Structural optimization of Go kart chassis by geometrical modifications based on modal analysis

STRUCTURAL OPTIMIZATION OF GO-KART CHASSIS

BY GEOMETRICAL MODIFICATIONS BASED ON

MODAL ANALYSIS

A. A. Dere

, M. Singh

, A. Thakan

and H. Singh

1

Automobile Engineering, Lovely Professional University, Jalandhar, India

2

Facilty of Mechanical Engineering, Lovely Professional University, Jalandhar, India

3

Department of Planning and Quality, Nettbuss AS, Oslo, Norway

E-Mail: [email protected]

ABSTRACT

A chassis houses various key components such as the power-train, suspension system and the bodywork of an automobile. Thus, the chassis must be robust enough to absorb the static and dynamic forces generated by these components to assure safety and superior rideability of an automobile. In this work, efforts were made to improve the static and dynamic characteristic of a Go-kart chassis by using the modal analysis. The results of the modal analysis were carefully examined and the members are geometrically modified on the basis of excessive deformation. At the termination of the of this analysis, it was observed that geometrical modification has not only improved the dynamic characteristic by 17.5% but also shown improvement in the stiffness and crashworthiness of the chassis.

Keywords: chassis structure, dynamic analysis, Go-Kart chassis, modal analysis, resonance.

1. INTRODUCTION

Go-kart is a racing vehicle with a single seat and four non-aligned wheels [1]. The chassis of the Go-kart is the major and inseparable component as it holds together all the critical components [2]. The chassis of the Go-kart is mainly responsible for absorbing various dynamic loads generated by the power-train and the road profile as the suspension system is absent in a Go-kart. If the frequency of dynamic load matches with the chassis natural frequency it may lead to the disastrous effect of resonance [3-5]. This devastating effect may lead to loosening of joints, stress generation and in severe cases may lead to structural failure [6,7]. Various techniques are available for measuring the dynamic characteristic but the studies reveal that the most reliable results can be achieved through modal analysis [8-11]. Modal analysis is the technique of determining the mathematical model of the system from its dynamic characteristic [9]. Determining the mathematical model analytically for a complicated structure like the tubular space frame proves to be a tedious task with large complication. This complication can be overcome by the computational method of modal analysis [12,13]. Considering the various findings, it can be seen that by varying the location of members and also by changing weight and thickness the dynamic properties show an improvement [6,7]. Mohammad et al. _{in their}

work on the truck chassis stated that variation in thickness the properties of the chassis can be improved [14]. Also, the various researches have changed the geometry of the chassis but neither took weight, strength and static characteristic into consideration [15-17].

In this work, the design modification of the Go-kart chassis has been carried out on the basis of results obtained for excessive deformation in modal analysis using computation FEM method. The final chassis was examined and compared with the initial chassis for the

dynamic and static strengths in which final chassis proves to better.

2. FINITE ELEMENT METHOD

Finite element method is a numerical methodology for solving the problem of engineering and mathematical physics that provides the designer remedy to the complex designing problems [18]. The Go-kart chassis have been a complex structure it is practically impossible to derive the set of differential equations and infeasible to solve it analytically. Thus, the process of FEM was implemented. The basic idea behind FEM is to discretize complex structure into small elements which are interconnected in a certain manner at nodes which intern helps in determining the specific solution [19].

3. CHASSIS ANALYSIS

A. Static Analysis

Static analysis is carried out to determine stress, strain and displacement by confining the motion a mechanical structure and applying known loads and forces [20]. In the problems of chassis torsional Stiffness test is generally carried out to know torsional rigidity of the structure.

Torsional Analysis: In the analysis of an automobile chassis the motion of the rear wheels mounting of a chassis are restricted and equal forces are applied on the front section in an opposite direction which helps to determine torsional stiffness of that chassis. The insufficiency in torsional stiffness may result in structural failure and sensation of insecurity to the occupant on the curve. The torsional stiffness for analysis is given by [20].

Where;

K = torsional stiffness; 𝑇 = Applied Torque;

∅ = deformation or the twist angle.

Twist angle ∅ is given by the formula,

∅ = sin−1₍2𝑉

𝐷) (2)

Where;

V is the vertical deflection of the point where the load is applied;

D is the distance between points on which the load is applied;

B. Dynamic Analysis

The dynamic analysis helps in determining the behavior of the mechanical structure when subjected to abrupt loading. Dynamic load removes the structure from the state of static equilibrium due to the action of acceleration. The Dynamic load can be determined by (Dynamic force=Mass*Acceleration) [19].

The linear dynamic response which is governed by the dynamic equilibrium equation mentioned below [20]:

[M]. {𝑋̈}+[C]. {𝑋̇}+[K]. {X}={F(t)} (3)

Where M, C, and K represent the system mass, system damping and system stiffness matrices of order n*n respectively. F(t) represents the external load vector and 𝑋̈,𝑋̇, and X represents the acceleration, velocities and displacement vectors. Dynamic property of a mechanical structure can be effectively determined with the help of modal analysis.

Modal Analysis: Modal analysis is a tool which helps to determine the natural frequencies, mode shape, and damping of a structure. The bottom line of modal analysis is to get the Frequency Response Function (FRF) of the mechanical structure. FRF provides the quantitative measure as a response to a stimulus of a mechanical structure. To calculate the FRF, an excitation needs to be applied to the space frame at a certain point and the responses will be measured at the elements of structure. In this study, the linear matrix mode extraction method “Lanczos Method” is implemented to determine the response in 3D using Finite Element Analysis. As this method provides a solution with less computation and is mentioned by the formula as below [12]:

[k]{∅_𝑖}=𝜔_𝑖2[M] {∅_𝑖} (4)

Where:

[k] = system Stiffness matrix; 𝜔𝑖 = natural Frequency of mode i; ∅𝑖 = eigenvector;

[M] = Mass matrix of the system.

Whenever the frequency of excitation matches the natural frequency of the mechanical system which leads to the devastating effect of resonance to occur. This leads to loosening of parts, noise, and partial or complete failure of the system also reduces the driving comfort of the occupant. To overcome this problem, the dynamic analysis of a chassis has become an inextricable tool during the design process [6].

4. THE CHASSIS GEOMETRY

The geometry under study was the chassis of the Go-kart used in the international Go-kart championship 2018. The design of geometry involves the raw material selection which was done by setting certain criteria such as weight and stiffness of the material. The material used for the chassis was robust and flexible to ensures speedy and safe drive. The tubes implemented in the original chassis are AISI 1020 with 1inch 2 mm wall thickness and its mechanical property are demonstrated in Table-1[10]. The geometry was drafted in Solid Edge CAD package and is represented in Figure-1.

Table-1. Mechanical properties of chassis.

AISI 1020 properties

Young’s modulus [N/𝐦𝟐] 2.00× 𝟏𝟎𝟏𝟏

Poisson ratio 0.29

Density [kg/m3] 7850

Yield strength [N/m2] 3.5× 108

Tensile strength [N/m2] 4.2× 108

Shear modulus [N/m2] 8.0× 108

Figure-1. Geometry of the initial Go-kart chassis.

5. COMPUTATIONAL ANALYSIS AND GEOMETRICAL MODIFICATION

subjected to excitation were fixed and the structure was tested till 8000 RPM as shown in Figure-2. After the analysis, certain members were found to have excessive deformation which are shown in Figure-3 to Figure-6.

Figure-2. Representation of applied condition on the initial chassis in ANSYS.

Figure-3. Excessive Deformation in the seat section.

Figure-4. Excessive deformation in the central section of the chassis.

Figure-5. Excessive deformation in the frontal bumper.

Figure-6. Excessive deformation in the rear bumper of the chassis.

Depending upon the excessive deformation in various sections of the chassis as seen in Figure-3 to Figure-6 the chassis was geometrically modified. The factors considered for modification are represented below and are shown in Figure-7. The dotted section in the Figure-7 was increased by 1mm of thickness. Also, the overall length of the front and the rear bumper was reduced to the minimal value which satisfies both the competition standard as well as our requirement of improving the dynamic characteristic. The location of the bumper support and the seat support also changed in order to reduce the effect of torque acting on the frontal and rear bumper. The dimensions were taken with respect to the axis which lies exactly at the centre of the chassis. The changes made in the dimensions along with the initial dimensions are mentioned in Table-2.

Figure-7b. Front view of the chassis with a parameter for geometrical modification.

Table-2. Dimensions of changed parameters of the initial and modified chassis.

Parameters for modification

Initial Chassis (meters)

Modified Chassis (meters)

A 611.10 406.04

A1 217.17 224.37

A21 217.17 336.55

A22 217.17 224.37

B 656.42 546.98

C 279.4 355.6

Torsional stiffness is carried out to check the structural strength of the structure against static load. In this test, the faces of the front two-wheel mounting were applied with the equal and opposite force of magnitude 2G (3532N) by keeping rear wheel mountings fixed as shown in Figure-8. The value of the load was decided on the basis rule book specifications by considering maximum weight of Go -Kart as 180 kg.

The impact test was followed by the torsional test in which the crashworthiness of the chassis was determined.

Figure-8. Conditions for Torsional Analysis of the Go-kart chassis.

In this test, an impact force of magnitude 4G (7064N) was applied to frontal and rear impact and 2G (3532N) to the side impact as rule book specifications. When applying the force on one face, the opposite faces wheel mountings were fixed. For example, when doing the

Figure-9. Conditions for frontal impact testing of the Go-kart chassis.

6. RESULTS AND DISCUSSIONS

The results of the modal analysis for the initial chassis and the final chassis are shown in Figures 10 and 11 respectively. The graphs represent maximum deformations obtained against their respective mode (Frequency).

From the graphs, it was observed that after modification the number of mode shapes has reduced from 14 to 8, which mean there are only 8 frequencies where problem of resonance would be affecting the structure. The starting mode frequency has also shifted to 54.773 Hz

from 45.674 Hz, which makes the chassis safe against resonance till 54.773 Hz frequency. Also, the modification has helped to reduce the maximum deformation from 1.1761 m to 0.9692 m with a reduction in the weight to 28.304 Kg from 29.907 Kg as an additional benefit.

The results of torsional analysis in terms of deformation for the final structure were obtained as 0.04237 m in comparison to initial structure as 0.073115 m. This proves the better strength in the final structure than that of the initial structure.

Figure-10. Result of modal analysis of the initial chassis.

1.1043

0.57076 0.48953 0.82537

0.69908 1.131

0.78261 1.1761

0.65025 1.0679

0.38728 1.048

0.80454 1.0483

0 0.2 0.4 0.6 0.8 1 1.2 1.4

DE

F

O

R

MA

T

IO

N

(

M)

Figure-11. Result of modal analysis of the modified chassis.

The deformations due to the impact test for the initial and final chassis are represented in Table-3. From Table-3, it can be observed that the final structure strength has also improved against initial structure.

Table-3. Results of impact tests.

Frontal impact

Rear

impact Side impact Initial

chassis 0.04364m 0.04179m 0.006215m Final

chassis 0.04196m 0.03588m 0.0052936m

CONCLUSIONS

In this work, modal analysis of the Go-kart chassis was carried out and on the basis of excessive deformation geometrical modifications were decided in the structure. The modification included increasing the thickness of the peripheral members and relocation of certain members. These modifications assisted in improving both the static as well as the dynamic characteristic of the Go-kart chassis in the following way:

 The maximum deformation in the final structure was found to be reduced by 17.5%, this ensures better strength of the structure against dynamic loading conditions.

 The mode shapes in the final structure was reduced to 8 from 14 that means the final structure would be facing a problem of resonance with only 8 frequencies.

 The initial mode frequency for the final structure was found to be shifted to 54.773 Hz from 45.674 Hz, which makes final structure free from resonance till 54.773 Hz frequency.

 The torsional rigidity was also found to be improved by 40.20%, this will make Go-Kart more stable against turnings.

 The crashworthiness was also improved for the final structure as frontal, rear and side impacts are showing

less deformation against the same impact loads.  The modification also helped in reducing the weight

of the chassis by 1.603 Kg, which will help in increasing the overall efficiency of the Go-kart.

REFERENCES

[1] Abdullah N.A.Z., Sani S., Rahman M. and Zaman I. 2017. Dynamics properties of a Go-kart chassis structure and its prediction improvement using model updating approach. Article in International Journal of Automotive and Mechanical Engineering. 14(1): 3887-3897.

[2] K. Singh. 2000. Automotive Engineering, Volume 1; Standard publication.

[3] Jena D.P., Singh M. and Kumar R. 2012. Radial Ball Bearing Inner Race Defect Width Measurement using Analytical Wavelet Transform of Acoustic and Vibration Signal. Measurement Science Review. 12(4):141-148.

[4] Singh M., Kumar R., Jena D.P. 2009. Detection of missing ball in bearing using decomposition of acoustic signal. Asian Journal of Chemistry. 21(10): 143-147.

[5] Singh M., Shoor S. and Singh H. 2018. Shannon entropy a better indice for local defect detection and to study the effect of variable loading conditions for taper roller bearing. International Journal of Mechanical Engineering and Technology (IJMET). 9(7):198–208.

[6] Rao S. and Bhattu A. 2018. Dynamic Analysis and Design Modification of a Ladder Chassis Frame

0.6256 0.63699

0.44665 0.68062

0.40469 0.74239

0.9692 0.86987

0 0.2 0.4 0.6 0.8 1 1.2

DE

F

O

R

MA

T

IO

N

(

M)

Using Finite Element Method. Amity Journal of Computational Sciences (AJCS). 1(2): 16-21.

[7] Rao S. and Bhattu A. 2019. Dynamic Analysis and Design Optimization of Automobile Chassis Frame Using FEM. Machines, Mechanism and Robotics, Lecture Notes in Mechanical Engineering, Springer Nature Singapore Pte Ltd: 671-680.

[8] Rajappan R. and Vivekanandhan M. 2013. Static and Modal Analysis of Chassis by Using FEA. The International Journal of Engineering and Science. 2(2): 63-73.

[9] J. He and Z. Fu. 2001. Modal Analysis, Butterworth-Heinemann publication.

[10]Rodrigues A.F.A., Poncio A.L.R., Cervieri A., Gertz L.C., Pereira M.D.S. and Oliveira A.B. 2015. Static and Dynamic Analysis of a Chassis of a Prototype Car. 24th SAE BRAZIL, International Congress and Display, Sao Pulo, Brazil. 22-24, 2015-36-0353.

[11]Tomar A. and Singh D. 2016. Static Analysis, Modal Analysis and Design Modification in Chassis Frame to Optimize Weight by Using Composite Material. International Journal of Mechanical Engineering. 5(1): 2319-2259.

[12]Silva F.M.A.D., Valle R.M., Faria M.T.C.D. and Duarte F.P. 2004. Modal Analysis of a Tubular Structure Vehicle Chassis. XIII Congresso e Exposico Internacionals da Tecnologia da Mobilidade Sao Paulo, Brazil. 16-18: 2004-01-3423.

[13]Sane S.S., Jadhav G. and Anandaraj H.Stress Analysis of Light Commercial Vehicle using ANSYS. Piaggio Vehicles Pvt. Ltd. HTC 08:1-5.

[14]Forouzan M. R. and Hoseini R. 2010. Dynamic Analysis of a Modified Truck Chassis. International Journal Advanced Design and Manufacturing Technology. 3(4).

[15]Fui T. H. and Rahman R. Abd. 2007. Statics and Dynamics Structural Analysis of A 4.5 Ton Truck Chassis. Jurnal Mekanikal. 2(24): 56-67.

[16]Khannukar K., Kallannavar V. and Manjunath B. S. 2018. Design Modification of Ladder Chassis Frame Based on Dynamic Analysis. International Research Journal of Engineering and Technology (IRJET). 2(9).

[17]Salunke P., Awadhani L. 2018. Design modification of ladder chassis frame based on dynamic analysis.

International research journal of Engineering and technology. 5(4): 3877-3882.

[18]Logan D. L.2011.A First Course in the Finite Element Method, 5th Edition. Cengage Learning.

[19]Gandhi U.N. and Hu S.J. 1995. Data Based Approch in Modeling Automobile Crash. International Journal Impact Engineering. 16(1): 95-118.