## 3 Vibration Analysis
### Mathematical Modeling and Derivation of Equations
#### 3.1.1. Forcing Conditions  
The proposed method aims to predict the force transmitted through the headlamp assembly. The assembly must be excited by external forces at one end to obtain the value of the force transmitted to the other end. In actual driving conditions, the vehicle moves across several even and uneven road surfaces. A series of random excitation forces is applied to the headlamp every second; thus, it is challenging to find a trend in the values. To reproduce the actual driving conditions in the simulation, an experiment was conducted using an accelerometer mounted on the dashboard of a vehicle. The accelerometer could not be mounted at the back side because the readings were to be obtained for the headlamps. There was a possibility to mount it right above the headlamp but it could not be fixed there properly. While driving through the non-uniform surfaces, it may get loose and record false readings. It should be as much near to the center of the vehicle as possible, so that the pitch of vehicle has minimum effect on the vertical acceleration readings. Other factors that could influence the accelerometer sensitivity are temperature and mount type. A DC-response accelerometer was used because it can perform well in shock applications where we need to integrate the acceleration data.  
Figure 5. Free-body diagrams of the housing.  
Other types will have an intrinsic decay function that can result in significant error. An accelerometer with sensitivity of 10 mV/g and frequency range of 1 to 10,000 Hz (\(-\)5%) was used in this experiment. The vehicle was driven through even and uneven roads, and two sets of time-acceleration data were collected in each case (case I and case II). The following directions were assumed: vertical to the road surface (z-axis), direction of vehicle travel (\(x\)-axis), and direction transverse to vehicle travel (\(y\)-axis). The data were collected in seconds, and the duration of the experiment varied for each case. Figure 6 illustrates the road surface conditions for the even and uneven cases.  
The data obtained for case I is shown in Figure 7. Figure 7 plots the magnitude of acceleration with respect to time, with values ranging between 9.9 and 10.2 m/s\({}^{2}\). The total mass of the four components under study is approximately 3 kg. The product of the total mass and acceleration provides the magnitude of the force applied. The obtained magnitude of force in case I reached 31 N, with the experiment lasting for 300 s and the vehicle driven along an even road at constant speed; the plot of the data is shown in Figure 7b.  
The data for case II is shown below in Figure 8. In case II, the acceleration reached a maximum of 20.72 m/s\({}^{2}\). The vehicle was driven across bumps and potholes, which led to this abrupt increase in magnitude. Figure 8b shows the force applied in case II. The maximum amplitude of the force is approximately 63 N, which is double than that in case I. The data for case II is shown below.  
Figure 6: Images from video recording of data collection experiments: (**a**) case I; (**b**) case II.  
Figure 7: Input data for case I: (**a**) acceleration data; (**b**) excitation force.