apply the DTM to segment the beam structure with the elastic constraint as the boundary, the boundary conditions of the riser are as (A1) in Appendix A.

2.2. DTM Analysis

The differential transformation of $\overline{\varphi }(\overline{x})$ is defined as

$$\mathsf{\Phi }(k)={\frac{1}{k!}\left(\frac{d^k\overline{\varphi }(\overline{x})}{d{\overline{x}}^k}\right)|}_{\overline{x}={\overline{x}}_0},0\le \overline{x}\le 1$$

(5)

where $\mathsf{\Phi }(k)$ is the k-order differential function of $\overline{\varphi }(\overline{x})$ after $\overline{x}={\overline{x}}_0$ transformation, called T function. The inverse differential transformation of $\mathsf{\Phi }(k)$ is defined as

$$\overline{\varphi }(\overline{x})=\sum _{k=0}^{\infty }{\left(\overline{x}-{\overline{x}}_0\right)}^k\mathsf{\Phi }(k)$$

(6)

Substitute (5) into (6), then obtain the Taylor series expansion of $\overline{\varphi }(\overline{x})$ at $\overline{x}={\overline{x}}_0$

$$\overline{\varphi }(\overline{x})=\sum _{k=0}^{\infty }\frac{{\left(\overline{x}-{\overline{x}}_0\right)}^k}{k!}{\left(\frac{d^k\overline{\varphi }(\overline{x})}{d{\overline{x}}^k}\right)|}_{\overline{x}={\overline{x}}_0}$$

(7)

The basic operation rules of differential transformation, such as linear properties, product properties, differential properties, etc., can be found in Appendix B.

Using the differential transformation method and the basic operational properties of DTM, ordinary differential Equation (5) can be transformed into the following algebraic equation

$$\begin{array}{l}{\displaystyle \sum _{r=0}^k}S(k-r)\left(r+1\right)\left(r+2\right)\left(r+3\right)\left(r+4\right)\mathsf{\Phi }(r+4)\\ +2{\displaystyle \sum _{r=0}^k}\left(k-r+1\right)S(k-r+1)\left(r+1\right)\left(r+2\right)\left(r+3\right)\mathsf{\Phi }(r+3)\\ +{\displaystyle \sum _{r=0}^k}\left(k-r+1\right)\left(k-r+2\right)S(k-r+2)\left(r+1\right)\left(r+2\right)\mathsf{\Phi }(\mathsf{r}+2)-{\displaystyle \sum _{r=0}^k}N(k-r)\left(r+1\right)\left(r+2\right)\mathsf{\Phi }(r+2)\\ -{\displaystyle \sum _{r=0}^k}\left(k-r+1\right)N(k-r+1)\left(r+1\right)\mathsf{\Phi }(r+1)={\displaystyle \sum _{r=0}^k}{\lambda }^{4}Q(k-r)\mathsf{\Phi }(r)\end{array}$$

(8)

where the corresponding primitive functions of $\mathsf{\Phi }(k)$, $S(k)$, $N(k)$ and $Q(k)$ are $\overline{\varphi }(\overline{x})$, $s(\overline{x})$, $n(\overline{x})$ and $q(\overline{x})$. Substituting (9) into boundary condition (A1), can get the differential transformation expression of boundary conditions as (A2) in Appendix A.

Equation (9) is a recurrence relation formula about the unknown quantity $\mathsf{\Phi }(0), \mathsf{\Phi }(1), \mathsf{\Phi }(2), \mathsf{\Phi }(3)$. According to the three vibration modes of segmented beam, assume

$$\begin{array}{l}{\mathsf{\Phi }}_1(0)={\mathsf{{\rm X}}}_{10},{\mathsf{\Phi }}_1(1)={\mathsf{{\rm X}}}_{11},{\mathsf{\Phi }}_1(2)={\mathsf{{\rm X}}}_{12},{\mathsf{\Phi }}_1(3)={\mathsf{{\rm X}}}_{13},\\ {\mathsf{\Phi }}_2(0)={\mathsf{{\rm X}}}_{20},{\mathsf{\Phi }}_2(1)={\mathsf{{\rm X}}}_{21},{\mathsf{\Phi }}_2(2)={\mathsf{{\rm X}}}_{22},{\mathsf{\Phi }}_2(3)={\mathsf{{\rm X}}}_{23},\\ {\mathsf{\Phi }}_3(0)={\mathsf{{\rm X}}}_{30},{\mathsf{\Phi }}_3(1)={\mathsf{{\rm X}}}_{31},{\mathsf{\Phi }}_3(2)={\mathsf{{\rm X}}}_{32},{\mathsf{\Phi }}_3(3)={\mathsf{{\rm X}}}_{33}.\end{array}$$

Furthermore, by using the recurrence relation formula, various types of boundary conditions in (A2) can be transformed into a set of algebraic equations about these 12 unknown quantities, written in the form of matrix as (10).

$${\left[A\right]}_{12\times 12}{\left\{{\mathsf{{\rm X}}}_{10},{\mathsf{{\rm X}}}_{11},{\mathsf{{\rm X}}}_{12},{\mathsf{{\rm X}}}_{13},{\mathsf{{\rm X}}}_{20},{\mathsf{{\rm X}}}_{21},{\mathsf{{\rm X}}}_{22},{\mathsf{{\rm X}}}_{23},{\mathsf{{\rm X}}}_{30},{\mathsf{{\rm X}}}_{31},{\mathsf{{\rm X}}}_{32},{\mathsf{{\rm X}}}_{33}\right\}}^T={\left\{0\right\}}_{12\times 1}$$

(9)

The condition that the above equation has nontrivial solution needs to be satisfied

$$Det({\left[A\right]}_{12\times 12})=0$$

(10)

Equation (11) is a higher-order algebraic equation about $\lambda$, which can be solved by numerical rooting method. According to $\omega =\frac{{\lambda }^2}{L^2}\sqrt{\frac{E(x_0)I(x_0)}{m(x_0)}}$, the vibration frequency of each order of the high riser pipe can be obtained, then as well the vibration mode of the structure. Firstly, assume ${\mathsf{{\rm X}}}_{10}$ = 1, and substitute the obtained $\lambda$ into Formula (10), then ${\mathsf{{\rm X}}}_{11}~{\mathsf{{\rm X}}}_{33}$ will all be obtained. Furthermore, use recursion Formula (9) to obtain ${\mathsf{\Phi }}_1(k)~{\mathsf{\Phi }}_3(k)$ successively. Finally, use the differential transformation (7) to obtain the vibration mode, then normalize the vibration mode according to the following:

$${\tilde{\varphi }}_i(\overline{x})=\frac{{\overline{\varphi }}_i(\overline{x})}{{\displaystyle {\sum }_{i=1}^3{\displaystyle {\int }_{x_{i-}}^{x_{i+}}\left|{\overline{\varphi }}_i(\overline{x})\right|d\overline{x}}}}$$

(11)

2.3. Constraint Stiffness on Vibration Mode Analysis

It is assumed that the section and material properties are consistent along the length, and only gravity and concentrated mass are considered. The physical parameters of the model are shown in

Table 1. Physical parameters of VNA.

Parameter	Value	Parameter	Value
Elastic modulus	1.4 × 1010 MPa	Low position constraint stiffness	1 × 105 N/m
Outer diameter	1.52 m	High position constraint stiffness	1 × 105 N/m
Inner diameter	1.5 m	Low position constraint elevation	30 m
Riser density	1856 kg/m3	High position constraint elevation	60 m
Additional mass	2000 kg	riser length	90 m
Additional rotational inertia	5000 kg·m²	Gas density	1.225 kg/m3

.

Table 2. First five natural frequencies of VNA.

	ω1 (rad/s)	ω2 (rad/s)	ω3 (rad/s)	ω4 (rad/s)	ω5 (rad/s)
DTM	2.0108	7.6238	20.2602	38.6768	62.5487
FEM	2.0086	7.5970	20.1703	38.4927	62.2990

shows the first five natural frequencies of the VNA riser under the two calculation methods, and the results are quite consistent. The calculation result of the finite element method will be easily affected by the number of divided elements. The calculation accuracy of the differential transformation method depends on the number of its expansion terms. The result of the differential transformation method is close to the exact solution in theory [38] and is convenient for parameter analysis.

In the engineering installation, EPDM rubber will be filled between the annular guide support and VNA riser to reduce vibration and local stress. The diversity of rubber elastic stiffness caused by different processes will significantly affect the change in vibration mode. Therefore, the system vibration mode can be obtained by using the DTM method for different constraint stiffnesses.

Figure 3a shows that stiffness has an obvious impact on the first mode shape. When the stiffness is 10⁵ N/m, the vibration mode of the cantilever riser pipe still shows characteristics similar to those of a cantilever beam. However, with the gradual increase in the elastic constraint stiffness, the displacement-limited effect of the elastic constraint point becomes obvious, and the arch appears between the constraint points. As seen in Figure 3b, the stiffness has a relatively small impact on the second-order vibration mode. When the stiffness is less than 10⁷ N/m, the vibration mode of the cantilever riser pipe is similar to that of a cantilever beam. Although the vibration mode collapses near the constraint point when the stiffness is equal to 10⁷ N/m, it can still be approximately equal to the vibration mode of the cantilever beam. Especially when the stiffness is equal to 10⁸ N/m, as a consequence of the displacement-limited effect, the second vibration mode of the structure is similar to the fourth vibration mode of low constraint stiffness. That is because the change in constraint increases the system stiffness and then changes the sequence of the system mode. By using this method, the precision instrument at the key position of the high-rise structure can be avoided from producing a large range of vibration due to resonance under second-order frequency excitation. It can be seen from Figure 3c,d that when the stiffness is less than 10⁶ N/m, the vibration mode of the cantilever riser pipe is very close to that of the cantilever beam. When the stiffness is equal to 10⁸ N/m, the vibration mode changes significantly.

2.4. Constraint Arrangement Analysis

In the actual project, the elastic constraint simulates the vertical guide support of the VNA system. Multiple guide supports, to some extent, limit the horizontal vibration of the system. Therefore, the layout of the guide supports has an important impact on the VNA system. The qualitative impact of different layout schemes on the natural vibration characteristics of the system is discussed below. In order to highlight the impact of layout, the constraint elastic stiffness is selected at 10⁸ N/m. The layout schemes are as follows:

→

Layout 1: Low constraint elevation is 30 m and high constraint elevation is 60 m,

→

Layout 2: Low constraint elevation is 15 m and high constraint elevation is 75 m,

→

Layout 3: Low constraint elevation is 40 m and high constraint elevation is 50 m,

→

Layout 4: Low constraint elevation is 15 m and high constraint elevation is 30 m,

→

Layout 5: Low constraint elevation is 60 m and high constraint elevation is 75 m.

Layout 1, Layout 2, and Layout 3 are symmetrical arrangements about the midpoint of the pipeline, while Layout 4 and Layout 5 are respectively low at the root and high at the top. Refer to Table 1 for other physical parameters.

The first four-order natural frequencies and vibration mode function curves under different layout schemes can be obtained through the DTM method, as shown in

Table 3. The first four natural frequencies of VNA system with different bracket arrangement.

	f1 (Hz)	f2 (Hz)	f3 (Hz)	f4 (Hz)
Layout 1	0.9249	5.8070	7.4536	8.7769
Layout 2	1.4401	3.0100	6.0971	11.303
Layout 3	0.6700	4.3996	5.0971	11.987
Layout 4	0.3239	2.1522	6.0335	11.529
Layout 5	2.4147	2.4887	6.5410	12.259

and Figure 4.

When the symmetrical layout (Layout 1 to 3) is further away from the midpoint, the higher the first-order frequency of the structure. But in the higher order modes, the frequencies of those symmetric arrangements farther away from and closer to the midpoint are approximate, and those relatively moderate symmetrical arrangements show significant increases or decreases in higher order frequencies.

The analysis of structural natural frequency in Layouts 1, 4, and 5 indicates that the closer the two constraints are arranged at the top, the higher the first-order natural frequency of the structure. When the two constraints are simultaneously located in the middle (Layout 1), the natural frequencies are larger compared with the other two arrangements at the second and third order frequencies, but smaller at the fourth order.

In Figure 4a, the first mode of vibration exhibits an arcing phenomenon between constraints due to the limiting effect of constraints. At the same time, the way of layout 3 is arranged symmetrically near the midpoint causes the high elastic constraint point to move downward; thus, the first, second, and third mode shapes of the structure all exhibit a vibration pattern characteristic of half being relatively stationary and half being substantially oscillating. For the restriction of large stiffness and the different arrangement schemes, the original system vibration modes of each order are changed due to the constraint force, and there are vibration mode curves similar to simple supported beams between the constraints. Through the analysis of the effects of different constraint arrangements on the natural vibration characteristics of structures, it provides important theoretical guidance for structural design optimization and seismic response spectrum calculation.

2.5. Outer Diameter Function on Vibration Mode Analysis

The slender and high-rise structure of the VNA system is made up of multi-segment fiberglass pipes, which are assembled together by flanges. The inevitable installation deviations would cause the outer diameter of the VNA system to not have a constant value. Therefore, it is necessary to discuss the impact of different outer diameter functions on system vibration modes and natural frequencies.

When the outer diameter is constant, the calculation parameters refer to the values in Table 1. In order to ensure the premise of similar costs, it should be noted that different outer diameter functions are designed to make the total weight of the structure equal. The first and second outer diameter functions can be obtained as follows:

$$D_{out}(x)=-0.000445x+3.06$$

(12)

$$D_{out}(x)=3.02+0.00133x-0.0000148x^2$$

(13)

where x is the coordinate along the length of the pipeline. Because the outer diameter function is no longer a constant value, the VNA system becomes a variable cross-section riser, and the bending stiffness and mass per unit length also change with x. That makes it difficult to solve the fundamental frequency and mode of vibration of the structure theoretically. Using piecewise linear processing will make the boundary conditions multiply and the dimensions of the solution matrix increase sharply, consuming too much time and energy. Using the finite element method can solve the natural vibration characteristics of a VNA system with a higher-order polynomial outer diameter, but it is not conducive to carrying out parametric analysis. Therefore, the DTM method introduced in this chapter would be a beneficial and efficient way to solve this kind of problem. Through calculation, the first four natural frequencies of the outer diameter under different conditions can be obtained, as shown in

Table 4. The first four natural frequencies of VNA system with different outer diameter functions.

External Diameter	f1 (Hz)	f2 (Hz)	f3 (Hz)	f4 (Hz)
Constant value	0.3200	1.2140	3.2245	6.1556
Linear function	0.3798	1.2645	3.2312	6.1269
Quadratic function	0.3367	1.1608	3.0977	5.9339

.

From Table 4, it can be seen that changes in the outer diameter function have a certain impact on the system structure’s fundamental frequency and a significant impact on its first-order frequency. Taking the constant value diameter as a benchmark, the first-order natural frequency of the VNA system increases by 18.7% when the outer diameter is linear and increases by 5.2% when the outer diameter is quadratic. The frequencies of other orders change slightly, all below 5%.

Figure 5a indicates the difference in the first-order mode shapes under the conditions of each outer diameter function is not significant, but in the second, third, and fourth-order mode shapes, the outer diameter of the quadratic function shows significant differences in mode shape, with wave peaks and valleys moving upward and the curve near the root changing more smoothly. In addition, the vibration modes of the linear function outer diameter and the constant outer diameter are relatively close. The analysis of the influence of different outer diameter functions on the natural vibration characteristics of structures has important theoretical guiding significance for structural design optimization and seismic response spectrum calculation.

3. Dynamic Analysis of VNA System under Combined Excitation

The VNA system is a high-rise building attached to the outer shell of a nuclear island. The VNA system must be able to withstand the once-in-a-century strong wind load, tornado load, earthquake load, and even a combination of these normative loads.

3.1. Mathematical Model

This chapter considers a VNA system with one end fixed and vertically placed. The fluid flow rate in the duct is shown in Figure 6. The pipeline is subjected to transverse external flow at a velocity of $V_{out}$, and external displacement-based excitation recorded as $w_b$. It should be noted that this article only considers the occurrence of transverse flow vibrations in the pipeline system, that is, perpendicular to the direction of the external flow velocity. Refer to the literature [19,40,41,42,43,44] for details; this assumption has been proven to be feasible. At the same time, the damping characteristics of EPDM rubber gaskets are considered in this chapter.

Due to the pipe’s large length-diameter ratio, the Euler-Bernoulli beam theory can be used for the pipe structure. Here we propose two assumptions: one is that the fluid inside and outside the pipeline is incompressible and flows at a constant rate, and the other is that the damping of the VNA system itself is not considered.

According to research by Timoshenko et al., the transverse absolute displacement of a beam, that is, the transverse displacement relative to a fixed inertial system, can be written as

$$w(x,t)=w_b(x,t)+w_{rel}(x,t)$$

(14)

where, $w_{rel}(x,t)$ is the transverse displacement relative to the fixed end, $w_b(x,t)$ is the base displacement, i.e.,

$$w_b(x,t)={\delta }_1(x)g_L(t)+{\delta }_2(x)h_L(t)$$

(15)

In the formula, ${\delta }_1(x)$ and ${\delta }_2(x)$ are respectively the influence functions of foundation transverse displacement and small rotation on beam transverse displacement. Simply set ${\delta }_1(x)=1$, ${\delta }_2(x)=x$ for a cantilever beam. The dynamic equations of VNA pipeline subjected to both foundation excitation and vortex-induced vibration are as follows:

$$\begin{array}{l}EI\frac{{\partial }^4{w}_{rel}}{\partial x^4}-\frac{\partial }{\partial x}\left(T\left(x\right)\frac{\partial w_{rel}}{\partial x}\right)+m_f{V_{in}}^2\frac{{\partial }^2{w}_{rel}}{\partial x^2}+2m_f{V}_{in}\frac{{\partial }^2{w}_{rel}}{\partial x\partial t}+m_{\sum }\frac{{\partial }^2{w}_{rel}}{\partial t^2}\\ -[\frac{EA}{2L}{{\displaystyle {\int }_0^L\left(\frac{\partial w_{rel}}{\partial x}\right)}}^{2}dx]\frac{{\partial }^2{w}_{rel}}{\partial x^2}+K_1{w}_{rel}\delta (x-l_a)+K_2{w}_{rel}\delta (x-l_b)+K_3{w_{rel}}^3\delta (x-l_a)\\ +K_4{w_{rel}}^3\delta (x-l_b)+c_{epdm}\frac{\partial w_{rel}}{\partial t}\delta (x-l_a)+c_{epdm}\frac{\partial w_{rel}}{\partial t}\delta (x-l_b)=f(x,t)-m_{\sum }{\ddot{w}}_b\end{array}$$

(16)

In this paper, the empirical model of vortex wake oscillator proposed by Facchinetti et al. is used, and the equation for simulating vortex-induced force is expressed as follows,

$$\frac{{\partial }^{2}q}{\partial t^2}+{\lambda }_s{\omega }_s\left(q^2-1\right)\frac{\partial q}{\partial t}+{{\omega }_s}^{2}q=\frac{P}{D}\frac{{\partial }^{2}w(x,t)}{\partial t^2}$$

(17)

where $q(x,t)$ is the equivalent lift coefficient used to describe the wake behavior, ${\omega }_s$ is the vortex shedding frequency, ${\omega }_s=2\pi S_t{V}_{out}/D$, ${\lambda }_s$ and P are empirical parameters. Considering the effect of foundation excitation, Equation (17) for simulating vortex-induced lift can be further written as

$$\frac{{\partial }^{2}q}{\partial t^2}+{\lambda }_s{\omega }_s\left(q^2-1\right)\frac{\partial q}{\partial t}+{{\omega }_s}^{2}q=\frac{P}{D}(\frac{{\partial }^2{w}_{rel}(x,t)}{\partial t^2}+{\ddot{w}}_b)$$

(18)

In addition, according to literature [45], the expression of vortex-induced force is

$$f(x,t)=\frac{C_{L0}{\rho }_f{V}_{out}^{2}D}{4}q(x,t)-\frac{C_D{\rho }_f{V}_{out}D}{2}{\dot{w}}_{rel}$$

(19)

$C_{L_0}$ and $C_D$ are respectively the stable lift coefficient and the average section drag coefficient, which are taken as 0.3 and 1.2 for ease of calculation.

For the convenience of analysis and calculation, the following dimensionless parameters are introduced:

$$\begin{array}{l}\eta =w/L; \xi =x/L; \tau =t/\left(L^2\sqrt{m_{\sum }/\left(EI\right)}\right); \nu =V_{in}L\sqrt{m_f/\left(EI\right)}; o=\frac{A{L}^2}{2I};\\ c_0=c_s{L}^2/\sqrt{m_{\sum }EI}; c_{1,2}=c_{epdm}{L}^2/\sqrt{m_{\mathsf{\Sigma }}EI}; \alpha =m_f/m_{\sum }; \beta =m_{s}g{L}^3/\left(EI\right); \gamma =\left(m_{t}g+m_{s}gL\right)L^2/\left(EI\right);\\ {\kappa }_1=K_1{L}^3/\left(EI\right); {\kappa }_2=K_2{L}^3/\left(EI\right); {\kappa }_3=K_3{L}^5/\left(EI\right); {\kappa }_4=K_4{L}^5/\left(EI\right);\\ {\mathsf{\Omega }}_s={\omega }_s{L}^2\sqrt{m_{\sum }/\left(EI\right)}; {\mathsf{\Omega }}_{in}={\omega }_{in}{L}^2\sqrt{m_{\sum }/\left(EI\right)}; f_D=\frac{C_D{\rho }_{f}D{V}_{out}{L}^2}{2\sqrt{m_{\sum }EI}}; f_L=\frac{C_{L0}{\rho }_{f}D{V_{out}}^2{L}^3}{4EI}; p=\frac{PL}{D}\end{array}$$

The dimensionless dynamic equation of the exhaust duct can be obtained as follows:

$$\begin{array}{l}\frac{{\partial }^4{\eta }_{rel}}{\partial {\xi }^4}+\frac{{\partial }^2{\eta }_{rel}}{\partial {\tau }^2}+{\nu }^2\frac{{\partial }^2{\eta }_{rel}}{\partial {\xi }^2}+2\nu \sqrt{\alpha }\frac{{\partial }^2{\eta }_{rel}}{\partial \xi \partial \tau }-\beta \frac{\partial {\eta }_{rel}}{\partial \xi }+\gamma \frac{{\partial }^2{\eta }_{rel}}{\partial {\xi }^2}-\beta \xi \frac{{\partial }^2{\eta }_{rel}}{\partial {\xi }^2}\\ -\left[o{\int }_0^1{\left(\frac{\partial {\eta }_{rel}}{\partial \xi }\right)}^2\mathrm{d}\xi \right]\frac{{\partial }^2{\eta }_{rel}}{\partial {\xi }^2}+{\kappa }_1{\eta }_{rel}\delta (\xi -{\xi }_a)+{\kappa }_2{\eta }_{rel}\delta (\xi -{\xi }_b)+{\kappa }_3{{\eta }_{rel}}^3\delta (\xi -{\xi }_a)\\ +{\kappa }_4{{\eta }_{rel}}^3\delta (\xi -{\xi }_b)+c_1\frac{\partial {\eta }_{rel}}{\partial \tau }\delta (\xi -{\xi }_a)+c_2\frac{\partial {\eta }_{rel}}{\partial \tau }\delta (\xi -{\xi }_a)=f(\xi ,\tau )-G{\ddot{\eta }}_b\end{array}$$

(20)

${\eta }_{rel}$ is the dimensionless relative displacement, $G=m_{\mathsf{\Sigma }}{L}^3/\left(EI\right)$, $c_{1,2}=c_{epdm}{L}^2/\sqrt{m_{\mathsf{\Sigma }}EI}$, $c_{epdm}$ is the damping coefficient of the rubber gasket, and the value in this chapter is 0.3 N/(m/s). $f(\xi ,\tau )$ is the dimensionless vortex-induced force. For ease of expression, this chapter will use $\eta$ instead of ${\eta }_{rel}$.

Equation (5-5) can be dimensionless and expressed as:

$$\frac{{\partial }^{2}q}{\partial {\tau }^2}+{\lambda }_s{\mathsf{\Omega }}_s\left(q^2-1\right)\frac{\partial q}{\partial \tau }+{{\mathsf{\Omega }}_s}^{2}q=p(\frac{{\partial }^2\eta }{\partial {\tau }^2}+G{\ddot{\eta }}_b)$$

(21)

3.2. Vibration Response of the VNA System under First Order Frequency Locking

To obtain the reduced-order model equations of the Hualong-1 VNA system, the Galerkin discretization method can be used. Perform modal expansion on the wake oscillator lift coefficient q, and if the vortex shedding frequency ${\omega }_s$ is close to the r-th order natural frequency of the VNA system pipe, the contribution to q only exhibits in the r-th order mode, which is also a manifestation of the frequency locking effect. Therefore, the VNA system pipe displacement and lift coefficient q are written in the following form:

$$\eta =\sum _{i=1}^n{\mathsf{\Phi }}_i(\xi ){\overline{\eta }}_i(\tau )$$

(22)

$$q={\mathsf{\Phi }}_r(\xi ){\overline{q}}_r(\tau )$$

(23)

where, ${\overline{\eta }}_i(\tau )$ is the time-dependent displacement, and ${\mathsf{\Phi }}_i(\xi )$ is a modal function. From Section 2, it can be seen that when the constraint spring stiffness is small, that is, less than 10⁷ N/m, the vibration mode of the pipe is approximately the vibration mode of a cantilever beam containing axial force, and the vibration mode function can be expressed as:

$${\mathsf{\Phi }}_i(\xi )=A_i\left(\sin ({\beta }_i\xi )+{\chi }_i\cos ({\beta }_i\xi )-\frac{{\beta }_i}{{\alpha }_i}\sinh ({\alpha }_i\xi )-{\chi }_i\cosh ({\alpha }_i\xi )\right)$$

(24)

where, ${\chi }_i=\frac{-EI{{\beta }_i}^3\cos ({\beta }_i)-EI{{\alpha }_i}^2{\beta }_i\cosh ({\alpha }_i)+m_t{{\omega }_i}^2\sin ({\beta }_i)-\frac{m_t{\beta }_i{{\omega }_i}^2\sinh ({\alpha }_i)}{{\alpha }_i}}{-EI{{\beta }_i}^3\sin ({\beta }_i)+EI{{\alpha }_i}^3\sinh ({\alpha }_i)-m_t{{\omega }_i}^2\cos ({\beta }_i)+m_t{{\omega }_i}^2\cosh ({\alpha }_i)}$;

${\alpha }_i=L{\left({\left({\left(\frac{T}{2EI}\right)}^2+\frac{m_{\mathsf{\Sigma }}{{\omega }_i}^2}{EI}\right)}^{\frac{1}{2}}+\frac{T}{2EI}\right)}^{\frac{1}{2}}$; ${\beta }_i=L{\left({\left({\left(\frac{T}{2EI}\right)}^2+\frac{m_{\mathsf{\Sigma }}{{\omega }_i}^2}{EI}\right)}^{\frac{1}{2}}-\frac{T}{2EI}\right)}^{\frac{1}{2}}$. $T=-m_{t}g+m_f{V_{in}}^2$, $A_i$ is the normalized amplitude satisfying ${\displaystyle {\int }_0^1{\mathsf{\Phi }}_i(\xi )\overline{{\mathsf{\Phi }}_i(\xi )}}\mathrm{d}\xi =1$. And the vibration mode function satisfies the following boundary conditions:

$$\begin{array}{l}\eta (0,\tau )=0; {\eta }^{\prime }(0,\tau )=0;\\ {{\eta }^{\prime }}^{\prime }(1,\tau )+j_t{\ddot{\eta }}^{\prime }(1,\tau )=0; {\eta }^{‴}(1,\tau )-{\mu }_{mt}\ddot{\eta }(1,\tau )+T{\eta }^{\prime }(1,\tau )=0\end{array}$$

(25)

where $j_t=\frac{J_t}{m_{0}L}, {\mu }_{mt}=\frac{m_t}{m_0}L$. ${\eta }^{\prime }(\xi ,\tau )$ represents the first order partial derivative for $\xi$. $\ddot{\eta }(\xi ,\tau )$ represents the second order partial derivative for $\tau$, and so on.

By substituting Equations (22)–(24) into Equations (17) and (21), the ordinary differential equations of the parameterized model under the combined action of vortex-induced vibration and foundation excitation of the VNA pipeline can be obtained:

$$\begin{array}{l}{\displaystyle \sum _{j=1}^n}{h}_{i,j}{\ddot{\overline{\eta }}}_j(\tau )+{\displaystyle \sum _{j=1}^n}{h}_{i,j^{(4)}}{\overline{\eta }}_j(\tau )+2\sqrt{\alpha }\nu {\displaystyle \sum _{j=1}^n}{h}_{i,j^{(1)}}{\dot{\overline{\eta }}}_j(\tau )+\left(\gamma +{\nu }^2\right){\displaystyle \sum _{j=1}^n}{h}_{i,j^{(2)}}{\overline{\eta }}_j(\tau )-\beta {\displaystyle \sum _{j=1}^n}h{\xi }_{i,j^{(2)}}{\overline{\eta }}_j(\tau )-\beta {\displaystyle \sum _{j=1}^4}{h}_{i,j^{(1)}}{\overline{\eta }}_j(\tau )\\ +f_D{\displaystyle \sum _{j=1}^n}{h}_{i,j}{\dot{\overline{\eta }}}_j(\tau )+o{\displaystyle \sum _{l=1}^n}{\displaystyle \sum _{m=1}^n}{\displaystyle \sum _{s=1}^n}{h}_{i,l^{(1)},m,s^{(1)}}{\overline{\eta }}_l(\tau ){\overline{\eta }}_m(\tau ){\overline{\eta }}_s(\tau )+{\kappa }_1{\displaystyle \sum _{j=1}^n}{H}_{i,j}({\xi }_a){\overline{\eta }}_j(\tau )+c_1{\displaystyle \sum _{j=1}^n}{H}_{i,j}({\xi }_a){\dot{\overline{\eta }}}_j(\tau )\\ +{\kappa }_3{\displaystyle \sum _{l=1}^n}{\displaystyle \sum _{m=1}^n}{\displaystyle \sum _{s=1}^n}{H}_{i,l,m,s}({\xi }_a){\overline{\eta }}_l(\tau ){\overline{\eta }}_m(\tau ){\overline{\eta }}_s(\tau )+{\kappa }_2{\displaystyle \sum _{j=1}^n}{H}_{i,j}({\xi }_b){\overline{\eta }}_j(\tau )+c_2{\displaystyle \sum _{j=1}^n}{H}_{i,j}({\xi }_b){\dot{\overline{\eta }}}_j(\tau )+{\kappa }_4{\displaystyle \sum _{l=1}^n}{\displaystyle \sum _{m=1}^n}{\displaystyle \sum _{s=1}^n}{H}_{i,l,m,s}({\xi }_b){\overline{\eta }}_l(\tau ){\overline{\eta }}_m(\tau ){\overline{\eta }}_s(\tau )\\ =f_L{h}_{i,r}{\overline{q}}_r(\tau )-G{h}_i{\ddot{\eta }}_b(\tau ),(i=1,2,\cdots ,n)\end{array}$$

(26)

$$h_{i,r}{\ddot{\overline{q}}}_r(\tau )+{{\mathsf{\Omega }}_s}^2{h}_{i,r}{\overline{q}}_r(\tau )-{\lambda }_s{\mathsf{\Omega }}_s{h}_{i,r}{\dot{\overline{q}}}_r(\tau )+h_{i,r^3}{\dot{\overline{q}}}_r(\tau ){{\overline{q}}_r}^2(\tau )=p(\sum _{j=1}^n{h}_{i,r}{\ddot{\overline{\eta }}}_j(\tau )+G{h}_i{\ddot{\eta }}_b(\tau )),(i=1,2,\cdots ,n)$$

(27)

where $h_{i,j}, h_{i,j^{(4)}}, h_{i,j^{(1)}}, h_{i,j^{(2)}}, h{\xi }_{i,j^{(2)}}, H_{i,j}({\xi }_a), h_{i,l^{(1)},m,s^{(1)}}, H_{i,j}({\xi }_b), H_{i,l,m,s}({\xi }_a), H_{i,l,m,s}({\xi }_b), h_{i,r}, h_i, h_{i,r^3}$ as shown in (A3) in Appendix A.

Through modal truncation analysis, it is easy to know that under frequency locking conditions, studying first-order main resonance is sufficient to predict the vibration response of VNA system pipelines. Taking I = 1 from Equations (26) and (27) obtained by the Galerkin discretization method, the following ordinary differential equation under first-order frequency locking can be obtained:

$${\ddot{\overline{\eta }}}_1(\tau )+a_1{\overline{\eta }}_1(\tau )+a_2{\dot{\overline{\eta }}}_1(\tau )+a_3{{\overline{\eta }}_1}^3(\tau )+a_5{{\mathsf{\Omega }}_s}^2{\overline{q}}_1(\tau )+a_4\cos ({\mathsf{\Omega }}_g\tau )=0$$

(28)

$$\begin{array}{l}{\ddot{\overline{q}}}_1(\tau )+{{\mathsf{\Omega }}_s}^2{\overline{q}}_1(\tau )+b_1{\dot{\overline{q}}}_1(\tau )+b_2{\dot{\overline{q}}}_1(\tau ){{\overline{q}}_1}^2(\tau )\\ +b_3{\overline{\eta }}_1(\tau )+b_4{\dot{\overline{\eta }}}_1(\tau )+b_6{{\overline{\eta }}_1}^3(\tau )+b_5{{\mathsf{\Omega }}_s}^2{\overline{q}}_1(\tau )=0\end{array}$$

(29)

where,

$$\begin{array}{ll}{a}_1=& {\int }_0^1{\mathsf{\Phi }}_1\left(\xi \right){\mathsf{\Phi }}_1^{‴\prime }(\xi )\mathrm{d}\xi +\left(\gamma +{\nu }^2\right)({\int }_0^1{\mathsf{\Phi }}_1\left(\xi \right){{\mathsf{\Phi }}^{″}}_1(\xi )\mathrm{d}\xi )-\beta ({\int }_0^1\xi {\mathsf{\Phi }}_1(\xi \left){{\mathsf{\Phi }}^{″}}_1\right(\xi )\mathrm{d}\xi )\\ & -\beta ({\int }_0^1{\mathsf{\Phi }}_1(\xi \left){{\mathsf{\Phi }}^{\prime }}_1\right(\xi )\mathrm{d}\xi )+{\kappa }_1{{\mathsf{\Phi }}_1}^2({\xi }_a)+{\kappa }_2{{\mathsf{\Phi }}_1}^2({\xi }_b)\\ a_2=& 2\sqrt{\alpha }\nu ({\int }_0^1{\mathsf{\Phi }}_1(\xi \left){{\mathsf{\Phi }}^{\prime }}_1\right(\xi )\mathrm{d}\xi )+f_D+c_1{{\mathsf{\Phi }}_1}^2({\xi }_a)+c_2{{\mathsf{\Phi }}_1}^2({\xi }_b)\\ a_3=& o({\int }_0^1{{\mathsf{\Phi }}_1}^2(\xi \left){{{\mathsf{\Phi }}^{\prime }}_1}^2\right(\xi )\mathrm{d}\xi )+{\kappa }_3{{\mathsf{\Phi }}_1}^4({\xi }_a)+{\kappa }_4{{\mathsf{\Phi }}_1}^4({\xi }_b)\\ a_4=& G({\displaystyle {\int }_0^1{\mathsf{\Phi }}_1(\xi )d}\xi )f_a, a_5=-\frac{D^3{C}_{L0}{\rho }_f}{16L{\pi }^2{m}_{\mathsf{\Sigma }}{S}_t^2}\end{array}$$

$$b_1=-{\lambda }_s{\mathsf{\Omega }}_s, b_2={\displaystyle {\int }_0^1{{\mathsf{\Phi }}_1}^4(\xi )d}\xi , b_3=p{a}_1, b_4=p{a}_2, b_5=-p{a}_5, b_6=p{a}_3$$

$f_a$ is the acceleration amplitude.

The multi-scale method can be used to solve Equations (28) and (29) to obtain the approximate periodic response of the system. Two tuning parameters ${\sigma }_1$ and ${\sigma }_2$ are introduced, and the following relationship is given,

$${{\mathsf{\Omega }}_g}^2={{\mathsf{\Omega }}_1}^2+{\epsilon }^2{\sigma }_1;{{\mathsf{\Omega }}_s}^2={{\mathsf{\Omega }}_1}^2+{\epsilon }^2{\sigma }_2$$

(30)

where, ${\mathsf{\Omega }}_1$ is the linear dimensionless resonance frequency of the first mode of the system, $\epsilon$ is a small quantity. In order to make the damping and forcing terms effects and nonlinear effects appear in the same perturbation equation and the damping term be a quadratic small quantity, select a small parameter $\epsilon$ to label the small quantity in the dynamic system. By sorting them out, the following dynamic equations can be obtained:

$$\begin{array}{l}{\ddot{\overline{\eta }}}_1(\tau )+{{\mathsf{\Omega }}_g}^2{\overline{\eta }}_1\left[\tau \right]=-{\epsilon }^2\widehat{a_1}{\overline{\eta }}_1(\tau )-{\epsilon }^2\widehat{a_2}{\dot{\overline{\eta }}}_1(\tau )-a_3{{\overline{\eta }}_1}^3(\tau )-a_5{\epsilon }^2{\widehat{{\mathsf{\Omega }}_1}}^2{\overline{q}}_1(\tau )\\ -a_5{\epsilon }^2{\sigma }_2{\overline{q}}_1(\tau )-{\epsilon }^3\widehat{a_4}\cos ({\mathsf{\Omega }}_g\tau )+{\epsilon }^2{\widehat{{\mathsf{\Omega }}_1}}^2{\overline{\eta }}_1(\tau )+{\epsilon }^2{\sigma }_1{\overline{\eta }}_1(\tau )\end{array}$$

(31)

$$\begin{array}{l}{\ddot{\overline{q}}}_1(\tau )+{{\mathsf{\Omega }}_g}^2{\overline{q}}_1(\tau )={\epsilon }^2\left({\sigma }_1-{\sigma }_2\right){\overline{q}}_1(\tau )-{\epsilon }^2\widehat{b_1}{\dot{\overline{q}}}_1(\tau )-b_2{\dot{\overline{q}}}_1(\tau ){{\overline{q}}_1}^2(\tau )\\ -{\epsilon }^2\widehat{b_3}{\overline{\eta }}_1(\tau )-{\epsilon }^2\widehat{b_4}{\dot{\overline{\eta }}}_1(\tau )-b_6{{\overline{\eta }}_1}^3(\tau )-b_5{\epsilon }^2{\widehat{{\mathsf{\Omega }}_1}}^2{\overline{q}}_1(\tau )-b_5{\epsilon }^2{\sigma }_2{\overline{q}}_1(\tau )\end{array}$$

(32)

where, $a_i={\epsilon }^2\widehat{a_i}(i=1,2)$, $a_4={\epsilon }^3\widehat{a_4}$, $b_i={\epsilon }^2\widehat{b_i}(i=1,3,4)$, ${{\mathsf{\Omega }}_1}^2={\epsilon }^2{\widehat{{\mathsf{\Omega }}_1}}^2$.

Assume the approximate solution form of Equations (31) and (32) as

$$\begin{array}{l}{\overline{\eta }}_1={\epsilon }^1{\overline{\eta }}_{11}(T_0,T_2)+{\epsilon }^3{\overline{\eta }}_{13}(T_0,T_2)\\ {\overline{q}}_1={\epsilon }^1{\overline{q}}_{11}(T_0,T_2)+{\epsilon }^3{\overline{q}}_{13}(T_0,T_2)\end{array}$$

(33)

where $T_n={\epsilon }^n\tau , (n=1,2,3,\dots )$.

By substituting Equation (33) into Equations (31) and (32) and comparing the coefficients of the same power at both ends of the equation, the following set of partial differential equations can be obtained:

$$\begin{array}{l}O\left({\epsilon }^1\right):D_0^2{\overline{\eta }}_{11}+{{\mathsf{\Omega }}_g}^2{\overline{\eta }}_{11}=0\\ D_0^2{\overline{q}}_{11}+{{\mathsf{\Omega }}_g}^2{\overline{q}}_{11}=0\end{array}$$

(34)

$$\begin{array}{ll}O\left({\epsilon }^3\right):& D_0^2{\overline{\eta }}_{13}+{{\mathsf{\Omega }}_g}^2{\overline{\eta }}_{13}=-\left(\begin{array}{c}\cos ({\mathsf{\Omega }}_g\tau )\widehat{a_4}+{\widehat{{\mathsf{\Omega }}_1}}^2{a}_5{\overline{q}}_{11}+a_5{\sigma }_2{\overline{q}}_{11}+\widehat{a_1}{\overline{\eta }}_{11}\\ -{\widehat{{\mathsf{\Omega }}_1}}^2{\overline{\eta }}_{11}+\widehat{a_2}{D}_0{\overline{\eta }}_{11}+2D_0{D}_2{\overline{\eta }}_{11}-{\sigma }_1{\overline{\eta }}_{11}+a_3{\overline{\eta }}_{11}^3\end{array}\right)\\ & D_0^2{\overline{q}}_{13}+{{\mathsf{\Omega }}_g}^2{\overline{q}}_{13}=-\left(\begin{array}{l}{\widehat{{\mathsf{\Omega }}_1}}^2{b}_5{\overline{q}}_{11}+\widehat{b_1}{D}_0{\overline{q}}_{11}+2D_0{D}_2{\overline{q}}_{11}-{\sigma }_1{\overline{q}}_{11}+{\sigma }_2{\overline{q}}_{11}\\ +b_5{\sigma }_2{\overline{q}}_{11}+b_2{\overline{q}}_{11}^2{D}_0{\overline{q}}_{11}+\widehat{b_3}{\overline{\eta }}_{11}+\widehat{b_4}{D}_0{\overline{\eta }}_{11}+b_6{\overline{\eta }}_{11}^3\end{array}\right)\end{array}$$

(35)

The general solution form of Equation (34) can be expressed as

$$\begin{array}{l}{\overline{\eta }}_{11}(T_0,T_2)=A_{11}(T_2){\mathrm{e}}^{\mathrm{i}{\mathsf{\Omega }}_{gr}{T}_0}+\overline{A_{11}(T_2)}{\mathrm{e}}^{-\mathrm{i}{\mathsf{\Omega }}_{gr}{T}_0}\\ {\overline{q}}_{11}(T_0,T_2)=B_1(T_2){\mathrm{e}}^{\mathrm{i}{\mathsf{\Omega }}_{gr}{T}_0}+\overline{B_1(T_2)}{\mathrm{e}}^{-\mathrm{i}{\mathsf{\Omega }}_{gr}{T}_0}\end{array}$$

(36)

where

$$\begin{array}{l}{A}_{11}(T_2)=\frac{1}{2}{u}_1(T_2){\mathrm{e}}^{\mathrm{i}{\theta }_1(T_2)}\\ B_1(T_2)=\frac{1}{2}{u}_2(T_2){\mathrm{e}}^{\mathrm{i}{\theta }_2(T_2)}\end{array}$$

(37)

Here, $u_1$ and $u_2$ respectively represent the time term amplitude of the VNA system and the time term amplitude of the vortex oscillator. ${\theta }_1$ and ${\theta }_2$ respectively represent the initial phase of the VNA system and the vortex oscillator. Substitute Equations (36) and (37) into Equation (25), eliminate the perpetual term, and collate to obtain the average equation for the amplitude and phase of the system as follows:

$$\begin{array}{l}{u_1}^{\prime }=\frac{\widehat{a_4}\sin ({\theta }_1)-\widehat{a_2}{\mathsf{\Omega }}_g{u}_1+\sin (\varphi )\left({\widehat{{\mathsf{\Omega }}_1}}^2{a}_5{u}_2+a_5{\sigma }_2{u}_2\right)}{2{\mathsf{\Omega }}_g}\\ {u_2}^{\prime }=-\frac{1}{8{\mathsf{\Omega }}_g}\left(4\cos (\varphi )\widehat{b_4}{\mathsf{\Omega }}_g{u}_1+\sin (\varphi )\left(4\widehat{b_3}{u}_1+3b_6{u_1}^3\right)+4\widehat{b_1}{\mathsf{\Omega }}_g{u}_2+b_2{\mathsf{\Omega }}_g{u_2}^3\right)\\ {{\theta }_1}^{\prime }=\frac{1}{8{\mathsf{\Omega }}_{gr}{u}_1}\left(4\cos ({\theta }_1)\widehat{a_4}+4\widehat{a_1}{u}_1-4{\widehat{{\mathsf{\Omega }}_1}}^2{u}_1-4{\sigma }_1{u}_1+3a_3{u_1}^3+\cos (\varphi )\left(4{\widehat{{\mathsf{\Omega }}_1}}^2{a}_5{u}_2+4a_5{\sigma }_2{u}_2\right)\right)\\ {\varphi }^{\prime }=\frac{1}{8{\mathsf{\Omega }}_g{u}_1}\left(4\cos ({\theta }_1)\widehat{a_4}+4\widehat{a_1}{u}_1-4{\widehat{{\mathsf{\Omega }}_1}}^2{u}_1-4{\sigma }_1{u}_1+3a_3{u_1}^3+\cos (\varphi )\left(4{\widehat{{\mathsf{\Omega }}_1}}^2{a}_5{u}_2+4a_5{\sigma }_2{u}_2\right)\right)\\ -\frac{1}{8{\mathsf{\Omega }}_g{u}_2}\left(-4\widehat{b_4}\sin (\varphi ){\mathsf{\Omega }}_g{u}_1+\cos (\varphi )\left(4\widehat{b_3}{u}_1+3b_6{u_1}^3\right)+4{\widehat{{\mathsf{\Omega }}_1}}^2{b}_5{u}_2-4{\sigma }_1{u}_2+4{\sigma }_2{u}_2+4b_5{\sigma }_2{u}_2\right)\end{array}$$

(38)

where $\varphi ={\theta }_1-{\theta }_2$. Let ${u_1}^{\prime }=0$, ${u_2}^{\prime }=0$, ${{\theta }_1}^{\prime }=0$, ${\varphi }^{\prime }=0$, then the equation for determining the steady solution can be obtained

$$\begin{array}{l}\widehat{a_4}\sin ({\theta }_1)-\widehat{a_2}{\mathsf{\Omega }}_g{u}_1+\sin (\varphi )\left({\widehat{{\mathsf{\Omega }}_1}}^2{a}_5{u}_2+a_5{\sigma }_2{u}_2\right)=0\\ 4\cos (\varphi )\widehat{b_4}{\mathsf{\Omega }}_g{u}_1+\sin (\varphi )\left(4\widehat{b_3}{u}_1+3b_6{u_1}^3\right)+4\widehat{b_1}{\mathsf{\Omega }}_g{u}_2+b_2{\mathsf{\Omega }}_g{u_2}^3=0\\ 4\cos ({\theta }_1)\widehat{a_4}+4\widehat{a_1}{u}_1-4{\widehat{{\mathsf{\Omega }}_1}}^2{u}_1-4{\sigma }_1{u}_1+3a_3{u_1}^3+\cos (\varphi )\left(4{\widehat{{\mathsf{\Omega }}_1}}^2{a}_5{u}_2+4a_5{\sigma }_2{u}_2\right)=0\\ -4\widehat{b_4}\sin (\varphi ){\mathsf{\Omega }}_g{u}_1+\cos (\varphi )\left(4\widehat{b_3}{u}_1+3b_6{u_1}^3\right)+4{\widehat{{\mathsf{\Omega }}_1}}^2{b}_5{u}_2-4{\sigma }_1{u}_2+4{\sigma }_2{u}_2+4b_5{\sigma }_2{u}_2=0\end{array}$$

(39)

Eliminate $\varphi , {\theta }_1$, then two bifurcation equations for amplitude $u_1$ and $u_2$ can be obtained:

$$\begin{array}{l}-16{\widehat{b_3}}^2{u_1}^2-16{\widehat{b_4}}^2{\mathsf{\Omega }}_g^2{u_1}^2-24\widehat{b_3}{b}_6{u_1}^4-9{b_6}^2{u_1}^6+16{\widehat{{\mathsf{\Omega }}_1}}^4{b}_5^2{u_2}^2-32{\widehat{{\mathsf{\Omega }}_1}}^2{b}_5{\sigma }_1{u_2}^2+\\ 16{\sigma }_1^2{u_2}^2+32{\widehat{{\mathsf{\Omega }}_1}}^2{b}_5{\sigma }_2{u_2}^2+32{\widehat{{\mathsf{\Omega }}_1}}^2{b}_5^2{\sigma }_2{u_2}^2-32{\sigma }_1{\sigma }_2{u_2}^2-32b_5{\sigma }_1{\sigma }_2{u_2}^2+16{\sigma }_2^2{u_2}^2\\ +32b_5{\sigma }_2^2{u_2}^2+16b_5^2{\sigma }_2^2{u_2}^2+16{\widehat{b_3}}^2{\mathsf{\Omega }}_g^2{u_2}^2+8\widehat{b_3}{b}_2{\mathsf{\Omega }}_g^2{u_2}^4+{b_2}^2{\mathsf{\Omega }}_g^2{u_2}^6=0\end{array}$$

(40)

$$\begin{array}{l}(-4\widehat{a_1}{u}_1+4{\widehat{{\mathsf{\Omega }}_1}}^2{u}_1+4{\sigma }_1{u}_1-3a_3{u_1}^3+(16\left({\widehat{{\mathsf{\Omega }}_1}}^2{a}_5{u}_2+a_5{\sigma }_2{u}_2\right)(4\widehat{b_3}{\widehat{{\mathsf{\Omega }}_1}}^2{b}_5{u}_2-4\widehat{b_3}{\sigma }_1{u}_2+4\widehat{b_3}{\sigma }_2{u}_2+4\widehat{b_3}{b}_5{\sigma }_2{u}_2+4\widehat{b_1}\widehat{b_4}{\mathsf{\Omega }}_g^2{u}_2\\ +3{\widehat{{\mathsf{\Omega }}_1}}^2{b}_5{b}_6{u_1}^2{u}_2-3b_6{\sigma }_1{u_1}^2{u}_2+3b_6{\sigma }_2{u_1}^2{u}_2+3b_5{b}_6{\sigma }_2{u_1}^2{u}_2+\widehat{b_4}{b}_2{\mathsf{\Omega }}_g^2{u_2}^3))/\left(u_1\left(16{\widehat{b_3}}^2+16{\widehat{b_4}}^2{\mathsf{\Omega }}_g^2+24\widehat{b_3}{b}_6{u_1}^2+9{b_6}^2{u_1}^4\right)\right){)}^2\\ +16(\widehat{a_2}{\mathsf{\Omega }}_g{u}_1-(\left({\widehat{{\mathsf{\Omega }}_1}}^2{a}_5{u}_2+a_5{\sigma }_2{u}_2\right)(-16\widehat{b_1}\widehat{b_3}{\mathsf{\Omega }}_{gr}{u}_1{u}_2+16\widehat{b_4}{\widehat{{\mathsf{\Omega }}_1}}^2{b}_5{\mathsf{\Omega }}_g{u}_1{u}_2-16\widehat{b_4}{\sigma }_1{\mathsf{\Omega }}_g{u}_1{u}_2+16\widehat{b_4}{\sigma }_2{\mathsf{\Omega }}_g{u}_1{u}_2\\ +16\widehat{b_4}{b}_5{\sigma }_2{\mathsf{\Omega }}_{gr}{u}_1{u}_2-12\widehat{b_1}{b}_6{\mathsf{\Omega }}_g{u_1}^3{u}_2-4\widehat{b_3}{b}_2{\mathsf{\Omega }}_g{u}_1{u_2}^3-3b_2{b}_6{\mathsf{\Omega }}_g{u_1}^3{u_2}^3))/\left({u_1}^2\left(16{\widehat{b_3}}^2+16{\widehat{b_4}}^2{\mathsf{\Omega }}_g^2+24\widehat{b_3}{b}_6{u_1}^2+9{b_6}^2{u_1}^4\right)\right){)}^2=16{\widehat{a_4}}^2\end{array}$$

(41)

It can be seen that the linear and nonlinear stiffness of high and low position elastic constraints affect the vortex-induced vibration response of the VNA system. In order to explore the impact of constraint stiffness parameters on the response of the VNA system bearing the combined action of vortex-induced vibration and foundation excitation in the first-order frequency locking circumstance, this section divides the constraint types into seven types, namely,

I:

linear stiffness 10³ N/m, nonlinear stiffness 10³ N/m³;

II:

Linear stiffness 10⁴ N/m, nonlinear stiffness 10³ N/m³;

III:

Linear stiffness 10⁵ N/m, nonlinear stiffness 10³ N/m³;

IV:

Linear stiffness 10³ N/m, nonlinear stiffness 10⁴ N/m³;

V:

Linear stiffness 10³ N/m, nonlinear stiffness 10⁵ N/m³;

VI:

Linear stiffness 10⁴ N/m, nonlinear stiffness 10⁴ N/m³;

VII:

Linear stiffness 10⁵ N/m, nonlinear stiffness 10⁵ N/m³.

The high and low constraint stiffnesses and damping are consistent. Applying the multiscale method to solve the bifurcation Equations (40) and (41) of the VNA system under the first-order frequency locking condition, the amplitude frequency response of the system can be obtained as shown in Figure 7, where the abscissa is the dimensionless basic excitation frequency and ${\eta }_A$ is the dimensionless vibration amplitude of the top of the VNA system pipeline.

As shown in Figure 7a, when the nonlinear stiffness of the elastic constraint remains unchanged, the resonance response amplitude of the system decreases slightly with the increase in the linear stiffness, and the resonance region also shifts to the right, all as a consequence of the fact that the increase in the linear stiffness makes the linearized natural frequency of the structure enlarged and the resonance sensitive region to the excitation frequency shift to the right. From Figure 7b, it can be seen that as the linearized stiffness remains constant and the nonlinear stiffness gradually increases, the frequency drift of the system becomes more and more obvious, and the rightward shift exhibits a significant “hard” characteristic. With the emergence of the drift phenomenon, the resonance response amplitude of the structural system decreases significantly. In summary, as the nonlinear phenomenon gradually becomes prominent, the multivalued response region of the VNA system under a single excitation frequency becomes wider, and the amplitude jump phenomenon becomes more obvious. As can be seen from Figure 7c, when the linearized stiffness of elastic constraints changes synchronously with the nonlinear stiffness, the resonance response amplitude of the VNA system also decreases significantly. However, due to the increase in linearized stiffness, the multivalued response region of the structure at a single excitation frequency does not widen as much as in Figure 7b, especially under Type VII constraints.

In order to study the impact of external input energy on the response of the VNA system, fundamental excitations of different amplitudes are applied to the VNA system. Under the frequency-locked condition of vortex-induced vibration (i.e., ${\sigma }_2=0$), the system response amplitudes of different elastic foundation constraint stiffnesses are qualitatively studied. The constraint type is selected from the previous section, that is,

I:

Linear stiffness 10³ N/m, nonlinear stiffness 10³ N/m³;

VI:

Linear stiffness 10⁴ N/m, nonlinear stiffness 10⁴ N/m³;

VII:

Linear stiffness 10⁵ N/m, nonlinear stiffness 10⁵ N/m³.

The relationship between the steady-state response amplitude of the VNA pipeline and the fundamental excitation amplitude is studied when the main resonance is fully tuned (${\sigma }_1=0$) and the main resonance is detuned (${\sigma }_1=490.33$). Results are shown in Figure 8, where ${\eta }_A$ is the dimensionless amplitude at the top of the VNA system pipeline.

As shown in Figure 8a, when the main resonance parameters of the VNA system are fully tuned with the gradual increase in the foundation acceleration excitation amplitude, the vibration response amplitudes at the top of the VNA pipeline are all unique, determined values, and the system exhibits stable periodic motion. furthermore, the amplitudes grow nonlinearly with the increase in the foundation acceleration excitation, and the growth rate gradually decreases when the foundation acceleration excitation amplitude is large. In Figure 8b, the main resonance parameters are completely detuned, and the steady-state amplitude of the VNA pipeline simultaneously exhibits multiple steady-state solutions within a certain range. Two of the three solutions are stable and the other is unstable, and the specific real response is determined by the initial conditions. There is only one stable solution for the system outside the multistable solution region. When the base acceleration excitation amplitude is below that of the corresponding jump point, the amplitude of the VNA system shows an approximate linear relationship with the acceleration excitation amplitude. When the acceleration excitation amplitude gradually increases and changes to the jump point, the amplitude of the VNA system jumps, jumping from the current amplitude to another higher, stable one. On the contrary, when the acceleration excitation amplitude gradually decreases and changes to the jump point, the amplitude of the VNA system also jumps, jumping from the current amplitude to another lower, stable one.

The magnitude of the constraint stiffness, which limits the amplitude of the VNA system, can be intuitively observed in Figure 8. However, from Figure 8b, it can be seen that as the elastic constraint stiffness increases, not only does the amplitude of the VNA system decrease, but also the region of the multistable solution narrows. Therefore, in practical engineering, in order to achieve the design goal of obtaining deterministic motion of the structure under various operating conditions, the stiffness parameters of the elastic constraint of the VNA system can be appropriately increased within the allowable range of practical engineering.

3.3. System Vibration Response Analysis under Design and Accident Conditions

When designing the VNA system of Hualong-1, the load combinations are divided into the following two types of working conditions for calculation [46,47]:

(1)

Design condition (level O, level A): dead weight + accessory load + OBE seismic load + 100 year return period strong wind load.

(2)

Accident condition (level D): (1) dead weight + accessory load + 71.4 m/s tornado load; (2) dead weight +accessory load + SSE seismic load.

In order to simplify calculations and provide design optimization ideas, this chapter simplifies and adjusts the working conditions. The design working condition combination remains unchanged. The once-in-a-century wind speed load is $V_{out}=68 \mathrm{m}/\mathrm{s}$, and the OBE seismic load is from the nuclear power ACP1000 reactor type using a standardized design. On the basis of eight foundation section parameters, the ground action peak value is 0.1 g, and the frequency is widened by 15%. The seismic response spectrum obtained from the calculation result envelope value is taken. Here, the basic acceleration excitation is simplified for convenience of calculation, ${\ddot{\eta }}_b(\tau )={\displaystyle \sum _{i=1}^5{m}_i{f}_{ai}\cos ({\mathsf{\Omega }}_{gi}\tau )}$. The basic excitation contains five frequency components, corresponding to the first to fourth order resonant frequencies and non-resonant frequency components of the system, ${\mathsf{\Omega }}_{g1}=0.5L^2\sqrt{m_{\mathsf{\Sigma }}/(EI)}$, ${\mathsf{\Omega }}_{g2}=1.128L^2\sqrt{m_{\mathsf{\Sigma }}/(EI)}$, ${\mathsf{\Omega }}_{g3}=7.174L^2\sqrt{m_{\mathsf{\Sigma }}/(EI)}$, ${\mathsf{\Omega }}_{g4}=20.328L^2\sqrt{m_{\mathsf{\Sigma }}/(EI)}$, ${\mathsf{\Omega }}_{g5}=40.177L^2\sqrt{m_{\mathsf{\Sigma }}/(EI)}$. $f_{ai}$ is the acceleration excitation amplitude corresponding to each frequency component, $f_{a1}=0.9 \mathrm{g}$, $f_{a2}=1.8 \mathrm{g}$, $f_{a3}=1.4 \mathrm{g}$, $f_{a4}=0.9 \mathrm{g}$, $f_{a5}=0.5 \mathrm{g}$. $m_i$ is the mass fraction of each frequency component of the base acceleration excitation, $m_1=5\%$, $m_2=70\%$, $m_3=10\%$, $m_4=10\%$, $m_5=5\%$, and the sum of them is 1. Due to the high wind speed in the external flow field, it is not within the frequency locking region of the system’s vortex induced vibration. Based on modal truncation analysis, it is easy to know that when considering the combined effects of vortex induced vibration and foundation excitation, the fourth order modal truncation should be selected for analyzing the vibration response of the VNA system structure. In this section, the Runge-Kutta method is used for numerical calculation of Equations (26) and (27), and the initial value of the calculation is 0. The physical parameters are shown in Table 1. The linear and nonlinear stiffness values of the elastic constraint are taken as 10³ N/m and 10³ N/m³. The dimensionless flow velocity in the pipe is taken as 0.5, regardless of structural damping, and the elastic constraint damping coefficient is 0.3 N/(m/s). The time history response curve and phase diagram of the free end of the VNA system structure can be obtained through calculation, as shown in Figure 9.

Figure 9a shows that the system exhibits a phenomenon similar to multistable vibration under the conditions of multi-frequency foundation excitation and high-velocity vortex-induced vibration. However, the motion changes around the time history response wave peaks and valleys are relatively severe, which can also be seen from the phase diagram of Figure 9b. The phase trace at the free end of the VNA system exhibits a quasi-multistable periodic motion. This is because, under the action of high wind speed in the external flow field, high-frequency vortex shedding causes VIV to superimpose on the basic excitation vibration, causing the system to experience slight shaking while undergoing periodic motion. This should be avoided in practical engineering as much as possible because such repeated and drastic changes will reduce the fatigue life of the structure, and the impact kinetic energy will be transmitted to the constrained end, causing repeated impacts on the constrained end and triggering great stress and local damage to the structure.

After trial and error calculations, it can be found that increasing the linear spring stiffness to 10⁵ N/m, keeping the nonlinear spring stiffness unchanged. At the same time, in order to stabilize the vibration of the free end, an additional elastic constraint is added at a height of 75m, which has a relatively better design effect. After numerical calculation, the time history response curve and phase diagram of the free end of the VNA system structure are shown in Figure 10.

Figure 10a shows that the multi-frequency fluctuations that previously existed near the peak vibration of the system have disappeared, and the system presents a monostable periodic motion. Slight jitters still exist during system motion as a consequence of the high-frequency vortex shedding frequency caused by strong wind, which makes the vortex excitation act upon the foundation excitation vibration. It is difficult to eliminate the vortex-induced vibration effect under the action of a certain foundation acceleration excitation amplitude. This effect is evident in Figure 10b, where the envelope outer contour of the phase diagram is similar to a limit cycle. As a result of slight vibration, the speed of the structure changes frequently with vibration displacement. This design is closer to the needs of engineering optimization.

Referring to the above method, it is easy to determine through numerical calculation that this optimization design approach is also effective for accident conditions. It can effectively increase the security of the system, thereby ensuring its service life.

4. Conclusions

A third-generation nuclear power VNA system model was investigated. Particularly, the complex vibration characteristics of the VNA system are studied by considering multistage limit constraints and the combined action of internal flow velocity, vortex-induced vibration, and foundation excitation. The study found that the linear stiffness of elastic constraints has a significant impact on the first-mode shape of VNA systems. According to the limiting effects of high and low elastic constraints, the structure undergoes modal reordering. The changes in the relative position and installation height of adjacent constraints can significantly change the low-order natural frequency and mode characteristics of the system, with little impact on the high-order. The vibration mode characteristics of a VNA pipeline with Quadratic function outer diameter obviously show that the peak point moves upward, and the vibration mode curve near the bottom of the pipeline is relatively gentle, which has a positive effect on mitigating the impact between the system and the bottom fixed support.

Under the first-order frequency locking condition, the nonlinear stiffness parameters of elastic constraints have a significant impact on the resonance response amplitude of the system. When the main resonance is fully tuned, the system exhibits stable periodic motion. When the main resonance is detuned, the steady-state amplitude of the VNA pipeline shows the phenomenon of multistability within a certain range. An optimization design concept for VNA systems has been proven feasible, which involves adding elastic constraints at the highest possible elevation to increase the stiffness parameters of the elastic constraints while ensuring that the linear spring stiffness parameters are as large as possible compared with the nonlinear stiffness parameters. These research results help to prevent severe oscillations near the amplitude caused by multi frequency excitation in VNA systems under design and accident conditions, ensuring equipment life and personnel safety.

Further research should focus on a more accurate VNA system dynamics model that includes larger slenderness ratios of inner tubes, structural asymmetry, and the influence of surrounding nuclear islands on the convective field in order to investigate its richer dynamic behavior.