# Escalator Fractal Behavior, Part Four ### Series conclusion details deployment of the concept of the fractal and its partial values in correlation to the mechanical performance and availability of escalators for passenger service.

The third part of this series established that the behavior of an escalator with a mechanical defect can be quantified against the behavior of another escalator without a mechanical
defect. The concept of the fractal dimension (Df) was estimated to quantify the recoded data from the smart step in the two escalators.[3 & 4] This article dives deeper into the recorded traces in the previous study by dividing the length of travel of the smart step inside the escalator into four regions. The Df value of each region has been determined, as when it was previously
established that the overall Df value is the product of the simple mathematical sum of the Df values of the individual regions.[5 & 6]

The study in this article reiterates the above fact again. Furthermore, it compares two operation scenarios for an escalator with and without a mechanical defect. The benefit behind establishing such a methodology is to help maintainers diagnose failure types quantitatively, as well as identify the location of the defect in the machine. These principles and fundamentals can be applied not only to escalators, but to any mechanical or electrical system.

### Introduction

So far, using the smart step and the data from it, we have examined and es established the following:

1. The smart step is an excellent detection tool for escalators that records and establishes the operation fingerprint of an escalator.
2. The data from the smart step are repeatable and consistent.
3. We have implemented the Df concept to quantify the mechanical behavior of an escalator.
4. The impact of passengers on the behavior of an escalator can be quantified using the Df concept.[5 & 6]

The length of travel of one cycle for the smart step in an escalator can be divided into individual regions. The Df of the overall behavior of a trace for an escalator is equal to the mathematical sum of the Df values of the individual regions of that escalator.

This article will reiterate points 4 and 5 by establishing that mechanical defects can be detected and quantified using the Df concept. Also, the overall Df value for an escalator with a mechanical defect is equal to the simple mathematical sum of the Df values of the individual regions in that escalator.

### Methodology Used to Determine Df

The scaling step technique was again used to determine Df. A computer program was written in Microsoft Excel to determine Df and plot the data. Our methodology is very similar to the Multiresolution Length Method, which has been used by many researchers.[7 & 9-12]

Part three of this series showed the data was collected from a smart step which has a strain gauge on each (left and right) side of the step. The step was run in two escalators, A and B. It was established that escalator A had a mechanical defect. Each trace for each strain gauge was divided into four regions (Figure 1):

1. Upper D region
2. Main incline region
3. Lower D region
4. Return incline
“D” is used to indicate the transition stages in the step band from the main to the return sides and from the return to the main side. Figure 2 shows how the trace from strain gauge 1, for example, was divided into the four regions. The Df value was calculated for each region.

### Results and Discussion

The Df values of all four divided regions were determined for the runs in escalators A and B (Figure 3) and listed in Tables 1 and 2. The results in Tables 1 and 2 confirm that the Df value for the overall trace of escalator A or B is equal to the simple mathematical sum of the Df values of the individual regions.

Comparing the Df values in Tables 1 and 2 shows that the estimated values produced from escalator A are higher than those produced from escalator B. The highest recorded Df value in Table 1 is in the return incline region. This finding agrees with our previous findings in part 3 and confirms the visual investigations we conducted on both escalators. Misalignment in the loading relieving ramps on the return side is the cause behind the increase in Df value.

### Conclusions

An important conclusion can be drawn in this study: the partial Df values can be determined separately, and their quantities can be used to identify whether a machine like an escalator is free of defect. There is also a very good chance to correlate the Df values to the stress level, which is generated in the critical component of the step structure.