For engineers who already know the math—but still lose projects. For the last few years, I’ve been sharing technical guides here on Mechanical Design Handbook —how to size a motor, how to calculate fits, and (as you recently read) how to choose between timing belts and ball screws. But after 25 years in industrial automation, I realized something uncomfortable: Projects rarely fail because the math was wrong. They fail because: The client changed the scope three times in one week. A critical vendor lied about a shipping date (and no one verified it). The installation technician couldn’t fit a wrench into the gap we designed. University taught us the physics. It didn’t teach us the reality. That gap is why I wrote my new book, The Sheet Mechanic . This is not a textbook. It is a field manual for the messy, political, and chaotic space between the CAD model and the factory floor. It captures the systems I’ve used to survive industrial projec...
Figure 1: The friction force (F) always acts in the opposite direction of the applied motion.
Friction is the resistance to motion that occurs when one body moves upon another. It is defined as the tangential force acting at the surfaces of contact that resists relative sliding.
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1. The Coefficient of Friction
For sliding motion, the friction force F is proportional to the normal force N (the force pressing the surfaces together). This relationship is expressed by the coefficient of friction, denoted by the Greek letter mu (μ):
F = μ × N
therefore
μ = F / N
Example 1: Imperial Units
A body weighing 28 lb rests on a horizontal surface. If a force of 7 lb is required to keep it in motion:
μ = 7 / 28 = 0.25
Example 2: SI Units (Newtons)
A steel block with a mass of 50 kg rests on a steel table. To find the Normal Force (N), we multiply mass by gravity (9.81 m/s²).
- Normal Force (N): 50 kg × 9.81 m/s² = 490.5 N
- Force to Slide (F): Experimentally measured as 150 N
Coefficient (μ): 150 N / 490.5 N = 0.306
2. Angle of Repose
When a body rests on an inclined plane, friction prevents it from sliding until a critical angle is reached. This angle is called the angle of repose, denoted by θ.
Figure 2: At the exact moment sliding begins, the tangent of the angle equals the coefficient of friction.
At this condition, a simple relationship exists:
μ = tan θ
This provides a practical experimental method for determining the coefficient of friction between two materials without needing complex force sensors.
3. Laws of Friction (Dry vs. Lubricated)
Dry Friction (Coulomb Friction)
- Load: Friction is directly proportional to the normal force (moderate pressures).
- Area: Friction is independent of the apparent area of contact (e.g., sliding a brick flat vs. on its side requires the same force).
- Velocity: Static friction (to start motion) is higher than Kinetic friction (to maintain motion).
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Lubricated Surfaces
When a film of oil or grease separates the surfaces, the physics change dramatically, often described by the Stribeck Curve:
- Independence: Friction becomes almost independent of pressure when surfaces are fully flooded (Hydrodynamic lubrication).
- Velocity Factor: At start-up (low speed), friction is high. As speed increases and a fluid wedge forms, friction drops to a minimum before rising again due to fluid drag.
- Temperature: Highly sensitive to temperature due to changes in lubricant viscosity.
4. Friction and Efficiency
Friction is the enemy of efficiency. Here are typical efficiency values for common machine elements:
Note: These are general approximations. Real-world efficiency varies based on load, speed, and lubrication quality.
| Component | Typical Efficiency |
|---|---|
| Ball Bearings | ~99% |
| Roller Chains | 95–97% |
| Spur Gears | ~99% |
| V-Belts | 96–98% |
| Plain Bearings | 95–98% |
5. Rolling Resistance
Rolling is much more efficient than sliding, but it still has resistance. This is caused by the deformation of the wheel or the surface (hysteresis).
Figure 3: Rolling resistance is caused by the deformation of the surface, which shifts the Normal Force (N) forward by a distance 'f'.
Formula:
Resistance Force = (W × f) / r
- W: Load on the wheel (lb or N)
- r: Wheel radius (in or m)
- f: Coefficient of rolling resistance (units of length!)
Engineering Insight: Why is 'f' a length?
Physics vs. Machine Design Conventions
You may see two different formulas in your career:
- Dimensionless (Physics): F = Crr × W. (Here Crr is a ratio, similar to sliding friction).
- Length-Based (Machine Design): F = (W × f) / r. (Here 'f' is the physical lever arm length shown in Figure 3).
Neither is wrong; they are just different ways to model the same torque. The length-based method highlights the physical deformation offset.
Relationship: Crr = f / r
Read more about Rolling Resistance Physics (Engineering Toolbox) »
Typical Values for 'f' (Length):
- Iron on Iron: 0.02 in (0.5 mm)
- Iron on Asphalt: 0.15 in (3.8 mm)
- Iron on Wood: 0.22 in (5.6 mm)
- Iron on Iron: 0.02 in (0.5 mm)
- Iron on Asphalt: 0.15 in (3.8 mm)
- Iron on Wood: 0.22 in (5.6 mm)
Source: Adapted from standard engineering texts and Google Books.
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