How to scale a velocity profile

[bradyj7]

Hello,

I am reading a paper in which the author describes how he scales the velocity profile of a car.

He is doing this so that he can compare the velocity profiles of cars.

This is an extract from the text:

The first methodology consisted of evaluating the similarities between velocity profiles by computing the differences between their velocity profiles. This is a direct method, since the velocity profiles are considered in their entirety. The aim was not to quantify the links between different velocity profiles, but simply to highlight the similarities that might be present. To apply this methodology, it was first necessary to scale the velocity profiles and to perform a linear interpolation (step of 0.1 s) since all velocity profiles do not all have the same duration and the same time resolution. This was achieved by using the ECE part of the NEDC cycle as a reference. Consequently, once scaled, all the cycles had the same duration of 195 s. A comparison of the original and scaled NEDC cycles is illustrated in Figure 1.

https://dl.dropbox.com/u/54057365/All/scaleNEDC.JPG



My question is, if I had the time and velocity values of the "original NEDC" velocity profile, how would you actually scale it to achieve the "scaled NEDC". How would you scale the numbers to fit in the 195 second time frame?

Thank you

Here is the data in an excel sheet:https://dl.dropbox.com/u/54057365/All/NEDC.xlsx

 

[Chestermiller]

Hello,

I am reading a paper in which the author describes how he scales the velocity profile of a car.

He is doing this so that he can compare the velocity profiles of cars.

This is an extract from the text:

The first methodology consisted of evaluating the similarities between velocity profiles by computing the differences between their velocity profiles. This is a direct method, since the velocity profiles are considered in their entirety. The aim was not to quantify the links between different velocity profiles, but simply to highlight the similarities that might be present. To apply this methodology, it was first necessary to scale the velocity profiles and to perform a linear interpolation (step of 0.1 s) since all velocity profiles do not all have the same duration and the same time resolution. This was achieved by using the ECE part of the NEDC cycle as a reference. Consequently, once scaled, all the cycles had the same duration of 195 s. A comparison of the original and scaled NEDC cycles is illustrated in Figure 1.

https://dl.dropbox.com/u/54057365/All/scaleNEDC.JPG



My question is, if I had the time and velocity values of the "original NEDC" velocity profile, how would you actually scale it to achieve the "scaled NEDC". How would you scale the numbers to fit in the 195 second time frame?

Thank you

Here is the data in an excel sheet:https://dl.dropbox.com/u/54057365/All/NEDC.xlsx

From the figure, it looks like the only parameter that has been scaled is the time. If t is the running time in the experiment, and t_max is the overall maximum time, then the scaled time is t x 195/ t_max

 

[bradyj7]

Thanks very much

 

[bradyj7]

Hello ChesterMiller,

You helped me scale the time parameter of velocity profiles above for a vehicle a number of weeks ago. Thank you.

I was wondering if you had the time to help me with something else?

When I scale the velocity profiles in excel the resulting times are same length (in seconds) but they occupy a different number of cells (in length).

I am trying to apply this methodology to them to compute the difference between the profiles. But because they occupy a different number of cells I cannot apply it because the "t" values are different lengths.

https://dl.dropbox.com/u/54057365/All/QD.JPG

I have attached a workbook here with two example original and scaled profiles

https://dl.dropbox.com/u/54057365/All/QD.xlsx

I would be grateful if you had the time to consider it.

Kind Regards

J

 

[Chestermiller]

It all depends on what you are trying to do, and what the data represents. It looks like you are trying to determine the rms velocity difference between the two velocity variations.

Do you want to smooth the differences between the two profiles first, or do you to include all the sturucture? If you want each of the profiles smoothed first, you can fit a Fourier series to the variation and include only the first few terms. Or fit another type of smooth curve.

If you want to include the sturucture, you can join all the sequential pairs points on each curve by straight lines, and interpolate between the points. Or you can represent each of the functions as a series of steps (like a bar chart), with constant values from half way between one pair of points, to half way between the next pair of points. In any event, you are fitting each of the velocity variations as a piecewise continuous function of time. Then you can evaluate the difference between the curves at any value of the time. Then you can integrate.

$$RMS = \frac{1}{t_{max}}\int_0^{t_{max}}(v_2(t)-v_1(t))^2dt$$

This equation is the continuous form of your discrete summation relationship.

 

[bradyj7]

Hello,

Thank you for your reply. I do not need to smooth the velocity profiles.

Yes, I am trying to compute rms velocity difference between the two velocity variations.

My problem is that velocity profiles have different time duration. Hence, I have scaled them to same duration.

My problem is that now that they are scaled, the time values are not the same so I cannot compute the difference between to corresponding velocity points.

For example, say profile A is 970 seconds and Profile B is 1075 seconds. So I scaled the time component of both profiles using your method above to 177 seconds.

t x 177/ tmax (970)

t x 177/ tmax (1075)

https://dl.dropbox.com/u/54057365/All/scale.JPG

This is an example of the time values after scaling.

https://dl.dropbox.com/u/54057365/All/values.JPG

Because the scaled time values are not the same, i cannot subtract the corresponding velocities.

I have been struggling with this for a few days but cannot seem to figure it out.

I'd appreciate any help you could offer.

Regards

 

[Chestermiller]

OK. Then just follow the advise I gave you in my previous posting. Fill in the regions between the grid points with straight lines or stepwise variations, and then integrate numerically. Use small time increments in the integration. If the times at the grid points on the scaled profiles had actually matched, doing this integration would have given you the exact same result as you summation equation.