Investigation of Dynamic Behavior and Durability of Double Cardan Drive Shafts with Composite Materials for High-Speed Vehicles

Drive shafts are essential components in various vehicles, playing a crucial role in ensuring efficient power transmission and smooth operation. As a key mechanical link between the engine and the wheels, a drive shaft's main function is to transfer the power generated by the engine to the vehicle's wheels, allowing for forward and backward motion. Additionally, it maintains proper alignment between the engine and the wheels, ensuring that power is transferred efficiently without significant misalignment. This alignment is necessary for a vehicle's smooth and controlled operation. Furthermore, drive shafts are important for reducing and absorbing vibrations generated during vehicle operation, contributing to a more comfortable and stable ride for the occupants. In vehicles with independent suspension systems, drive shafts must also accommodate the vertical movement of the wheels while maintaining constant power transmission, allowing each wheel to move independently without compromising overall performance [1].

The versatility of automotive drive shafts extends beyond passenger vehicles, finding applications in various domains such as rear-wheel drive vehicles, four-wheel drive (4WD) and all-wheel drive (AWD) vehicles, trucks and commercial vehicles, performance and racing vehicles, off-road and utility vehicles, construction and agricultural equipment, industrial applications, and even marine and boating. In each of these applications, drive shafts are designed to handle specific power and torque requirements, often utilizing lightweight and durable materials to optimize performance. As a result, the study and optimization of drive shafts have become increasingly important in the automotive industry, with researchers and engineers continuously seeking ways to enhance their efficiency, durability, and overall performance to meet the ever-growing demands of modern vehicles [2].

Despite the advancements in drive shaft technology, conventional steel drive shafts face limitations in terms of their ability to cope with the demands of high-speed applications. The inherent characteristics of steel, such as its weight and sensitivity to vibrations, can hinder vehicle performance and fuel efficiency. Moreover, the increasing complexity of modern drive systems necessitates the development of advanced design and analysis techniques that can accurately predict and optimize drive shaft behavior under various operating conditions.

This section presents the design and analysis of a Double Cardan Driveshaft for high-speed vehicles. It covers the fundamental aspects of machine design, stress analysis, fatigue failure criteria, composite materials, and their failure criteria. The theory of modal analysis and harmonic response analysis is also discussed, essential for understanding the drive shaft's dynamics. The design and numerical analysis of the Double Cardan Driveshaft is the focus of this section. The design uses SOLIDWORKS software, and numerical analysis is performed using ANSYS Workbench in two stages. The first stage examines the steel shaft, while the second and third stages incorporate composite materials. The analysis includes meshing, boundary condition application, loading, and subjecting the model to various operating conditions to simulate real-world scenarios. Stress, vibration, and dynamic analyses are conducted to evaluate the drive shaft's performance and structural integrity. The results are presented along with the final driveshaft model, mesh results, boundary conditions, and applied loads. This section provides a comprehensive approach to developing a high-efficiency Double Cardan Driveshaft by combining theoretical concepts with practical design and analysis techniques.

3.1 Machine design

Machine design is a complex process that combines creativity and analytical skills to develop functional machines. It involves planning and designing individual components and integrating them into a system that performs a specific function. Designing a machine requires understanding mechanical principles and considering practical constraints such as material properties, manufacturing techniques, sustainability, and cost. The process starts with conceptualization, followed by detailed engineering and analysis to optimize components for performance and reliability. It often involves simulation and prototyping to validate design choices and refine the machine before final production. Effective machine design is crucial for innovation in technology and industry, enabling the development and improvement of machinery [12].

3.1.1 Axial stress

Axial stress is the stress that happens when normal forces are applied along the length of a structure, such as in columns or struts. This stress could be either (tensile or compressive) nature. It can be calculated using the equation below [13]:

$\sigma_{\text {axial }}=\frac{F}{A}$    (1)

In this equation, $\sigma_{\text {axial }}$ represents the axial stress, F is the axial force (tensile or compressive), and A is the cross-sectional area of the object. This is essential for determining whether a material will withstand loading conditions without failing under tension or compression [13].

3.1.2 Shear stress

Shear stress is caused by shear forces that make the material particles of an object slip past each other, such stress is most often observed in beams and shafts. It can be calculated by the equation below [13]:

$\tau_{\text {shear }}=\frac{V}{A}$      (2)

Here, $\tau_{\text {shear }}$ is the shear stress, V is the shear force, and A is the area over which the shear force is acting. Calculating shear stress is critical for ensuring components like beams, bolts, and pins can handle such forces without shearing [13].

3.1.3 Flexural stress (from bending moment)

A material is under flexural stress, or bending stress when it is subjected to a moment of flexure. It is widely analyzed in beams. It can be calculated by the equation below [13]:

$\tau_{\text {shear }}=\frac{M \cdot y}{I}$    (3)

σ bending is the bending stress, M is the bending moment, y is the distance from the neutral axis to the outermost fiber of the beam (the height of the cross-section), and I is the moment of inertia of the cross-section. This equation helps designers ensure that beams are capable of sustaining loads without bending excessively or failing [13].

3.1.4 Flexural stress (from bending moment)

Torsional stress occurs when torque is applied to a cylindrical object, for example, a shaft, and causes it to twist. It can be calculated by the equation below [13]:

$\tau_{\text {torsional }}=\frac{T \cdot r}{I}$     (4)

The given formula expresses τ torsional as torsional stress, T as the torque, r as the radius of the shaft, and J as the polar moment of inertia of the cross-section. Shafts and other rotating elements should be designed so as not to fail because of twisting under load.

3.2 Composite materials

Composite materials are made up of two or more separate materials combined to produce enhanced mechanical or physical properties. These materials are typically produced from a blend of metals, polymers, ceramics, and reinforcement materials like fibers. As technology has advanced, the scale of these materials has been improved to include nano-sized components such as nanotubes and nanofibers, which exhibit excellent properties. Composite materials have become increasingly important, particularly when conventional materials cannot offer the required performance. They are preferred in industries such as aerospace, automotive, and civil infrastructure, where strength-to-weight ratio, endurance, and resistance to environmental factors are crucial. Composites can display properties that are unattainable with individual constituent materials alone. The introduction of nanocomposites, which are reinforced at the nanoscale, has further advanced the field of composite materials, making them irreplaceable in engineering applications with lightweight structures in aircraft and high-performance uses in sports equipment [14].

3.2.1 Composite material failure criteria

Maximum strain criterion

The maximum strain criterion is a failure theory primarily employed in the case of orthotropic materials like composite laminates. It fails when the strain in any major direction of the material in question violates its corresponding allowable strain limit. This parameter generalizes the hypothesis of maximum normal strain, which is usually employed for isotropic materials, to handle the anisotropic behaviour of composites by determining strains in the directions of principal material axes. It can be calculated using the equations below [14]:

$-\epsilon_L^{-}<\epsilon_1<\epsilon_L^{+}$   (5)

$-\epsilon_T^{-}<\epsilon_2<\epsilon_T^{+}$     (6)

$\left|\gamma_{12}\right|<\gamma_{L T}$    (7)

Maximum stress criterion

In a basic failure theory designed for orthotropic materials such as composites, the maximum stress criterion predicts failure when the stress in any principal material axis equals and exceeds the corresponding permissible stress limit. This criterion applies the maximum normal stress theory used for isotropic materials to the peculiarities of composites by analysis of stresses in main material directions [14]. It can be calculated using the equations below [14]:

$-s_L^{-}<\sigma_1<s_L^{+}$     (8)

$-s_T^{-}<\sigma_2<s_T^{+}$     (9)

$\left|\tau_{12}\right|<\tau_{L T}$        (10)

where, $\sigma_1, \sigma_2$: Stress components along the longitudinal and transverse directions of the lamina, respectively, $s_L^{-}+, s_L^{+}$: Ultimate tensile and compressive strengths in the longitudinal direction, $s_T^{-}, s_T^{+}$: Ultimate tensile and compressive strengths in the transverse direction. $\tau_{12}$: Shear stress in the plane of the lamina. $\tau_{L T}$: Ultimate shear strength allowable before failure.

Quadratic interaction criteria

The quadratic interaction criteria are complex failure theories of composite materials with consideration of the interaction between different stress components. Contrary to the more straightforward failure theories which consider stress or strain to be applied independently, these criteria define failure by the combined effect of stresses through a quadratic formula. This method is more all-encompassing because it accounts for the intricate relationship’s characteristics in anisotropic materials such as composites [14]. It can be calculated using the equations below [14].

$F_i \sigma_i+F_{i j} \sigma_i \sigma_i=1$    (11)

where, $\sigma_i$: Stress components, $i$: represents different directions (e.g., longitudinal, transverse, shear). $F_i$: Coefficients representing the linear influence of the stress components. $F_{i j}$: Coefficients representing the interaction between different stress components.

3.3 Modal analysis

Modal analysis is a vital engineering concept, which deals with identifying the natural vibrational properties of systems. It deals with the determination of the natural frequencies, mode shapes and damping factors of a structure or mechanical system. The natural modes determine the vibrations the system will produce when left to its own devices free from any external forces. It is critical to know that these characteristics duly affect the dynamic response of the system to external forces. Modal analysis helps engineers to forecast the responses of structures to various types of loads and vibrations, which is very important for engineering and safety evaluations. The process utilizes theoretical and experimental techniques whereas the theoretical analysis uses numerical methods such as the finite element method for simulations. In contrast, the experimental modal analysis relies on physical testing to validate the numerical models [15].

3.3.1 Natural frequency equation

$\omega_n=\sqrt{\frac{k}{m}}$     (12)

The above equation is the fundamental equation for calculating the undamped natural frequency (ω_n) of a single-degree-of-freedom (SDOF) system, where k is the stiffness of the system, and m is the mass.

3.3.2 Damped natural frequency

$\omega_d=\omega_n \sqrt{1-\zeta^2}$      (13)

In systems where damping is present, the damped natural frequency $\left(\omega_d\right)$ is used. Here, $\zeta$ is the damping ratio, and $\left(\omega_n\right)$ is the undamped natural frequency.

3.3.3 Basic equation of motion in modal space

$[M\rceil[\psi] \ddot{y}+[K][\psi\rceil y=0$        (14)

where, $[M]$ is the mass matrix, $[K]$ is the stiffness matrix, $[\psi]$ represents the mode shapes, and $y$ denotes the modal coordinates.

3.4 Harmonic response analysis

Harmonic response analysis, also called frequency response analysis, is a dynamic linear analysis method employed to determine a system’s response to excitation at certain frequencies. It uses the output of a previously performed modal analysis to get the required input frequencies. Modal superposition is used in harmonic response analysis to determine the system response to steady-state sinusoidal loads at specified frequencies that can be in and out of phase with each other. This method is often used to evaluate stresses in machines, vehicles, or process equipment under continuous harmonic loading, for instance, rotor imbalance or cyclic loading in combustion engines [16].

3.4.1 Equation of motion

$M \ddot{u}+C \dot{u}+K u=F$    (15)

This equation is the generalized dynamic equilibrium equation, where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, and F is the forcing function.

3.4.2 Forcing function and displacement

$F=F_{\max } e^{i \psi} e^{i \Omega t}$       (16)

$u=u_{\max } e^{i \phi} e^{i \Omega t}$     (17)

Here, $F_{\max }$ and $u_{\max }$ are the amplitude of the force and displacement respectively, $\psi$ and $\phi$ are their phase shifts, and $\Omega$ is the imposed circular frequency.

3.4.3 Harmonic equations of motion

$-\Omega^2 M+i \Omega C+K\left[u_1+i u_2\right]=\left[F_1+i F_2\right]$        (18)

This equation converts the complex notation into a form that includes both the real and imaginary parts of displacement and force vectors.

3.4.4 Modal superposition method

$y_{j c}=\frac{f_{j c}}{\left(w_i^2-\Omega^2\right)+i\left(2 w_j \Omega \zeta_j\right)}$    (19)

This equation is employed in the modal superposition method, with $y_{j c}$ representing complex modal coordinates, $f_{j c}$ denoting complex modal forces, $w_j$ representing natural circular frequencies, $\Omega$ denoting the excitation frequency, and $\zeta_j$ is the damping ratio [17].

3.4.5 Modal complex displacements calculation

$u_c=\sum_{j=1}^n \phi_j y_{j c}$       (20)

In this formulation, $u_c$ represents the complex displacement vector, $\phi_j$ are the mode shapes, and $y_{j c}$ is the complex modal coordinates computed previously [17].

3.5 Drive shaft design and numerical analysis

This section focuses on the design and numerical simulation of a Double Cardan Driveshaft for high-speed vehicles to develop a new design that improves performance, durability, and reliability. The section is divided into two main parts. The first part deals with the design of the driveshaft using SOLIDWORKS software, where a 3D model is created with unique modifications. The second part involves numerical simulations using ANSYS Workbench, conducted at high rotational speeds to replicate the demanding operating conditions of high-speed vehicles. The numerical simulations are carried out in two stages. The first stage simulates the driveshaft using steel, without composite materials, to provide a baseline for the driveshaft's behavior. The second stage includes composite materials in the driveshaft design, with a thickness of 5mm, consisting of 10 layers, each 0.5mm thick, to explore their potential benefits in performance enhancement. During the simulations, various operating conditions, such as loads, speeds, and vibrations, are applied to the model to closely resemble real-world scenarios. Stress analysis, vibration analysis, and dynamic analysis are performed to predict the driveshaft's performance under these conditions. The results are analyzed to assess the design's structural integrity, load-bearing capacity, and dynamic stability at high rotational speeds. By using SOLIDWORKS for design and ANSYS Workbench for numerical simulations, this section aims to develop an optimized Double Cardan Driveshaft design that improves performance and reliability in high-speed applications. The findings from this approach contribute to the advancement of driveline systems and provide insights for future design improvements in the automotive industry.

3.5.1 Double Cardan Driveshaft design

This component was designed using SOLIDWORKS software, as clarified before due to its accuracy and ease of use for such applications [18]. The figures below illustrate the final model, along with the dimensions of the component.

The 3D shape of the Driveshaft that used in this study is shown in Figure 1.

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Figure 1. Final 3D model of Double Cardan Driveshaft

All dimensions of the driveshaft are shown in Figure 2.

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Figure 2. Double Cardan Driveshaft dimensions

3.5.2 Double Cardan Driveshaft numerical analysis

Numerical analysis without composite material

In this study, two modules in the ANSYS software will be used: (Modal analysis) and (harmonic response analysis). Modal analysis will be performed to determine the natural frequencies of the drive shaft. These natural frequencies will then be linked to the harmonic response analysis to investigate the stresses and deformations on the shaft at different frequencies while applying the external load.

After completion of the design with SOLIDWORKS, the next phase is numerical simulation and analysis employing ANSYS Workbench. Meshing is the first stage in numerical analysis to achieve correct solutions. In this work, the body sizing with a 10mm element size was used to get an accurate mesh. Figure 3 illustrates the meshing results.

The process of meshing includes splitting the driveshaft into smaller segments with nodes in between. The mesh quality impacts the accuracy of the numerical solutions. After the finishing of the mesh, the remaining analysis steps can be performed like imposing boundary conditions, specifying the material properties, and configuring the required analysis types. The next sections will cover the detailed analyses conducted to determine the performance and durability of the Double Cardan Driveshaft under different loading conditions.

Regarding the boundary conditions, a fixed support will be applied to one end of the drive shaft, as shown in Figure 4 below.

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Figure 3. Double Cardan Driveshaft mesh results

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Figure 4. Fixed support on Double Cardan Driveshaft

For the applied load, torque will be applied to the other end of the drive shaft. This torque will be determined at high speeds using the relationship between torque, horsepower, and RPM, as shown in the equation below [19].

Horsepower $(H P)=\frac{\text { Torque }(\mathrm{T} \text { in lb-ft }) \times \mathrm{RPM}}{52.52}$    (21)

Since the study will be conducted at high speeds, a car with 600 horsepower and 6000RPM will be considered. Based on the equation above, the resulting load (torque) applied to the drive shaft will be:

Torque(lb.ft)=(5252×Horsepower)/RPM=(5252×600)/6000=525.2lb.ft.

Figure 5 shows the calculated moment on the driveshaft.

The project schematic of the first case study is shown in Figure 6.

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Figure 5. Torque applied on Double Cardan Driveshaft

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Figure 6. Project schematic for the first case study

Numerical analysis with composite material

Composite materials will be added to the second and third case studies. In the second case study, the composite material will be (Carbon matrix, carbon fibre reinforced (Vf50%)), and in the third case study, it will be (Composite, Epoxy CF, resin-infused fabric, biaxial). Modal Analysis and Harmonic Response Analysis will be used to investigate the frequencies, and the ACP (ANSYS Composite PrepPost) module will be utilized to add the composite materials to the middle section of the Double Cardan Drive Shaft. Ten layers of composite materials will be added, each with a thickness of 0.5mm, resulting in a total thickness of 5 mm. Each layer will have a specific orientation, with the stacking sequence of [0, 45, -45, 0, 90]2s, representing the fiber orientation in each layer of the composite material.

Table 1 shows the composite material layers and directions.

The properties of the two composite materials used in this research are shown in Tables 2 and 3.

The area where the composite material was added is shown in Figure 7.

Table 1. Carbon matrix, carbon fiber reinforced (Vf: 50%) properties [20]

Table 2. Carbon matrix, carbon fiber reinforced (Vf: 50%) properties [20]

Table 3. Composite, epoxy CF, resin-infused fabric, biaxial [20]

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Figure 7. Double Cardan Driveshaft after composite material layers are added

The meshing process was performed for the part to which the material composite material was added to concentrate the solution more on it, and the meshing results are shown in Figure 8.

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Figure 8. Mesh results for part of the driveshaft that the composite material added on it

The project schematic of the second and third case study is shown in Figure 9.

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Figure 9. Project schematic for the second and third case studies

For clarification, the boundary conditions remain the same as in the previous section and are not discussed in this section. The results of all three case studies, including the analysis with composite materials, will be presented in the next chapter.

In conclusion, this study demonstrates the significant benefits of integrating composite materials with traditional steel in Double Cardan Driveshafts for high-speed vehicles. Utilizing SOLIDWORKS and ANSYS Workbench, the research reveals that the Composite, EpoxyCF, resin-infused fabric, biaxial material substantially enhances driveshaft performance. Key findings include a 338% increase in safety factor and a 99.6% reduction in damage factor compared to traditional steel designs, along with improved stiffness and reduced stress under high-speed conditions. The novel incorporation of Harmonic Response Analysis provides crucial insights into dynamic loading behavior, enhancing result accuracy and reliability. These improvements underscore the potential of composite materials in creating stronger, more durable driveshafts for high-performance vehicles. Future research should focus on experimental validation, composite layup optimization, vehicle integration, and manufacturing optimization to further advance the design and efficiency of driveline systems for high-speed applications.