2.3 Natural frequency and resonance

Natural frequency and resonance are key concepts in vibrating systems. They describe how a system oscillates without external forces and what happens when an applied force matches that frequency. Understanding these phenomena is crucial for engineers designing structures and machines.

In single degree-of-freedom systems, natural frequency depends on mass and stiffness. Resonance occurs when the forcing frequency matches the natural frequency, causing large oscillations. These concepts help explain system behavior and guide design choices to avoid unwanted vibrations or harness resonance for specific applications.

Natural frequency of SDOF systems

Understanding natural frequency

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• Natural frequency represents the rate at which a system oscillates without external forces or damping
• For SDOF systems, calculate natural frequency (ωn) using mass (m) and stiffness (k): $ωn = \sqrt{\frac{k}{m}}$
• Measure natural frequency in radians per second (rad/s), convert to Hertz (Hz) by dividing by 2π
• Period of oscillation (T) relates inversely to natural frequency: $T = \frac{2π}{ωn}$
• Undamped systems have natural frequency equal to resonant frequency
• Systems with multiple degrees of freedom exhibit unique natural frequencies for each vibration mode

Factors influencing natural frequency

• Mass changes affect natural frequency inversely (increase mass, decrease frequency)
• Stiffness changes affect natural frequency directly (increase stiffness, increase frequency)
• Geometric modifications alter natural frequency by changing mass distribution or stiffness
• Material properties impact natural frequency through their effect on system stiffness
• Temperature fluctuations can influence natural frequency by altering material properties
• Boundary conditions affect natural frequency by constraining or freeing system movement

Resonance in vibrating systems

Fundamentals of resonance

• Resonance occurs when external force frequency matches or approaches system's natural frequency
• Small periodic forces produce large amplitude oscillations during resonance
• Rapid amplitude increase characterizes resonance as driving frequency nears natural frequency
• Undamped systems theoretically experience infinite amplitude at resonance (not physically possible)
• Resonance manifests beneficially in some applications (musical instruments) and detrimentally in others (structural failures)
• Quality factor (Q) inversely relates to resonance peak width, indicating system's energy storage capacity
• Resonance understanding proves crucial for designing structures, machines, and electrical circuits

Applications and implications of resonance

• Acoustic resonance enhances sound production in musical instruments (guitar strings, organ pipes)
• Electrical resonance circuits form basis for radio tuning and wireless communication
• Mechanical resonance causes unwanted vibrations in machinery (rotating equipment, bridges)
• Seismic resonance amplifies earthquake damage in buildings with matching natural frequencies
• Magnetic resonance imaging (MRI) utilizes resonance for medical diagnostics
• Resonant power transfer enables wireless charging of electronic devices
• Resonant mass sensors detect minute mass changes in chemical and biological applications

Resonance effects on system response

Amplitude and phase response

• System response amplitude reaches maximum at resonance, limited by damping in real systems
• Amplification factor at resonance relates inversely to system's damping ratio
• Frequency response curve illustrates amplitude ratio variation with frequency ratio
• Phase angle between input force and system response shifts rapidly near resonance
• Typically, phase shift of 180 degrees occurs as system passes through resonance
• Half-power bandwidth method estimates damping ratio by analyzing resonance peak width
• Lightly damped systems exhibit slightly lower resonant frequency than natural frequency

Analyzing resonant behavior

• Quality factor (Q) indicates resonance peak sharpness and energy dissipation rate
• Higher Q-factor systems display narrower, more pronounced resonance peaks
• Forced response of system near resonance depends on damping ratio and frequency ratio
• Resonance curves for different damping ratios show varying peak amplitudes and widths
• Bode plots visualize both magnitude and phase response across frequency spectrum
• Nyquist plots provide alternative representation of system's frequency response
• Time domain analysis reveals transient and steady-state behavior during resonance

Avoiding resonance in design

Resonance mitigation strategies

• Implement frequency detuning to separate natural frequency from expected excitation frequencies
• Add damping to reduce resonance amplitude and widen resonance peak
• Utilize dynamic vibration absorbers to counteract primary system vibrations at specific frequencies
• Modify structure by adding stiffeners or changing mass distribution to alter natural frequencies
• Apply active vibration control systems using sensors and actuators for real-time counteraction
• Identify and avoid critical speeds in rotating machinery through proper design and speed control
• Employ finite element analysis and experimental modal analysis to predict and measure resonant behavior

Design considerations for resonance avoidance

• Conduct thorough frequency analysis of system and potential excitation sources
• Incorporate safety factors in natural frequency calculations to account for uncertainties
• Design structures with multiple load paths to distribute forces and reduce resonance risk
• Use materials with inherent damping properties (viscoelastic materials) in critical components
• Implement isolation systems to decouple sensitive equipment from vibration sources
• Regularly maintain and monitor equipment to detect changes in natural frequencies over time
• Develop operational procedures to avoid prolonged operation at or near resonant frequencies