High-throughput characterization of snap-through stability boundaries of bistable beams in a programmable rotating platform

TL;DR

This work addresses dynamic stability in bistable beams by mapping snap-through thresholds in the phase space of angular velocity and acceleration, $$(\Omega,\dot{\Omega})$$, using a high-throughput rotating platform that tests six beams in parallel under programmable unsteady loading. The authors demonstrate that stability boundaries can be empirically captured by parabolic functions $\dot{\Omega}(\Omega)=C_0+C_2\Omega^2$, and they show how geometry and boundary conditions–notably tilt $\theta$, thickness $h$, pre-compression $\varepsilon$, and clamp angles–systematically tune the offset $C_0$ and curvature $C_2$. A modal analysis reveals mode-switching phenomena in antisymmetric clamp configurations, where the deformation pathway shifts between symmetric and antisymmetric modes depending on loading, explaining observed non-parabolic features. The resulting framework enables scalable, data-rich exploration of nonlinear dynamic instabilities with potential applications in bistability-based metamaterials, mechanical memory, and sensing systems, and the accompanying large dataset supports data-driven design and modeling efforts.

Abstract

We introduce a high-throughput platform that enables simultaneous, parallel testing of six bistable beams via programmable motion of a rotating disk. By prescribing harmonic angular dynamics, the platform explores the phase space of angular velocity and acceleration $(Ω,\,\dotΩ)$, producing continuously varying centrifugal and Euler force fields that act as tunable body forces in our specimens. Image processing extracts beam kinematics with sub-pixel accuracy, enabling precise identification of snap-through events. By testing six beams in parallel, the platform allows systematic variation of beam thickness, pre-compression, tilt angle, and clamp orientations across 65 distinct configurations, generating 23,400 individual experiments. We construct stability boundaries and quantitatively parameterize them as parabolic functions, characterized by a vertical offset and a curvature parameter. Tilt angle provides the most robust mechanism for tuning the curvature parameter, while beam thickness and pre-compression modulate vertical offset. Modal decomposition analysis reveals that antisymmetric clamp configurations can trigger mode switching, in which competing geometric and inertial effects drive transitions through different deformation pathways. Our work establishes a scalable experimental framework for high-throughput characterization of dynamic nonlinear instabilities in mechanics. The complete experimental dataset is made publicly available to support data-driven design and machine learning models for nonlinear mechanics with applications to bistability-based metamaterials, mechanical memory, and electronics-free sensing systems.

Figures (8)

• Figure 1: High-throughput characterization of stability boundaries via programmable unsteady rotation. A. Schematic of the rotating platform showing centrifugal $\mathbf{f}_{\Omega}$ (purple) and Euler $\mathbf{f}_{E}$ (pink) force distributions acting on a bistable beam. B. Imposed angular velocity profiles $\Omega(t)$ of varying amplitude produce C. time-varying forces scaling as $\mathbf{f}_{\Omega}\sim \Omega^2$ (centrifugal) and $\mathbf{f}_{E}\sim \dot{\Omega}$ (Euler). D. Snap-through transitions between stable states ($s=0$ and $s=1$) occur when loading trajectories cross stability thresholds. E. Varying loading profiles trace elliptical orbits in the $(\Omega,\,\dot{\Omega})$ phase space, serving to map the stability boundaries (shaded regions).
• Figure 2: Experimental apparatus. A. Side- and B. top-view schematics. C. Photograph of the apparatus. A high-torque motor (1) rotates a disk (2) with grooves for up to six sample holders (3). A camera (4) records the disk from above, illuminated by LED panels (5), all inside a safety enclosure (6). D. Modular sample holders enable independent control of (D$_1$.) pre-compression $\varepsilon$ and radial position $R$, (D$_2$.) tilt angle $\theta$, and (D$_3$.) clamp orientations $\varphi_i$, $\varphi_e$.
• Figure 3: Image processing pipeline.A. Representative raw experimental image. White circles serve as fiducial markers, and their centroids define the instantaneous rotation angle $\psi$. B. Schematic of the detection and spline-fitting workflow. A neural network locates the clamp corners (yellow squares), from which the beam's clamp-to-clamp axis (dashed vertical line) and the normal vectors (green arrows) to the internal and external clamps, $\mathbf{n}_i$ and $\mathbf{n}_e$, are established. Three search lines, perpendicular to the clamp-to-clamp axis, detect intensity maxima corresponding to the spline interpolation nodes (green circles). C. Time-sequence montage of fitted splines (red solid lines) overlaid on raw beam images exhibits excellent agreement.
• Figure 4: Definition of snap-through events.A--B. From top to bottom, measurements of angular velocity $\Omega(t)$, angular acceleration $\dot{\Omega}(t)$, midspan displacement $\delta(t)$, and midspan velocity $\dot{\delta}(t)$ of a representative bistable beam ($h=1.85\,\mathrm{mm}$, $\varepsilon=0.035$, $\theta=5^\circ$) for two harmonic loadings of varying (A) amplitude ($f=1.5\,\mathrm{Hz}$ with black lines: $a=32.5\,\mathrm{rad\,s}^{-1}$; blue lines: $a=65.0\,\mathrm{rad\,s}^{-1}$), and (B) frequency ($a=32.5\,\mathrm{rad\,s}^{-1}$ with black lines: $f=1.5\,\mathrm{Hz}$; green lines: $f=3.0\,\mathrm{Hz}$). Snap-through events (triangles) are defined as the mean value between the point where $\delta=0$ (crosses) and when the beam midspan reaches velocity $|\dot{\delta}| = 0.15\cdot|\dot{\delta}_{\mathrm{Max}}|$ (diamonds). The snapping direction is indicated by triangle orientation ($s=0{\rightarrow} s=1$: up-triangle; $s=1{\rightarrow} s=0$: down-triangle). C. Representation of the three harmonic drives from panels A (black and blue lines) and B (black and green lines) in the $(\Omega,\,\dot{\Omega})$ phase space. Snapping points are overlaid for each loading orbit.
• Figure 5: Statistical construction of stability boundaries in the $(\Omega,\,\dot{\Omega})$ phase space.A. A set of loading orbits (light gray lines) generates a dense, structured cloud of snap-through events (pink triangles) for a representative specimen ($h=1.85\,\mathrm{mm}$, $\varepsilon=0.035$, $\theta=5^\circ$). B. Snap-through data are mirrored about $\Omega=0$ to construct the full boundary. Parabolic functions (solid and dashed black lines) are fitted to the binned data (black circles with error bars) to define a 95% confidence region for the stability boundary.