To collect raw histogram data in a real vehicle in a real working environment, an automated data acquisition system has to be temporarily installed on the vehicle. For the speed histograms of Figures 1 and 2, a speed sensor also must be attached to the output shaft of the HST. Once everything is in place, the data-acquisition system then will read and log the speed data over a predetermined amount of time.

This method, called a sampling process, takes speed samples at precisely regular intervals (samples are also called observationsin data-acquisition jargon). The regular interval may be one second. The exact sampling time interval isn’t important at this time. However, suffice to say, more samples are better than fewer.

At each second of the machine’s operation, the speed is sampled and stored in the data-acquisition/computer memory for later retrieval. During this stage, a statistician can help with the selection of a proper sample interval, the correct number of data buckets, and the total time of sampling. Details regarding those processes are beyond the scope of this brief analytical study.

The example vehicle data (which are not real statistical data) had 73,980 samples. If a 1-sec sampling interval was used, this would cover an operating period of just under 21 hr, which is usually expected in three eight-hour work shifts.

Provided that future vehicle operation resembles its operation during the 21-hr sampling period, expectations are that a new histogram would be similar to that during the sampling period. In other words, the transmission output shaft speed should be between, for example, 2800 and 2900 rpm about 6% of the time, Figure 2.

You could go a step further and combine this data with a Power Lost curve, Figure 3. It can be seen that the transmission power loss is almost 44 hp at the interval midpoint of 2850 rpm.

Continuing, consider the interval centered at 1250 rpm. Lost power is about 28 hp; Figure 2 indicates that the machine operates within that speed interval about 2.4% of the time. It should be obvious that lost energy depends not only on the lost power characteristics, but also the amount of time that the machine spends at any chosen operating condition.

The next step is to establish a total time interval for determining the amount of energy that goes into lost power. Perhaps the task is to determine the total energy lost in a typical 8-hr shift, and the machine is in actual operation for 7.5 hr. The total energy lost while operating between 2800 and 2900 rpm is simply the product of the histogram amplitude in that interval (about 6%) times the power lost when operating at that speed (roughly 42 hp) times the total time of operation (7.5 hr). The result is approximately18.9 hp-hr.

Now let’s look at the speed interval centered on 1250 rpm, Figure 3. The lost power in that interval is about 28 hp. Referring to the histogram of Figure 2, the transmission spends about 2.4% of the time at speeds between 1200 and 1300 rpm, so the total energy lost while operating in that speed range over the 7.5 hr is:

7.5 hr ×0.024 ×28 hp = 5.04 hp-hr

The energy lost while operating in the 1250-rpm interval and the 2850-rpm interval is the sum of the two — about 24.84 hp-hr. So, total lost energy is simply the sum of:

(time) ×(probability) ×(lost power)

Summing up the summing up

It should be apparent that calculating the total power lost during the 7.5-hr operating period merely involves multiplying the probability of a speed occurring at each of the 35 histogram intervals, times the power that’s lost at the center point of that interval, times the total operational time of interest. Then all 35 values are added together.

It’s important to note that all of the bar amplitude percentages in the probability histogram, when added together, will equal 100%. This is a requirement of such a histogram. At this point, it’s safe to assume that the probability of finding a speed between 0 and 3500 rpm is 100% for the hydrostatic transmission used in this study. The formula for determining the total energy lost during an arbitrary period of time is:

where:

T= the total time interval for evaluating the lost energy (units of time are arbitrary)

Pi= the probability of a speed occurring in the ith probability histogram interval (numeric)

WLi= the power lost when operating at the midpoint speed of the ith probability histogram interval (hp)

KN= the total number of speed intervals in the probability histogram (numeric)

ELT= the total energy lost during the time interval of T(units depend on those used for T)

Overall, the process is relatively simple if there’s a realistic and typical histogram, and the starting point is the power loss diagram for the transmission. It yields information that relates directly to fuel costs and, ultimately, money down the drain. Such information is not available from the efficiency data.

Part two of this article delves deeper into histogram statistical analysis, concentrating on the low-speed, high-torque region of transmission operation.