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Transcript

Rigid Ramp Walker

Pranav Thiyagarajan

Problem Statement

Problem Statement

Create a stable four-legged structure for traversing a rough incline, and explore how its shape and certain factors impact it's terminal velocity.

Thesis

The terminal velocity of the ramp's descent is directly influenced by the incline of the ramp, while the density and friction of the surface, along with the surface area of the ramp, play indirect roles in affecting the terminal velocity.

The structure of the legs affect the movement of the walker.

POINTS OF INTERESTS

POI

  • What is the mechanism of the walker? (Q.1)

  • Can the walker be stopped by steep/flat angles? (Q.2)

  • Will extreme weights halt the walker's function? (Q.3)

  • Do rough or smooth inclines affect the walker's operation? (Q.4)

BONUS SLIDE

BONUS

Before getting in to the answers to the questions, this is an explanatory slide to give the viewer a brief idea of the concept(s) of torque & toppling

Torque

Torque

Definition:

Torque is the rotational force applied to an object around an axis, causing it to rotate.

Torque depends on both the force applied and the distance from the axis of rotation.

Mathematically,

Torque ( ) = Force (F) × Lever Arm (r).

Toppling

TOPPLING

Toppling occurs when an object loses its balance and starts to rotate or fall.

It is influenced by the distribution of mass, base area, and the center of gravity.

A stable object has a low center of gravity and a wide base.

Toppling before sliding

TOPPLING BEFORE SLIDING

Mechanism of the walker

The walking, as we call it, is actually a step function. It is somewhat similar to an animal walking on a ramp, except the hind legs drag on.

Q.1

Surprisingly so, there is not an instant where both legs are in contact with the ground! (during the movement).

The above is true because of the curved feet of the walker.

We shall go more in detail about the step function and the structure & geometry of the legs

Leg Geometry & Structure

Geometry and Structure

  • The feet's bases curvature is the reason why the walker even walks.
  • The curvature is outward from the centre of the two legs.
  • The front legs also have to be slightly bent more than 90° before being set on the ramp.
  • All the legs must be flexible to carry out the step function (seen in next topic).
  • The hind legs are for support, and regulate the speed of the walker.
  • Two hind legs provide more stability, and only one increases the risk of the walker toppling to one side.

Step function

STEP FUNCTION

  • The step function is an event due to the curvature of the legs.

  • The walker leans to one side.
  • This compresses the leg on the side where the walker leaned.
  • The other leg is then freed to move a bit forward.
  • The weight of the walker is shifted to the leg that moved forward.
  • The walker then tilts to the new side.
  • The process repeats: the leg on the newly tilted side compresses, and the other leg moves forward.

Can the walker be stopped by steep/flat angles?

Q.2

Theorical proof

There are two main conditions:

  • Very small angle.
  • Very large angle.

Theoretical proof

Small θ

Small angle

If the angle is too small, the ramp walker won't walk. How small? Let's explore when the walker begins to slide or move on the ramp, with the smallest angle possible.

-1

A slightly higher angle than tan will make the walker move, but anything lower will keep it still.

Large θ

Large angle

If the angle is too large, the ramp walker won't walk either.How large?

Let's explore when the walker begins to topple off the ramp, with the largest angle possible.

-1

A slightly lower angle than tan (b/h) will make the walker move, but anything above will result in it toppling.

sliding = walking (here)

Q.3

Extreme weights

Mass is irrelevant in all derivations; the walker's weight doesn't matter, as long as there are:

  • Constant environmental conditions
  • Proportional dimensions
  • Identical centers of mass and gravity across all walkers.

Friction of incline

Q.4

Again there would be two extreme conditions:

  • Very rough incline
  • Very smooth incline

Rough

Rough incline

For the walker to move, enough force parallel to the incline must exceed the opposing friction.

There also is the possiblility of the walker toppling, just like anything with a rubber base toppling when applied with a force on the top

Smooth

Smooth incline

On a smooth incline, the walker moves uncontrollably, risking falls due to the lack of controlled sliding.

At a certain angle, the walker can move, but it won't actually "walk".

On smooth inclines, all four legs of the ramp walker will be at contact with the ground, as it slides down.

APPARATUS USED

Apparatus

  • Cardboard
  • Rigid Ramp Walker (made of cardboard)
  • Measuring tape
  • Protractor
  • Stopwatch
  • Bottle with detachable rubber base

EXPERIMENT

The experimental analysis will either confirm that the thesis is correct, OR, partially incorrect, or even completely wrong.

Experiment

The walker "walks" a certain distance, in a certain amount of time, at a certain angle.

Multiple angles have been tested to check Questions 2 and 4 from the POI (points of importance)

Can the walker be stopped by steep/flat angles?

Mathematically, above a certain angle, or below, the walking is NOT observed.

Q.2 experiment

Experiment

Experiment

x-axis = TRIALS

y-axis = velocity

Do rough or smooth inclines affect the walker's operation?

Mathematically, depending upon the coefficient of friction, the walker will either slide, walk properly, topple, or stay motionless.

Q.4 experiment

Experiment

ROUGH/SMOOTH INCLINES

x-axis = TRIALS

y-axis = velocity

Different surfaces are used

Velocity formula

Velocity Formula

A generalized walker velocity is crucial, enabling future predictions without the need for time-consuming experiments.

This helps save time when determining the velocity over extensive distances covered by the walker.

Distance

Distance.

We shall calculate the distance covered by ONE leg per step function.

Time

The time.

Velocity

Velocity!

Conclusion

Conclusion

Thesis

Result

Incline of ramp directly

increases velocity

TRUE

Density indirectly affects velocity

FALSE

Surface area of ramp is indirectly proportionate to velocity

TRUE

Friction is indirectly proportionate to velocity

Structure of legs affect the movement

TRUE

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