[38]Can the Vibration of Onboard Devices Be Modeled by Underdamped Oscillation

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1. Siau Hong Ng Yi-Ta Chuang Chih-Wei Yi Can the Vibration of Onboard Devices Be Modeled by Underdamped Oscillation Method Three methods are used to show the experimentally collected oscillations correspond with the underdamped oscillation model (Fig. 1), which are: A. RMSE Between Predicted Waveform and Exact Waveform This method was used to measure the differences between the predicted and exact waveforms. The exact waveform came directly from experimental data, and the predicted waveform was constructed from the predicted parameters , and . B. Logarithmic Decrement Method This method is used to find the damping ratio of an underdamped system in the time domain. If the damping ratio is between 0 and 1, the oscillation is called underdamped oscillation. C. Half Power Bandwidth Method This method uses data from the frequency domain to estimate the damping ratio. A Fast Fourier Transform was done with discrete points in the time domain, Computer Science Department, National Chiao Tung University siauhong100@hotmail.com ytchuangcs96g.cs96g@g2.nctu.edu.tw yi@cs.nctu.edu.tw Experiment Result In the experiment, the HTC Desire was used to collect the oscillation data. The smart phones were fixed on two different kinds of rack in a vehicle. During the experiment, the vehicle moved over a certain number of abnormal points, and the data were obtained during each point. The results showed that the RMSE was within 0~3 and the damping ratio was also found in the time and frequency domains which fell within range of 0 and 1 in both Rack 1 and Rack 2. Conclusion As experiments were done to prove that onboard vibrations could be modeled by underdamped oscillation through a novel parameter: damping ratio ζ. The results showed that all of the experimental damping ratios fell within the range, so it can be said that onboard vibration can be modeled by underdamped oscillation. Once onboard vibration can be modeled by underdamped oscillation, applications such as road abnormality detection can use this model to create newer and simpler algorithms than the existing applications, which need very specific setups. We believe that this model can be used in an auto-adjusting car seat, which can use negative underdamped oscillation to create a more comfortable and smoother ride. Abstract The vehicular applications which make use of the vibration of onboard devices are useful but the use of accelerometers could cause development issues. Therefore, we try to use a mathematical model to describe vehicle vibrations in order to solve the discrepancies in the accelerometer data collected under different conditions. Experiment were done to prove that onboard vibrations could be modeled by underdamped oscillation. Fig. 1. Underdamped Oscillation Model Fig. 2. RMSE result between predicted waveform and exact waveform Fig. 3. Damping ratio of time domain and frequency domain (a) Rack 1 (b) Rack 2
2. Can the Vibration of Onboard Devices Be Modeled by Underdamped Oscillation Chih-Wei Yi Department of Computer Science, National Chiao Tung University, 1001 University Road, Hsinchu City 30010, Taiwan. Email: yi@cs.nctu.edu.tw Yi-Ta Chuang Department of Computer Science, National Chiao Tung University, 1001 University Road, Hsinchu City 30010, Taiwan. Email: ytchuangcs96g.cs96g@ g2.nctu.edu.tw Siau Hong Ng Department of Computer Science, National Chiao Tung University, 1001 University Road, Hsinchu City 30010, Taiwan. Email: siauhong100@ hotmail.com Abstract- The vehicular applications which make use of the vibration of onboard devices is useful but the use of accelerometers could cause greater amounts of trivial preliminary work or development issues. Therefore, we found a mathematical model to describe vehicle vibrations in order to solve the discrepancies in the accelerometer data collected under different conditions. Experiment were done to prove that onboard vibrations could be modeled by underdamped oscillation. Keyword: underdamped oscillation; onboard vibration; acceleration I. INTRODUCTION Nowadays, many vehicular applications make use of the vibration of onboard devices. For example, vehicle airbag systems [1] use a MEMS device with built-in accelerometers, which are then employed as a switch to trigger the airbags in the vehicle when the accelerometer detects data that exceeds a certain threshold and indicates a crash. Another application is in road abnormality detection [2], which uses accelerometers to detect the vibration of a vehicle. Abnormality positions are recorded when the vibration of the vehicle exceeds a specific threshold, which occurs when the vehicle passes through an abnormality point on the road such as a pothole. These applications are all very useful, but the use of accelerometers could potentially cause greater amounts of trivial preliminary work or development issues. Problems that occur when using an
3. accelerometer to measure vibrations are usually caused by issues in the setup position of the accelerometer on the vehicle or the sealing of the deployment threshold. The airbag deploys when the accelerometer senses that the vehicle is in a situation of large shocks, which is determined by the deployment threshold. One of the reasons that airbags don’t deploy during crashes is because of incorrect setting of the air bag deployment threshold, which is often due to inadequate testing. There are several applications that use accelerometers to do road abnormality detection. As mentioned earlier, the application system records abnormal positions as the accelerometer detects vibrations that exceed a specified abnormality threshold. The threshold is not suitable in every condition, due to different setup positions and vehicle types, but it will influence the vibration of a vehicle in any situation. For example, raw vibration data for movements over potholes is different for each type of vehicle, so the threshold changes as well. Therefore, it is necessary to find a mathematical model to describe vehicle vibrations in order to solve the discrepancies in the accelerometer data collected under different conditions. In this paper we describe a mathematical model, realized through experiments, and show that underdamped oscillation models the vibration of onboard devices. First, the oscillation data was collected on vehicles by setting up smartphones on different vehicles and racks. Next, these data were used to figure out the unknown parameters λ and ω of the underdamped model. After calculating these values, we then showed that the oscillation corresponds to the underdamped oscillation in three ways: the RMSE between predicted waveform and exact waveform in time domain, the logarithmic decrement method, and the half-power bandwidth method. II. UNDERDAMPED OSCILLATIONS This section introduces the basic model for damping oscillation. Fig. 1 illustrates the Kelvin model, in which a string with spring constant k and a damper with damping coefficient b are connected in parallel to a mass m. When the mass is given a force F to push the string and damper, the displacement of the mass is x. According to Newton’s second law, = − − = . As shown previously, a forced damped oscillator follows the equation ̈ + ̇ + = 0, with m > 0, ≥ 0 and k > 0. The characteristic equation is then m2 + + = 0, with characteristic roots −±√2−4 2 . When 2 < 4, the system is underdamped. If ω = √|2 − 4|/2 is the frequency of the damped oscillator, then the equation gives characteristic roots of − 2 ± ω. The general solution of the real part is x(t) = −/2 cos( + ). After taking two time derivatives, a simple solution of a(t) = 2−cos ( + ) is found for acceleration, where = −/2, A is the real initial
4. amplitude, and determines the relative phase of the oscillator. Fig. 1. Kelvin Model Fig. 2. The waveform of an underdamped system III. FINDING MODEL PARAMETERS To verify that the vibration of onboard device follows the underdamped model, it is necessary to calculate the related unknown parameters, λ and ω. A typical waveform of the acceleration of an underdamped system is illustrated in Fig. 2. Since ω = 2π/T, it is possible to find ω when given T. The average of time interval between two peaks is the predicted period of oscillation T. Therefore, ω is calculated directly using the equation. Since −λt describes exponential decay, λ could be calculated through the following equation: 2−λ 2−λ+1 = −λ(−+1) λ is then calculated by root mean square (RMS) with n-λ: λ = √ 1 (λ1 2 + λ2 2 + ⋯ + λ 2) IV. VERIFICATION BY EXPERIMENT Based on the properties of the damping system, there are three ways to show that the experimentally collected oscillations correspond with the underdamped oscillation model. The first method uses the RMSE between the predicted and experimental waveforms in the time domain. The second is the logarithmic decrement method, which obtains the damping ratio of a system using time domain data. The last method, the half-power bandwidth method, uses data in the frequency domain to determine the damping ratio. A. Experiment Configuration In the experiment, the HTC Desire was used to collect the oscillation data. The smart phones were fixed on two different kinds of rack in a vehicle. During the experiment, the vehicle moved over a certain number of abnormal points, and the data were obtained during each point. B. RMSE Between Predicted Waveform and Exact Waveform Root-mean-square error (RMSE) is frequently used to observe the differences between values predicted by a model. Since it is a good measure of accuracy, this method was used to measure the differences between the predicted and exact waveforms. The exact waveform came directly from experimental data, and the predicted waveform was constructed from the predicted parameters T, λ and . Fig. 3 showed that the RMSE is within 0~3.
5. Fig. 3. RMSE result between predicted waveform and exact waveform C. Logarithmic Decrement Method The logarithmic decrement method is used to find the damping ratio of an underdamped system in the time domain. The logarithmic decrement, δ, is the natural log of the ratio of the amplitudes of any two successive peaks: δ = 1 ln () (+nT) , where x(t) is the amplitude at time t and x(t+nT) is the amplitude of the peak n periods away, with n as any integer number of successive, positive peaks. The damping ratio, ζ, is then found from the logarithmic decrement equation: ζ = 1 √1+( 2 ) 2 . If the damping ratio is between 0 and 1, the oscillation is called underdamped oscillation. Fig. 4 and Fig.5 shows that our data collected from experiment, the counted damping ratio of time domain is all in the range. Fig. 4. Damping ratio of time domain and frequency domain in Rack 1 Fig. 5. Damping ratio of time domain and frequency domain in Rack 2 D. Half-power Bandwidth Method The half-power bandwidth method uses data from the frequency domain to estimate the damping ratio. A Fast Fourier Transform was done with discrete points in the time domain, and the peak amplitude was obtained in the frequency domain. There are two half-power points in the waveform, each at /√2. Thus, the damping ratio can be found by ζ = − 2 , where is the frequency corresponding to the peak amplitude, while and are the frequencies of the two half power points. Fig. 4 and Fig. 5 shows that either we found damping ratio from time domain or frequency domain, the result is all in range.
6. V. CONCLUSION In this paper, experiments were done to prove that onboard vibrations could be modeled by underdamped oscillation through a novel parameter: damping ratio ζ. First, raw data were collected when a vehicle moved on a road and passed through abnormality points; these data were used to find the underdamped model’s parameters T, λ and . These parameters were then used to simulate the predicted underdamped oscillation waveform, which was compared with the exact waveform using the RMSE method. The results showed that the RMSE was within 0~3. In addition, the damping ratio was also found in the time and frequency domains to see if the calculated values fell within the range of 0 and 1. The results showed that all of the experimental damping ratios fell within the range, so it can be said that onboard vibration can be modeled by underdamped oscillation. Once onboard vibration can be modeled by underdamped oscillation, applications such as road abnormality detection can use this model to create newer and simpler algorithms than the existing applications, which need very specific setups. We believe that this model can be used in an auto-adjusting car seat, which can use negative underdamped oscillation to create a more comfortable and smoother ride. VI. REFERENCES [1] Binghamton University, Smart Switch – Accelerometer Sensor for Airbag Deployment [2] Yi-Ta Chuang, Chih-Wei Yi, Chia Sheng Nian iPave: Intelligent Pavement Detection System via Croudsourcing