Continuous Time-Varying Stiffness Models for the Harmonic Response of Single-Degree-of-Freedom Systems: Calibration, Resonance Transition, and Uncertainty Analysis

Abstract This study investigates the continuous representation of time-varying effective stiffness in structural systems and the influence of the selected functional form on harmonic dynamic response. Exponential, rational, and shape-parameterized generalized exponential models were calibrated using three stiffness–time values obtained from a previously published study. To ensure a comparable level of model complexity, only one rate parameter was calibrated for each model, while the shape parameter of the generalized exponential model was fixed at a representative value of b=1.50. Calibration performance was evaluated using the mean absolute error, root-mean-square error, and maximum relative error. The generalized exponential model produced the lowest calibration error, whereas the exponential model provided a favorable balance among accuracy, parsimony, and physical interpretability. The calibrated stiffness functions were incorporated into a single-degree-of-freedom equation of motion with time-varying coefficients, and the system response was computed using an adaptive-step Runge–Kutta method. Parametric analyses demonstrated that stiffness degradation does not necessarily increase the dynamic response. When the excitation frequency coincided with the initial natural frequency, stiffness degradation shifted the system away from resonance. Conversely, when the excitation frequency approached the reduced natural frequency, the maximum response increased substantially. For a total stiffness loss of 20%, the largest responses occurred near a frequency ratio of r_Ω≈0.90. Local sensitivity analysis identified the initial damping ratio as the most influential parameter governing the maximum dimensionless displacement. A Monte Carlo analysis conducted using a fixed random seed showed that the sensitivity intervals of the calibrated rate parameters widened with increasing assumed data uncertainty. The rational-model parameter βexhibited the greatest sensitivity to perturbations in the reference data. The functional form and temporal evolution of stiffness degradation should be considered when evaluating resonance transitions and dynamic demand. Nevertheless, the results are limited to theoretical scenarios calibrated using three reference values and require independent experimental validation.