Non-linear Stiffness Estimation using Two-Tone Excitation

Abstract

This thesis studies how to estimate cubic non-linear stiffness in dynamic systems using two-tone excitation. When a structure with cubic stiffness is driven at two frequencies that lie close to each other, it produces inter modulation peaks at sum and difference frequencies. By relating the amplitudes at these frequencies to the system parameters, the cubic stiffness can in principle be extracted without full-scale numerical modelling or fits with convoluted functions. An analytic relation between the amplitude at an excitation tone (A1) and the first-order sum frequency amplitude (A3) is derived using the harmonic balancing method, expressed in terms of the resonance frequency ω0, detuning δ, and cubic stiffness γ. The derivation uses a Ansatz with the corresponding sum frequencies and orders terms to obtain a analytic relation. This analytic curve is then validated in two ways: (i) by numerically solving the harmonic-balance equations, and (ii) by direct time-domain integration (RK45) with amplitude ex traction at the relevant frequencies. Symbolic regression is also used to fit compact additional formulas where the analytic relation loses accuracy. Experiments are carried out on Silicon-Nitride beams using a Polytec MSA-500 LDV and a Moku: Lab for signal generation and spectrum analysis. The main result is that the analytic A1–A3 relation agrees well with harmonic-balance and numeric integration simulations, but only within a certain region: low to moderate drive and sufficiently separated tones so that higher-order products remain small. Outside this region, solution branching and neglected terms reduce accuracy. In measurements, the expected near-linear relation between the two drive amplitudes (A1, A2) is observed, but the measured A1–A3 curves rise more steeply than predicted. This could be caused by additional effects such as higher-order non-linearities, damping, or resonance shifts under strong drive. The work delivers: (1) an analytic and numerical framework that maps where two-tone inter modulation can identify cubic stiffness, and (2) a robust experimental workflow. It also outlines future improvements: adding (non-linear) damping and quadratic stiffness to the model, lock-in based detection, better input conditioning, and resonance tracking to improve the differences between theory and experiments.