Modal Random Response Analysis

What Is It?

Modal random response analysis extends the SDOF random vibration method to multi-degree-of-freedom structures. The structure is decomposed into its modes of vibration. Each mode is treated as an SDOF system excited by the input PSD. The modal responses are computed independently and then combined statistically to produce the total response PSD and RMS at each location.

Why It Matters

Most engineering structures have many modes within the excitation frequency range. Modal random response analysis is the standard method for computing the response of these structures to random excitation. It is the method used by commercial FEA solvers for random vibration analysis. Understanding the modal approach is essential for setting up, running and interpreting random vibration FEA.

Modal random response analysis is the standard method for multi-degree-of-freedom random vibration. Every commercial FEA solver uses this approach.

Modal Extraction

The first step in modal random response analysis is modal extraction — computing the natural frequencies, mode shapes and modal parameters of the structure. This is done with a modal analysis in FEA. The number of modes extracted must be sufficient to cover the excitation frequency range and to capture the effective mass in the excitation direction. Modes beyond the PSD frequency range may still contribute if they have significant modal mass.

• Modal analysis extracts natural frequencies and mode shapes
• Number of modes must cover the PSD frequency range
• Cumulative effective mass should be checked — typically >90% in excitation direction
• Modes beyond the PSD range may contribute if effective mass is significant

Modal Coordinates

The structural response is expressed in modal coordinates — each mode has a modal coordinate that describes how much that mode contributes to the total response. The modal coordinate for each mode is computed as the SDOF response of that mode to the input PSD, using the modal natural frequency and damping. The physical response at any location is the sum of modal contributions weighted by the mode shape at that location.

Modal decomposition:

Physical response  =  Σ_n  φ_n(x) · q_n(t)

where:
φ_n(x)  =  mode shape at location x for mode n
q_n     =  modal coordinate for mode n

Each modal coordinate responds as an SDOF system:
  G_qn(f)  =  |H_n(f)|² · G_in(f)

where H_n(f) is the SDOF transfer function for mode n.

Mode Shapes

Mode shapes describe the deformation pattern of each mode. They determine how strongly each mode contributes to the response at each location. A location near an antinode of a mode (large mode shape value) has a strong response from that mode. A location near a node (zero mode shape value) has no response from that mode. Mode shapes also determine the modal mass and the participation of each mode in the excitation direction.

• Mode shapes describe the deformation pattern of each mode
• Antinode — large mode shape value, strong response from that mode
• Node — zero mode shape value, no response from that mode
• Mode shapes determine modal mass and directional participation

Modal Participation

The modal participation factor describes how strongly each mode is excited by the input. For base excitation in a given direction, the participation factor depends on the mode shape and the modal mass. Modes with high participation in the excitation direction contribute strongly to the response. Modes with low participation contribute little, even if their natural frequency is within the PSD range.

Modal participation factor:

Γ_n  =  (φ_n^T · M · d) / (φ_n^T · M · φ_n)

where:
Γ_n  =  participation factor for mode n
M    =  mass matrix
d    =  excitation direction vector
φ_n  =  mode shape for mode n

Effective modal mass:
  m_eff,n  =  Γ_n² · (φ_n^T · M · φ_n)

Frequency Response

The frequency response for each mode is the SDOF transfer function evaluated at the modal natural frequency and damping. The total frequency response at a location is the superposition of all modal frequency responses, weighted by mode shapes and participation factors. This produces a transfer function with peaks at each natural frequency — the FRF that filters the input PSD into the response PSD.

Modal analysis → Mode shapes + natural frequencies + damping → SDOF transfer function per mode → Superposition weighted by participation → Total transfer function → × Input PSD → Response PSD

Response PSD

The response PSD at each location is computed by applying the input-output PSD relationship: the input PSD is multiplied by the total transfer function magnitude squared. This produces a response PSD with peaks at the natural frequencies, whose heights depend on the damping and the input PSD level at those frequencies. The response PSD is then integrated to obtain the RMS response.

Response PSD at location x:

G_out(f, x)  =  |H(f, x)|² · G_in(f)

where H(f, x) is the total transfer function at location x:

H(f, x)  =  Σ_n  Γ_n · φ_n(x) · H_n(f)

H_n(f)  =  SDOF transfer function for mode n

RMS response:
  x_rms(x)  =  √( ∫ G_out(f, x) df )

Modal Combination

The modal responses must be combined to produce the total response. Because all modes are excited simultaneously by the random input, the combination is statistical. The most common methods are SRSS (square root of sum of squares) and CQC (complete quadratic combination). SRSS is simple and accurate for well-separated modes. CQC accounts for modal coupling and is preferred for closely spaced modes.

Cross-Correlation Between Modes

For closely spaced modes (natural frequencies within about 10% of each other), the modal responses are correlated — the SRSS method, which assumes uncorrelated modes, may underpredict the response. CQC includes cross-correlation terms that account for the coupling between closely spaced modes. For well-separated modes, the cross-correlation is negligible and SRSS and CQC give similar results.

For closely spaced modes, use CQC rather than SRSS. SRSS assumes uncorrelated modes and may underpredict the response when modes are within ~10% of each other.

Key Takeaways

• Modal random response analysis decomposes the structure into modes, each treated as an SDOF system
• Mode shapes and participation factors determine how strongly each mode contributes at each location
• Response PSD = |H(f)|² × Input PSD, where H is the superposition of modal transfer functions
• Modal combination is statistical — SRSS for well-separated modes, CQC for closely spaced modes
• Modal extraction must capture sufficient effective mass in the excitation direction