Modified Schedule

• 1:00-2:00 Lecture:Engineering Fundamentals

• 2:00-3:00 Final Vehicle Modifications

• 3:00-3:30 Break

• 3:30-5:00 Soccer Competition

• 5:00-6:00 Design Planning (Adventure Racing)

Xtreme Robot OlympiadEngineering Fundamentals

Peter Laz Associate Professor

Department of EngineeringUniversity of Denver

Outline

• Vectors

• Forces

• Torque

• Mechanical Advantage

• Levers

• Gears

Scalars versus Vectors

A Scalar has a magnitude (no direction)• Examples: speed, time, mass

• Notation: s, t, m

A Vector has both magnitude and direction• Examples: distance, velocity, acceleration,

weight, force

• Notation: f,w,a,v,x

Unit Vectors

y

Unit vectors have magnitudes = 1 and directions along the coordinate axes

xi

j

Vector ComponentsA vector can be broken into components

Use Trigonometry !!!

AA

jAiAA yx

)sin(AA

)cos(AA

y

x

y

Ax

Ay

x

A

Vector ComponentsA vector has magnitude and direction

jAiAA yx

x

y

2y

2x

A

AtanArc

AAA

y

Ax

Ay

x

A

Exercise

1. Determine the components Ax and Ay

if the magnitude A = 80 N and = -40°

2. What are the magnitudes and directions of the vectors?

j6i2B

j10i5A

Vector Addition and Subtraction

• Two Vectors

• Vector Addition• add components in the same direction

jBiBB

jAiAA

yx

yx

j)BA(i)BA(BAR yyxx

Vector Addition and Subtraction

• Vector Subtraction• subtract components in the same direction

j)BA(i)BA(BAR yyxx

Multiplication by a Scalar

• Multiplication by a scalar• multiply each component by the scalar

jmAimAAmR yx

Exercise

• Two Vectors

• Find the following

j6i2B

j10i5A

B2A3R

BAR

BAR

a.

b.

c.

Force

• Units of force• Newtons (N = kg*m/s2) SI system• Pounds (lb = lb*ft/s2) US system

• Force = mass * acceleration

• Weight = mass*g• Mass (kg), g = 9.81 m/s2 SI system• Mass (slugs), g = 32.2 ft/s2 US system

Torque

A torque or moment is equal to a force x distance at which it acts

FrT

F

r

= perpendicular distance

Torque

The direction a torque acts is determined by the right hand rule.• Point your hand in the direction of r• Then bend your fingers in the direction of F• Your thumb points to the direction of the torque

For your unit vectors: but note:

kji kji

kij kij

0ii

0jj

Exercise

Find the magnitude and direction of the torque for each of the conditions

j6F

i2r

a.

j6F

i2r

b.

j6i2F

i2r

c.

Sir Isaac Newton(1642-1727)

Three Laws of Mechanics1. A body continues in its state of rest or motion until a force

is applied

2. The change of motion is proportional to the force applied

3. For every action there is an equal and opposite reaction

Static Equilibrium

• Newton’s First Law

• The sum of the forces and moments acting on a body are zero (0)

0M

0F

0F

o

y

x

0F

Levers

• Consist of three parts• Effort• Resistance• Fulcrum (pivot point)

W

Effort Force

Levers

• First class lever – fulcrum between the weight and the effort

• What happens to the effort• if the fulcrum moves to the left?• if the fulcrum moves to the right?

W

Effort Force

Levers

• Second class lever

• Third class lever

W

Effort Force

W

Effort Force

Static Equilibrium

• Moments caused by effort and resistance are equal

resistresistefforteffortfulcrum

fulcrum

FrFrM

0M

resistresistefforteffort FrFr

Mechanical Advantage

• Measure of the ability of a machine to amplify force

M.A. =Resistance (Force)

Effort (Force)

M.A. =Effort Arm

Resistance Arm

Gears

• Some examples include• Can opener• Cork screw• Transmission on your car• Bicycle

• Gears are used to• Change the direction of motion• Increase or decrease speed• Increase or decrease torque

• Gears are commonly used in power transmission applications because of their high efficiency (~98%)

Gears Configurations• Spur gears

• Wheels with mating teeth

• Rack and pinion gears• Changes rotational motion to

linear motion

• Worm gears

• Bevel gears• Connects shafts lying at angles

Gear Ratio

• A gear will rotate with an angular velocity () with units of radians/second

• Gears have teeth that must mesh• Same pitch = same distance between teeth • There is a fixed ratio between the teeth and the gear

radius

22

1r

r

NN 1

N = Number of teeth, r = radius

Gear Ratio - Velocity

• Velocity of pitch point C on both bodies must be equal

Driven

Driver or Pinion

C2211

rrVc

1 2

221 N

N

r

r112

= angular velocity

Gear Ratio - Torque • Force of gear 1 on gear 2 is equal and

opposite to force of gear 2 on gear 1

Driven

Driver or Pinion

C2r

T

r

TF 2

1

1 1 2

2221 T

T

N

N

r

r1112

= angular velocity

T1

T2

Gear Problems

• Master equation

• Small gear to large gear• Slower angular velocity, increased torque

• Large gear to small gear• Faster angular velocity, reduced torque

2221 T

T

N

N

r

r1112

ExerciseWhat are the gear ratios?

Let:rgreen = 6 inchesrblue = 10 inchesrred = 15 inches

green = 10 rad/sec

What is red?

Is Tred < or > Tgreen?

1

2

1

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