Study on Local Vibration Control of the 100 m X-BOW Polar Exploration Cruise Ship

Abstract

A finite element model of a 100 m X-BOW polar exploration cruise ship has been developed. The ship’s frequency response analysis was conducted, with the simulated results closely matching the test data. The maximum discrepancy was 22%, equating to a negligible 0.24 mm/s difference in terms of comfort. This indicates that the simulation meets the standards of engineering precision and validates the model’s accuracy. Utilizing a global modal equivalent mass solution approach, in conjunction with the spatial distribution of local modal mass, a method for calculating the equivalent mass of a single local mode in mixed modes has been devised. This method was applied to determine the equivalent mass of the local vibration region of the 100 m X-BOW ship. Tuned mass dampers (TMDs) were then designed based on this equivalent mass. Analysis reveals that the TMDs achieve a 31 dB vibration absorption effect at a frequency of 13.4 Hz with a mass ratio of 0.05. They also provide a control effect at 10 Hz and 18.8 Hz, corresponding to 3 dB and 2 dB reductions, respectively. The control frequency band is broad, flat, and robust, indicating the effectiveness of the TMDs in mitigating vibrations across a wide range of frequencies.

1. Introduction

The 100 m X-BOW polar expedition cruise ship is the first domestic luxury polar expedition cruise ship. This type of ship has the characteristics of diverse equipment, large power, complex hull structure and layout in the engine room, etc. In the design stage, the main frequency of the main excitation force is known, so the main modal frequency of the whole ship and the main excitation frequency can be staggered by optimizing the design of the main hull, so as to prevent the whole ship from resonating. However, the hull structure is intricate and complex, the natural frequency of the whole ship can be adjusted through the early optimization design, and it is impossible to make the natural frequency of each local structure avoid the excitation frequency. In the structure of the 100 m X-BOW type polar expedition cruise ship, there are large-area local structures, such as the sun platform, viewing platform, etc., whose main natural frequencies are concentrated in the range of 10 Hz~18 Hz [1], which is easily connected with the propeller and the main engine, causing resonance and affecting passenger comfort, so it is necessary to control local structural vibration. At present, the local vibration of the hull structure is mainly accomplished by optimizing the local structure and changing the structural stiffness to avoid the intrinsic frequency of the main excitation [2,3,4,5], such as modifying the properties of the beam–column section to strengthen the local structural strength, but these methods only can be implemented during the construction stage. When the whole ship enters the commissioning or sea trial stage, it is no longer possible to perform local optimization of the structure. Tuned mass dampers (TMDs) can effectively control structural vibration. Zong Zhi-xiang, Wen Yongpeng [6] installed multiple dynamic vibration absorbers in the body of urban rail vehicles. According to the principle of multiple dynamic vibration absorbers and the characteristics of urban rail vehicles, including multiple dynamic vibration absorbers is established, and a design method of multiple power absorbers applicable to the body of urban rail vehicles is proposed. Zhang Ya-xin, Lei Xiaoyan [7] designed and made a similar test model of the rail box girder structure by taking the railway 32 m simply supported box girder bridge as a prototype. On the basis of verifying the reliability of the test modal, add the additional control of TMDs of the first-order and second-order modal vibration. The vibration response of the measurement point before and after the vibration reduction is obtained by the performance of a single-point vibration excitation test on the test model, and the vibration reduction performance of the TMD on the box girder bridge is analyzed. Tang Si-cong, Wang Hailong [8] analyzed the position-related terms in the multiple tuned mass dampers (MTMDs) equation of motion, studied the participation of position in the MTMD vibration reduction control, and proposed an MTMD position optimization method based on the finite element method and the mode shape decomposition method; on this basis, a joint optimization strategy of MTMD parameters and position is formulated. Lian Ji-jiang, Zhaoyue [9] studied the feasibility of the application of a new type of eddy-tuned mass damper (EC-TMD) based on offshore wind torch-type infrastructure from tow aspects of theoretical reasoning and engineering application, aiming at the problem of large vibrations detrimental to the engineering safety of offshore wind power structures under extreme wind load. The results showed that under extreme wind load conditions, EC-TMDs can effectively reduce the vibration displacement amplitude of the tower cylinder top by 21~33%, which indicates that EC-TMD has engineering application value for vibration reduction of offshore wind torch-type infrastructure. Yang Ying, Jin Licheng [10] used TMD as a vibration control measure to reduce the dynamic response of suspension tunnel tubes under moving loads.

Since 1909, when Frahm first used TMD to control lateral tumbling vibrations in ships, TMD has been used extensively in the vibration control of ship structures and equipment [11]. Li Bing, Liu Guang-shuo [12] studied the effect of the mass ratio of TMD on the vertical vibration damping effect of steel plants and concluded that the damping effect of the plant with TMDs placed at the maximum response is better than that of the plant placed close to the excitation. Chen Ling-shuang, Li Shu-jin [13] investigated the effect of pendulum tuned mass damper (PTMD) placed in the nacelle on the in-plane vibration response of Barge floating wind turbines and optimized the collision parameters. Zhang Kaidong, Tian Shizhu [14] investigated the effect of air-damped TMDs formed by replacing the oil damping in conventional tuned mass dampers with air damping on the vibration control of simply supported girder bridges under pedestrian loads. Zhiwei Su, Zheng Z, Huang X, et al. [15] propose a tuned mass damper with negative stiffness using TMD applied to the vibration control of a propulsion shaft system, and provides a theoretical and design approach. The control effect of a tuned mass damper with negative stiffness was compared with that of a conventional tuned mass damper, and it was found that the control effect of a tuned mass damper with negative stiffness was 10 dB better than that of a conventional tuned mass damper. Mohammad Reza Ghasemi, Shabakhty N, Enferadi M H [16] propose a TMD using shape memory metals as dampers; a theoretical design method is provided in the paper, and it is used for wave-induced vibrations of jack-up marine platforms. At the same time, four calculation conditions were designed: 2-year regression period, 5-year regression period, 10-year regression period, and 100-year regression period. Based on the calculation results of the four conditions, it can be seen that the average reduction of vibration displacement of the platform reaches 43%. Jiang-hai Wu, Zhu H, Sun Y, et al. [17] applied TMD to pipeline vibration control. Since the control band of a single TMD is narrow, multiple TMDs were installed on the pipeline in order to improve the control bandwidth, which was modeled and simulated using finite elements and compared experimentally; the method effectively improved the control bandwidth of TMDs. At present, TMD is mainly used for the overall vibration mode control, and there are few studies on the local vibration mode control. When the local vibration of hull structure occurs, it is generally the coupling of multiple local modes, and it is difficult to directly judge the participation degree of each local mode.

When the 100 m X-BOW polar expedition cruise was in the commissioning or sea trial phase, tests revealed local structural vibrations at the sun deck on deck 7, but it was not possible to optimize the local structural adjustments at this time, so this paper proposes the use of a tuned mass damper (TMD) to control local structural vibration. TMD can be subsequently installed without extensive modification to the hull structure.

The primary objective of this study is to introduce an innovative approach to control local structural vibrations on the 100 m X-BOW using TMDs. Our methodological contribution lies in the development of a local mode equivalent mass identification technique, which is crucial for the design of TMDs that target specific local modes within the mixed-mode vibration spectrum of the hull structure. This approach is significant as it allows for the effective control of vibrations in critical areas, such as the sun deck on deck 7, which was identified to have a local resonance frequency of 13.4 Hz during the commissioning phase.

To address the problem that it is difficult to judge the degree of participation of a single local mode when local vibration occurs in the hull structure, this paper proposes a local mode equivalent mass identification method based on the intrinsic mode method, which combines the spatial distribution of the mode masses in the mixed mode, identifies the single local mode in the mixed mode and calculates the equivalent mass of the local mode, and solves for the optimal TMD parameters suitable for the local mode based on the equivalent mass of the local mode. To analyze the optimum mass ratio, the mass ratio was set to 0.005, 0.01, 0.02, 0.05, 0.1, and 0.2, and the frequency response was calculated for each of these mass ratios. The analysis indicates that a tuned mass damper, with a mass ratio of 0.05, achieves a significant 31 dB absorption effect at a frequency of 13.4 Hz. Furthermore, it demonstrates effective control over the amplitude response at frequencies of 10 Hz and 18.8 Hz, with respective control levels of 3 dB and 2 dB. This tuned mass damper also offers a more uniform control band and enhanced robustness across a range of frequencies.

2. Method

2.1. Optimal Parameter Design of TMD Based on Fixed Point Theory

The mechanical model of TMD is shown in Figure 1, which is mainly composed of a mass element, elastic elements, and damping elements. According to the literature [18], its vibration suppression mechanism is as follows: by adjusting the natural frequency of the tuned mass damper to make it consistent with the excitation main frequency, the vibration energy of the main structure is transferred to the mass block, and the TMD gradually absorbs the vibration energy, while the vibration of the main structure is gradually reduced, which can be reduced to zero from the theoretical point of view.

If the displacement of the main systems is $x_1$, and the mass displacement of the dynamic vibration absorber is $x_2$, then the equation of motion can be expressed as

$$\left\{\begin{array}{l}M\ddot{x}+c\dot{x}+\left(K+k\right)x_1-c{\dot{x}}_2-k{x}_2=f=F\sin \omega t\hfill \\ m{\ddot{x}}_2-c{\dot{x}}_1-k{x}_1+c{\dot{x}}_2+k{x}_2=0\hfill \end{array}\right.$$

(1)

Assume that f = F$e^{i\omega t}$, $x_1=\overline{X_1}{e}^{i\omega t}$, and $x_2$=$\overline{X_2}{e}^{i\omega t}$. Here, $\overline{X_1}$ and $\overline{X_2}$ are the complex amplitude of $x_1$ and $x_2$, respectively. The solution of Equation (1) can be expressed as

$$\frac{X_1}{X_{st}}\left(\omega \right)=\sqrt{\frac{{\left({\gamma }^2-{\lambda }^2\right)}^2+{\left(2\zeta \lambda \right)}^2}{{\left[\left(1-{\lambda }^2\right)\left({\gamma }^2-{\lambda }^2\right)-\mu {\gamma }^2{\lambda }^2\right]}^2+{\left[1-\left(1+\mu \right){\lambda }^2\right]}^2{\left(2\zeta \lambda \right)}^2}}$$

(2)

$$\frac{X_2}{X_{st}}\left(\omega \right)=\sqrt{\frac{{\left({\gamma }^2\right)}^2+{\left(2\zeta \lambda \right)}^2}{{\left[\left(1-{\lambda }^2\right)\left({\gamma }^2-{\lambda }^2\right)-\mu {\gamma }^2{\lambda }^2\right]}^2+{\left[1-\left(1+\mu \right){\lambda }^2\right]}^2{\left(2\zeta \lambda \right)}^2}}$$

(3)

where ${\Omega }_n$ = $\sqrt{\frac{K}{M}}$ represents the natural angular frequency of the main vibration system, ${\omega }_n$ = $\sqrt{\frac{k}{m}}$ represents the natural angular frequency of TMD, $X_{st}$ = $\frac{F}{K}$ represents the static deformation of the main vibration system, µ = $\frac{m}{M}$ represents the mass ratio, ζ = $\frac{c}{2m{\Omega }_n}$ represents the damping ratio, λ = $\frac{\omega }{{\Omega }_n}$ represents the ratio of the excitation frequency to the natural frequency of the main structure, and γ = $\frac{{\omega }_n}{{\Omega }_n}$ represents the ratio of the TMD natural frequency to the natural frequency of the main structure.

According to the literature [19], the optimal homology conditional of TMD is

$$\gamma =\frac{{\omega }_n}{{\Omega }_n}=\frac{1}{1+\mu }$$

(4)

The optimal damping ratio of TMD is

$${\zeta }_{opt}=\sqrt{\frac{3\mu }{8{\left(1+\mu \right)}^3}}$$

(5)

According to Equations (4) and (5), the mass ratio is crucial for solving the optimal parameters.

2.2. Local Mode Equivalent Mass Solution Method

A discontinuous discrete multi-degree of freedom system is established, as shown in Figure 2.

After decoupling the multi-DOF system, let the mode mass of the i-th mode be $M_i$ and the vibration velocity be $v_i$, then the kinetic energy of the mode is

$$T_i=\frac{1}{2}{M}_i{v_i}^2$$

(6)

The kinetic energy of the mode i-th is a constant value, but the values of the mode mass $M_i$ and vibration velocity $v_i$ are not unique and are related to the scaling factor of the mode shape vector, assuming the shape vector of the i-th mode is

$$X_i={\left\{x_{1i},x_{2i},\cdots ,x_{ni}\right\}}^T$$

(7)

In this mode, the vibration velocity of each point is proportional to the mode shape, such that

$$V_i={\left\{\rho x_{1i},\rho x_{2i},\cdots ,\rho x_{ni}\right\}}^T=\rho {\left\{x_{1i},x_{2i},\cdots ,x_{ni}\right\}}^T\left(m/s\right)$$

(8)

Then, the total kinetic energy of the system in this mode can be expressed as

$$T_i=\frac{1}{2}{\rho }^2\left(m_1{x}_{1i}^2+m_2{x}_{2i}^2+\cdots +m_n{x}_{ni}^2\right)$$

(9)

The mode amplitude of the $j$-th particle is taken as the scaling factor, such that

$$\begin{array}{l}{T}_i=\frac{1}{2}\left(m_1{\left(\frac{x_{1i}}{x_{ji}}\right)}^2+m_2{\left(\frac{x_{2i}}{x_{ji}}\right)}^2+\cdots +m_j+\cdots +m_n{\left(\frac{x_{ni}}{x_{ji}}\right)}^2\right){\left(\rho x_{ji}\right)}^2\\ =\frac{1}{2}{M}_{i-j}{\left(\rho x_{ji}\right)}^2\end{array}$$

(10)

This equation is compared with (9); it can be known that $M_{i-j}$ is the overall modal mass of the $i$-th mode with $x_j$ as the scaling factor, and point $j$ is the control target point of the $i$-th mode; $M_{i-j}$ can be used as the structural reference quality when TMD controls the $i$-th mode; and $m_a{\left(\frac{x_{ai}}{x_{ji}}\right)}^2$ is the mass distribution of modal $M_{i-j}$ at each point in space, also called the contribution of $m_a$ to the overall modal mass $M_{i-j}$.

When the k-th mode of the multi-DOF system is assumed to be the local mode, two local modes simultaneously occur; as shown in Figure 3, only $m_{j-2}, m_{j-1}, m_j, m_{j+1}, m_{j+2}$ and $m_{l-1}, m_l, m_{l+1}$ vibrate locally.

So, the total kinetic energy is

$$\begin{array}{l}{T}_k=\frac{1}{2}{M}_{k-j}{\left(\rho x_j\right)}^2\\ =\frac{1}{2}\left(m_{j-2}{\left(\frac{x_{(j-2)k}}{x_{jk}}\right)}^2+m_{j-1}{\left(\frac{x_{(j-1)k}}{x_{jk}}\right)}^2+m_j+m_{j+1}{\left(\frac{x_{(j+1)k}}{x_{jk}}\right)}^2+m_{j+2}{\left(\frac{x_{(j+2)k}}{x_{jk}}\right)}^2+\right.\\ \left.m_{l-1}{\left(\frac{x_{(l-1)k}}{x_{jk}}\right)}^2+m_l{\left(\frac{x_{lk}}{x_{jk}}\right)}^2+m_{l+1}{\left(\frac{x_{(l+1)k}}{x_{jk}}\right)}^2\right){\left(\rho x_{jk}\right)}^2\end{array}$$

(11)

Assuming

$$M_{local1,k-j}=m_{j-2}{\left(\frac{x_{(j-2)k}}{x_{jk}}\right)}^2+m_{j-1}{\left(\frac{x_{(j-1)k}}{x_{jk}}\right)}^2+m_j+m_{j+1}{\left(\frac{x_{(j+1)k}}{x_{jk}}\right)}^2+m_{j+2}{\left(\frac{x_{(j+2)k}}{x_{jk}}\right)}^2$$

(12)

$$M_{local2,k-j}=m_{l-1}{\left(\frac{x_{(l-1)k}}{x_{jk}}\right)}^2+m_l{\left(\frac{x_{lk}}{x_{jk}}\right)}^2+m_{l+1}{\left(\frac{x_{(l+1)k}}{x_{jk}}\right)}^2$$

(13)

Then

$$T_k=\frac{1}{2}\left(M_{local1,k-j}+\right.\left.M_{local2,k-j}\right){\left(\rho x_{jk}\right)}^2$$

(14)

2.3. Conversion Method of Concentration Quality Matrix

By the finite element method, the internal displacement of the cell is generally expressed by interpolation, so that the displacement of each node of the cell is ${\left\{\delta \right\}}^e$, then the internal displacement of the cell is

$$\left\{u\right\}=\left[N\right]{\left\{\delta \right\}}^e$$

(15)

where $\left[N\right]$ is the shape function matrix; the in-cell acceleration is

$$\left\{\ddot{u}\right\}=\left[N\right]{\left\{\ddot{\delta }\right\}}^e$$

(16)

where ${\left\{\ddot{\delta }\right\}}^e$ is the acceleration matrix of the cell nodes.

Using the inertial force ${\left\{\ddot{\delta }\right\}}^e$ in the cell as a volume-distributed load and distributing it to the nodes of the cell according to the distribution principle, we obtain

$${\left\{Q\right\}}_{\rho }^e=-{\int \left[N\right]}^T\rho \left\{\ddot{u}\right\}dv$$

(17)

Substituting Equation (15) into Equation (16) yields

$$\begin{array}{cc}\hfill {\left\{Q\right\}}_{\rho }^e& =-{\int \left[N\right]}^T\rho \left[N\right]{\left\{\ddot{\delta }\right\}}^{e}dV\hfill \\ \hfill & =-{\int \left[N\right]}^T\rho \left[N\right]dV{\left\{\ddot{\delta }\right\}}^e=-{\left[M\right]}^e{\left\{\ddot{\delta }\right\}}^e\hfill \end{array}$$

(18)

where ${\left[M\right]}^e$ is a consistent mass matrix, which is not diagonal; there is some inertial coupling between cells.

The consistent mass matrix can be equated to a diagonal matrix, and the equivalent matrix is the concentrated mass matrix.

Let the total kinetic energy of the structure be T, and the total kinetic energy of the sum of the kinetic energies of the cells, such that

$$T=\sum T^e$$

(19)

The kinetic energy of the unit is

$$\begin{array}{c}{T}^e=\frac{1}{2}{{\int }_{V^e}\left\{\dot{u}\right\}}^T\rho \left\{\dot{u}\right\}dV=\frac{1}{2}{{\int }_{V^e}\left(\left[N\right]{\left\{\dot{\delta }\right\}}^e\right)}^T\rho \left[N\right]{\left\{\dot{\delta }\right\}}^{e}dV\hfill \\ =\frac{1}{2}{\left\{{\dot{\delta }}^e\right\}}^T{{\int }_{V^e}\left[N\right]}^T\rho \left[N\right]dV{\left\{\dot{\delta }\right\}}^e=\frac{1}{2}{\left\{{\dot{\delta }}^e\right\}}^T{\left[M\right]}^e{\left\{\dot{\delta }\right\}}^e\end{array}$$

(20)

When the overall mass is equally distributed among all cells, the kinetic energy of each cell is

$$T^e={\sum }_{k=1}^n\frac{1}{2}{m}_k{\dot{\delta }}_k^2$$

(21)

where $m_k$ is the nodal mass corresponding to the displacement component of node $k$, and $n$ is the number of cell degrees of freedom, at which point the cell mass matrix is diagonal, such that

$${\left[M\right]}^e=\left[\begin{array}{ccccc}{m}_1& 0& 0& \cdots & 0\\ 0& m_2& 0& \cdots & 0\\ 0& 0& m_3& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & m_n\end{array}\right]$$

(22)

For thin beam and thin plate and shell units, the kinetic energy of rotation of the nodes can be neglected when using the concentrated mass matrix to represent the structural mass. Assume a two-node straight beam cell, $i$ and $j$ being the two nodes, and $M$ being the mass of the cell. However, when the unit masses are equally distributed by the two nodes, the nodal displacements can be expressed as

$${\left\{\delta \right\}}^e=\left[{\omega }_i{\theta }_i{\omega }_j{\theta }_j\right]$$

(23)

The kinetic energy is

$$T^e=\frac{1}{2}\frac{M}{2}{\dot{\omega }}_i^2+\frac{1}{2}\frac{M}{2}{\dot{\omega }}_j^2$$

(24)

The matrix of concentrated masses is

$${\left[M\right]}^e=\left[\begin{array}{cccc}\frac{M}{2}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& \frac{M}{2}& 0\\ 0& 0& 0& 0\end{array}\right]$$

(25)

3. Results

3.1. The 100 m X-Bow Local Vibration Test

In order to analyze the mode of the polar expedition cruise, a finite element model is established. The total length of the model is 104 m, the ship’s width is 18 m, the waterline height is 5.1 m, and the weight of the ship is 4265 tons. A centralized mass point simulation device is used, with a coupling between the centralized mass point and the deck picking, and an increase in material density is used to simulate the distribution of additional mass. The added mass and the center of gravity of the model strictly follow the report of weight and center of gravity, as shown in Figure 4. The mesh size is mainly 400 $\times$ 400 mm, the plate structure is quad 4 element and tria3 element, the beam structure is bar2 element, and the damping factor of the structure is 2% [1]. The 100 m X-BOW Polar Expedition Cruiser has twin propellers and twin rudders with a 3100 mm oar diameter and 80% rated working speed of 192 RPM; the main engine parameters are shown in

Table 1. Host parameters.

Specification	6L20	8L20
Rated power	1200 kW	1600 kW
Rated speed	1000 RPM	1000 RPM
Weight	9.3 t	11 t
Quality	2	2

. According to the literature [20], the main excitation forces of the ship are main engine excitation and propeller pulsation excitation, and the excitation spectrum is shown in Figure 5.

The excitation in Figure 5 was loaded into the finite element model to obtain the vibration velocity of each cabin. Several typical regions were selected to compare the vibration response spectrum curve of the central node in these regions with the test value. The comparison curve is shown in Figure 6. It can be seen from the figure that the dominant frequency positions of the simulation curve and the test curve can correspond well, and the trend of the curves is basically the same. The effective value pair of vibration velocity is shown in

Table 2. Comparison of vibration response simulation value and test value effective value.

Deck	Area	Test Value (mm/s)	Simulation Value	Error
DECK7	The sun deck	1.07	0.83	22.43%
DECK6	Passenger compartment	0.046	0.04	13.04%
DECK5	The restaurant	0.31	0.3	3.23%
DECK4	Passenger compartment	0.22	0.18	18.18%
DECK3	Passenger compartment	0.11	0.094	14.55%
DECK2	Laundry room	0.09	0.08	11.11%

. The area with the largest error is the sunshine deck on the seventh deck, which is 22%, but the difference is only 0.24 mm/s. According to the literature [21] and the literature [22], from the perspective of comfort, this error can be ignored, so the simulation results meet the engineering accuracy requirements. It shows that the 100 m X-Bow polar cruise ship modal can reflect the actual vibration characteristics of the hull. The reasons for this error mainly come from two aspects: (1) Dressing, outfitting, and interior components are omitted in the finite element model, such as the floating bottom plate and interior partition in the four-deck and passenger compartments, which affect the admittance of the transfer path to a certain extent; (2) In the actual measurement, the cruise ship is also affected by the excitation of wind and waves, pumps, steering engines, and fans. For example, the second deck–laundry room is affected by the air compressor in the adjacent cabin, so the error is relatively large compared with other areas.

3.2. Local Vibration Control Analysis of TMD

In order to carry out TMD control on the local vibration of the 100 m polar expedition cruise ship, the main frequency of the excitation source should be determined first, and the frequency search range should be determined according to the main frequency of the excitation source, which is generally ±15% of the main frequency of the excitation source. The local modes of the 100 m polar expedition cruise ship should be searched in the frequency search range, and the local modes to be controlled should be selected. The modal amplitude and modal mass of the local structure were extracted by Nastran, and the equivalent mass of the local mode was calculated according to Equations (12) and (13). Using the equivalent mass, the optimal frequency ratio of TMD was obtained from Equation (4), and the optimal damping ratio of TMD was obtained from Equation (5). The vibration reduction effect of TMD was analyzed by the finite element method.

The X-Bow polar expedition cruise vessel, with a length of 100 m, exhibits a primary excitation frequency that is pivotal for its propeller blade excitation. This frequency is identified at 12.5 Hz, with a search bandwidth extending from 11.25 Hz to 13.75 Hz. Within this specified bandwidth, a comprehensive modal analysis of the ship’s structural integrity was conducted. Notably, at a frequency of 13.4 Hz, pronounced local modal responses were observed at the stern section of the seventh deck, as depicted in Figure 7. A detailed examination of the data, as presented in Table 2, reveals that the measured vibration velocity at this specific location reached a peak value of 1.07 mm/s. This value surpasses all other recorded measurements within the scope of Table 2. Consequently, it is inferred that the localized structural dynamics at the 13.4 Hz mode are particularly susceptible to the resonant effects induced by the propeller’s blade frequency excitation.

Because this mode shape is coupled with other local mode shapes, the equivalent mode mass of this local mode shape cannot be calculated by the whole modal. Therefore, it is necessary to extract the centralized mass matrix and modal deformation of each node in the mode shape and use Equation (14) to calculate the equivalent mass of the local mode. It can be seen from the test that vertical vibration is the main factor affecting the comfort level. So, the vertical vibration mode analysis is given priority to.

There are a total of 1018 nodes in this local structure, and the MATLAB R2019b (9.7.0.1190202) program is used to extract the centralized mass matrix and modal deformation matrix of nodes, as shown in Equations (26) and (27).

$$M=\left.\left[\begin{array}{cccccccccc}0.011& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& 0.0288& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& 0.0348& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& 0.0374& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0.0341& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0.037& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& 0& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0.0641& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0.0323& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \ddots & \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0.0059\end{array}\right]\right\}\left(1018\times 1018\right)$$

(26)

$$D=\left.\left[\begin{array}{cccccccccc}-0.0637& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& -0.0342& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& -0.053& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& -0.0279& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0.0604& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0.0888& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& 0& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0.1127& \hspace{1em}& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& 0.1397& \hspace{1em}& \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \ddots & \hspace{1em}\\ \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& \hspace{1em}& -0.0458\end{array}\right]\right\}\left(1018\times 1018\right)$$

(27)

By substituting the diagonal elements in Equations (26) and (27) into Equation (14), Equation (28) can be obtained, such that

$$T={{\displaystyle \sum _{i=1}^{1018}{M}_i\left(\frac{D_i}{D_{max}}\right)}}^2$$

(28)

where $D_{max}$ is the maximum value in the modal deformation.

The equivalent mass of the local structure at 13.4 Hz is 3.46 tons. In order to analyze the influence of the mass ratio on the vibration absorption effect, the mass ratio was set as 0.005, 0.01, 0.02, 0.05, 0.1, and 0.2, and the mass ratio was substituted into Equations (4) and (5) to obtain the frequency ratio γ and the optimal damping ratio ${\xi }_{opt}$ under the optimal homology condition, through calculation; the parameters are shown in

Table 3. Parameters of TMD.

Mass Ration	Total Quality of TMD (T)	Total Stiffness of TMD (N/m)	$\mathit{\gamma }$	${\mathit{\xi }}_{\mathit{o}\mathit{p}\mathit{t}}$
0.50%	0.017	122.8	0.995	0.043
1%	0.035	243	0.99	0.06
2%	0.069	476.7	0.98	0.085
5%	0.173	1124.7	0.95	0.13
10%	0.346	2049.5	0.91	0.185
20%	0.69	3444.3	0.83	0.25

. Since the mode is the second-order mode with double peaks, two TMDs need to be set, and one TMD is loaded on each peak. The total mass and total stiffness of the two TMDs correspond to the parameters in Table 3, and the TMD loading diagram is shown in Figure 8, which is connected with node 2566 and node 41,807, respectively. In this case, a point cell is used to represent the mass element, a Beam cell is used to represent the spring element, and the spring and damping properties are provided.

The two nodes in the figure indicate the positions of the TMDs. Their respective identification numbers are node 2566 and node 41,807.

The frequency response analysis of the unit force of the whole ship is carried out, and the frequency response curve under each working condition at node 2566 is taken, as shown in Figure 9. Form Figure 9a, the local structure has three orders of modes within 20 Hz: the first order is 10 Hz, the second order is 13.4 Hz, and the third order is 18.8 Hz, where the second-order mode is the main control target. After loading the TMD, the control effect of the response at 13.4 Hz is more obvious, but some working conditions have a certain amplification effect on the response at 10 Hz and 18.8 Hz. The response of the frequency band from 12 Hz to 15 Hz is amplified, as shown in Figure 9b, which shows that when the mass ratio is small, as the mass ratio increases, the vibration absorption effect of the TMD becomes obvious, and the response curve is flat.

In order to obtain the most suitable TMD scheme for this structure from Table 3, the control effect of the TMD response at 13.4 Hz and the amplification effect at 10 Hz and 18.8 Hz need to be comprehensively evaluated. The control effect can be evaluated through Equation (29), such that

$$\Delta L=20\lg (X_0/X_1)$$

(29)

where $X_0$ is the vibration amplitude at 13.4 Hz when TMD is not installed, $X_1$ is the vibration amplitude at 13.4 Hz after TMD is installed, and ∆L is the control effect.

When evaluating the control effect of TMD, the larger ∆L is, the better. When evaluating the magnification effect of TMD, the smaller ∆L is, the better.

The control effect of each working condition is evaluated. Figure 10 shows the control effect at 13.4 Hz. It can be seen from Figure 10 that when µ is in the range of 0.005~0.05, the control effect of TMD is significantly improved from 24 dB to 31 dB. When µ > 0.05, the control effect of TMD does not increase significantly, increasing from 31 dB to 33 dB. When µ = 0.1 and µ = 0.2, the control effect is basically not different; therefore, it is recommended the quality ratio µ ≤ 0.05. Figure 11 and Figure 12 show the control effect at 10 Hz and 18.8 Hz, respectively. As can be seen from the figures, at these two frequency points, there is always a mass ratio ${\mu }^{\prime }$, which makes ∆L reach the peak; when $\mu =\mu \prime$, TMD has the worst control effect at 10 Hz and 18.8 Hz, and even has the amplification effect. At 10 Hz, $\mu \prime =0.01$, the TMD amplitude effect is 4 dB; at 18.8 Hz, $\mu \prime =0.02$, the TMD amplification effect is 3.7 dB; and when $\mu \ne \mu \prime$, TMD has a certain control effect on the amplitude at 10 Hz and 18.8 Hz.

To sum up, for this local structure, it is recommended to take $\mu =0.05$; at this time, the control effect at 10 Hz is 31 dB, which has the highest control cost performance. At the same time, the amplitude response at 10 Hz and 18.8 Hz also has a certain control effect, which is 3 dB and 2 dB, respectively.

After TMD parameters are selected, TMD is installed on the model loaded with the excitation source, and the vibration velocity response is calculated. The vibration velocity response before and after TMD installation is compared, as shown in Figure 13. It can be seen from Figure 13 that after TMD installation, the response near 13.4 Hz decreases from 0.8 mm/s to 0.13 mm/s. It shows that TMD has a good control effect.

4. Conclusions

According to the global mode equivalent mass solution method and the spatial destruction principle of local mode mass, a solution method of single local mode equivalent mass in mixed modes is proposed, and the single local mode equivalent mass of the seventh-layer sun deck in the 100 m X-Bow polar exploration cruise ship is solved by using this method.

The finite element model of the 100 m X-Bow polar cruise ship was established to predict the structural vibration response. Six typical cabins were selected, and the simulation value of the vibration response of these six cabins was compared with the test value. The frequency response curve trend was basically the same, and the effective value error of the vibration response was within 22%, but the difference was only 0.24 mm/s. In the authors’ opinion, the error can be ignored. So, the simulation results meet the engineering accuracy requirements and verify the effectiveness of the model, which can be used for vibration control analysis.

The main excitation frequency was determined to be 12.5 Hz. Based on this frequency, the local mode search of the whole ship model was carried out, and the search range was 11.25–13.75 Hz. Finally, the natural frequency of the seventh-layer sun deck was determined to be in the range of 11.25–13.75 Hz, which was consistent with the propeller excitation blade frequency, and there was a resonance risk. This area is the area with the largest test response, and the vibration speed is 1.07 mm/s. It is necessary to control the vibration in this area.

Using the local mode equivalent mass solution method, the local equivalent mass of the seventh layer of the sun deck is solved. The mass ratio µ is set as 0.005, 0.01, 0.02, 0.05, 0.1, and 0.2. According to the six mass ratios, the six optimal parameters of TMD are determined. The optimal mass ratio of TMD in this region is determined to be 0.05. When the mass ratio of TMD is 0.05, the TMD has the vibration absorption effect of 31 dB at 13.4 Hz, and the amplitude responses at 10 Hz and 18.8 Hz also have certain control effects, which are 3 dB and 2 dB, respectively. The overall control frequency band is flat, and the robustness is good. After TMD was installed on the model loaded with the excitation source, the response around 13.4 Hz decreased from 0.8 mm/s to 0.13 mm/, indicating that TMD has a good control effect.

In summary, this study not only enhanced the understanding of local vibration control for polar exploration cruise ships but also effectively reduced the vibration levels in key areas by applying TMD technology, thereby improving the safety and comfort of the vessel. This work demonstrates that significant improvements in the vibration control effectiveness of ship structures can be achieved through precise modal analysis and parameter optimization, which has important practical application value for ship design and operational maintenance.