Minimum-weight frames under subresonant harmonic excitation: Robust constraints on dynamic compliance, peak input power, and eigenfrequencies

Abstract

This work addresses minimum-weight design of undamped Euler-Bernoulli frame structures under subresonant single-frequency harmonic excitations, focusing on (robust) dynamic compliance and (robust) peak input power with ellipsoidal load uncertainty. We develop a semidefinite reformulation of robust dynamic compliance for subresonant single-frequency excitation and prove its equivalence to robust peak input power. We show that both these response measures admit an exact reformulation as a free-vibration eigenvalue constraint with design-independent mass augmentation, unifying static, dynamic, and modal requirements. Despite the nonconvex polynomial dependence on cross-sectional areas, certified bounds on global minimizers are obtained via the moment-sum-of-squares hierarchy of semidefinite relaxations. Benchmark studies on 10- and 35-segment frames corroborate the theory. For the 10-segment problem, we obtain the global optimum; for the 35-segment frame, we find high-quality locally-optimal designs that substantially improve on the best known one. We further fabricate and experimentally validate an additional design that closely matches the predictions of the model.

Figures (9)

• Figure 1: Flowchart of the global solution approach. $\overline{w}_{\min}$ is the tightest upper bound found so far.
• Figure 2: Schematic illustration of the relationships between different constraint types.
• Figure 3: $10$-segment frame problem: design domain and boundary conditions. For the fundamental free-vibration eigenvalue constraint, we additionally include the nonstructural mass shown in red, while for the dynamic compliance and peak input power scenarios the uniform-phase forces are drawn in blue.
• Figure 4: $10$-segment frame problem: (a) Best design reported in Ni2014 ($195.59$ kg), (b) global minimum weight design ($w^*=148.44$ kg) and the (c) corresponding eigenmode associated with fundamental free-vibration eigenvalue which concurrently constitutes the scaled amplitudes of worst-case displacements in the robust dynamic compliance constraint and of worst-case velocities in the robust peak input power constraint. For the optimal design, we further show (d) instantaneous force-displacement product, with peaks at the maxima corresponding to dynamic compliance values, and (e) instantaneous force-velocity product, where both maxima and minima attain the same absolute value, corresponding to peak input power. Note that the peak values in (d) and (e) are related by $d_\mathrm{R} = 2p_\mathrm{R}/\omega$, but the time instants of their occurrence differ.
• Figure 5: $35$-segment frame problem: design domain and boundary conditions. For the fundamental free-vibration eigenvalue constraint, we additionally include the nonstructural mass shown in red, while for the dynamic compliance and peak input power scenarios the uniform-phase forces are drawn in blue.

Theorems & Definitions (17)

• Definition 1: Moore-Penrose pseudoinverse Boyd2004
• Lemma 1: Schur complement Horn2005
• Lemma 2: Generalized Schur complement Horn2005
• Proposition 1: Dynamic compliance matrix inequality
• proof
• Lemma 3: Worst-case load and dynamic compliance
• proof
• Proposition 2: Peak input power matrix inequality
• proof
• Lemma 4: Displacement-velocity relation