Design of a Submarine Vehicle for Higher Natural Frequency Using U ∗ Index Theory Approach

,e natural frequency of a vehicle structure should be far higher than the road excitation frequency in order to avoid dynamic problems caused by structural resonance. Submarine vehicles or robots are facing the same issue as mentioned above. ,is paper proposed a means to effectively increase the natural frequency of submarine vehicles or robots by using U∗ index approach. ,e U∗ theory was originally introduced to investigate load transfer paths within a structure. It is a new design paradigm for vehicle structures. In this study, the U∗ index theory was applied to evaluate the internal stiffness distribution of a submarine vehicle structure and then a modified design was achieved based on the U∗ analysis. Compared to the original design, the natural frequency of the modified vehicle structure was increased by more than 40% while the weight of the modified vehicle was only raised by 4.6%.


Introduction
Recently, ocean engineering technologies have increasingly been focused by many countries and areas of the world.Submarine vehicles or robots are important to the investigation of ocean ecosystem, exploration of ocean energy, and heath monitoring of ocean platform and submarine pipelines.However, compared to ground vehicles, the development of design and manufacturing techniques of submarine vehicles are still in their primary stage [1].Structural vibration and control is always one of the principal problems in vehicle design which significantly affects structural performance such as structural strength, durability, kinematic accuracy, and dynamic stability [2,3].To minimize the dynamic problems caused by the road excitation, natural frequency of ground vehicles needs to be far higher than 10-20 Hz which is a typical road excitation frequency [3].To avoid the resonance phenomena, reasonable matching of stiffness between different parts or assemblies is also necessary in vehicle design [2].Although road excitation data from the seabed are limited to date, a means to design submarine vehicles or robots with higher natural frequencies is imperative.
Structural stiffness is a determinant of modal shapes and frequencies of a structure.However, traditional finite element analysis (FEA) only gives structural stresses and displacements instead of a stiffness distribution.U * index theory which was firstly introduced by Takahashi in 2005 is a method to investigate stiffness distribution of a structure [4].
e original purpose for proposing the U * theory method is to calculate load transfer paths of a load-bearing structure.Load paths analysis is essential for designing better engineering structures in both mechanical performance and weight efficiency [5].
e U * index theory was initially proposed to solve static and linear elastic problems.Recently, the U * index theory has been extended to dynamic loads [6], nonlinear elasticity, composite materials [7], and six degrees of freedom systems [8].Load paths analysis based on the U * index theory is a new design paradigm because it can provide useful information for decision-making which cannot be achieved from traditional finite element analysis (FEA) [9].e U * theory has been applied in vehicle design by automotive manufacturers like Honda and Nissan [10,11].
In this paper, instead of load paths analysis, the U * index theory was applied as a means to increase the natural frequency of submarine vehicles or robots to reduce the dynamic problems caused by excitations with lower frequencies.A submarine vehicle designed by our research group was used as the sample structure to demonstrate the effectiveness of the proposed method.Firstly, the internal relative stiffness distribution of the submarine vehicle structure was computed by the U * index theory.en, a U * -based design criterion was employed to improve the overall stiffness of the submarine vehicle structure by only adding a small amount of mass.Finally, due to the improvement of the stiffness distribution, the natural frequency of the submarine vehicle structure was raised by 43.5% while the structural weight was only increased by 4.6%.
In the following parts of the paper, the U * index theory and the U * -based design criterion are presented in Section 2; the computational design method of the submarine vehicle structure is introduced in Section 3; the computational results of both original and modified designs are compared in Section 4; and the conclusion of the paper is drawn in Section 5.

Theoretical Preliminary: U * Index Theory
Figure 1(a) shows a linear elastic body with the loading point A, supporting point B, and an arbitrary point C. e beams represent the internal stiffness between any two points of the system.e internal stiffness is the coupling stiffness between two specific points, whose meaning is totally different from the stiffness matrix in FEA. e following equation describes the force-displacement relationship among points A, B, and C: where  B,C) are the internal stiffness tensors, and they, respectively, characterize the relationships between forces and translations, moments and rotations, forces and rotations, and moments and translations.
In Figure 1 e external work of the system caused by the force F A and the bending moment M A is presented in Equations ( 4) and ( 5), respectively: ( Figure 1(b) shows the arbitrary point C (D C � R C � 0) of the system is restrained while same displacements D A and R A are applied at point A. Equations ( 6) and ( 7) indicate the external work of the constrained system induced by the force F A ′ and the bending moment M A ′ , respectively: e U * index under six DOFs is defined as where U is the total strain energy of the original system (Figure 1(a)) and U ′ is the total strain energy of the constrained system (Figure 1(b)).Here, the total strain energy of the system is transformed from the external work [12].Equation ( 8) can be rewritten as By subtracting 1 and then adding 1, Equation ( 9) equals to By substituting Equations ( 4)- (7) into Equation (10), we get 2

Shock and Vibration
It can be seen from Equation ( 11) that the U * index can be expressed in terms of the internal sti ness between the loading point and the arbitrary point of the system.In a structure, forces are mainly transferred through the sti est parts in series with the loading point.erefore, the lines with the highest U * values (i.e., the ridge line of the U * distribution) is de ned as the main load paths.If a vector g (−) is de ned as the main load path is the line with the smallest g (−) .e vector g (−) is regarded as the "sti ness decay vector" [11].
Figures 2(a) and 2(b) elaborate U * and von Mises stress distributions of a plate with a hole, respectively.e solid black lines shown in Figure 2(a) are the main load paths predicted by the U * index, which satisfy the expectation form the engineering point of view.As can be seen in Figure 2(b), the higher stresses are formed in the vicinity of the hole.However, since the hole is not important in transferring the forces, it is unreasonable to conclude the main load paths pass the locations with those higher stresses.
Normally, the U * index decreases from the maximum value of one (1) at the loading point to the minimum value of zero (0) at the supporting point.In Figure (3), the solid line elaborates the variation of the U * index along a path of a load-bearing structure and the dashed line depicts the U * variation of the desirable condition.A uniform decay of the U * index means the structural sti ness distribution is perfect, whereas, in reality, the U * variation is nonuniform because of the structural issues.An increase or decrease of the rate of the U * decay indicates that the sti ness in the corresponding part is higher or lower than other parts.e uniformity of the U * decay is de ned as the criterion to evaluate the rationality of the sti ness distribution of the structure.Structural modi cations are required when the U * curve is not uniform such as the solid line in Figure 3. e e ective structural modi cations shall reduce the shade area shown in Figure 3.

Computational Design Method
ere are mainly three steps in the proposed computational design method.Firstly, a dynamic model is built, and a modal analysis is conducted on the vehicle structure.is is to obtain the natural frequency of the original design.
en, the vehicle structure is evaluated by using the U *based design criterion, while modi cation directions are suggested based on the U * analysis.
irdly, the natural frequency is recalculated to see the improvement from the design modi cation.If extra improvement is expected, the above procedures would be repeated until the result is desired.
Figure 4 shows the sample structure, a submarine vehicle (or robot) designed by our research group, which is used to demonstrate the usefulness of the proposed computational design method.
e duties of the submarine vehicle are investigations of seabed ecosystems and explorations of ocean energy.Hence, minimizing the dynamic problems can e ectively improve the accuracy and stability of data collection and dynamic control to the submarine vehicle.In the current study, the dynamic analysis and the design modi cation focuse on the vehicle suspension system.e material properties of the major components of the vehicle are given in Table 1.
Figure 5 illustrates the dynamic model of the submarine vehicle for modal analysis.e model is created in commercial FEA code ANSYS (release 18).e dynamic model includes the vehicle body and the major parts of the suspension system ignoring the crawler belt systems and the propellers.
e properties of each component of the dynamic model are elaborated in Table 1.e boundary conditions can also be seen from Figure 5, where all the wheel holes in the vertical section of the suspension are fully constrained.All the components in the FEA model are modeled using Shell 181 Element with full integration.e block Lanczos method is selected as the mode extraction method [13].
Figure 6 demonstrates the static computer model for U * calculation of the vehicle suspension system.In this model, the vehicle body is not included because the U * analysis focuses on the load transfer behavior of the vehicle suspension system.e body weight of the vehicle body component is applied to the vehicle suspension system as the external force.e boundary conditions are the same as the dynamic model shown in Figure 6. e U * analysis gives how the force of the vehicle body gravity is transferred through the vehicle suspension system.Meanwhile, the insight of the suspension structure behind the load transfer pattern is revealed through the U * -based design criterion introduced in Section 2. Finally, design modi cation can be proposed to the suspension structure based on the U * analysis.
In this study, an in-house FEA-based program was developed for U * calculation in Ansys Parametric Design Language [13].Figure 7 shows the logic and the procedure e rst step is the calculation of the total strain energy of the original system as shown in Figure 1(a).Secondly, by xing an arbitrary node of the system and applying the same displacement to the loading point, the total strain energy of the modi ed system (Figure 1(b)) is calculated.en, the U * value of the arbitrary node can be computed through Equation (8).By repeating this step for each node of the system, the U * distribution of the whole system is obtained.e total strain energy of the original system is only calculated once.

Results
Figure 8 presents the rst three modal frequencies and shapes of the submarine vehicle.e rst modal frequency of the vehicle, which is regarded as the natural frequency, is 37.6 Hz.
e rst modal shape of the vehicle is pitching oscillation, the second modal shape is heaving oscillation, and the third modal shape is rolling oscillation.e frequencies of the second and third modes can be also seen from Figure 8.
e U * distribution of the suspension structure is shown in Figure 9(a).It shows the gravity force of the vehicle body is transferred through the horizontal part to the vertical part of the suspension.In Figure 9(b), the dashed white line illustrates the ideal situation of the U * variation, while the solid blue line represents the actual U * values along the marked load path in Figure 9(a).We can see the real U * values along the load path deviate from the ideal situation, especially the circled area.Figure 9(b) indicates the U * index decays sharply along the horizontal section of the suspension, while it drops slowly along the vertical section.e reason of this phenomenon is that the sti ness of the horizontal section of the submarine vehicle suspension is far lower than the vertical section.erefore, increasing the sti ness of the horizontal section can e ectively improve the overall sti ness of the suspension and thus raising the natural frequency of the whole vehicle.As discussed above, to increase the sti ness of the horizontal section of the suspension, its thickness was raised from 20 mm to 30 mm.
Figure 10(a) presents the U * contour of the modi ed suspension, while Figure 10(b) shows the comparison of the U * values along the load path between the modi ed and the original designs.From Figure 10(b) we can see that, in the horizontal section, the U * values of the modi ed suspension are larger than the original design.It also can be seen that, at the joint between the horizontal and the vertical sections of the suspension, the U * value of the modi ed structure is much closer to the ideal value compared to the original structure, i.e., with respect to the ideal condition, the deviation of the modi ed design is smaller than the original design.To quantitatively show the design improvement, ratios of the U * values of the modi ed or the original design to the ideal U * 4 Shock and Vibration values (i.e., U * design /U * ideal ) were obtained using the data points from Figure 10(b) and the mean and the coe cient of variation (COV) of the ratios were then computed, respectively, and illustrated in Table 2. e mean for the modi ed design is 1.01 comparing with 0.86 for the original design and the COV for the modi ed design is 17% instead of 33% for the original design.
e improvement in terms of the U * distribution indicates that the design modi cation narrowed the gap of the sti ness between the horizontal and the vertical sections of the vehicle structure.As a result, the sti ness distribution of the whole suspension is more uniform and the overall sti ness of the whole suspension is improved at the same time.
To see the e ectiveness of the design modi cation, a new modal analysis was conducted to the submarine vehicle with the modi ed thickness in the horizontal section of the suspension.e modal shapes and the corresponding frequencies are depicted in Figure 11, where we can see that the rst three modal frequencies of the modi ed vehicle were, respectively, improved by 43.5%, 40%, and 48% compared to the original design.Due to the design modi cation, the total weight of the

Fixing node i
No Yes U of original system

Conclusion
e paper proposed a means for designing submarine vehicles or robots with the higher natural frequency in order to decrease the potential dynamic problems due to the road or seabed excitations.
With the help of the U * index theory, the distribution of relative sti ness of a structure can be obtained, which provides designers with useful information in addition to the traditional FEA approach.e design modi cation based on the U * -based approach considerably raised the natural frequency of the vehicle by adding a small amount of weight.
e results indicate that upgrade of overall sti ness of a structure can increase the structural natural frequency while the former can be improved by raising the uniformity of the sti ness distribution of the structure.Subjected to external forces or excitations, a structure with uniform sti ness distribution can realize the maximum overall structural sti ness based on a constant usage of material.
e design criterion based on the U * index theory is effective to evaluate and improve the uniformity of sti ness distribution of a structure.
Simply adding materials to each part of the vehicle structure can also improve its natural frequency.However, the increase of the structural weight may cause many other problems, e.g., fuel e ciency, weight limit, or control accuracy.is point can further highlight the usefulness of the proposed U * -based approach in designing submarine vehicle structures with better dynamic performance.e work of this paper also implies that the U * index theory is able to be applied for sti ness matching between di erent components of a vehicle system for addressing vibration problems.

Data Availability
All the computational data used to support the ndings of this study are included within the article.It includes all the data related to the CAD model and the FEA model and the date of the modal analysis and U * analysis.All the original data in this study are available upon request.Shock and Vibration 7

Figure 2 :Figure 3 :Figure 4 :
Figure 2: Load transfer analysis of a plate with a hole: (a) U * distribution and main load paths; (b) von Mises stress distribution [9].

Figure 5 :Figure 6 :
Figure 5: FEA model and mesh of the submarine vehicle structure for dynamic analysis.

Figure 7 :
Figure 7: Flowchart to develop computer program for U * calculation.

Figure 10 :
Figure 10: U * analysis of the modi ed submarine vehicle structure: (a) U * contour of the structure; (b) U * variation along a load path (white arrow).

6
Shock and Vibration submarine vehicle was only raised by 4.6% which was calculated based on the dynamic model shown in Figure 5.

Table 1 :
Material properties of the major parts of the submarine vehicle.

Table 2 :
Quantitative comparison with the ideal condition.