Random Vibration Response of a Single-Degree-of-Freedom System
How mass, stiffness, damping and natural frequency combine with the input PSD and transfer function to produce the output PSD and RMS response of an SDOF system.
What Is It?
The single-degree-of-freedom (SDOF) system is the simplest structural model that captures the essential physics of random vibration response. It consists of a mass, spring and damper excited by a random input. Understanding the SDOF random vibration response provides the foundation for understanding multi-degree-of-freedom and modal random response analysis.
Why It Matters
Every mode in a modal random vibration analysis is treated as an SDOF system. The total response is the combination of many SDOF modal responses. Understanding the SDOF case — how the input PSD is filtered by the transfer function to produce the output PSD — is the key to understanding the full structural response. It is also the basis for Miles equation.
Every mode in a random vibration analysis is an SDOF system. Understanding the SDOF response is the foundation for understanding all modal random response.
Mass, Stiffness and Damping
The SDOF system is defined by its mass m, stiffness k and damping c. These determine the natural frequency and damping ratio — the two parameters that control the dynamic response. The natural frequency determines where the system resonates; the damping ratio determines how strongly it responds at resonance.
SDOF parameters: Natural frequency: f_n = (1 / 2π) · √(k / m) [Hz] Damping ratio: ζ = c / (2 · √(k · m)) = c / (2 · m · ω_n) where: m = mass [kg] k = stiffness [N/m] c = damping coefficient [N·s/m] ω_n = 2π · f_n = natural circular frequency [rad/s]
Transfer Function
The transfer function H(f) relates the input to the output of the SDOF system. For base excitation (acceleration input at the base, acceleration output at the mass), the transfer function magnitude depends on the frequency ratio f/fn and the damping ratio ζ. Near resonance (f = fn), the transfer function magnitude is approximately 1/(2ζ) — the amplification factor.
SDOF transfer function (base excitation, acceleration output): |H(f)| = √[ 1 + (2ζr)² ] / √[ (1−r²)² + (2ζr)² ] where: r = f / f_n = frequency ratio ζ = damping ratio At resonance (r = 1): |H(f_n)| ≈ 1 / (2ζ) [amplification factor]
Input PSD to Output PSD
The output PSD is the input PSD multiplied by the transfer function magnitude squared. This is the fundamental input-output relationship for random vibration. The structure acts as a filter — amplifying the input near resonance and attenuating it away from resonance. The output PSD describes how the response energy is distributed across frequency.
Input-output PSD relationship: G_out(f) = |H(f)|² · G_in(f) where: G_out(f) = output (response) PSD G_in(f) = input (excitation) PSD |H(f)|² = transfer function magnitude squared This is the fundamental relationship of random vibration analysis.
The output PSD equals the input PSD multiplied by the transfer function squared. The structure filters the input — amplifying near resonance, attenuating away from it.
RMS Response
The RMS response is obtained by integrating the output PSD across frequency. This gives the overall statistical level of the response. For an SDOF system, the RMS response depends on the input PSD level near the natural frequency, the damping ratio and the bandwidth of the resonance.
RMS response: x_rms = √( ∫ |H(f)|² · G_in(f) df ) For a flat input PSD near resonance (Miles equation approximation): x_rms ≈ √( π · f_n · G_in(f_n) / (4ζ) ) This is the Miles equation — valid when the input PSD is approximately flat across the resonance bandwidth.
Response Near Resonance
Near resonance, the transfer function magnitude is large — the input is strongly amplified. For lightly damped systems, the response PSD is dominated by a narrow peak centred at the natural frequency. The width of this peak is the half-power bandwidth, approximately 2·ζ·fn. Most of the response energy is concentrated in this narrow band.
- Near resonance, the transfer function magnitude ≈ 1/(2ζ) — strong amplification
- The response PSD has a narrow peak centred at fn
- Peak width ≈ 2·ζ·fn (half-power bandwidth)
- Most response energy is concentrated in this narrow band
Response Away from Resonance
Away from resonance, the transfer function magnitude is small — the input is attenuated. Below resonance, the response follows the input (the structure is stiff enough to track the base motion). Above resonance, the response decreases (the structure is isolated from the high-frequency input). The response away from resonance contributes little to the overall RMS for lightly damped systems.
| Frequency Region | Transfer Function Behaviour | Response Character |
|---|---|---|
| Well below fn (f << fn) | |H| ≈ 1 — structure tracks input | Response ≈ input — little amplification |
| Near fn (f ≈ fn) | |H| ≈ 1/(2ζ) — strong amplification | Response dominated by resonance peak |
| Well above fn (f >> fn) | |H| → 0 — structure isolated | Response attenuated — little contribution |
Effect of Damping
Damping is the most influential parameter for the resonant response. The amplification factor at resonance is 1/(2ζ) — halving the damping doubles the amplification. Since the response PSD is proportional to |H|², halving the damping quadruples the peak response PSD. The RMS response is proportional to 1/√ζ — halving the damping increases the RMS by a factor of √2. This strong sensitivity makes damping a critical parameter.
Damping sensitivity: the resonant amplification is 1/(2ζ). Halving the damping doubles the amplification and quadruples the peak response PSD. The RMS increases by √2. Damping is the most critical parameter in random vibration response.
Key Takeaways
- The SDOF system is the building block of all modal random vibration analysis
- Output PSD = |H(f)|² × input PSD — the structure filters the input
- Near resonance, amplification ≈ 1/(2ζ) — strongly dependent on damping
- RMS response = √(∫ output PSD df) — integrates the filtered PSD
- Miles equation gives the approximate RMS for flat input PSD near resonance