Frequency Response Functions in Random Vibration

What Is It?

A frequency response function (FRF) — also called a transfer function — describes how a structural system modifies an input signal at each frequency. In random vibration, the FRF relates the input PSD to the output PSD: the output PSD is the input PSD multiplied by the FRF magnitude squared. The FRF is the link between the excitation and the response.

Why It Matters

The FRF determines how the input PSD is filtered into the response PSD. It encodes all the structural information — mass, stiffness, damping, mode shapes, boundary conditions — that controls the response. Understanding FRFs is essential for interpreting random vibration results: why the response is high at some frequencies and low at others, and how changes to the structure affect the response.

The frequency response function is the link between input PSD and output PSD. It encodes all the structural information that controls the random vibration response.

Transfer Function

The transfer function H(f) is a complex-valued function of frequency that relates the input and output of a linear system. In random vibration, the key relationship is that the output PSD equals the input PSD multiplied by the magnitude of the transfer function squared. The phase of the transfer function is not directly used in PSD-based random vibration analysis — only the magnitude matters for the PSD relationship.

Transfer function:

H(f)  =  complex transfer function

|H(f)|  =  magnitude
∠H(f)  =  phase

Output PSD:

G_out(f)  =  |H(f)|² · G_in(f)

Note: only |H(f)|² appears in the PSD relationship.
The phase does not affect the output PSD directly.

Magnitude

The magnitude |H(f)| describes how much the system amplifies or attenuates the input at each frequency. Near resonance, the magnitude is large — the input is amplified. Away from resonance, the magnitude is small — the input is attenuated. The magnitude squared |H(f)|² is what appears in the PSD relationship, so a doubling of the transfer function magnitude produces a fourfold increase in the output PSD.

• |H(f)| — amplification or attenuation at each frequency
• |H(f)|² appears in the PSD relationship — squared effect on output PSD
• Near resonance: |H| is large — strong amplification
• Away from resonance: |H| is small — attenuation

Phase

The phase ∠H(f) describes the time lag between input and output at each frequency. Below resonance, the response is approximately in phase with the input. At resonance, the phase shifts by 90 degrees. Above resonance, the response is approximately out of phase (180 degrees). The phase is not directly used in PSD-based random vibration analysis, but it is important for understanding the physical behaviour and for correlated multi-input analysis.

Phase behaviour of SDOF:

f << f_n:  ∠H ≈ 0°    (in phase)
f = f_n:   ∠H = −90°   (quadrature — at resonance)
f >> f_n:  ∠H ≈ −180°  (out of phase)

The phase does not affect the output PSD directly,
but matters for correlated multi-input analysis.

Modal Amplification

For a multi-degree-of-freedom structure, the transfer function is the superposition of modal contributions. Each mode contributes a resonant peak to the FRF. The height of each peak is controlled by the modal damping (approximately 1/(2ζn) for mode n), and the width by the damping and natural frequency. The transfer function magnitude at any frequency is the result of all modal contributions at that frequency.

Modal transfer function:

H(f)  =  Σ_n  φ_n · φ_n^T / (k_n − m_n · (2πf)² + i · c_n · 2πf)

where:
φ_n    =  mode shape vector for mode n
k_n    =  modal stiffness
m_n    =  modal mass
c_n    =  modal damping

Each mode contributes a resonant peak to the FRF.

Input/Output PSD Relationship

The input-output PSD relationship is the central equation of random vibration analysis. It states that the output PSD at any location equals the input PSD multiplied by the transfer function magnitude squared between the input and that location. This relationship is applied at each frequency and produces the output PSD curve.

Input PSD → × |H(f)|² (transfer function magnitude squared) → Output PSD → ∫ df → RMS response

Output PSD = |H(f)|² × Input PSD. This is applied at every frequency. The transfer function encodes all structural information — modes, damping, mode shapes, boundary conditions.

Multiple Transfer Functions

For a structure with many response locations, each location has its own transfer function from the input. The same input PSD produces different response PSDs at different locations because the transfer functions differ. A location near a mode shape antinode (large modal displacement) has a large transfer function at that mode's frequency. A location near a node (zero modal displacement) has a small transfer function.

• Each response location has its own transfer function from the input
• The same input PSD produces different response PSDs at different locations
• Antinode locations — large transfer function, high response
• Node locations — small transfer function, low response

Key Takeaways

• The FRF (transfer function) relates input PSD to output PSD via |H(f)|²
• Magnitude controls amplification/attenuation; phase does not affect PSD directly
• Each mode contributes a resonant peak to the FRF — height controlled by damping
• Different response locations have different transfer functions — different response PSDs
• The FRF encodes all structural information that controls the random vibration response