Equivalent-SDOF iteration for base isolator response

Four-bearing isolation system

The problem considers a rigid superstructure of mass \(m = 200\times 10^3\,\mathrm{kg}\) supported by 4 high-damping rubber (HDR) bearings. The hysteresis curves (Figure 1 in the problem) describe the cyclic behavior of one bearing; shear strain is the horizontal displacement divided by the total rubber thickness \(H = 3.25\,\mathrm{in}\) for that bearing.

For the equivalent SDOF, the four bearings act in parallel, so the system storage stiffness is \(k = 4\times K_1\), where \(K_1\) is the per-bearing storage stiffness from the hysteresis curves.

The effective damping ratio \(\zeta_{\mathrm{eff}}\) from the curves is preserved at the system level for the following reason. Each hysteresis test is performed with axial load \(M/4\) on that bearing (one quarter of the total isolated mass). The per-bearing equivalent viscous coefficient \(c_1\) inferred from \(\zeta_{\mathrm{eff}}\) therefore incorporates this factor (it is the \(c\) that gives \(\zeta_{\mathrm{eff}}\) for a SDOF with stiffness \(K_1\) and mass \(M/4\)). When the four bearings are in parallel, \(c_{\mathrm{system}} = 4 c_1\) and \(k_{\mathrm{system}} = 4 K_1\). In the ratio \(\zeta = c\big/(2\sqrt{km}\big)\), the factor 4 in \(c\) and the factor 4 in \(k\) cancel, so \(\zeta_{\mathrm{system}} = \zeta_{\mathrm{eff}}\). We therefore use \(c = 2\zeta_{\mathrm{eff}}\sqrt{km}\) with the system stiffness \(k = 4 K_1\) and full mass \(m\). All iteration and response calculations use this system-level \(k\) and \(c\).

Solution methods overview

In relative coordinates \(u(t)\), the isolated mass is modeled as a single degree of freedom (SDOF) with equation of motion \[ m\,u''(t) + c\,u'(t) + k\,u(t) = -m\,\ddot{u}_g(t), \] where \(m\) is the effective mass, \(k = 4 K_1\) is the system storage stiffness (four times the per-bearing \(K_1\) from the hysteresis curves), \(c\) is a viscous coefficient chosen to reproduce \(\zeta_{\mathrm{eff}}\), and \(\ddot{u}_g(t)\) is the ground acceleration record.

The dashboard compares two ways of solving this equation:

1. Time-domain Newmark method. The continuous equation is discretized on a uniform time grid with step \(\Delta t\). Newmark's constant-average-acceleration scheme (\(\beta = 1/4\), \(\gamma = 1/2\)) is used to advance displacement, velocity, and acceleration from time step \(i\) to \(i+1\), using the standard effective-stiffness and effective-load formulas. This is implemented in NumPy in the function newmark_sdof, which stores \(u_i\), \(u'_i\), and \(u''_i\) in arrays and loops once over all time steps.
2. Frequency-domain FFT method. The same SDOF equation is solved by taking the discrete Fourier transform of the ground motion with numpy.fft.fft, forming the transfer function \[ H(\omega) = -\frac{m}{k + i c \omega - m\omega^2}, \] and computing \(U(\omega) = H(\omega)\,\ddot{U}_g(\omega)\). Velocity and absolute acceleration follow from spectral derivatives: \(V(\omega) = i\omega U(\omega)\), \(\ddot{U}_{\mathrm{rel}}(\omega) = -\omega^2 U(\omega)\), \(\ddot{U}_{\mathrm{abs}}(\omega) = \ddot{U}_{\mathrm{rel}}(\omega) + \ddot{U}_g(\omega)\). An inverse FFT (numpy.fft.ifft) with zero-padding is used to return \(u(t)\), \(u'(t)\), and \(\ddot{u}_{\mathrm{abs}}(t)\) in the time domain.

Both methods use identical SDOF parameters \((m, k, c)\), so differences between the blue (Newmark) and red (FFT) traces in the plots quantify only numerical effects, not changes in the underlying physical model.

Key metrics after convergence

Iteration history

Each iteration proceeds as a fixed-point update:

1. Start from the current peak displacement estimate \(U_{\max}\) and compute the corresponding shear strain using the known rubber thickness \(H = 3.25~\mathrm{in}\): \(\gamma = 100\,U_{\max}/H\).
2. Interpolate the per-bearing storage stiffness \(K_1(\gamma)\) and effective damping ratio \(\zeta_{\mathrm{eff}}(\gamma)\) from the isolator hysteresis library.
3. Set system stiffness \(k = 4 K_1\) (four bearings in parallel), form \(c = 2\,\zeta_{\mathrm{eff}}\,\omega_n m\) with \(\omega_n = \sqrt{k/m}\), and solve \(m u'' + c u' + k u = -m \ddot{u}_g(t)\) using Newmark's method in relative coordinates.
4. Extract a new peak displacement from the response, \(U_{\max}^{\text{new}} = \max_t |u(t)|\), and repeat until the relative change \(|U_{\max}^{\text{new}} - U_{\max}|/U_{\max}\) falls below the convergence tolerance.

The plot below shows how the assumed shear strain \(\gamma\) and the associated damping ratio \(\zeta_{\mathrm{eff}}\) evolve and settle as the iteration converges.

Input ground motion

The equivalent-SDOF system is driven by the ground acceleration time history shown below. When a CSV file is present in input_ground_motion/, it is used directly; otherwise a synthetic pulse-like motion is generated for demonstration purposes.

Equivalent-SDOF response histories

The calibrated equivalent-SDOF model is then solved using both methods described above. Newmark's scheme provides the reference time histories in blue, while the FFT-based convolution produces the overlaid red curves. For each quantity the two traces share the same axes, allowing direct visual comparison of displacement, velocity, and absolute acceleration of the isolated mass under the Kobe ground motion.

Isolated vs fixed-base absolute acceleration

To quantify the benefit of isolation, we compare the calibrated isolated system with a fixed-base SDOF having period \(T = 0.30~\mathrm{s}\) and damping ratio \(\zeta = 0.05\). Both are driven by the same ground acceleration record.

For this motion, the isolated mass reaches a peak absolute acceleration of approximately 0.14 g, while the fixed-base system reaches about 0.58 g. In other words, the isolated system experiences roughly 0.24 times the peak acceleration of the fixed-base structure (a reduction of about 76%).

The price paid for this reduction in acceleration is larger relative displacement in the isolator levels (visible in the response histories above), which is precisely what base isolation is designed to trade: less force and acceleration in the superstructure, more controlled motion in the isolation layer.