# Mechanical Systems

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## Mechanisms (Robives) ## Simple Machines

### Lever Based

• Lever
• Wheel and Axle
• Block and Tackle (pulley)

### Incline Plane Based

• Inclined plane
• Wedge
• Screw

# Mechanical Advantage

The real mechanical advantage of any machine is the load force divided by the effort force. Usually, the mechanical advantage is greater than 1 (there is an advantage). In some cases (third class lever, wheel being driven) the mechanical advantage is less than 1 (there is a dis-advantage). This is done to get a greater range of motion at the expense of needing a bigger applied force.

### MA = R / E

For each type of simple machine, the ideal mechanical advantage is the ratio of two distances. Which two distances depends on the simple machine in question.

### MA = LE / LR or L / H

Often, the system is assumed to be friction-less and hence free of loss. In this case, the real mechanical advantage is equal to the ideal mechanical advantage

## Lever

1. First Class Lever
2. Second Class Lever
3. Third Class Lever

## Block and Tackle (Pulley)

For a pulley system, the mechanical advantage is the number of lines (ropes) on the moving block (pulley).

### MA = Number of lines on moving block

When you lift a weight, the distance the weight is lifted becomes LR and the amount of rope you have to pull becomes LE.

## Inclined Plane

If you push an object up a ramp (inclined) plane. The amount of force you have to exert (E) is less than the weight (R) of the object by the ratio of the ramp length (L) to its height (H).

## Wedge

A wedge is an inclined plane that is stuck on the short side to produce a splitting force.

### Splitting Force = Striking Force X (L / H)

Where: L is the length of the long side and H is the length of the side being stuck

## Screw

A screw is an inclided plane wrapped around an axle. The Pitch of the screw is how far it advances into the wood (or the nut) for one complete revolution. The Thread Count is the number of threads per inch on the screw. So:

### MA = R / E = L / H

The long side (L) of the inclined plane is the circumference of the screw, or π times the diameter

The pitch (P) becomes the height in the equation.

## Catapults

Three types of catapults are the Mangonel, the Ballista, and the Trebuchet

About

A Floating Arm Trebuchet out of Popsicle Sticks

## Exterior Ballistics — Projectile Motion

In absence of wind resistance (a very real phenomenon) baseballs, basketballs, and other projectiles travel in a parabola.

### Basic Terms

Velocity (V); or in X or Y it is VX or VY
Initial Velocity (V0); in X or Y it is V0X V0Y
Elevation Angle (θ)
Initial Altitude (Y0)
Distance (D); Initial Distance is then D0
or in X and Y coordinates; initial position is X0 and Y0
Time (t) in seconds
Acceleration (a); acceleration due to gravity (g, use -32.2 ft/sec)

### General Equation

a = constant

V = V0 + a t

D = D0 + V0 t + ½ a t2

#### In X

X0 = 0
a = 0
V0X = V0 cosine (θ)

#### In Y

a = g = -32.2 ft/sec2
V0Y = V0 sine (θ)

#### Equations for this Case

X = X0 + V0X t

Y = Y0 + V0Y t + ½ g t2

VX is constant

= V0Y + g t

#### Position as a function of time

X = V0 cosine (θ) t

Y = Y0 + V0 sine (θ) t + ½ g t2

#### General Problem Types

• Find X when Y = 0 feet (range of shot)
• Find maximum Y (altitude)
• Given range and angle, find Initial Velocity
• Find Elevation Angle for maximum range

#### Sine and Cosine for common Angles

 Sine Cosine 1/cosine 30° 1/20.5 3½/20.87 2/3½1.15 45° 1/2½0.71 1/2½0.71 2½1.41 60° 3½/20.87 1/20.5 22.0

### Problems

1. A skydiver jumps from 10,000 ft. How fast is she going after:
a. 1 second?
b. 2 seconds?
c. 10 seconds?
d. What is her altitude after 10 seconds?

2. A baseball is thrown straight up at 64 ft/sec.
a. How long does it take to get to its highest point?
b. How high does it go?
c. How long does it take to get back to your glove?
d. How fast is it going when you catch it?

3. A projectile is thrown at 100 ft/sec at an elevation angle of 45°.
a. What is the speed in the horizontal (X) direction?
b. What is the speed in the vertical (Y) direction?
c. What is the time of flight (t for Y=0)?
d. What is the time of maximum altitude (t for VY=0)?
e. How far does it go?

4. A projectile is thrown at 100 ft/sec at an elevation angle of 30°.
a. What is the speed in the horizontal (X) direction?
b. What is the speed in the vertical (Y) direction?
c. What is the time of flight (t for Y=0)?
d. What is the time of maximum altitude (t for VY=0)?
e. How far does it go?

5. A projectile is thrown at 100 ft/sec at an elevation angle of 60°.
a. What is the speed in the horizontal (X) direction?
b. What is the speed in the vertical (Y) direction?
c. What is the time of flight (t for Y=0)?
d. What is the time of maximum altitude (t for VY=0)?
e. How far does it go?

6. Which of the last three went farthest? Why don't hit baseballs fly the farthest at this angle?

7. A projectile thrown at 30° goes 150 yards in 4.66 seconds
a. What is the velocity in the horizontal (X) direction in ft/sec?
b. What is the Initial Velocity?
c. How high did the projectile go?

A floating arm Trebuchet from popsicle sticks

## Real Tractor Pulling

See how tractor engineering is applied full-scale to real life competitions. Though model tractors and simple designs provide a basic understanding of how tractors work, here you will find some enormous, professionally built models with real brute strength.

Just to get a glimpse of how popular tractor pulling has become, check out some of the many tractor pulling clubs and organizations across the world.

## About Gears and Gear Reduction

It is common knowledge that a good gear reduction is essential to a good tractor. Without it, the axles would spin with much less torque and the tractor would be unable to pull nearly as much weight. Here you will find some general information about how gears work and how to construct some functional gear trains.

For a fundamental explanation of gears and how they work please visit www.howthingswork.com/gears.htm

Here is another, easier, explaination at Badlinks

Here are some examples of some amazing, real-life gears and gear trains that are used by companies today: http://www.dgsteel.com/

Another effective method of system reduction involves the usage of pulleys. For information on pulley reduction and an example of a modern pulley system, please visit http://www.robotics.com/reducers.html and http://www.robotics.com/pr23.html

## Micro-Pulling and Legos

If you are interested in a local high school engineering challenge, your project will probably involve the design and construction of what is known as a micro-tractor. Just as the term tractor pulling refers to real-life, full-scale tractors, mini-tractor pulling refers to more of a lawnmower-sized, medium tractor. Smaller than normal tractors and beyond a mini-tractor comes the micro-tractor: a model tractor usually no larger than a football. Here you will find general information, concepts, and possible designs related to micro-tractors and the realm of micro-pulling.

The Lego Design PageWill help you get started

For information about Legos™ visit www.lego.com or "My Lego Projects"

Here's something that you can't do Nine to One Gear Reduction

Last updated on 22 January 2009 by P. A. Wiedorn 