## 4 Methodology
### Test Tracks  
Accelerated fatigue tracks are widely used to test vehicle suspensions for ride comfort and durability. These tracks consist of a variety of surfaces from low-to-high severity and are designed to produce accelerated ageing of the vehicle's structure and components. Fatigue tracks contain a variety of surfaces like Belgium pave, pot holes, twist track, washboard, resonance track, herringbone, rough road, sine sweep, etc. There are a total of 28 different profiles that are available at the National Automotive Test Tracks (NATRAX) where the experimental study was performed. These tracks enable manufacturers to decide their own course of durability with various options.  
The test tracks and tests performed for the proposed parameter identification procedure at the NATRAX facility are given in Table 3. These tracks have a simple displacement profile.  
1. _Chassis-Twist Track_: This is a 4 m-wide track. The left and right track ends are sinusoids separated in phase by \(\pi\) radians (see Figure 7b). Between the two extreme profiles on the right and the left, the variation is linear. During testing, a narrow vehicle can be driven on one side of the track to increase the intensity of excitation at one wheel at the cost of reducing the intensity at the other.
2. _Herringbone Track_: In this track, the bumps are inclined at an angle of 20 degrees across the width of the track (Figure 7a). The road inputs for the left and right wheels are no longer identical; the phase difference between them depends on the vehicle width.
3. _Washboard Track_: This is the simplest sinusoidal track. Inputs to the left and right wheels are identical.  
The vehicle used for field testing has an unloaded tire radius of 300 mm. Since the wavelength of the Herringbone sinusoidal track is comparable to the tire radius, although the road profile is sinusoidal, the actual displacement input to the wheel is not sinusoidal. The road surface geometry has to be converted into an equivalent vertical input to be applied at the wheel-ground contact points \(C_{i}\) of the vehicle ride model. A simple road contact kinematic model that calculates the effective wheel displacement is employed.  
_Road Contact Kinematic Model_: Consider a rigid wheel going over a sinusoidal track as shown in Figure 8a. The track amplitude is \(A\), the wavelength is \(L\), the wheel radius is \(R\), the coordinates of the wheel centre O are \((\bar{x},\bar{u})\), and the wheel-ground contact point P is \((x,u)\). From geometrical considerations,  
\[x=\bar{x}+R\sin\theta, \tag{20}\] \[u=A\sin\left(\frac{2\pi x}{L}\right),\] (21) \[\frac{du}{dx}=\tan\theta. \tag{22}\]  
Figure 7: (a) Chassis-twist track; (b) Herringbone track.  
Differentiating (20) gives  
\[\dot{x}=\dot{x}+R\dot{\theta}\cos\theta, \tag{23}\]  
where \(\dot{x}\) is the longitudinal velocity \(V\) (taken as a constant). Substituting (21) into (22) and differentiating, we get  
\[\dot{\theta}=-A\left(\frac{2\pi}{L}\right)^{2}\sin\left(\frac{2\pi x}{L}\right) \dot{x}\cos^{2}\theta. \tag{24}\]  
(23) and (24) yield  
\[\dot{x}=\frac{V}{1+AR\cos^{3}\theta\left(\frac{2\pi}{L}\right)^{2}\sin\left( \frac{2\pi x}{L}\right)},\quad\dot{\theta}=\frac{-Av\left(\frac{2\pi}{L} \right)^{2}\sin\left(\frac{2\pi x}{L}\right)\cos^{2}\theta}{1+R\cos^{3} \theta A\left(\frac{2\pi}{L}\right)^{2}\sin\left(\frac{2\pi x}{L}\right)} \tag{25}\]  
Equation set (25) is solved numerically in MATLAB using ode45 with initial conditions \(x=\ddot{x}=\frac{L}{4},\theta=0\) at \(t=0\).  
A typical solution is shown in Figure 8b.  
This ground excitation is used as input for solving the differential equations of motion of the ride model to compute vehicle responses.