A fractional-order model of viscoelastic damping in seismically isolated structures

Abstract

Viscoelastic dampers and high-damping isolation bearings are strongly frequency, amplitude, and temperature dependent, yet seismic response analyses often represent them with integer-order Kelvin-Voigt or Maxwell elements calibrated at one frequency. We present a fractional-order damping model for seismically isolated structures and calibrate it against component tests and shake-table experiments on a one-third-scale three-storey steel frame supported by laminated rubber bearings and auxiliary viscoelastic shear dampers. The damper force is represented by a fractional standard-linear-solid law with Caputo derivative order alpha, while the bearings are modelled by a smooth hysteretic element with axial-load-dependent post-yield stiffness. Model parameters were identified from sinusoidal shear tests at 0.15-2.5 Hz, 25-150 mm displacement amplitude, and 10-30 deg C, then validated without re-fitting against twelve earthquake-table runs. The fractional model reproduced component hysteresis loops with a median force error of 5.6%, compared with 14.8% for the best integer-order Maxwell model. In table tests, it predicted peak isolation displacement, roof acceleration, and damper energy dissipation with mean absolute percentage errors of 6.9%, 8.7%, and 9.4%, respectively. Errors increased for long-duration near-fault pulses because self-heating and bearing scragging were only approximated. The results support fractional damping as a compact engineering representation of broadband viscoelastic memory, provided that calibration spans the frequency and temperature range expected in design earthquakes.

Introduction

Seismic isolation reduces structural accelerations by shifting the dominant period of a building or bridge away from the high-energy range of many earthquake records. The price of this period shift is larger displacement demand in the isolation plane, so isolation systems commonly include damping from high-damping rubber, lead cores, sliding interfaces, or supplemental dampers. The role of damping is not monotonic: additional damping can reduce bearing displacement but increase floor acceleration and force transfer when it is too stiff or too velocity sensitive [10,11]. Accurate modelling of damping is therefore a design issue, not a cosmetic refinement.

Viscoelastic devices are attractive because they provide stable, distributed energy dissipation without the force discontinuities associated with some yielding or friction devices. Their constitutive behaviour is not purely viscous. Storage stiffness, loss stiffness, and equivalent damping depend on excitation frequency, strain amplitude, temperature, and previous loading. Early structural-control work established practical design procedures for viscoelastic dampers and showed that they can reduce seismic response in steel frames [5,6,7]. However, a Kelvin-Voigt or integer Maxwell element calibrated at one frequency often predicts the wrong phase angle and dissipated energy when used across a broad earthquake spectrum.

Fractional derivatives provide a compact way to represent the power-law memory that many polymers exhibit. Bagley and Torvik gave the theoretical and structural-dynamics basis for applying fractional calculus to viscoelasticity [1,2]. Koh and Kelly then applied fractional derivatives to the seismic analysis of base-isolated models, using shaking-table data to show that fractional-order descriptions can match isolation-system behaviour over a wider frequency band than simple viscous damping [3]. Related fractional Maxwell models have been used for viscous dampers [4] and for seismic analysis of structures with viscoelastic dampers [8,9].

The present study revisits this idea with a deliberately narrow question: can a fractional-order damper model calibrated from component tests predict the shake-table response of an isolated structure without table-run-specific re-fitting? The article is fictional, but the modelling choices, measurement programme, and performance claims are kept within the scale and uncertainty of published isolation and damper studies.

Experimental programme

The test structure was a one-third-scale three-storey steel moment frame with plan dimensions of 3.6 m by 2.4 m and total seismic mass of 18.4 t. It was mounted on four laminated natural-rubber isolation bearings and equipped with four replaceable viscoelastic shear dampers installed diagonally between the foundation platen and the isolation deck. The target fixed-base fundamental period was 0.46 s; after isolation it shifted to 1.72 s in low-amplitude white-noise tests. The isolation deck had mechanical stops at +/-190 mm, but no table run in the calibration set reached impact.

Each damper consisted of two 10 mm thick acrylic-based viscoelastic layers bonded between steel plates, with active shear area of 0.032 m^2 per layer. Full-device cyclic tests were performed before table installation using imposed sinusoidal displacements of 25, 75, 110, and 150 mm at frequencies of 0.15, 0.3, 0.6, 1.0, 1.6, and 2.5 Hz. The laboratory temperature was controlled at 10, 20, and 30 deg C. The tested frequency range brackets the dominant isolation-mode frequencies and the higher-frequency deformation content that appears when the superstructure rocks or higher modes participate. Low-temperature tests were included because viscoelastic damper properties can change sharply near service-temperature limits [17].

The frame was then subjected to twelve horizontal shake-table runs: four far-field records, four pulse-like near-fault records, and four broadband synthetic records matched to the site-compatible design spectrum. Intensities were scaled to 25%, 50%, 75%, and 100% of the design basis earthquake for selected records, with peak table accelerations from 0.12 g to 0.54 g. Instrumentation included table and floor accelerometers, string potentiometers across the isolation plane, load cells in each damper, thermocouples embedded in the viscoelastic layers, and vertical load cells under two bearings. The table runs were ordered from low to high intensity, with at least 40 min between high-intensity runs to limit temperature carry-over.

Fractional damping formulation

The viscoelastic damper was represented by a fractional standard-linear-solid law. In scalar shear form, F(t) + tau^alpha D_t^alpha F(t) = k_0 x(t) + k_inf tau^alpha D_t^alpha x(t), where F is damper force, x is relative displacement, tau is a relaxation timescale, alpha is the fractional order between zero and one, and D_t^alpha is a Caputo derivative. The limiting stiffnesses k_0 and k_inf describe low- and high-frequency response. For harmonic input, the model gives a complex stiffness with both storage and loss terms varying smoothly with frequency, which is the behaviour observed in polymeric dampers and high-damping elastomers [1,2,6].

Temperature dependence was represented by a single horizontal shift factor applied to tau, using a log-linear approximation over the tested 10-30 deg C range. We did not use a full Williams-Landel-Ferry fit because the data span was too narrow to identify it robustly. Amplitude dependence was handled by allowing k_0 and k_inf to vary weakly with shear strain through a saturating scalar factor. The fractional order alpha was held constant for each damper type because preliminary fits showed that allowing alpha to vary with amplitude improved component-loop error by less than 1% but degraded extrapolation to earthquake records.

The rubber bearings were not represented by the fractional law. Instead, their lateral force was modelled with a smooth hysteretic element similar in purpose to established elastomeric-isolation bearing models [13,16]. The model includes pre-yield stiffness, post-yield stiffness, characteristic strength, axial-load sensitivity, and a Mullins-type scragging factor for the first large-amplitude excursion. This separation is intentional. The fractional element is used for broadband viscoelastic shear in the supplemental dampers; the bearings retain a conventional hysteretic description because their response includes geometric stiffness, axial-load effects, and cyclic softening not captured by a single linear fractional element.

The fractional derivative was integrated by a history-recursive quadrature with adaptive truncation of weights whose contribution fell below 0.2% of the current force. This reduced the cost of nonlinear response-history analysis while preserving the long-memory behaviour over the 30-70 s duration of the earthquake records. A direct convolution implementation was used as a benchmark for three runs and changed peak displacement by less than 0.6%.

Parameter identification

Damper parameters were identified in the frequency domain from the component tests. For each temperature and amplitude, the measured force and displacement histories were converted into first-harmonic storage stiffness, loss stiffness, and equivalent viscous damping ratio. The objective function combined errors in storage stiffness, loss stiffness, and loop area, with higher weight assigned to the 0.3-1.6 Hz band because it contains the isolation-mode response. Parameters were estimated by differential evolution followed by local least-squares refinement. Confidence intervals were obtained by bootstrapping the cyclic-test repetitions rather than by assuming independent pointwise force errors.

The identified fractional order was alpha = 0.37 +/- 0.03 at the reference temperature of 20 deg C. The relaxation time tau was 0.21 s at small amplitude and shifted to 0.28 s at the largest amplitude, reflecting mild softening under high shear strain. The ratio k_inf/k_0 was 3.6 +/- 0.4. These values produced the observed phase angle plateau between 0.5 and 1.8 Hz. The best integer-order Maxwell model could match either the 0.3 Hz loop area or the 1.6 Hz phase angle, but not both simultaneously.

Bearing parameters were calibrated from low-amplitude white-noise tests and from isolated bearing shear tests conducted at the design vertical load. The axial-load sensitivity was modest in the table set because overturning demands were low, but it was retained because axial-load effects can be important in rubber and lead-rubber bearings [15,16]. The final structural model therefore combines a fractional damper law, conventional hysteretic bearings, elastic frame members, and measured table acceleration input.

Shake-table validation

The fractional model was validated against the twelve table runs without changing damper parameters. Across all records, the mean absolute percentage error was 6.9% for peak isolation displacement, 8.7% for peak roof acceleration, 7.8% for peak base shear, and 9.4% for total damper energy dissipation. The best integer-order Maxwell model, calibrated on the same component data, gave corresponding errors of 13.5%, 16.9%, 14.1%, and 22.3%. A Rayleigh-damped linear isolation model performed worse for damper force and energy because it could not reproduce the measured force-displacement phase lag.

The improvement was clearest in records with broadband velocity content. For a synthetic spectrum-compatible run with peak table acceleration 0.38 g, the measured peak isolation displacement was 132 mm. The fractional model predicted 127 mm, while the integer Maxwell model predicted 111 mm because its high-frequency calibration over-stiffened the damper at the isolation period. The same run produced measured peak roof acceleration of 0.31 g; the fractional model predicted 0.33 g. The force loops retained their rounded shape and asymmetric unloading slope, features that a single-frequency equivalent damping ratio cannot reproduce.

Near-fault pulse records exposed the model limitations. In the strongest pulse-like run, the embedded thermocouples showed a 7.5 deg C damper temperature rise during the first 18 s. The fractional model, using the measured initial temperature and a simplified heat correction, under-predicted damper softening after the pulse and over-predicted residual isolation displacement by 14%. Lead-rubber bearing studies have shown that heating and loading history can materially change isolation-device behaviour [15], and the present viscoelastic dampers show a milder but measurable analogue. A fully coupled thermo-viscoelastic model would be required for long-duration or repeated pulses.

Clustered sensitivity analysis indicated that alpha and tau controlled most uncertainty in roof acceleration, while k_0 and bearing post-yield stiffness controlled isolation displacement. The temperature shift factor was unimportant in moderate far-field runs but became the second-largest uncertainty source in the two strongest records. This reinforces the need to calibrate fractional models over the service temperature range rather than treating temperature as a secondary correction.

Discussion

The practical value of the fractional model is that it describes broadband viscoelastic memory with only a small number of parameters. It is not more accurate because it is mathematically exotic; it is more accurate because the damper material does not have one relaxation time. Integer-order models can approximate this behaviour by using many Maxwell branches, but that increases parameter correlation and can make earthquake-history analysis cumbersome. A fractional element compresses the same power-law relaxation into alpha and tau, which are identifiable from standard harmonic tests when the frequency range is chosen carefully.

The study also shows why component calibration alone is not enough. A model that fits damper loops in a test machine can still fail in a structure if bearing flexibility, frame higher modes, damper temperature rise, or table-control artefacts shift the deformation history. The present validation uses shake-table data for that reason. This follows the logic of earlier isolation and supplemental-damping experiments, where model credibility came from comparing both device forces and structural responses rather than from matching one equivalent damping ratio [3,7,12,14].

For design practice, the model should be used with restraint. Fractional damping is most useful when the expected response spans a broad frequency band or when temperature variation is important. For a narrowband isolation system at a well-controlled temperature, a calibrated integer model may be adequate and easier to communicate in code checks. Conversely, using a fractional law outside its calibration envelope is risky. The derivative order is a phenomenological representation of material memory, not a universal constant of the polymer.

Several limitations remain. The test structure was planar, with limited torsion and no pounding. The bearings were natural-rubber devices rather than lead-rubber bearings, friction-pendulum bearings, or high-damping rubber bearings with strong scragging. The damper temperature range was modest. Ageing, ultraviolet exposure, construction tolerances, and long-term creep were not considered. The table records were unidirectional, so bidirectional shear interaction in the dampers was outside the study. These limitations should be addressed before transferring the identified parameters to full-scale buildings.

Conclusion

We developed and validated a fractional-order model for viscoelastic dampers in a seismically isolated structure. Calibrated from component tests, the model predicted shake-table displacement, acceleration, base shear, and energy dissipation more accurately than integer-order Maxwell and equivalent-viscous alternatives. The fitted derivative order alpha = 0.37 +/- 0.03 reflects broadband polymer memory over the tested frequency range, while the relaxation time and stiffness parameters capture temperature and amplitude dependence.

The central conclusion is methodological: fractional models are credible for seismic isolation when they are calibrated over the expected frequency, amplitude, and temperature range and then validated at the structural level. They should not be used as black-box curve fits. Future work should couple the fractional law to heat generation, extend the validation to bidirectional excitation, and compare compact fractional models against multi-branch hereditary models in full-scale isolation-system design.

Data and code availability

Processed force-displacement histories, shake-table acceleration records, identified model parameters, response-history outputs, and analysis scripts are included in the supplementary archive. The fractional-derivative routines were implemented in MATLAB R2022a and cross-checked against a Python 3.10 implementation using NumPy 1.23 and SciPy 1.9. Raw high-rate table-control files are stored separately because of size but are represented by downsampled acceleration histories in the archive.