Modelling and Dynamic Analysis of a Vehicle

Abstract

In this topic, modelling of vehicles, dynamic analysis and verification of both vehicle model and dynamic analysis will be elaborated. Vehicle modelling can be performed by either using simple discrete models, finite element models or rigid bodies that are connected with elastic elements. Element types of each modelling method and in which circumstances can be used will be explained. Advantages and disadvantages will be encompassed. After that, vehicle model will be evaluated in terms of dynamic behaviours such as mode shapes and natural frequencies. Moreover, damping parameter selection methodology will be elaborated. Consequently, model verification methodologies will be handled such as modal tests and dynamic load tests. How the model can be updated according to the tests will be briefly explained.

1. Introduction

In this chapter, modelling of vehicle, methods of dynamic analysis of a vehicle and verification tests of the methods will be explained. Moreover, industrial experience in modelling, analysis and verification will be shared.

The vehicle market evolution requires reduction of time to market. Moreover, it is inevitable to decrease the time of concept to mass production. Furthermore, new vehicle models are revealed every couple of years in order to the survival of vehicle companies. It is merely possible by performing most of the verifications by analyses. Building models and relying on analyses for verification requires experience. Therefore, modelling methods, verification of models, analysis methods, verification of analysis and test methods are elaborated to share theoretical and industrial experience with the readers.

2. Modelling of a vehicle

In order to produce a vehicle, design phase is required to be completed. After the vehicle design verification, production phase starts. Some modifications will be needed to be performed before the final product such as design improvements, customer requests, etc. Performing verification tests after every modification will cause overbudget. Moreover, the production of each prototype will cause overschedule. Therefore, other verification methods will be needed such as verification of models and analyses. Models should be created before analyses are performed. There are two types of models such as analytical models and finite element models.

2.1 Analytical models

In analytical methods, there are two methods such as creating simple discrete models and creating flexible models. After creating these two mathematical models, analytical solutions can be calculated.

Advantages of analytical methods can be summarized as follows:

1. Analytical methods provide faster solutions due to the fact that it has fewer degrees of freedom when compared with finite element methods.

2. Modification of the mathematical model is faster for analytical methods when compared with finite element methods.

3. Modeling and solving an analytical model requires less computational cost.

4. Analytical methods provide insight into the solution and provide an order of magnitude for the solution.

Disadvantages of analytical methods can be summarized as follows:

1. Analytical methods provide less accurate solutions due to the fact that it has fewer degrees of freedom when compared with finite element methods.

2. Analytical methods require more assumptions and simplifications when compared with finite element methods.

3. Analytical methods are inadequate for detailed analysis.

2.1.1 Simple discrete model

Simple discrete models can be used at the beginning of the design phase. Simple discrete models are particularly utilized for a quick and approximate solution since they have tens of degrees of freedom. Rigid body and flexible joint assumptions can be utilized in order to obtain analytical model. Chassis and the upper structure of the vehicle can be modelled as a single rigid body that has mass and inertia or a set of point masses that have inertia. In this approach, suspension systems and tires can be modelled as point mass and inertia with their stiffness and damping. Dynamic behaviour of a vehicle can be calculated by using Lagrange’s equation shown in Eqs. (1) and (2) [1].

$$\frac{d}{\mathit{dt}}\left(\frac{\partial \text{£}}{\partial \dot{q_r}}\right)-\frac{\partial \text{£}}{\partial q_r}=Q_r^{\ast }\mathrm{r}=0,\dots ,\mathrm{n}$$E1

$$\text{£}=E_k-E_p$$E2

where,

£ is Lagrangian

q_r, r=1,…,n are generalized coordinates

Q_r^*_, r=1,…,n are generalized forces representing all non-conservative forces

E_k is Kinetic Energy

E_p is Potential Energy

There are three types of simple discrete vehicle modes such as quarter vehicle model, half vehicle model and full vehicle model. Quarter simple discrete vehicle model, which is one-fourth of the whole vehicle, is shown in Figure 1. Quarter simple discrete vehicle model is the simplest version of a vehicle model. It contains only one tire, suspension system and one-fourth of the vehicle mass as a point mass.

Figure 1.

Quarter simple discrete vehicle model [2].

Half simple discrete vehicle model, which is more complicated, is shown in Figure 2. It contains left or right half of the vehicle model.

Figure 2.

Half simple discrete vehicle model [3].

Full simple discrete vehicle model is the most complicated version of the simple discrete models and is shown in Figure 3.

Figure 3.

Full simple discrete vehicle model [4].

In simple discrete models, M represents the mass of the vehicle body without suspension system and tires, I represents the inertia of the vehicle without suspension system and tires. m_i2 and I_i2 represent relevant suspension system’s mass and inertia, respectively. m_i1 and I_i1 represents relevant tire’s mass and inertia, respectively. Suspension system’s and tire’s stiffness and damping properties are represented with k and c, respectively. In this simple discrete model, main body of the vehicle is represented by a rigid body that has a single mass and inertia. Each one of the four suspension systems is represented with a point mass and inertia. Similarly, each one of the four tires is represented with a point mass and inertia. Suspension system and tires are represented by single stiffness and damping coefficients. Q_i1 is the excitation applied from the road [5, 6].

Quarter simple discrete vehicle model should be used for initial analysis. For further analysis, a half simple discrete vehicle model should be used for more accurate results which are also closer to the full simple discrete vehicle model when compared to dynamic responses [2]. Subsequently, a full simple discrete vehicle model should be used for more accurate results. MATLAB can be preferred as a solver of equations of motion.

2.1.2 Flexible multi-body model

Flexible multi-body models are more complicated when compared with simple discrete models since they have rigid and flexible bodies. In this method, system is modelled with rigid and flexible bodies connected by joints. In this method, solution can be calculated by deriving below fundamental equations [7].

Newton’s equation using D’Alambert’s principle for particle i is shown in Eq. (3).

$$\overrightarrow{F_i}-m_i\overrightarrow{\ddot{r_i}}=0$$E3

where,

$\overrightarrow{r_i}$: Position vector of particle i

$\overrightarrow{F_i}$: Sum of forces acting on the particle

Virtual work which is done by these forces is also zero.

$$\delta V_i=\left(\overrightarrow{F_i}-m_i\overrightarrow{\ddot{r_i}}\right)\delta \overrightarrow{r_i}=0$$E4

where,

$\delta V_i$: Virtual work

$\delta \overrightarrow{r_i}$: Virtual displacements of the particle i

$$\overrightarrow{F_i}=\overrightarrow{F_i^e}+\overrightarrow{F_i^s}+\overrightarrow{F_i^r}+\overrightarrow{F_i^d}$$E5

where,

$\overrightarrow{F_i^e}$: External forces acting on the particle i

$\overrightarrow{F_i^s}$: Internal forces from other particles in the same body

$\overrightarrow{F_i^r}$: Joint reaction force from another movable body through a joint

$\overrightarrow{F_i^d}$: Reaction forces due to joints with the fixed frame

$$\sum _{i=1}^{\infty }\left(\overrightarrow{F_i^e}-m_i\overrightarrow{\ddot{r_i}}\right)\delta \overrightarrow{r_i}-\sum _{j=1}^n\delta s_j=0$$E6

where,

n: number of flexible bodies

$\overrightarrow{F_i^e}$: External forces acting on the particle i

$m_i$: Mass of the particle i

$\overrightarrow{\ddot{r_i}}$: Acceleration vector of the particle i

$\delta \overrightarrow{r_i}$: Virtual displacements of the particle i

$\delta s_j$: Virtual change in strain energy of flexible body j

Lagrange form of D’Alambert’s principle is shown in Eq. (7).

$$\sum _{j=1}^n\left(f_j+f_j^s\right)\delta q_i+\sum _{i=1}^{\infty }\overrightarrow{F_i^{\ast }}\delta \overrightarrow{r_i}=0$$E7

where,

$f_j$: Generalized external force corresponding to general coordinate j

$f_j^s$: Generalized structural stiffness forces

$\delta q_i$: Virtual displacement in independent coordinate i

$\overrightarrow{F_i^{\ast }}$: Inertia method of particle i

$\delta \overrightarrow{r_i}$: Virtual displacements of the particle i

For a system of rigid or flexible bodies generalized inertia forces can be calculated as follows.

$$f_j^{\ast }=-\sum _{k=1}^N\int \rho \ddot{\overrightarrow{r^k}}\frac{\partial \dot{\overrightarrow{r^k}}}{\partial \dot{q_j}}\mathit{dV}$$E8

$f_j^{\ast }$: Generalized inertia forces j

N: Number of bodies

$\rho$: Density

$\overrightarrow{r^k}$: Position vector of an arbitrary point in body k

$q_j$: Independent coordinate j

V: Volume

2.2 Finite element model (FEM)

After analytical models created and detailed analyses are required, detailed models should be created such as finite element models. Finite element method is a powerful and popular method that provides numerical solutions of differential equations. In this section, finite element method will be handled for dynamic and structural analysis. FEM divides whole domain into smaller divisions which are named finite elements in order to solve the problem. This method converges a solution by minimizing an error function.

Advantages of FEM are as follows:

1. FEM represents a complicated model more accurately.

2. FEM represents the total solution of the whole model more accurately.

3. FEM provides more detailed solutions for structural and dynamic analysis.

4. FEM requires less assumption and simplification.

Disadvantages of FEM are as follows:

1. FEM causes more time and computational cost.

2. FEM causes more modelling effort.

FEM of a vehicle can be classified in three groups such as simplified FEM, reduced degrees of freedom FEM and detailed FEM. At the beginning of the design phase, obtaining faster results is the most important thing since lots of iterations are needed to get the final design. Therefore, simplified and less degrees of freedom models should be preferred at the beginning. When desired response is achieved, detailed and more degrees of freedom models should be used for more accurate results. Detailed models are particularly required for structural analysis. Simplified and approximate models are inadequate for structural analysis. On the other hand, detailed models are not convenient for iterative solutions.

2.2.1 Simplified FEM

Simplified FEM generally consists of simple beam, truss, shell elements (As a rule of thumb smallest dimension is at least 20 times less than the largest dimension), point masses and rigid body elements to bind them. When creating shell elements, mid-surfaces should be created at first. Then mid-surfaces should be attached to each other at the same level for continuity. Subsequently, shell elements should be created by defining the thickness and mid-surface layer position. This process is critical for creating shell models.

The purpose of creating simplified FEM is to obtain quick and approximate results to anticipate how close the first approach to the final one is. Dynamic response of the vehicle can be approximately calculated by using these models. Moreover, simplified FEM has particularly huge advantage over detailed FEM when the past experience over vehicle design is weak or the dynamic load acting on the system is quite new.

Simplified FEM should be created by using an automated code since making modifications for every iteration will take time. Therefore, simplified FEM should be built as parametric. Easily modifiable parts should be modelled as parametric, and modelling and post processing codes should be automated.

Cicek et al. [8] created a simplified FEM to investigate the individual effects of a heavy vehicle’s design parameters. Simplified Fem is shown in Figures 4 and 5.

Figure 4.

Simplified finite element model [8].

Figure 5.

Simplified finite element model’s mesh [8].

There are other ways to create and analyse the system response such as gathering mass and stiffness matrix of a vehicle by using a FEA software and calculate system response by using analytical methods and these matrices. This method should be preferred for much simpler models since gathering these matrices will be much harder according to the complexity of the vehicle model. These matrices can be used to perform modal analysis only which is the first step of figuring out of the dynamic behaviour of the vehicle. On the other hand, computation time will increase due to matrix inversion when vehicle model has more degrees of freedom. Moreover, for further dynamic analysis numerical approach is better.

The other way is using superelements. A superelement is a special type of finite element which consists of a group of finite elements. It is particularly employed to simplify the model when most of the model remains the same after modifications. A simple example of a superelement is shown in Figure 6. Left part of the example represents the parts that will be modified during design iterations. Right part of the example represents the superelement that will remain the same during the design iterations. Exterior nodes represent the connection points. By using superelement approach, the number of degrees of freedom of the superelement part will be reduced which will provide a significant computation time due to the fact that analysis of the superlement part will not be calculated for every iteration [10]. This will provide a huge advantage during the first part of the vehicle design phase. On the other hand, reduction process will cause a computation cost. Superelement approach is applicable when the number of exterior elements is much less than the interior elements in the superelement. Furthermore, this method can be only utilized for figuring out the dynamic behaviour of the vehicle system like other simplified models. Moreover, the theory behind this technique should be comprehended before deciding to employ it.

Figure 6.

A simple superelement example [9].

2.2.2 Reduced DOF FEM

Reduced DOF FEM is another method to obtain faster results since it has hundreds of DOF when simplified FEM has thousands of DOF. Therefore, component modal synthesis (CMS) [11] which is the most popular reduction method is handled. This method employs highly truncated mode set by converting finite element model’s equations of motion into modal domain in other words generalized coordinates space. The usage of this method is an improved version of superelement method. CMS is developed by using Craig Bampton method [12].

$$u=\mathit{Tq}$$E9

where,

$u$: System response which consists of many DOF

$T$ : Transformation matrix

$q$: Modal coordinates consist of less coordinates

$$u=\mathit{\Phi q}$$E10

where,

$\Phi$: Mode shape matrix which is obtained by below equation

$$det\left(K-{\lambda }^{2}M\right)=0$$E11

$$M\ddot{u}+\mathit{Ku}=F$$E12

where,

$M$: Mass matrix

$K$: Stiffness matrix

$F$: Force matrix

$${\Phi }^T\mathit{M\Phi }\ddot{q}+{\Phi }^T\mathit{K\Phi q}={\Phi }^{T}F$$E13

By using above equations, N DOF finite element model can be converted into n DOF model where N is much greater than n. The main issue is defining n since if it is selected closer to N, reduction will be pointless. When n is selected as smaller there is risk for truncated model that cannot represent the main model adequately. Therefore, modal effective mass approach can be utilized. This method calculates the contribution of each mode on rigid body motion. As a rule of thumb, modal effective mass should be more than 90%. Moreover, highest natural frequency should be selected more than 2 or 3 times higher than excitation frequency for reduction. Furthermore, number of attachment nodes should be taken into account when calculating the minimum number of modes required for reduction. It should be noted that reduction will bring truncation some errors [13].

Semi-trailer finite element model is shown in Figures 7 and 8 as an example.

Figure 7.

Semi-trailer finite element model.

Figure 8.

Semi-trailer chassis finite element model.

In semi-trailer finite element model, suspension system will be fixed to the lower chassis.

In this example semi-trailer model, suspension system is connected from reduced DOF FEM from its external nodes. Suspension system is modelled as rigid bodies in this example that is shown in Figure 9. Moreover, spring, damper and tire elements are utilized to represent stiffness and damping characteristic of the suspension system and tires. Tire properties can be acquired from the producer or can be obtained by conducting tests. Furthermore, road should be modelled for a complete dynamic analysis. Road can be modelled as 2D or 3D. Tire and road properties and their total dynamic behaviour should be verified for more realistic results. Pure and coupled reaction forces applied from road to tire and slip conditions are also should be taken into account when defining road and tire properties for an analysis model.

Figure 9.

Reduced degrees of freedom finite element model.

3. Dynamic analysis of a vehicle

After modelling of a vehicle process completed dynamic analysis process starts. All assumptions, simplifications and modifications effect on the dynamic response of the system. Therefore, after verification of the dynamic analysis models should be reviewed and next iteration should be initiated after necessary modifications made on the vehicle model.

Particularly for the finite element models, at the very beginning of the dynamic analysis vehicle model should be elaborately reviewed such as finite element mesh continuity, common nodes, element shape and size, material properties, thickness definition for the . In order to reveal that firstly free-free modal analysis should be performed. As a result of the modal analysis local modes appear if mesh continuity could not accomplished. Moreover, rigid body modes can be seen if connection of some parts are missing. As a result of modal analysis, aforementioned mistakes should be resolved for a proper dynamic analysis.

When the finite element model is convenient for the dynamic analysis, modal analysis should be performed. Modal analysis will provide overview about dynamic characteristic of the vehicle model [14]. Bending mode of a truck chassis is shown in Figure 10.

Figure 10.

Modal analysis of a truck chassis.

Furthermore, modal analysis results should be compared with the analytical model results. Natural frequency order of magnitude and mode shape should be close to the analytical model results particularly for the first bending mode. After getting experience about the vehicle, the need for the analytical model will disappear. Moreover, modal effective mass results should be reviewed. This will give the idea of which mode will contribute when reduction is applied.

After modal analysis, static analysis should be performed by applying boundary conditions and gravity load on the system. Static analysis is commonly considered the first step of the structural analysis. The static analysis of vehicle parts is utilized to obtain the effects of constant gravity loads on the vehicle without taken into account the inertia and changing dynamic loads. After static analysis stress, strain and deformation of the parts can be obtained under a constant load. To figure out the deformation results better exaggerated views can help. Simple discrete models and simplified models can talk less about the static analysis results since they have less details. On the other hand, detailed finite element model should be employed to investigate stress, strain and deformation of the parts. In the detailed FEM, elaborated structural analysis can be performed on parts such as bolts, nuts and welding points. Every detail will take some time for modelling and will cause a computation cost. Therefore, static analysis of the detailed FEM should be performed for the last iterations.

Up to now, natural frequencies and mode shapes are calculated by using simple discrete model and simplified FEM. Modal analysis results are compared for the simple discrete model and simplified FEM. Modifications on the simplified FEM are applied if needed. Static analysis is performed for the simple discrete model and simplified FEM. Results are compared. Now we are sure about our models, assumptions and simplifications and it is time to perform dynamic analysis.

First step of the dynamic analysis is to model the dynamic load on the vehicle. Dynamic load can be caused by the road, engine, transmission and etc. Dynamic load can be acquired by test results or can be modelled. After dynamic load is acquired, damping characteristic of the vehicle should be defined as a second step. Damping characteristic can alter material properties and joints of the vehicle model. For a linear elastic system proportional damping can be employed [8].

$$\left[C\right]=\propto \left[M\right]+\beta \left[K\right]$$E14

where,

$\left[C\right]$: Damping matrix

$\left[M\right]$: Mass matrix

$\left[K\right]$: Stiffness matrix

$\propto$ and $\beta$ are the constants of proportionality

Dynamic analysis should be performed by using simple discrete model, simplified FEM and reduced DOF FEM as a third step. Dynamic analysis results should be compared to figure out the dynamic response of the system. Simple discrete model and simplified FEM can provide dynamic response of the system as an order of magnitude. For a more realistic solution reduced DOF FEM should be used. In order to use the reduced DOF FEM, vehicle model should be continuous, linear elastic and deformations should be small (about 10%) due to the fact that CMS is based on linear theory. Reduced DOF FEM is a commonly preferred model to calculate the dynamic response of a vehicle. On the other hand, for larger deformations and nonlinear structures, reduced DOF FEM is not valid. Moreover, to calculate stress and strain, detailed FEM should be preferred since reduced DOF FEM is not a powerful method since it solves the matrices in modal domain.

After reduced DEF FEM dynamic analysis is performed, dynamic characteristic of the vehicle is obtained and for elaborated analysis detailed FEM should be utilized. Detailed FEM and reduced DOF FEM dynamic analysis results should be compared in terms of deformation, reaction forces and transmissibility. Transmissibility is the ratio of the output to the input. Transmissibility equals to 1 when the model is rigid body. However, our vehicle model has some damping, transmissibility will change according to the frequency. Therefore, reduced DOF FEM and detailed FEM should be compared in terms of transmissibility in order to obtain more realistic results. According to the system response of a vehicle, some modifications can be required to decrease the deformation and transmissibility. Same rule applies here, too. Detailed FEM should be created, and dynamic analysis should be performed as multi body simulation particularly for the last iterations. Multi body simulation is a dynamic analysis method where multi body model has all necessary structural elements. Multi body truck model is shown in Figure 11.

Figure 11.

Multi body truck model [15].

As a last step, dynamic structural analysis results should be obtained by using detailed FEM. Deformations, stress, strain, transmissibility and reaction forces should be calculated by using detailed FEM. Particularly for stress results, mesh size and mesh quality should be reviewed. In some situations, stress results can change according to the mesh size. As a rule of thumb, big mesh size causes bad results and small mesh size causes more computation time. Therefore, different mesh sizes should be compared to have experience on the vehicle FEM. Likewise, mesh quality is another factor that causes misleading results. Thus, mesh quality should be kept high particularly in stress concentration areas.

4. Verification of the analysis

In order to rely on the analysis results, models should be verified. Testing is the most common method to verify the models. Verifying modal analysis results is the first step of the verification of dynamic analysis. Modal updating is the mostly applied method to verify the modal analysis. Modal updating works by updating mass, stiffness and damping parameters of the model to decrease the difference between the analysis and the test results. Decreasing the difference between analysis and test results more will provide a better and more realistic models. Therefore, modal test should be performed. Modal testing is applied to determine the natural frequencies, mode shapes, modal masses, modal damping ratios of a system under a test condition. Modal testing is a method that is used for characterizing the structure by delivering a known force to the structure and measuring both the force and the response of the structure. Structure’s response should be measured from multiple locations to figure out the response of the system. After acquiring the measurements, the data should be processed to determine the natural frequencies and the movement of the structure. There are two famous ways to perform modal analysis such as impact hammer modal testing and shaker modal testing.

The hammer impact test is a cost efficient and the simplest method to gather the system’s frequency response functions (FRFs). The main purpose of the impact hammer modal testing is to apply the structure a perfect impulse that will cause all modes of vibration by exciting the structure with equal energy. On the other hand, the hammer impact test has some restrictions. The acquired data may have noise that cause inaccurate calculations of modal parameters. It is impossible to perform a perfect impulse since there is a duration of contacting the hammer with the structure. This time period effects the frequency content of the impact. Therefore, this impact should be measured and recorded with a load cell. Impact hammer testing is more convenient for small and lightweight structures. Thus, small parts of the vehicle should be preferred instead of the whole vehicle for impact hammer testing. Hammer impact test set-up is shown in Figure 12.

Figure 12.

Hammer impact test set-up [16].

For complicated structures and obtaining more accurate results modal shaker testing is better than the impact hammer testing to identify coupled modes for closer frequencies. Full vehicle modal test set-up is shown in Figure 13.

Figure 13.

Full vehicle modal test set-up [17].

After modal updating procedure is completed, static test verification should be performed. Reaction forces of the structure under its own weight should be calculated and reaction forces of the structure should be compared with the test results. Load cells are mostly used test equipment for the static tests. Moreover, deformations should be measured and compared with the static analysis results.

After static analysis, dynamic load can be applied on the vehicle. Different dynamic load scenarios that are generated from user experiences and use cases should be applied to the vehicle model to assess subsystems and components effects on the whole vehicle model. Vehicle performance is obtained by applying these scenarios. Dynamic load can be either driving loads or dynamic force that acts on the static vehicle. Dynamic load is advised directly applied on the structure by omitting tire and road effect as a first step. Deformations and strain data should be acquired to figure out the system’s dynamic response. After that, dynamic test results should be compared with the dynamic analysis results. Modifications should be made on the vehicle model for verification. As a second step, tire and road effects should be added to the tests. Tire and road models are critical for verification. Therefore, tire and road model should be verified separately and together before dynamic analysis. Commercial analysis software’s experience can be utilized to model tire and road. However, some modification may be needed for verification. During dynamic tests load, torque, velocity and displacement data can be acquired. Measurement locations is the critical point for verification. Therefore, main target and other aims of this vehicle should be determined at the beginning. If the main target of a vehicle is to carry passengers then comfort of the passengers should be maximized. Vibration acting on the passengers should be measured, and design should be modified to minimize the disturbance of the passengers. In this scenario, vehicle design should be modified according to passenger riding comfort and verifications should rely on riding comfort. Moreover, control algorithms can be added to the vehicle dynamics to increase passenger’s riding comfort. In another scenario, vehicle can be truck that carries critical cargo. In this situation, main target of the vehicle is cargo. In another scenario, vehicle can be a working machine. In this case, dynamic load applies on the vehicle when vehicle is static. As a result, dynamic test measurements should be both acquired by considering verification of the analysis model and verification of the targets of the vehicle. Once the vehicle model and analysis results are verified, modifications on the design or test conditions can be changed and there will be no need to verify vehicle model again unless these modifications are out of verification. Dynamic test of a vehicle is shown in Figure 14.

Figure 14.

Dynamic test of a vehicle [18].

In this chapter, modelling of a vehicle, implicit dynamic analysis with linear elastic deformation assumption and verification of the model and analysis is examined. Explicit analysis, fatigue analysis, buckling analysis, large or non-linear deformations are out of scope of this chapter. In order to solve these analyses, different kind of mathematical equations must be solved, and different kind of assumptions must be made. Moreover, verification of these analyses and test methods of these conditions are totally different.

5. Conclusion

In this chapter, modelling of a vehicle, dynamic analysis methods and verification of both models and analysis are investigated. Modelling methods of a vehicle are classified in two types such as analytical models and finite element models. Analytical models are grouped as simple discrete model and flexible multi-body model. Finite element models are classified as simplified FEM, reduced DOF FEM and detailed FEM. Aforementioned models are examined before dynamic analysis process. After that dynamic analysis methods of a vehicle are elaborated. Modal analysis, static analysis and dynamic analysis of a vehicle are elaborated. Furthermore, verification methods of models and analyses are examined. Industrial experiences are employed when assessing the methods.

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Nomenclature

£

Lagrangian

q_r

r = 1,…,n are generalized coordinates

Q_r^*

r = 1,…,n are generalized forces representing all non-conservative forces

E_k

kinetic energy

E_p

potential energy

M

mass of the vehicle without suspension system and tires

I

inertia of the vehicle without suspension system and tires

m_i2

relevant suspension system’s mass

I_i2

relevant suspension system’s inertia

m_i1

relevant tire’s mass

I_i1

relevant tire’s inertia

q_i1

excitation applied from the road

k

stiffness coefficient

c

damping coefficient

$\overrightarrow{r_i}$

position vector of particle i

$\overrightarrow{F_i}$

sum of forces acting on the particle

$\delta V_i$

virtual work

$\delta \overrightarrow{r_i}$

virtual displacements of the particle i

$\overrightarrow{F_i^e}$

external forces acting on the particle i

$\overrightarrow{F_i^s}$

internal forces from other particles in the same body

$\overrightarrow{F_i^r}$

joint reaction force from another movable body through a joint

$\overrightarrow{F_i^d}$

reaction forces due to joints with the fixed frame

n

number of flexible bodies

$\overrightarrow{F_i^e}$

external forces acting on the particle i

$m_i$

mass of the particle i

$\overrightarrow{\ddot{r_i}}$

acceleration vector of the particle i

$\delta \overrightarrow{r_i}$

virtual displacements of the particle i

$\delta s_j$

virtual change in strain energy of flexible body j

$f_j$

generalized external force corresponding to general coordinate j

$f_j^s$

generalized structural stiffness forces

$\delta q_i$

virtual displacement in independent coordinate i

$\overrightarrow{F_i^{\ast }}$

inertia method of particle i

$\delta \overrightarrow{r_i}$

virtual displacements of the particle i

$f_j^{\ast }$

generalized inertia forces j

N

number of bodies

$\rho$

density

$\overrightarrow{r^k}$

position vector of an arbitrary point in body k

$q_j$

independent coordinate j

V

volume

$u$

system response which consists of many DOF

$T$

transformation matrix

$q$

modal coordinates consists of less coordinates

$\Phi$

mode shape matrix which is obtained by below equation

$M$

mass matrix

$K$

stiffness matrix

$F$

force matrix

$\propto$

constant of proportionality

$\beta$

constant of proportionality

Thanks

I would like to thank to ROKETSAN for funding this chapter. I am indebted to Bülent ACAR, principle engineer at ROKETSAN, for his support.