Understanding Forced Vibration
Definition: What is Forced Vibration?
Forced vibration is oscillatory motion caused by an external periodic force applied to a mechanical system. The vibration occurs at the frequency of the applied force (forcing frequency), and the amplitude is proportional to the magnitude of the forcing function and inversely proportional to the system’s resistance to motion at that frequency. Most vibration in rotating machinery is forced vibration, with common forcing sources including unbalance (rotating centrifugal force), misalignment (coupling forces), and aerodynamic/hydraulic pulsations.
Forced vibration is fundamentally different from self-excited vibration (where the system generates its own sustained oscillation) and free vibration (transient response after impulse). Understanding forced vibration principles is essential because it explains how vibration amplitude relates to fault severity and how vibration can be controlled by reducing forcing or modifying system response.
Characteristics of Forced Vibration
Frequency Matching
- Vibration frequency equals forcing frequency
- If forcing at 30 Hz, vibration at 30 Hz
- Unlike self-excited vibration which occurs at natural frequency
- Predictable frequency based on forcing source
Amplitude Proportionality
- Vibration amplitude proportional to forcing magnitude
- Double the force → double the vibration (linear system)
- Remove forcing → vibration stops
- Controllable through force reduction
Phase Relationship
- Definite phase relationship between force and response
- Phase depends on frequency relative to natural frequency
- Below resonance: vibration in-phase with force
- At resonance: 90° phase lag
- Above resonance: 180° phase lag
Stability
- System is stable—vibration bounded
- Doesn’t grow without bound
- Amplitude limited by forcing and system response
- Contrasts with unstable self-excited vibration
Common Forcing Functions in Machinery
1. Unbalance (1× Forcing)
- Force: Rotating centrifugal force from mass eccentricity
- Frequency: Once per revolution (1× shaft speed)
- Magnitude: F = m × r × ω² (proportional to speed squared)
- Most Common: Primary vibration source in most rotating equipment
2. Misalignment (2× Forcing)
- Force: Coupling forces from angular/parallel offset
- Frequency: Twice per revolution (2× shaft speed)
- Characteristic: High axial component
3. Aerodynamic/Hydraulic (Blade/Vane Passing)
- Force: Pressure pulsations from blade-stator interaction
- Frequency: Number of blades × shaft speed
- Examples: Fans, pumps, compressors
4. Gear Mesh Forces
- Force: Tooth engagement creating periodic loading
- Frequency: Number of teeth × shaft speed
- Magnitude: Related to transmitted torque and tooth quality
5. Electromagnetic Forces
- Force: Magnetic field pulsations in motors/generators
- Frequency: 2× line frequency (120/100 Hz)
- Independent: Of mechanical speed (asynchronous forcing)
Response to Forcing: System Behavior
Below Natural Frequency (Stiffness-Controlled)
- Vibration amplitude ≈ Force / Stiffness
- Response in-phase with forcing
- Amplitude increases with speed for speed-dependent forces
- Typical operating region for most rigid rotors
At Natural Frequency (Resonance)
- Vibration amplitude ≈ Force / (Damping × Natural Frequency)
- Amplitude amplified by Q-factor (typically 10-50×)
- 90° phase lag
- Small forces create large vibration
- Damping is only limiting factor
Above Natural Frequency (Mass-Controlled)
- Vibration amplitude ≈ Force / (Mass × Frequency²)
- 180° phase lag (vibration opposite to force direction)
- Amplitude decreases with increasing frequency
- Operating region for flexible rotors above critical speeds
Forced Vibration vs. Other Types
Forced vs. Free Vibration
- Forced: Continuous forcing, vibration sustained, at forcing frequency
- Free: Impulse response, vibration decays, at natural frequency
- Example: Bump test produces free vibration; running machine produces forced vibration
Forced vs. Self-Excited Vibration
- Forced: External force, amplitude proportional to force, stable
- Self-Excited: Internal energy source, amplitude limited by nonlinearity, unstable
- Examples: Unbalance is forced; oil whirl is self-excited
Control and Mitigation
Reduce Forcing
- Balancing: Reduces unbalance forcing directly
- Alignment: Reduces misalignment forces
- Repair Defects: Fix mechanical problems creating forces
- Most Effective: Eliminate or minimize the forcing source
Modify System Response
- Change Stiffness: Shift natural frequencies away from forcing frequencies
- Add Damping: Reduce resonance amplification
- Change Mass: Modify natural frequencies
- Isolation: Reduce force transmission to structure
Avoid Resonance
- Ensure forcing frequencies don’t match natural frequencies
- Separation margin typically ±20-30%
- Design-phase analysis to verify
- Speed restrictions if resonance unavoidable
Practical Significance
Most Machinery Vibration is Forced
- Unbalance, misalignment, gear mesh—all forced vibration
- Predictable and controllable through forcing reduction
- Standard maintenance actions (balance, align) address forcing
Diagnostic Approach
- Identify forcing frequency from spectrum
- Match to known forcing sources (1×, 2×, gear mesh, etc.)
- Diagnose forcing source
- Reduce forcing through appropriate maintenance
Forced vibration is the fundamental vibration type in rotating machinery, arising from external periodic forces acting on the system. Understanding forced vibration principles—frequency matching, amplitude proportionality, and response characteristics—enables proper diagnosis of vibration sources, appropriate corrective actions (reducing forcing or modifying response), and design strategies that minimize vibration through forcing reduction and resonance avoidance.
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