CE223 – Friction Pendulum (FPS)

CE223 – Friction Pendulum (FPS) — Kobe & Sylmar

This report compares a rigid‑superstructure friction pendulum (FPS) idealization under two horizontal records: Kobe KBU090 and Sylmar 360°. Part (a) solves a nonlinear bilinear regularization with Newmark and return mapping; part (b) uses an equivalent viscously damped linear model; part (c) reports floor spectra for linear nonstructural oscillators ($\zeta_p=2\%$) driven by absolute deck acceleration $u_t(t)=\ddot u(t)+\ddot u_g(t)$.

Model definition

The rigid-superstructure FPS system is modeled as one SDOF in relative coordinates, and the governing equation is:

$$M\ddot u(t) + F_r(u,\dot u) = -M\ddot u_g(t), \quad F_r \approx K_p u + Q\,\mathrm{sgn}(\dot u).$$

To avoid force discontinuity at velocity reversals, a bilinear regularization with kinematic hardening and very small yield displacement is used. The matching relations are:

$$F_y = \mu Mg, \quad k = \frac{F_y}{u_y}, \quad K_p=\frac{Mg}{R}, \quad H=\frac{kK_p}{k-K_p}.$$

Inputs: $M=1.470e+06$ kg, $R=1.00$ m, $\mu=0.030$, $u_y=0.0300$ mm, $n_b=15$.

Computed: $K_p=14.416$ MN/m, $Q=0.432$ MN, $k=14.416$ GN/m, $H=14.430$ MN/m.

Numerical procedure

Part (a) Nonlinear Bilinear: At each time step $t_{n+1}$, the algorithm performs Newton iterations on displacement $u_{n+1}$. For each iterate it computes trial force $F^{tr}=k(u-u_n^p)$, trial yield check $f^{tr}=|F^{tr}-q_n|-F_y$, then applies either elastic update ($f^{tr}\le 0$) or plastic correction ($f^{tr}>0$) with return mapping. The algorithmic tangent is $k$ (elastic) or $kH/(k+H)$ (plastic). Convergence is checked on both residual and displacement correction.

$$\Delta\gamma = \frac{f^{tr}}{k+H}, \quad F_{n+1} = F^{tr} - k\Delta\gamma\,\mathrm{sgn}\!\left(F^{tr}-q_n\right), \quad q_{n+1} = q_n + H\Delta\gamma\,\mathrm{sgn}\!\left(F^{tr}-q_n\right).$$

Part (b) Equivalent Linear: The effective stiffness and damping are iteratively updated from response amplitude. At each outer iteration, linear Newmark is solved with updated $(k_{eff}, c_{eff})$; then $u_{max}$ is re-estimated and properties are updated until stable.

$$k_{eff} = K_p + \frac{Q}{u_{max}}, \qquad \zeta_{eff} = \frac{2Q}{\pi k_{eff}u_{max}}, \qquad c_{eff} = 2\zeta_{eff}\sqrt{M k_{eff}}.$$

Part (c) Floor Spectra: The input to NSC oscillators is the absolute floor acceleration from parts (a) and (b). For each $T_p$ with fixed $\zeta_p=2\%$, an SDOF is solved and the peak absolute component acceleration is extracted.

$$\ddot z + 2\zeta_p\omega_p\dot z + \omega_p^2 z = -u_t(t), \qquad a_{p,abs}(t)=\ddot z(t)+u_t(t), \qquad S_{a,floor}(T_p)=\max|a_{p,abs}(t)|.$$

Peak response comparison

Motion	Model	Peak \|u\| [mm]	Peak \|u̇\| [m/s]	Peak \|F\| [MN]	Peak \|üt\| [g]
Kobe	Nonlinear	111.997	0.477	2.047	0.142
Kobe	Equivalent Linear	106.023	0.487	2.007	0.139
Sylmar	Nonlinear	658.157	2.085	9.920	0.688
Sylmar	Equivalent Linear	678.572	2.171	10.226	0.709

Part (a) — Nonlinear bilinear response

For each ground motion, the time histories of isolation displacement, isolation velocity, restoring force, absolute structural acceleration, and ground acceleration are presented together with the corresponding force–displacement relation.

Part (b) — Equivalent linear response

These plots use the same layout as part (a) for direct comparison. Differences reflect the limitations of a single linearized pair $(k_{eff},c_{eff})$ under transient loading and unloading.

Part (c) — Floor spectra ($\zeta_p = 2\%$)

Each NSC is a linear 2% oscillator; the only difference between curves is the floor motion $u_t(t)=\ddot u(t)+\ddot u_g(t)$ from part (a) versus part (b). Oscillator periods $T_p$ are spaced uniformly in log-space from 0.01 s to 10.0 s (a logarithmic period axis cannot include $T_p=0$). The dotted reference marks $f_p=2$ Hz ($T_p=0.5$ s), a band often used for stiff acceleration-sensitive components.

Note for $f_p$ above 2 Hz: The equivalent-linear isolation model does not guarantee a larger floor spectrum than the nonlinear one. Bilinear hysteresis and reversals can inject high-frequency content into the true $u_t(t)$, while the fitted viscously damped oscillator tends to produce a smoother deck motion with less energy above a few Hz. The NSC spectrum then often lies above the equivalent-linear curve in that band even though every oscillator is linear.