This paper attempts to analyze the formula modeling the motion of a freight elevator:

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Thi Ngoc Hanh Nguyen

Introduction

This paper attempts to analyze the formula modeling the motion of a freight elevator:

The formula above is then differentiated to obtain the velocity and acceleration equations. Based on the strengths and weaknesses of the original model, a new model is founded to fix all the problems of the original model while maintaining its strengths and usefulness. Later, the founded model is modified to fit into other applications such as residential elevators, pulleys and cranes.

1. Use these functions to:

a) Interpret the original vertical line motion simulation

Graph 1

Graph 2

Graph 3


From the three graphs above, the motion of the freight elevator can be described as below.

From the graph of displacement versus time, the elevator starts moving from t=0 to t=1. It then speeds up from t = 1 to t = 4. It stopped at -80m below the ground at t = 4. From t = 4 to t = 6, the elevator moves back up to the ground.

From the graph of velocity versus time, the elevator speeds up at first from t = 0 to t = 2. From t = 2 and t = 4, the velocity slows down, comparing with the displacement graph, the elevator has slowed down when it comes near the shaft. From t = 4 to t = 6, the velocity increases, which is parallel to the displacement graph where the elevator moves towards the ground from t = 4 and t = 6.

From the graph of acceleration versus time, at first, the elevator speeds up at first from t = 0 and t =2. Comparing to the velocity graph, it can be seen that both acceleration and velocity are negative from t = 0 to t = 2.  From t = 2, a(2) = 0, this is when the velocity decreases, so it reflects the changes in the concavity of the velocity curve. From t = 2 to t = 4, the acceleration increases to positive values while the velocity values are still negative, this shows a decrease in speed of the elevator. From t = 4 to t = 6, the elevator speeds up again when both acceleration and velocity are positive.

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b) Explain the meaning of the negative, positive and zero values of the velocity graph

The velocity function is derived by differentiating the displacement function, in other words, the velocity is the first derivative function of the displacement function.

Table 1 Values of displacement (y), velocity (v) and acceleration(a) based on the original model formula

At t = 0 and t = 4, v(0) = v(4) = 0. This indicates that the elevator is instantaneously at rest. These values fit with the displacement graph since at t = 0, the elevator is at the origin while at ...

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