Rotational Motion and RPM

Understanding rotational motion is essential for motor selection, gear design, velocity control, and predicting robot performance. This section covers the physics and practical applications of rotational motion in robotics.

Revolutions Per Minute (RPM)

RPM is the most common way to specify motor speed. It represents the number of complete 360° rotations an object makes in one minute.

Key Conversions

Converting between different rotational speed units:

Practical RPM Ranges in Robotics

Rotational Kinematics

Just as linear motion has kinematic equations, rotational motion has parallel equations that describe angular position, velocity, and acceleration.

Rotational Motion Equations

ω = ω₀ + αt

θ = ω₀t + ½αt²

ω² = ω₀² + 2αθ

θ (radians) = θ (degrees) × π / 180

α = dω/dt

ω = ∫α dt = α·t + C

With initial condition ω(0) = ω₀:
ω = ω₀ + αt

θ = ∫ω dt = ∫(ω₀ + αt) dt = ω₀t + ½αt² + C

With initial condition θ(0) = 0:
θ = ω₀t + ½αt²

θ = ω₀(ω - ω₀)/α + ½α[(ω - ω₀)/α]²
θ = (ω₀ω - ω₀²)/α + (ω² - 2ωω₀ + ω₀²)/(2α)
θ = (ω² - ω₀²)/(2α)

Rearranging:
ω² = ω₀² + 2αθ

Practical Examples

Example 1: Robot Arm Joint Acceleration

A robot arm joint accelerates from rest (ω₀ = 0) with angular acceleration α = 10 rad/s² for 2 seconds.

Find final angular velocity and displacement:

ω = ω₀ + αt = 0 + 10 × 2 = 20 rad/s

θ = ω₀t + ½αt² = 0 + ½ × 10 × 2² = 20 radians

Converting to degrees: 20 × 180/π = 1146° ≈ 3.2 full rotations

Example 2: Spinning Up a Wheel

A wheel with angular velocity of 50 rad/s must stop in exactly 1 rotation (2π radians) with constant deceleration.

Find the required deceleration:

Using ω² = ω₀² + 2αθ:
0² = 50² + 2 × α × 2π
0 = 2500 + 4πα
α = -2500/(4π) = -198.9 rad/s²

Time to stop:
ω = ω₀ + αt
0 = 50 + (-198.9) × t
t = 0.251 seconds

Gear Ratios and Speed Reduction

Gears are the primary method to trade motor speed for usable torque and to match motor characteristics to load requirements.

Understanding Gear Ratios

Simple Gear Ratio:

Gear Ratio = Number of Teeth on Driven Gear / Number of Teeth on Drive Gear
Gear Ratio = Output Speed / Input Speed

Example: Two Gears

Driver gear: 20 teeth, spinning at 300 RPM Driven gear: 100 teeth

Gear Ratio = 100 / 20 = 5:1

Output Speed = 300 RPM / 5 = 60 RPM
Output Torque = Input Torque × 5

Compound Gear Systems

For larger speed reductions (like 50:1 or 100:1), multiple gear stages are used:

Total Ratio = Ratio₁ × Ratio₂ × Ratio₃ × ...

Example: 3-Stage Gearbox

• Stage 1: 10:1 (20 teeth → 200 teeth)
• Stage 2: 8:1 (25 teeth → 200 teeth)
• Stage 3: 5:1 (20 teeth → 100 teeth)

Total Ratio = 10 × 8 × 5 = 400:1

5000 RPM motor → 5000/400 = 12.5 RPM output
1 N·m input → 1 × 400 × 0.95 ≈ 380 N·m output (95% efficiency)

Efficiency in Gearboxes

Real gearboxes lose energy to friction:

Output Power = Input Power × Efficiency
Output Torque = Input Torque × Gear Ratio × Efficiency

Typical Gearbox Efficiencies:

Motor-Load Matching

Proper gear selection ensures the motor operates in its efficient range:

Example: Mobile Robot Wheel Motor

Requirements:

• Desired wheel speed: 2 m/s
• Wheel radius: 0.1 m
• Required wheel RPM: (2 × 60) / (2π × 0.1) ≈ 191 RPM

Motor selection:

• Use 5000 RPM motor with 26:1 gearbox
• Output: 5000 / 26 ≈ 192 RPM ✓
• Motor runs at ~96% of max (efficient)
• Good acceleration response

Angular Motion in Robot Mechanics

Velocity Conversion: RPM to Linear Speed

For a rotating wheel or drum:

v = (RPM × 2πr) / 60

Where:

• v = Linear velocity (m/s)
• RPM = Motor/wheel speed
• r = Wheel/drum radius (meters)

Example: Drive Wheel

3000 RPM motor with:

• 50:1 gearbox: Output = 60 RPM
• 0.1 m wheel radius

v = (60 × 2π × 0.1) / 60 = 0.628 m/s ≈ 0.63 m/s or 2.3 km/h

Acceleration Conversion

Angular acceleration translates to linear acceleration at the wheel edge:

a = α × r

Where:

• a = Linear acceleration (m/s²)
• α = Angular acceleration (rad/s²)
• r = Radius (m)

Practical Considerations

Selecting RPM for Your Robot

For Speed Priority:

• Use higher RPM motor (3000+ RPM)
• Lower gear ratio (5:1 to 20:1)
• Fast acceleration
• Example: Racing robot, delivery robot

For Torque Priority:

• Use lower RPM motor (500-1500 RPM)
• Higher gear ratio (50:1 to 100:1)
• Strong climbing/lifting
• Slower response
• Example: Heavy loader, climbing robot

For Balance:

• Use medium RPM (1500-3000 RPM)
• Medium gear ratio (20:1 to 50:1)
• Good acceleration and torque
• Example: Most general-purpose robots

Key Takeaways:

✓ RPM describes motor speed, but gearing transforms it to useful speed/torque combinations ✓ Rotational kinematics follows the same principles as linear kinematics with analogous variables ✓ Gear ratios multiply torque and divide speed (or vice versa) through the mechanical advantage ✓ Efficiency matters - multi-stage gearboxes lose significant power ✓ Match motor RPM and gearing to your actual application requirements