^{1}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

^{4}

The vibration transmission performance of a floating raft system with attached pipes is investigated in this paper. The frequency response function-based (FRF-based) substructure synthesizing method whose accuracy has been verified by numerical simulations and experiment is applied for modeling the system. The power flow through the transmission paths is used for exploring the additional vibration transmission path provided by the attached pipes. The results show that the existence of the additional transmission paths caused by the pipes breaks the symmetries of the system, which leads to the enhancement of the coupling between each substructure. Consequently, it degrades the vibration isolation performance of the raft system. Moreover, a parametric study is performed to investigate the effects on the mean-square velocity of the hull of the attached pipes, which gives a brief guideline for designing the attached pipes.

Floating raft system is widely used in ships and submarines due to its excellent performance on vibration isolation and noise reduction. The prediction and control of vibration isolation performance are the critical issues in designing a floating raft system. With the rapid developments of the acoustic design of submarine, the demand for more precise and efficient modeling of floating raft system is becoming urgent.

In the past decades, many researches have been done on the floating raft system, and most of these works were based on two-stage isolation theory [

Transmissibility has often been used in the evaluation of vibration performance. Beyond mobility, the use of power flow in the problem of floating raft system is very valuable because it combines both force and velocity in a single quantity. The concept of power flow incorporating both force and velocity characteristics was proposed by Goyder and White [

From a review of the literature, it appears that the traditional modeling of the floating raft system only concerns the vibration transmission along the supporting path (machineries-raft-base) while neglecting the influence of the nonsupporting attached equipment like pipes, cables, and so on. A great amount of engineering practices indicate that the attached equipment becomes the major vibration transmission path in the frequency domains dominated by the resonances of it; thus they cannot be ignored. Therefore, with the introduction of the disturbance we mentioned before, a 40 dB/decade vibration isolation performance possessed by ideal two-stage isolation system usually cannot be achieved in practical floating raft system [

The aim of this paper is therefore to develop a generalized FRF-based dynamic modeling method of a practical floating raft system with attached pipes, attaining a deeper understanding on its vibration performance. Numerical simulations by the FEM and experimental study are then conducted to validate the present method, respectively. Subsequently, a general discussion is provided to reveal the power flow characteristics of the system with and without attached pipes. Furthermore, a parametric study is performed to investigate the influence of the mass and connecting stiffness of the attached pipes on the mean-square velocity of the hull. The work of this paper may provide a reference for the design of attached equipment in a practical floating raft system.

As shown in Figure

Sketch of the floating raft system with attached equipment.

In this paper, a FRF-based substructure synthesizing method is chosen for modeling the floating raft system with attached pipes. The whole system can be divided into five substructures: the base-hull

(a) Sketch of the FRF coupling of the synthesis; (b) synthesizing process.

The isolators are analytically described as impedance matrix

The FRF representation of base-hull

And continuing the synthesizing process twice, one can obtain the whole FRF representation of the system as follows:

Since the FRF matrix of the whole system has been obtained, locating the external force to the corresponding position in the FRF matrix, then we can get the vibration response of arbitrary coupling points. Taking the coupling points of base-hull, raft, and attached pipes, for example, when the external force is exerted on the centroids of machineries, these vibration response expressions are given as follows:

The response of coupling points on base-hull

where

The response of coupling points on raft

where

The response of coupling points on attached pipes

where

After obtaining the vibration response of the coupling points, one can get the isolator transmission force by using the upper and lower vibration response of the isolator and its impedance matrix. The transmitted force on the base via the lower isolator of the floating raft can be expressed as

And the transmitted force on the hull via the attached pipes can be expressed as

Based on the obtained force and vibration response, the transmitted power flow on base-hull via supporting path and nonsupporting path can be written as

In addition, based on the transmission force, one can get the vibration response on the surface of the hull which can be expressed as

Then, its normal mean-square velocity can be expressed as

It should be noted that the calculated results subsequent are represented as decibel (dB). The physical quantities such as displacement, velocity, and acceleration can be expressed as

In this section, a typical model of floating raft system is built to verify the present modified FRF-based substructure synthesizing method. As illustrated in Figure ^{4} N/m, 6 × 10^{4} N/m, 1.45 × 10^{4} N/m, 6 × 10^{5} Nm/rad, 6 × 10^{5} Nm/rad, 6 × 10^{5} Nm/rad, and 0.01, respectively. The stiffness and the structural damping coefficients of the lower isolator are 2.22 × 10^{4} N/m, 2.42 × 10^{4} N/m, 5.8 × 10^{4} N/m, 2.42 × 10^{5} Nm/rad, 2.42 × 10^{5} Nm/rad, 2.42 × 10^{5} Nm/rad, and 0.01, respectively. The masses of the three machines are 65 kg, 50 kg, and 65 kg, respectively.

Sketch of the floating raft system: (a) front view, (b) sectional view.

The dynamic model of the floating raft system is built towards two methods. First one is FRF-based synthesizing method proposed by this paper, which obtains the FRF matrix of the system by combining the FRF of the substructures; the second one is FEM which solves the vibration response of the system by building the FEM model. What comes into notice is that when modeling the system by using FRF-base synthesizing method, the FRF matrices of the substructures can also be obtained by FEM method. Based on the FRF matrices of the substructures, the FRF matrix of the floating raft system is obtained by the FRF-based synthesis method.

Firstly, considering the operating condition that the external force is single-sourced and single-direction, the vertical unit force is applied on the centroid of machine 2 which is also the centroid of whole system. Secondly, in consideration of the operating condition that the external forces are multisourced and multidirection, the three machines are all excited by an

Comparison of the synthesized FRFs and the simulated results excited by single-sourced single-direction force: (a) displacement of the lower isolator in

Comparison of the synthesized FRFs and the simulated results excited by multisourced multidirection forces: (a) displacement of the lower isolator in

In this section, two kinds of test models are established to verify the correctness of the method proposed in this paper. The first one includes floating raft, hull with base, and rigid body mass and based on that the second one includes the attached pipes.

For the test model with no attached pipes shown in Figure

(a) Test model of the raft; (b) test model of the hull with base; (c) test model of the floating raft system without attached pipes.

A vertical exciting force was applied at the center of mass 2. It can be observed from Figure

Comparison of the measured FRFs and the synthesis results for floating raft system without attached pipes: (a) driving FRF of the exciting point; (b) transfer FRF from exciting point to one point on top surface of the raft; (c) transfer FRF from exciting point to one point on bottom surface of the raft.

Besides that, a floating raft system with attached pipes was also tested to verify the validity of this synthesized method. In the experimental model, a circular pipe is taken as the attached pipes, and its outer diameter is 0.06 m, and its inner diameter is 0.0475 m. The experimental model is shown in Figure

(a) FRF test setup of the floating raft system with attached pipes; (b) a close view of pipes and masses connection.

A vertical exciting force was applied at the center of mass 3. Other test conditions are the same with previous test. The mobility of the connecting points on the hull was tested. There are two connecting points: point 1 connects attached pipe and hull; point 2 connects attached pipe and hull. The tested results are shown in Figure

Comparison of the measured FRFs and the synthesis results for floating raft system with attached pipes: (a) transfer FRF from exciting point to point 1; (b) transfer FRF from exciting point to point 2.

Figure

Effects of the attached pipes on power flow transmitted to the hull: (a) power flow with/without attached pipes; (b) power flow via different transmission paths.

Based on the research on the influence rule of the attached pipes in the raft floating system, a parametric study was performed in this section targeting at the primary design variables of the attached pipes (e.g., mass and connecting stiffness). Then the vibration performance of the system is analyzed quantitatively by comparing the results under different parameters.

Figure

Effects of the mass of attached pipes on the synthesized results.

Figure

Effects of the connection stiffness of attached pipes on the synthesis results: (a) the stiffness of isolator between attached pipes and machine; (b) the stiffness of isolator between attached pipes and hull; (c) the stiffness of isolator between attached pipes and raft.

In this paper, the FRF-based substructure synthesizing method is used to model the complex dynamic system with multiple transmission paths. The whole FRF matrix of a representative floating raft system is built by using the developed method. The vibration responses of the system considering six degrees of freedom under different exciting conditions are obtained. Numerical simulations and experiment study have been carried out to verify this method which is applicable to model floating raft system. Then a research on the influence of attached pipes is developed on the basis of transfer path analysis. After the parametric study on the vibration transmission and response of the design parameters of the attached pipes, some conclusions which can be drawn from this work are summarized as follows. Firstly, the attached pipes can not only change the symmetry of the system, but also add the mass and stiffness to power plant. This can enhance the coupling effect of the substructures. Secondly, the attached pipe is the second transmission path of the system; the vibration energy can also be transmitted to the hull via them. The vibration isolation performance will be significantly deteriorated by the short-circuiting. According to that, a reasonable mass and connecting stiffness of attached pipes should be chosen through parameter design to diminish the transmitted vibration in the required frequency range.

The authors declare that there are no conflicts of interest regarding the publication of this paper.