What is Centrifugal Force in Rotating Machinery? • Portable balancer, vibration analyzer "Balanset" for dynamic balancing crushers, fans, mulchers, augers on combines, shafts, centrifuges, turbines, and many others rotors What is Centrifugal Force in Rotating Machinery? • Portable balancer, vibration analyzer "Balanset" for dynamic balancing crushers, fans, mulchers, augers on combines, shafts, centrifuges, turbines, and many others rotors

What is Centrifugal Force in Rotating Machinery?

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Understanding Centrifugal Force in Rotating Machinery

Definition: What is Centrifugal Force?

Centrifugal force is the apparent outward force experienced by a mass moving in a circular path. In rotating machinery, when a rotor has unbalance—meaning its mass center is offset from the axis of rotation—the eccentric mass creates a rotating centrifugal force as the shaft spins. This force is directed radially outward from the center of rotation and rotates at the same speed as the shaft.

Centrifugal force from unbalance is the primary cause of vibration in rotating machinery and is the force that balancing procedures aim to minimize. Understanding its magnitude and behavior is fundamental to rotor dynamics and vibration analysis.

Mathematical Expression

Basic Formula

The magnitude of centrifugal force is given by:

  • F = m × r × ω²
  • Where:
  • F = centrifugal force (Newtons)
  • m = unbalance mass (kilograms)
  • r = radius of mass eccentricity (meters)
  • ω = angular velocity (radians per second) = 2π × RPM / 60

Alternative Formulation Using RPM

For practical calculations using RPM:

  • F (N) = U × (RPM/9549)²
  • Where U = unbalance (gram-millimeters) = m × r
  • This form directly uses unbalance units common in balancing specifications

Key Insight: Speed-Squared Relationship

The most important characteristic of centrifugal force is its dependence on the square of rotational speed:

  • Doubling the speed increases force by 4× (2² = 4)
  • Tripling the speed increases force by 9× (3² = 9)
  • This quadratic relationship explains why unbalance that’s acceptable at low speeds becomes critical at high speeds

Effect on Vibration

Force-to-Vibration Relationship

Centrifugal force from unbalance causes vibration through the following mechanism:

  1. Rotating centrifugal force applied to rotor
  2. Force transmitted through shaft to bearings and supports
  3. Elastic system (rotor-bearing-foundation) responds by deflecting
  4. Deflection creates measured vibration at bearings
  5. Relationship between force and vibration depends on system stiffness and damping

At Resonance

When operating at a critical speed:

  • Even small centrifugal forces from residual unbalance create large vibration
  • Amplification factor can be 10-50× depending on damping
  • This resonant amplification is why critical speed operation is dangerous

Below Resonance (Rigid Rotor Operation)

  • Vibration approximately proportional to force
  • Therefore vibration ∝ speed² (since force ∝ speed²)
  • Doubling speed quadruples vibration amplitude

Practical Examples

Example 1: Small Fan Impeller

  • Unbalance: 10 grams at 100 mm radius = 1000 g·mm
  • Speed: 1500 RPM
  • Calculation: F = 1000 × (1500/9549)² ≈ 24.7 N (2.5 kgf)

Example 2: Same Impeller at Higher Speed

  • Unbalance: Same 1000 g·mm
  • Speed: 3000 RPM (doubled)
  • Calculation: F = 1000 × (3000/9549)² ≈ 98.7 N (10.1 kgf)
  • Result: Force increased by 4× with 2× speed increase

Example 3: Large Turbine Rotor

  • Rotor Mass: 5000 kg
  • Permissible Unbalance (G 2.5): 400,000 g·mm
  • Speed: 3600 RPM
  • Centrifugal Force: F = 400,000 × (3600/9549)² ≈ 56,800 N (5.8 tonnes force)
  • Implication: Even “well-balanced” rotors generate significant forces at high speeds

Centrifugal Force in Balancing

Unbalance Force Vector

Centrifugal force from unbalance is a vector quantity:

  • Magnitude: Determined by unbalance amount and speed (F = m × r × ω²)
  • Direction: Points radially outward toward the heavy spot
  • Rotation: Vector rotates at shaft speed (1× frequency)
  • Phase: Angular position of force at any instant

Balancing Principle

Balancing works by creating an opposing centrifugal force:

  • Correction weight placed 180° from heavy spot
  • Creates equal and opposite centrifugal force
  • Vector sum of original and correction forces approaches zero
  • Net centrifugal force minimized, vibration reduced

Multi-Plane Balancing

For two-plane balancing:

  • Centrifugal forces in each plane create both forces and moments
  • Correction weights must cancel both force unbalance and couple unbalance
  • Vector addition of forces from both planes determines net force

Bearing Load Implications

Static vs. Dynamic Loads

  • Static Load: Constant bearing load from rotor weight (gravity)
  • Dynamic Load: Rotating load from centrifugal force (unbalance)
  • Total Load: Vector sum varies around circumference as rotor rotates
  • Maximum Load: Occurs where static and dynamic loads align

Bearing Life Impact

  • Bearing life inversely proportional to load cubed (L10 ∝ 1/P³)
  • Small increases in dynamic load significantly reduce bearing life
  • Centrifugal force from unbalance adds to bearing loads
  • Good balance quality essential for bearing longevity

Centrifugal Force in Different Machine Types

Low-Speed Equipment (< 1000 RPM)

  • Centrifugal forces relatively low
  • Static loads from gravity often dominant
  • Looser balance tolerances acceptable
  • Large absolute unbalances can be tolerated

Medium-Speed Equipment (1000-5000 RPM)

  • Centrifugal forces significant and must be managed
  • Most industrial machinery in this range
  • Balance quality grades G 2.5 to G 16 typical
  • Balancing important for bearing life and vibration control

High-Speed Equipment (> 5000 RPM)

  • Centrifugal forces dominant over static loads
  • Very tight balance tolerances required (G 0.4 to G 2.5)
  • Small unbalances create enormous forces
  • Precision balancing absolutely critical

Centrifugal Force and Critical Speeds

Force Amplification at Resonance

At critical speeds:

  • Same centrifugal force input
  • System response amplified by Q-factor (typically 10-50)
  • Vibration amplitude far exceeds below-critical operation
  • Demonstrates why critical speeds must be avoided

Flexible Rotor Behavior

For flexible rotors above critical speeds:

  • Shaft bends under centrifugal force
  • Deflection creates additional eccentricity
  • Self-centering effect above critical speed reduces bearing loads
  • Counterintuitive: vibration may decrease above critical speed

Relationship to Balancing Standards

Permissible Unbalance and Force

Balance quality grades in ISO 21940-11 are based on limiting centrifugal force:

  • Lower G-numbers allow less unbalance
  • Limits proportional force at any speed
  • Ensures centrifugal forces remain within safe design limits
  • Different equipment types have different force tolerances

Measurement and Calculation

From Vibration to Force

While force isn’t directly measured in field balancing, it can be estimated:

  • Measure vibration amplitude at operating speed
  • Estimate system stiffness from influence coefficients
  • Calculate force: F ≈ k × deflection
  • Useful for assessing bearing load contributions from unbalance

From Unbalance to Force

Direct calculation if unbalance known:

  • Use formula F = m × r × ω²
  • Or F = U × (RPM/9549)² where U in g·mm
  • Provides expected force for any unbalance amount and speed
  • Used in design calculations and tolerance verification

Centrifugal force is the fundamental mechanism by which unbalance causes vibration in rotating machinery. Its quadratic relationship with speed explains why balance quality becomes increasingly critical as rotational speeds increase and why even small unbalances can generate enormous forces and destructive vibration in high-speed equipment.

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