Forced vibration response in MDOF systems is all about how structures with multiple moving parts react to outside forces. It's like understanding how a multi-story building shakes during an earthquake or how a car's suspension handles bumpy roads.

This topic digs into the math behind these complex motions, showing how different parts of a system interact. We'll learn to predict how structures behave under various forces, which is crucial for designing safe and comfortable buildings, vehicles, and machines.

Steady-state response of MDOF systems

Harmonic excitation analysis

• Particular solution of the system's equations of motion characterizes the steady-state response of an MDOF system to harmonic excitation
• Complex algebra represents the harmonic excitation and response simplifying the mathematical analysis
• Solve a set of linear algebraic equations derived from the system's equations of motion to determine the steady-state response amplitude and phase
• Dynamic magnification factor (DMF) extends to MDOF systems relating the response amplitude to the static displacement
• Express steady-state response as a linear combination of the system's mode shapes using modal superposition
  • Example: In a two-story building model, combine the first mode (in-phase motion) and second mode (out-of-phase motion) to obtain the total response
• Analyze damping influence on the steady-state response
  • Effect on response amplitude (typically reduces peak amplitudes)
  • Impact on phase shift (introduces lag between excitation and response)
• Consider coupling between different degrees of freedom in steady-state response calculation
  • Example: Motion of one floor in a multi-story building affects the response of other floors

Response characteristics and applications

• Frequency-dependent behavior of steady-state response amplitude and phase
  • Peaks occur near system's natural frequencies
  • Troughs (anti-resonances) appear between peaks
• Utilize steady-state analysis for vibration isolation design
  • Example: Optimizing engine mounts in vehicles to minimize transmitted vibrations
• Apply steady-state response calculations in structural dynamics
  • Predict building response to harmonic wind loads
  • Analyze machine foundation vibrations due to rotating equipment

Frequency response functions for MDOF systems

FRF fundamentals and representation

• Frequency Response Functions (FRFs) describe the input-output relationship of an MDOF system in the frequency domain
• Derive FRF matrix from system's equations of motion relating complex amplitudes of response to complex amplitudes of excitation
• Each FRF matrix element represents the response of one degree of freedom due to excitation at another degree of freedom
• Express FRF matrix in terms of receptance (displacement/force), mobility (velocity/force), or accelerance (acceleration/force)
• Graphically represent magnitude and phase information in FRFs using Bode plots and Nyquist plots
  • Bode plots show magnitude and phase separately versus frequency
  • Nyquist plots display real and imaginary parts of FRF on complex plane
• Extend transfer function concept to MDOF systems relating Laplace transform of output to Laplace transform of input
• Use experimental modal analysis techniques with measured FRFs to identify dynamic properties of MDOF systems
  • Natural frequencies (peaks in FRF magnitude)
  • Mode shapes (from relative amplitudes at different DOFs)
  • Damping ratios (from width of resonance peaks)

FRF applications and analysis

• Employ FRFs for structural health monitoring
  • Changes in FRFs indicate potential damage or altered system properties
• Utilize FRFs in vibration testing and analysis of complex structures (aircraft, spacecraft)
• Apply FRF analysis to optimize sensor and actuator placement in active vibration control systems
• Investigate cross-coupling effects between different DOFs using off-diagonal FRF matrix elements
• Analyze FRF coherence to assess measurement quality and system linearity
• Use FRFs to validate and update finite element models of MDOF systems

Transient response of MDOF systems

Modal analysis and system transformation

• Transform coupled equations of motion into uncoupled modal equations using modal analysis simplifying MDOF system analysis
• Compose modal matrix using system's eigenvectors (mode shapes) to diagonalize mass and stiffness matrices
• Introduce generalized coordinates to express system's response as a linear combination of its mode shapes
• Apply Duhamel integral (convolution integral) to each uncoupled modal equation determining response to arbitrary excitation
• Obtain total transient response by superposing contributions from each mode in the physical coordinate system
• Introduce modal participation factors to quantify contribution of each mode to overall system response
  • Example: In a multi-story building, lower modes typically have higher participation factors for base excitation
• Solve modal equations for complex excitation functions using time-domain numerical integration methods
  • Newmark's method (commonly used in structural dynamics)
  • Runge-Kutta methods (versatile for various differential equations)

Transient response analysis techniques

• Evaluate importance of higher modes in transient response
  • Higher modes contribute more to short-duration, high-frequency excitations
• Analyze effect of damping on transient response decay
  • Higher damping leads to faster decay of free vibrations
• Investigate beat phenomena in lightly damped MDOF systems with closely spaced natural frequencies
• Apply transient response analysis to impact and shock loading scenarios
  • Example: Analyze vehicle suspension response to road bumps
• Utilize state-space formulation for transient response analysis of MDOF systems
  • Especially useful for systems with non-proportional damping
• Examine transient response envelopes to assess maximum system displacements and forces over time

Resonance frequencies in MDOF systems

Resonance characteristics and mode shapes

• Resonance frequencies in MDOF systems correspond to natural frequencies where response amplitude maximizes
• Number of resonance frequencies in an MDOF system equals the number of degrees of freedom
• Associate each resonance frequency with a specific mode shape describing relative motion of different system parts at that frequency
• Dominate forced response near a resonance frequency by contribution of corresponding mode
  • Example: In a two-mass system, excitation near the first natural frequency primarily excites the first mode (in-phase motion)
• Occur anti-resonances between resonance frequencies minimizing response amplitude specific to each input-output pair
• Change phase relationship between excitation and response rapidly near resonance frequencies typically shifting by 180 degrees
• Relate width of resonance peaks in frequency response to system damping with higher damping resulting in broader peaks

Resonance effects and analysis

• Investigate mode coupling and energy transfer between different DOFs at resonance
• Analyze effect of structural modifications on resonance frequencies and mode shapes
  • Example: Adding mass or stiffness to specific locations in a structure
• Examine resonance amplification factors for different modes and excitation locations
• Study resonance avoidance techniques in MDOF system design
  • Frequency tuning of components
  • Introduction of dynamic absorbers
• Investigate non-linear effects on resonance behavior in MDOF systems
  • Frequency shifting
  • Modal interactions
• Apply modal filtering techniques to isolate and analyze individual mode contributions at resonance
• Evaluate resonance effects on fatigue life and structural integrity of MDOF systems