Harmonic excitation is a key concept in forced vibrations. It involves applying a sinusoidal force to a system, causing it to oscillate. Understanding how systems respond to these forces is crucial for engineers designing everything from buildings to machines.

Frequency response functions (FRFs) are powerful tools for analyzing system behavior. They show how a system reacts to different input frequencies, helping predict resonance and optimize designs. Mastering FRFs is essential for tackling real-world vibration problems.

Harmonic Excitation of Single-DOF Systems

Fundamentals of Harmonic Excitation

• Harmonic excitation characterized by sinusoidal waveform with specific amplitude and frequency
• General equation of motion for SDOF system under harmonic excitation $mẍ + cẋ + kx = F₀sin(ωt)$
  • m represents mass
  • c represents damping coefficient
  • k represents spring stiffness
  • F₀ represents forcing amplitude
  • ω represents forcing frequency
• Steady-state response consists of two components
  • Transient response decays over time
  • Particular solution persists indefinitely

Response Analysis and Magnification Factor

• Particular solution for harmonic excitation takes form $x(t) = X sin(ωt - φ)$
  • X represents response amplitude
  • φ represents phase angle
• Response amplitude and phase angle depend on
  • System's natural frequency
  • Damping ratio
  • Forcing frequency
• Magnification factor (M) defined as ratio of response amplitude to static deflection
  • Function of frequency ratio (r = ω/ωn) and damping ratio (ζ)
• Frequency response curve graphically represents relationship between magnification factor and frequency ratio for various damping ratios
  • Helps visualize system behavior across different excitation frequencies (resonance peaks)

Frequency Response Functions

Displacement, Velocity, and Acceleration FRFs

• Frequency response functions (FRFs) describe relationship between input (forcing function) and output (response) in frequency domain
• Displacement FRF, H(ω), defined as ratio of steady-state displacement amplitude to force amplitude $H(ω) = X/F₀$
• Velocity FRF, H'(ω), obtained by differentiating displacement FRF with respect to time $H'(ω) = iωH(ω)$
• Acceleration FRF, H''(ω), obtained by differentiating velocity FRF with respect to time $H''(ω) = -ω²H(ω)$
• Magnitude and phase of each FRF expressed in terms of
  • System's natural frequency
  • Damping ratio
  • Forcing frequency

Graphical Representation and Applications

• Bode plots used to graphically represent magnitude and phase of FRFs as functions of frequency
  • Typically utilize logarithmic scales for clearer visualization
• FRFs used to analyze system's response to various input frequencies
  • Identify resonance conditions (peaks in magnitude plot)
  • Determine system stability (phase margins)
• Applications of FRFs
  • Structural analysis (building response to seismic excitations)
  • Mechanical design (vibration isolation in machinery)

Resonance, Damping, and Phase Shift

Resonance and System Response

• Resonance occurs when forcing frequency matches system's natural frequency
  • Results in maximum response amplitude and energy transfer
• Resonance frequency for undamped system equals natural frequency
• Resonance frequency for damped systems slightly lower than natural frequency
• Importance of understanding resonance
  • Avoid catastrophic failures in structures (Tacoma Narrows Bridge collapse)
  • Optimize energy harvesting in vibration-based energy harvesters

Damping Characteristics and Effects

• Damping dissipates energy in vibrating system
  • Affects amplitude of response and width of resonance peak
• Damping ratio (ζ) characterizes level of damping in system
  • ζ < 1 for underdamped systems (oscillatory response)
  • ζ = 1 for critically damped systems (fastest return to equilibrium)
  • ζ > 1 for overdamped systems (slow return to equilibrium)
• Half-power bandwidth method estimates damping ratio from frequency response curve
  • Measures width of resonance peak at 1/√2 of its maximum value
• Examples of damping in real systems
  • Shock absorbers in vehicles
  • Tuned mass dampers in tall buildings (Taipei 101)

Phase Shift and Frequency Dependence

• Phase shift (φ) represents time lag between applied force and system's response
• Varies with frequency ratio and damping ratio
• Frequency-dependent behavior
  • Low frequencies (ω << ωn) response in phase with excitation
  • High frequencies (ω >> ωn) response out of phase by approximately 180 degrees
• Importance of phase shift
  • Vibration isolation design
  • Control system stability analysis

Complex Representation for Harmonic Excitation

Complex Exponential Method

• Complex representation method simplifies analysis of harmonic excitation problems
• Forcing function expressed as $F(t) = F₀e^(iωt)$
  • i represents imaginary unit
  • ω represents forcing frequency
• Steady-state response assumed to have form $x(t) = Xe^(iωt)$
  • X represents complex amplitude including magnitude and phase information
• Substituting expressions into equation of motion yields complex frequency response function $H(ω) = X/F₀ = 1 / (k - mω² + icω)$

Analysis and Applications of Complex FRF

• Magnitude of complex frequency response function gives amplification factor
• Argument of complex frequency response function gives phase angle
• Real and imaginary parts of complex frequency response function separated to obtain
  • In-phase component of response
  • Quadrature component of response
• Complex representation method extended to analyze
  • Multi-degree-of-freedom systems (coupled oscillators)
  • Systems with multiple excitation frequencies (superposition principle)
• Applications of complex FRF analysis
  • Modal analysis in structural engineering
  • Vibration control in precision manufacturing equipment