Dynamic versus quasi-static response of a cantilevered beam rotated harmonically

Abstract

We investigate a cantilevered elastic beam subjected to harmonic rotational motion. In the rotating frame, the beam experiences centrifugal and Euler fictitious forces, with negligible Coriolis effects. We validate a reduced-order \textit{elastica} model through precision experiments on slender beams rotating with a controlled sinusoidal angular velocity. Systematically exploring the parameter space, we identify regimes where inertial effects are negligible, enabling a quasi-static treatment despite harmonic driving. We characterize the transition to dynamic response using two dimensionless parameters, the Euler and centrifugal numbers, which compare centrifugal and Euler forces to bending forces. Counterintuitively, the quasi-static regime expands as rotational speed increases: faster rotation produces less dynamic response. The critical Euler number separating these regimes remains constant at low centrifugal numbers but follows square-root scaling at higher rotation rates, a transition driven by centrifugal stiffening. Our results establish the conditions under which quasi-static approximations remain valid for rotating flexible beams under harmonic driving.

Figures (10)

• Figure 1: Schematic of a rotating cantilevered beam. (a) The slender beam (arclength $L$) is clamped at a distance $R_o$ from the center of rotation, $O$, with the system rotating about $\mathbf{e}_z$ at angular velocity $\Omega(t)$ and acceleration $\dot{\Omega}(t)$. Inset: Infinitesimal beam element at arclength $s$ showing fictitious body force $\mathbf{f}_B=\mathbf{f}_c+\mathbf{f}_\omega+\mathbf{f}_e$; cf. Eqs. (\ref{['eq:ForceC']}-\ref{['eq:ForceE']}), internal tension $\mathbf{n}$ and moment $\mathbf{m}_z$. (b) Representative sinusoidal angular velocity profile $\Omega(t)$ with mean $\Omega_0 = 15\,\text{rad}/\text{s}$, amplitude $a = 1\,\text{rad}/\text{s}$, and frequency $f_d = 7\,\text{Hz}$. (c) Corresponding angular acceleration $\dot\Omega$.
• Figure 2: Experimental apparatus. (a) Photograph of our setup: torque-controlled motor ①, rigid disk ②, cantilever beam ③, acrylic top plate ④ and bottom dish ⑤, and camera ⑥. (b) Top view showing fixed ⑦ and rotating ⑧ markers for image stabilization. Inset: Zoom-in of the sample. (c) Rotating beam frames with overlaid symbols at midspan position: experiments (magenta diamonds) and simulations (green circles) for $\Omega_0=15\,\text{rad}/\text{s}$ and $f_d = 7\, \text{Hz}$.
• Figure 3: Experimental validation of the numerical model (parameters in \ref{['table:params']}). (a) Resonance curve: RMS of dimensionless midspan displacement $\text{RMS}(y_m)$ versus drive frequency $f_d$ at mean angular velocity $\Omega_0 = 5\,\text{rad}/\text{s}$. Each experimental data point is the average of five repetitions; error bars are within the symbol size. (b) Time series of dimensionless midspan displacement, $y_m(\tau)$, at $f_d = 7\,\text{Hz}$. (c) Power spectrum of the time series in (b), showing resonance at 7 Hz. Symbols/lines: (a) experiments (solid diamonds), simulations (open circles); (b, c) experiments (dashed), simulations (solid).
• Figure 4: Comparison of dynamical and quasi-static simulations for the reference beam. (a, b) $\text{RMS}(y_m)$ versus normalized drive frequency for $\Omega_0 = 10\,\text{rad/s}$ and $100\,\text{rad/s}$, respectively. Circles: dynamical ($\text{RMS}_{\text{Dyn.}}$); stars: quasi-static ($\text{RMS}_{\text{QS}}$). (c, d) Relative RMS difference, $\Delta \text{RMS}$, for the same conditions. Vertical lines: critical normalized drive frequency $\tilde{f}_d^\star$ (dotted); normalized resonance frequency $\tilde{f}_0$ (dashed). Horizontal line in (c, d): threshold $\Delta \text{RMS}_\text{th}=0.1$. Open triangle: first point where $\Delta \text{RMS}\geq \Delta \text{RMS}_\text{th}$. Shaded region: quasi-static response; unshaded: dynamic response.
• Figure 5: Phase diagram of quasi-static and dynamic regions for the reference beam. (a) Critical maximal Euler number $\mathcal{E}^\star$ (triangles) versus maximal centrifugal number $\mathcal{C}_{\text{max}}$ separates quasi-static (shaded) and dynamic (unshaded) regions. (b, c) Time series of dimensionless midspan displacement: dynamical (solid) and quasi-static (dashed) for $\mathcal{C}_{\text{max}} \approx 50$ with (b) $\mathcal{E}_{\text{max}} \approx 2$ (above $\mathcal{E}^\star$) and (c) $\mathcal{E}_{\text{max}} \approx 1$ (below $\mathcal{E}^\star$).