Modal analysis of free vibration of an extremely lightweight panel model for bridge bearing applications Pasakorn Sengsri 1 Sakdirat Kaewunruen 1 1 Laboratory for Track Engineering and Operations for Future Uncertainties (TOFU Lab), School of Engineering, The University of Birmingham, Edgbaston B15 2TT, Birmingham, United Kingdom

ABSTRACT This paper reports a novel extremely lightweight panel model under free vibration. This novel model is likely to be used for railway/highway bridge bearing applications due to its high performance to weight ratio, which offers superior mechanical properties, such as sound and vibration attenuation, rigidity, and energy absorption. The structure of the model is based on triply periodic minimal surfaces (TPMS) conceived by observing the scales of butterflies’ wings. The vibration behaviours of this novel panel model used as bridge bearings are not well-known and have never been fully investigated under free vibration. It is important to comprehend the free vibration behaviours of the model and to identify its dynamic modal parameters. In term of modelling, a TPMS sandwich panel finite element (FE) model and a typical bridge bearing FE model under free vibration for bridge bearing applications are designed and examined with a computational method. In general, FEA predictions of the free vibration behaviours of the novel panel model compared to a conventional bridge bearing model provide very good results. These results can be implemented to better generate design standards of extremely lightweight sandwich structures under different vibrations for bridge bearing applications in the near future. Keywords : novel extremely lightweight panel model; free vibration; Triply periodic minimal surfaces (TPMS)

1. INTRODUCTION

Elastomeric bridge bearings (EBB) have been widely used in entire bridge system since they provide a good performance to cost ratio under vibration loading among other bridge bearings. EBBs can be reinforced with or without steel plates/fibre sheets. However, these bearings are still heavy and costly due to their base solid material and labour costs for fabrication [1].

Therefore, the concept of using a lightweight structure fabricated by polymer/elastomer based additive manufacturing (AM) has been considered for bridge bearing applications [2-6]. They would not only be designed for resisting compression, shear, rotation, and vibration loading, but also for reducing these loads. From our previous work in [2], we presented a novel meta-functional auxetic unit (MFAU) cell designed to improve performance and weight ratio for structural bridge bearing applications. The new findings have shown that the model can possibly fail in local buckling before yielding induced by compression because of the slenderness ratios of the model structs. Also, the proposed sandwich core which exhibited an auxetic mechanism under compression has been compromising for an alternative structure core for bridge bearing applications. Later, we presented another novel model using TPMS lattice structure made by a combination of schwarz primitive (SP)

unit cell models, the model was subjected to compression-shear loading. The results indicated that the proposed novel model with at least 6 SP unit cell models can achieve the mechanical properties and specific energy absorption of a common bridge bearing unit cell model under compression-shear loading.

Regarding the performance of a bridge bearing model using a lattice sandwich structure under vibration loading, Sengsri et. al. [4, 5] investigated the vibration characteristics of a novel engineered bridge bearing model with honeycombs under free-free and fixed-fixed boundary conditions. The results indicated that resonant frequencies of the model are influenced by the different boundary conditions and its worse case can be found in fixed-fixed condition which provides lower natural frequency when compared to that in the other condition.

In term of a failure mode of a bridge under vibration, when a bridge is subjected to a highly different vibration, it can immediately fail without the observation [7-8]. This phenomenon is referred to the resonance of the bridge. It is important to comprehend the modal properties of a bridge bearing in order to predict its vibration behaviour under resonance.

Based on many reviews related to bridge bearing applications, there is no study conducted to comprehend the dynamic mechanical characteristics of lattices/TPMS structures fabricated by AM processes. In this research, we assess the dynamic mechanical properties and deformation mechanism of an extremely lightweight panel model for bridge bearing applications numerically. Finite element method (FEM) is employed to investigate the conceptual mechanism of the proposed model and compare it with another model which is a typical bearing having a solid block structure without any reinforcement (plain bearing). The theoretical background of modal analysis will be described in the following section. Then, detailed designs of the two bridge bearing models will be presented in section 3. The finite element models to simulate the modal testing are introduced in section 4 followed by numerical results. The last section will provide conclusions obtained by this paper.

2. Theoretical Background of Modal Analysis

Modal analysis is the most commonly used approach to characterise and determine the dynamic characteristics of systems in the frequency domain for engineering. This analysis can provide engineers an overview of the object’s natural frequencies, mode shapes as well as damping parameters. It is important for engineers to comprehend these dynamic parameters of a structure under vibration. This leads to allowing them to modify and optimise the object’s design to be less sensitive to applied forces without its failure and before manufacturing without material waste. It is worth to note that in this paper the only single degree of freedom system of the model is considered through finite element method (FEM) and the equation of motion of a single-degree-of-freedom system can be expressed as follows [9]:

[𝒎]{𝒅 ̈ } + [𝒌]{𝒅} = {𝟎} , (1)

Free vibration solution is mathematically considered as a non-trivial solution. It should take the form as:

{𝒅} = {𝑫} 𝐬𝐢𝐧𝝎𝒕 , (2)

By substituting Eq. (2) into Eq. (1), the formula becomes a common algebraic matrix equation:

([𝑲] − 𝝎 𝟐 [𝑴]){𝑫} = {𝟎} , (3)

As {D} cannot be zero in Eq. (4), hence:

𝒅𝒆𝒕|[𝑲] − 𝝎 𝟐 [𝑴]| = {𝟎} , (4)

Where, 𝜔 2 is the eigenvalue that determines the natural frequency of the system and {D} expresses the eigenvector that determines the mode shape of the system.

3. Finite Element Analysis of PE and SP Bridge Bearing Models

Two finite element models of a plain block (PB) and a TPMS structure using schwarz primitive (SP) unit cells have been modelled and investigated to obtain their vibration behaviour and to identify their modal parameters (natural frequencies and mode shapes) using software Abaqus. Both the two material models are identical and considered as isotropic behaviour. The material used for both the models is a TPU material having its rubber-like material properties with extremely high bulk modulus, nevertheless providing low shear modulus.

Regarding the condition boundaries of both models, they are in free-free condition without any excitation force, in order to observe the influence of the different structures on the modal parameters. Figures 1a and 1b show the dimensions of the plain block and the TPMS CAD models, which are the same with 150 mm x 100 mm x 50 mm, based on the STANDARD DRAWINGS for Thai highway design and constructure [10]. The materials of the PE and SP models used in the analysis are provided in Table 1.

(a) (b)

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Figure 1: Illustrating CAD models of (a) the PB structure and (b) the SP structure.

Table 1: Engineering properties used in the modal analysis.

Parameter PB model and SP model

Material TPU

Elastic modulus (MPa) 26

Poisson’s ratio 0.39

Density (tonne/mm 3 ) 9.05e-10

4. Results and Discussion

Figure 2 and Table 2 show the results of numerical modal analysis for both the PE model and the SP model under free condition, such as natural frequencies and mode shapes. Their first 6 rigid modes under free vibration are shown in Figure 2. Regarding flexible modes of both the PE model and the SP model, the minimum oscillation of the two models corresponded to the fundamental torsional mode, the second mode to the bending mode, the third mode to the second torsional mode, and lastly the fourth mode to the second bending mode. Nevertheless, there is a slight difference in

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the natural frequencies of each mode between the plain elastomeric block (PB) model and schwarz primitive (SP) model. The natural frequency of the proposed SP model is 0.9 times less than that of the PB model. This means that the proposed model could better experience a resonance phenomenon than the other model. Moreover, the SP model provides a superior vibration behaviour due to the high performance to weight ratio.

Figure 2: First 6 rigid modes of the PB model and the SP model under free vibration.

Table 2: Comparison of the frequencies of the flexible modes between the PB model and the SP model.

PB model SP model with 50 mm unit cells Mode

Mode shape Frequency

Mode shape Frequency

∆% Behaviour

No.

(Hz)

(Hz)

1

209.49

179.06

14.52

1st Torsion

2

250.30

209.69

16.22

2nd Bending

3

415.07

388.99

6.28

2nd Torsion

4

486.85

456.75

6.18

2nd Bending

5. Conclusions

In this paper, the numerical modal analysis of free vibration has been conducted to investigate the dynamic behaviour of a plain elastomeric block (PB) model and schwarz primitive (SP) model used as bridge bearings. The model materials used in both the simulations are TPU offering rubber- like properties. The new findings have shown that their fundamental torsional mode certainly dominates the resonance mode of free vibration. Furthermore, the SP model can better behave in vibration with decreasing its natural frequencies, compared to those of the PB model. This leads to the SP model which provides vibration attenuation due to its higher performance to weight ratio.

Theses insights will be helpful to the design standards of bridge bearings to predict their modal dynamic parameters and behaviours under different vibrations. Additional work should be performed on the model validation for use in reality.

6. Acknowledgements

The first author wishes to thank Royal Thai Government for his PhD Scholarship at the University of Birmingham. The last author wishes to gratefully acknowledge the Japan Society for Promotion of Science (JSPS) for his JSPS Invitation Research Fellowship (Long-term), Grant No L15701, at the Track Dynamics Laboratory, Railway Technical Research Institute and at Concrete Laboratory, the University of Tokyo, Tokyo, Japan. The JSPS financially supports this work as part of the research project, entitled “Smart and reliable railway infrastructure.” Special thanks to European Commission for H2020-MSCA-RISE Project No. 691135 “RIS-EN: Rail Infrastructure Systems Engineering Net- work” (www.risen2rail.eu). Partial support from H2020 Shift2Rail Project No 730849 (S-Code) is acknowledged. In addition, the sponsorships and assistance from LORAM, Network Rail, RSSB (Rail Safety and Standard Board, UK) are highly appreciated.

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