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Designing a Freight Elevator

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Introduction

Introduction

A heavy-duty freight elevator is used to raise and lower equipment and minerals in a mineshaft. Models are created to represent the movement of the elevator in terms of position. There are a number of specifications to take into consideration when creating a model and these will depend on the situation that the model is supposed to represent. To accommodate different specifications, different models are created with these specifications in mind.  

Analyzing a possible model

The formula y = 2.5t3 − 15t2 represents the position of the elevator, y, measured in meters (y = 0 represents ground level) and t represents time measured in minutes (t = 0 is the starting time). The trip up and down the shaft, ignoring time spent at the foot of the shaft, is approximately six minutes and that the depth of the shaft is no more than 100 meters.

image00.png

By graphing the function (x = −1, y = 2.5t3 − 15t2) in parametric mode, one can better visualize the movement of the elevator down the shaft. The dot indicates where the function starts and stops as it runs from t = 0 to

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Middle

6

0

By changing to function mode, we can take a better look at the position, velocity, and acceleration of the elevator.

Position Function:

image01.png

In the position graph above, we see that the elevator starts at ground level,  reaches a height of 80 meters below, and goes back up to up to ground level, all in the course of 6 minutes.

Velocity Function:

image02.png

By taking the derivative of the position function (y = 2.5t3 − 15t2), we get the velocity function of y = 7.5x2 – 30x.  Velocity is the rate of change in position of the elevator. In the velocity graph, we see that the negative velocity means that the elevator is below ground level, which is arbitrarily labeled as y = 0. The velocity function matches the position function in indicating that the elevator goes down and then goes back up. The zero value of the velocity graph means that the elevator is stopped. The negative value means it is moving down away from ground level and the positive value means that it is going up moving towards ground level.  


Acceleration Function:

image03.png

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Conclusion

y = 0 and y = 100. The number in front of the sin term determines the altitude and in turn how far down the elevator goes. For a different freight elevator, this number could to adjusted to fit how far that elevator will travel. The term in sin also determines the period, which would be how long a trip up and down would take. In the case of my model, the trip would take 4 minutes without any stops. This value multiplied by the variable determines this and can be changed to make the elevator go faster or slower. Making the model a piecewise function allows for sudden stops, which is hard to model in a single continuous function.

Applying the model

        This model may be further modified to fit a number of other situations. It can represent the up and down movement of other objects in life such as an airplane or rocket. One can use the strategies mentioned above to modify the period, altitude, and maximum and minimum to fit the situation that it calls for.

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