The Spherical Shock Factor Theory of a FSP with an Underwater Added Structure

A floating shock platform (FSP) is important experimental equipment for the antishock assessment of large-scale shipborne equipment. Generally speaking, the FSP is a square barge structure with a flat bottom. In order to satisfy the key index of the horizontal-to-vertical ratio of the impact environment, a special added structure is mounted to the bottom of the platform, and the impact environment is more complicated due to the added structure. ,erefore, the spherical shock factor theory is proposed to analyze the impact environment, and the validity of the theory is proved by numerical experiments. ,e results show that the average impact spectrum response of the platform is essentially consistent under the same shock factor. Meanwhile, the spherical shock factor builds a linear relationship between the input parameters and structural response, which is convenient for the prediction of impact response.


Introduction
e antishock ability of shipborne equipment on a warship is directly related to a warship's combat effectiveness and survivability.In order to reduce the possibility of shipborne equipment being damaged after it is subjected to an explosion, countries all over the world attach great importance to the impact resistance ability of shipborne equipment, and corresponding experimental methods have been developed.In terms of standard formulations, German standard BV043/85 [1] stipulates the equipment impact resistance standard from the point of the installation site, and US military standard MIL-S-901D [2] stipulates in detail the equipment impact resistance standard with 8 aspects that include weight, impact level, and category.At the same time, a special experimental device for shipborne equipment impact resistance has been designed by the US Navy.Fluidstructure coupling subject to an underwater explosion is a very complex problem.
erefore, in the study of fluidstructure coupling theory, the transient fluid-structure interaction mechanisms of a three-dimensional spherical shell and a ship's stiffened plate were studied when subjected to an underwater explosion [3].
rough numerical simulation and experimentation, the fluid characteristics and fluidstructure coupling interactions of damaged ships in the process of submergence are analyzed [4].Considering the complex input-output relationship of structure impact resistance subject to an underwater explosion, the KSF [5] (keel shock factor) is used to describe the severity of impact.
e impact environment of a cylindrical shell subject to an underwater impact was compared and analyzed with different shock factors.It shows that, as a similarity parameter, the new shock factor can effectively reflect the response of an underwater explosion load, and the shock factor also has a certain feasibility for being used to describe the impact environment of a submarine [6].e improved shock factor indicates that cylindrical and spherical shells have the same impact response [7].Literature [8] uses the new shock factor to certify the impact environment similarity of a SWATH (small waterplane area twin hull) subject to an underwater explosion.When it comes to impact environment prediction, one author [9] uses the shock factor to predict the impact environment but does not take the difference of the hull and the equipment into account in the installation, and the shock factor has greater limitations itself.According to the type of ship and device, the position and the angle of incidence of the charge, and other factors, one researcher [10] summarizes the empirical formula of the impact environment, but the accuracy of the formula is not strictly checked, and the scope of application is not strictly defined.
In this paper, to satisfy the key index of the impact environment (shock spectrum [11,12]) that the horizontalto-vertical ratio of equipment antishock examination in UNDEX, the added structure is added to the bottom of the FSP, while the impact environment of the added structure is not considered.Since the added structure makes the impact environment of the FSP more complicated, the spherical shock factor theory is used to analyze the complex inputoutput relationship of the IFSP (irregular floating shock platform) impact environment, and the shock factor is used to predict the impact environment of the IFSP, which will guide the design, manufacture, and installation of the equipment.

The Spherical Shock Factor Theory of the IFSP
When the structure is subject to UNDEX, for the sake of reflecting the impact environment characteristics, different regularities of shock spectrum change with the parameters such as conditions, geometric conditions, medium, and etcetera.One researcher [6] thoroughly illustrates the shock factors C 1 and C 2 .e keel shock factor C 1 is difficult to use to correctly reflect the underwater explosion impact environment of the cylinder.C 2 is based on the plane wave hypothesis and defined from the angle of energy shielded by the structure, which reflects the similarity of underwater explosion loads in the far field.However, the plane wave assumption cannot be held in the near field.erefore, in this paper, a new shock factor C 3 is considered to describe the shock energy absorbed by platform, namely, the impact strength received by platform.

Basic Model and eoretical Analysis of the Shock Factor.
In an explosives experiment, charge weight, explosive distance, and structural characteristics are closely related to the impact environment.In order to study the similarity parameter which reflects the impact environment of the underwater explosion, one can assume that the shock wave is a spherical wave, so the energy born in an explosion can be expressed as follows [13]: where ρ e is the chemical energy per unit mass of charge, η e is the conversion of chemical energy to shock wave energy, and W is the charge weight (kg).
In an infinite fluid field, the shock wave energy is considered to be distributed over the entire spherical shock wave surface.us, the spherical shock wave energy absorbed by the structure is given by where E s is the shock wave energy absorbed by structure, S e is the projection area of the structure on the wave surface, and R is the shortest distance from the charge to the hull.e shock factor C 3 is defined as follows: where S e is the area truncated by the solid angle and Ω [14] is the solid angle.
It can be seen from formula (2) that E s is in proportion to C 2  3 , and the shock factor is the unique variable in the expression.C 3 includes the information of the charge weight, the detonation distance, and the structural geometry characteristics of the platform.erefore, if C 3 is equal under different conditions when the charge is relative to different positions of the structure, the average shock spectrum response of the structure should be essentially consistent and C 3 should fully reflect the characteristics of the impact environment.According to formula (3), the calculation method of Ω under a spherical wave is explained as follows, using a simple rectangular plate.
Figure 1 shows the physical model and geometric relationship of a simple rectangular plate under a spherical shock wave.e vertex Z 0 of the angle Ω is located on the central axis of the rectangular plate.From the symmetry, we can calculate the angle Ω of the whole rectangle using the quarter of the rectangle in the graph.
e length of the rectangle is 2a, and the width is 2b.e length of the small rectangle is a, and the width is b.Arbitrarily taking the area microelement dS in the small rectangle, the angle of axis Z and the line between the area microelement and vertex Z 0 is given by e solid angle of the area microelement to the vertex Z 0 is given by e solid angle of the small rectangle to the vertex Z 0 is given by e integral of formula (6) needs to be solved by numerical methods.It can be seen from formula (6) that the integral function is known, so the calculation of the solid angle is transformed into the determination of integral range.If the integral range is determined, the solid angle of the structure corresponding to the spherical shock wave can 2 Shock and Vibration be calculated; consequently, the corresponding shock factor can be obtained.

Spherical Shock Factor of the IFSP.
For the IFSP, the solid angle of the structure on a spherical surface with radius R can be solved similarly to the method used for the rectangular plate.When calculating the shock factor, the charge is located in the transverse section of the platform.Due to the existence of the underwater added structure, the accurate calculation of the solid angle is very complicated, and it needs to be integrated separately along the boundary of the platform.In order to simplify the boundary conditions according to different locations of the charge, the calculation of the shock wave area shielded by the structure is divided into 6 regions as shown in Figure 2.
Taking region S 3 as an example, the local coordinate system xyz is established at the contact point between the platform and the shock wave, and the global coordinate system XYZ is established at the intersection points of the starboard side and the waterline plane, where axes x and X lie along the longitudinal line of the platform.Due to the manufacturing technology, the actual added structure is a little shorter than the platform, but this has no effect on the calculation of the solid angle, so the added structure can be regarded as the same length as the platform.e geometric relationship of the IFSP is shown in Figures 3  and 4, and the fixed geometric parameters in the graph are defined in Table 1.

Starboard Conditions.
When the charge is located on the starboard side of the platform, according to the position relationship between the charge and platform, there are three cases: From Figure 5, the geometric relation can be given as follows: e integral function of the other cases is the same as formula (8), so it is only necessary to give the integral limits h 1 and h 2 .
(b) In Figure 6, if (c) In Figure 7, if H ≤ T, there is h 1 � H and R � L:

Port Conditions.
When the charge is located in the port side of the platform, according to the position relationship between the charge and the platform, there are three cases.
(d) In Figure 8, if H > H 1 , there is (e) In Figure 9, if H ≤ H 1 and  4 Shock and Vibration (f ) In Figure 10, if H ≤ H 1 and In all of the above cases, the basic theory is similar to formula (8), so it is only necessary to modify the parameters of various conditions according to the definition.

The Numerical Model Establishment and Simulation Analysis of the IFSP
Literature [15][16][17][18][19][20] shows that numerical methods make it easier for researchers to understand the response characteristics of structures.In order to study the impact of the environmental characteristics of the IFSP subject to an underwater explosion, an underwater explosion numerical model of the platform is established, and the experimental data are used to verify the accuracy of the model.In this paper, the model is prepared for the experimental data that are needed for the prediction of the platform's impact environment.

e Establishment of Numerical Model.
In order to make the numerical model approximate to the real structure and simplify the calculation, the model is established using a three-dimensional finite element model.e numerical simulation algorithm uses the acoustic-structure coupling method, the acoustic medium is used to describe the fluid, and the shock wave propagates in the acoustic element.e platform is in the form of a double bottom structure in the finite element model, the inner bottom and outer bottom are connected by a longitudinal girder and rib, and the surrounding walls are shock resistant bulkheads.Based on the platform width B � 9 m, the nondimensional principal dimensions of the model are shown in Table 2; the fluid field is composed of a half cylinder and two quarter spheres.
All mesh sizes of the platform are 5 cm.e girder of the platform and the strengthened beam are composed of first order shear deformation spatial beam elements, and there are about 11,000 elements.e stiffened plate between the girders of the platform and the stiffened beam ,and the outer plates are all composed of quadrilateral or triangular shell elements, which have a linear reduction integral, and there are about 611,000 elements.e fluid field is composed of the linear tetrahedron acoustic medium element, AC3D4.
e internal mesh size of the fluid field is 25 cm, external mesh is four times as large as the internal mesh, and there are about 700,000 elements.Finally, a coupling constraint "tie" is added to the underwater interface between the IFSP and fluid field in ABAQUS, which ensures the loads act on platform effectively, and the nonreflecting boundary conditions are established at the external boundary surface of the fluid field.e finite element model of the IFSP subject to an underwater explosion is shown in Figures 11 and 12.

Model Test on the Basis of the Experiment.
As we know, in the course of the model and actual underwater explosion tests, the acceleration response can directly reflect the stress of the local structure which is easy to obtain.In order to describe the underwater explosion impact environment of the FSP, generally, the shock spectrum in a four-dimensional coordinate system is used to analyze the impact.According to structural mechanics, for a single degree-of-freedom system, the relationship between the spectral displacement, spectral velocity, and spectral acceleration of a mass block is given by It can be seen from formula (14) that the spectral velocity is a link parameter between spectral displacement, spectral acceleration, and vibrator frequency.erefore, this paper will focus on the spectral velocity in the impact environment.
According to the established numerical model of the IFSP, the charge is located to the side of the added structure in the cross section of the IFSP, and the equivalent quantity of TNT is 150 kg for the charge, with a depth of 6.5 m and a transverse distance of 16 m.e position of detonation point specific conditions of the test and the numerical simulation Figure 8: e condition at S 4 .e design spectrum in shock spectrum is shown in Figure 16 on the basis of experimental data; it can be seen that the spectral velocity of the simulation is very close to the test at points A, B, and C, and the gap of point D is a little bigger.To estimate the errors between the simulation and test, the comparison of the specific values is shown in Table 3.
e following conclusions are drawn from Table 3: (1) the maximum error of the horizontal spectral velocity is 25.33% between the simulation and the test, the average error of the inner bottom is 7.13%, the maximum error of the vertical spectrum velocity is 26.75%, and the average error of the inner bottom is 19.33%.(2) e error of the vertical and horizontal spectral velocities at the same point is extremely close, and the results show that the simulation results are consistent for the whole model.( 3) Except for individual points, the error between the simulation and test is less than 15%, and the error of the impact environment on bottom is within 20%.erefore, the finite element model can effectively simulate the impact environment of the platform.

Analysis of the Impact Environment on the Basis of the Spherical Shock Factor
As a similarity parameter of underwater explosion, the problem of the complex input-output relationship of the impact environment on IFSP can be solved by the shock factor.is section is going to examine whether the average impact response spectrum of the IFSP is consistent and use the definition of the shock factor to predict the impact environment.

Analysis of the Relationship between the Shock Factor and the Impact Environment.
Based on the definition of the spherical shock factor, in order to reflect the effect of various parameters to impact environment, the conditions of C 3 � 6. 45 are shown in Tables 4 and 5 by changing the value of various parameters which include the charge weight, depth, and transverse distance.e established numerical model is used to calculate the average response of the platform under various conditions, and the results shown in Tables 6 and 7 are used to investigate whether the shock factor reflects the average shock spectrum response of the structure.e comparison of impact response is shown in Figures 17 and 18.It can be seen that when the shock factor is equal, with changes of W, H, and L, the maximum error of the

Waterline
Wa ter lin e     Shock and Vibration horizontal and vertical average response on the starboard side is 5.31% and 4.84%, respectively.us, the average shock spectrum response of the platform is essentially consistent on starboard with the same shock factor that effectively describes the shock energy absorbed by platform or the impact strength received by the platform.e maximum error of the horizontal and vertical average response on the port side is 16% and 10%, respectively, which shows that the effect of shock factor is relatively poor on the port side.

e Prediction of the Impact Environment
Based on the Spherical Shock Factor.It has been proved that the shock factor is able to reflect the impact strength of the charge.erefore, the shock factor is going to be used as the input parameter to study the prediction of the IFSP's impact environment.According to the characteristics of the spherical shock factor, Table 8 presents various conditions set at the port and starboard areas to investigate the impact environment prediction error by shock factor.e error between the fitting spectrum parameters and actual values is calculated by fitting the data for all conditions; Figures 19 and 20 show the fitting results about spectrum velocity for all conditions, which present a linear relationship between spectrum velocity and shock factor.Table 9 shows all the results about fitting formula and error; it can be seen from the table that except for the longitudinal spectrum velocity and displacement, the shock factor can predict the impact environment well in starboard and port conditions, and the accuracy is within 10%.
To date, no distinct relation or conclusion has been given about the existence of shock factor theory and the structure's response, while the shock factor proposed in this article effectively converts the nonlinear relation into a linear relationship between the input parameters, such as the mass of charge, the explosive distance, and the structural response, so the shock factor presented in this paper has a great significance to the research of the structure's input-output subject to underwater explosion, which can be used to predict the impact environment on IFSP, and the more conditions there are, the higher the prediction accuracy is.

Conclusion
(1) In order to study the impact environment characteristics of IFSP subject to underwater explosion, a  10 Shock and Vibration spherical shock factor about IFSP is defined from the point of energy, and its calculation method is given by the structural characteristics of the platform.(2) Based on a large number of numerical experiments, the results show that the average response of the platform is essentially consistent with the same shock factor; therefore, the spherical shock factor, as a similarity parameter, can effectively describe the impact environment, that is, the shock energy absorbed by the platform or the impact strength received by the platform.(3) After changing the charge parameters of various conditions, the results prove that the shock factor proposed in this thesis effectively builds a linear relationship between the input parameters and structural response, and the spectrum parameters are predicted effectively by the shock factor with an error of 10%.

Figure 16 :
Figure 16: Comparison of the design spectrum of the test points.(a) Horizontal design spectrum at A. (b) Horizontal design spectrum at B. (c) Horizontal design spectrum at C. (d) Vertical design spectrum at C. (e) Horizontal design spectrum at D. (f ) Vertical design spectrum at D.

Figure 18 :
Figure 18: Comparison of the impact response on the port side.

Figure 19 :
Figure 19: Linear fitting of the impact environment on the starboard side.

Figure 20 :
Figure 20: Linear fitting of the impact environment on the port side.

Table 2 :
e principal dimensions of the model.

Table 3 :
Spectral velocity comparison of the platform.

Table 4 :
e starboard condition parameters under the same shock factor.

Table 5 :
e port condition parameters under the same shock factor.

Table 6 :
Comparison of the structure's average response on the starboard side.

Table 7 :
Comparison of the structure's average response on the port side.

Table 8 :
Condition setting of impact environment forecast.

Table 9 :
Fitting formula and error of spectral parameters.