Response Spectrum Analysis of a Single-Degree-of-Freedom System
A worked example showing how spectral acceleration is obtained from a response spectrum for a single-degree-of-freedom system, and how peak displacement and equivalent static force are calculated.
Problem definition
A single-degree-of-freedom (SDOF) system represents a piece of equipment mounted on a rigid frame. The equipment has a mass of 500 kg supported by a frame with lateral stiffness of 2.0 x 10^6 N/m. The system is located on a floor where the floor response spectrum specifies a spectral acceleration at the system natural frequency. Determine the peak displacement, peak velocity and equivalent static force.
Assumptions
- The system is linear-elastic
- Damping ratio is 2% of critical
- The frame is represented as a single lateral spring
- The floor response spectrum is the input
- The mass is lumped at the floor level
Inputs
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Mass | m | 500 | kg |
| Lateral stiffness | k | 2.0 x 10^6 | N/m |
| Damping ratio | zeta | 0.02 | - |
| Spectral acceleration at fn | Sa | 0.35g | - |
Step 1: Calculate natural frequency
The natural frequency of the SDOF system is: fn = (1 / 2pi) x sqrt(k / m). fn = (1 / 2pi) x sqrt(2.0 x 10^6 / 500) = (1 / 2pi) x sqrt(4000) = (1 / 2pi) x 63.25 = 10.07 Hz. The natural period is: Tn = 1 / fn = 1 / 10.07 = 0.0993 s.
Step 2: Convert spectral acceleration to SI units
Sa = 0.35g = 0.35 x 9.81 = 3.43 m/s^2
Step 3: Calculate peak displacement
For a SDOF system, the peak displacement is: xd = Sa / omega_n^2 where omega_n = 2pi x fn = 2pi x 10.07 = 63.25 rad/s. xd = 3.43 / (63.25)^2 = 3.43 / 4001 = 8.57 x 10^-4 m = 0.857 mm.
Step 4: Calculate peak velocity
The peak pseudo-velocity is: Sv = Sa / omega_n = 3.43 / 63.25 = 0.0542 m/s = 54.2 mm/s.
Step 5: Calculate equivalent static force
The equivalent static force (base shear) is: F = m x Sa = 500 x 3.43 = 1715 N. Alternatively, F = k x xd = 2.0 x 10^6 x 8.57 x 10^-4 = 1714 N (consistent).
Results
| Quantity | Value | Unit |
|---|---|---|
| Natural frequency, fn | 10.07 | Hz |
| Natural period, Tn | 0.0993 | s |
| Spectral acceleration, Sa | 3.43 | m/s^2 |
| Peak displacement, xd | 0.857 | mm |
| Peak velocity, Sv | 54.2 | mm/s |
| Equivalent static force, F | 1715 | N |
Checks
- Order of magnitude: a 500 kg mass at 0.35g gives F approximately 1715 N - consistent
- Displacement: 0.86 mm on a frame with k = 2 MN/m gives F = k*x = 1714 N - consistent
- Frequency: 10 Hz is in a typical range for stiff equipment frames - plausible
Interpretation
The equipment frame has a natural frequency of approximately 10 Hz. At this frequency, the floor response spectrum specifies a spectral acceleration of 0.35g. The resulting peak displacement is less than 1 mm, and the equivalent static force is approximately 1.7 kN. This force would be used to design the anchorage and assess the frame stresses.
Limitations
- The SDOF idealisation assumes the frame moves in a single mode - if the frame has significant higher modes, a multi-mode assessment is needed
- The damping ratio of 2% must be justified - if the frame is bolted, damping may be higher; if welded, lower
- The floor response spectrum must correspond to the actual mounting location of the equipment