An official website of the United States government
Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers ( $${Ra}_{cr}$$ ) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The $${Ra}_{cr}$$ value depends on the non-dimensional frequency $$\omega$$ of the boundary heat-flux modulation. Floquet theory is used to find $${Ra}_{cr}$$ for linear stability, and the energy method is used to find $${Ra}_{cr}$$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $$\omega$$ , with only the latter at large $$\omega$$ . For a given frequency, the linear stability $${Ra}_{cr}$$ is generally higher than the nonlinear stability $${Ra}_{cr}$$ , as expected. For large $$\omega$$ , $${Ra}_{cr} \omega ^{-2}$$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $$\omega$$ . The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or non-modulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.
more »
« less
Topology optimization of stability‐constrained structures with simple/multiple eigenvalues
Abstract This work focuses on topology optimization formulations with linear buckling constraints wherein eigenvalues of arbitrary multiplicities can be canonically considered. The non‐differentiability of multiple eigenvalues is addressed by a mean value function which is a symmetric polynomial of the repeated eigenvalues in each cluster. This construction offers accurate control over each cluster of eigenvalues as compared to the aggregation functions such as ‐norm and Kreisselmeier–Steinhauser (K–S) function where only approximate maximum/minimum value is available. This also avoids the two‐loop optimization procedure required by the use of directional derivatives (Seyranian et al.Struct Optim. 1994;8(4):207‐227.). The spurious buckling modes issue is handled by two approaches—one with different interpolations on the initial stiffness and geometric stiffness and another with a pseudo‐mass matrix. Using the pseudo‐mass matrix, two new optimization formulations are proposed for incorporating buckling constraints together with the standard approach employing initial stiffness and geometric stiffness as two ingredients within generalized eigenvalue frameworks. Numerical results show that all three formulations can help to improve the stability of the optimized design. In addition, post‐nonlinear stability analysis on the optimized designs reveals that a higher linear buckling threshold might not lead to a higher nonlinear critical load, especially in cases when the pre‐critical response is nonlinear.
more »
« less
- Award ID(s):
- 1762277
- PAR ID:
- 10486372
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- International Journal for Numerical Methods in Engineering
- Volume:
- 125
- Issue:
- 3
- ISSN:
- 0029-5981
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Composite laminates with negative Posson's ratios (i.e., auxetic composite laminates) were experimentally found to demonstrate a three-fold increase in buckling strength under uniaxial compression in comparison with the equivalent non-auxetic ones. To investigate whether the enhancement is genuinely due to the negative Poisson's ratio (i.e., the auxeticity) or merely caused by the concurrent change in the bending stiffness matrix as the composite layup changes, a novel monoclinic plate-based composite laminate approach is proposed, which for the first time, allows to isolate the auxeticity effect from the concurrent change of the stiffness matrix. Results provided theoretical proof that the auxeticity plays an active role in enhancing the critical buckling strength of layered composite structure. However, such a role is dynamically sensitive to elements in the bending stiffness matrix, especially the bending-twisting ratio and the anisotropy of the bending stiffness between the longitudinal and lateral directions. Insights are expected to provide guidance in exploiting negative Poisson's ratio for improving the stability of layered composite structures.more » « less
-
We present a nonlinear stability analysis of quasi-static slip in a spring-block model. The sliding interface is governed by rate-and-state friction, with an intermediate state evolution law spanning ageing and slip laws. We examine the robustness of prior results to changes in the evolution law, including the unconditional stability of the ageing law for spring stiffnesses above a critical value. We investigate two scenarios: a spring-block model with stationary and non-stationary point loading rate. In the former scenario, deviations from the ageing law lead to only conditional stability for spring stiffnesses above a critical value: finite perturbations can trigger instability, consistent with prior results for the slip law. For a given supercritical stiffness, the perturbation size required to induce instability grows as the state evolution law approaches the ageing law. By contrast, for a stationary point loading rate, there exists a maximum critical stiffness above which instability can never develop, for any perturbation size. This critical stiffness vanishes as the slip law is approached. For a limited extension of the results found for a spring-slider to continuum faults, we derive relations for an effective spring stiffness as a function of elastic moduli and a characteristic fault dimension or perturbation wavelength.more » « less
-
In this paper two nonlinear effects are investigated. One is the effect of the static stiffness nonlinearity in changing the linear dynamic natural frequency and the other is the combination of nonlinear stiffness and nonlinear inertia effects in changing the nonlinear dynamic transient response due to a change in the initial release state of the system. A theoretical model has been developed for a cantilevered thin plate with a range of length to width ratio using beam theory and considering both stiffness and inertial nonlinearities in the model. Lagrange’s equation was used to deduce nonlinear inertia and stiffness matrices for a modal representation. Some insights into how these nonlinear components influence the beam response are presented. Measurements with both a hammer test and also a release test of cantilevered thin plates were done using different configurations and tip mass values. Results from static and dynamic analysis using the linear and the nonlinear theoretical model show good agreement between theory and experiment for natural frequencies and the amplitude displacements versus time.more » « less
-
The use of Command governors (CGs) to enforce pointwise-in-time state and control constraints by minimally altering reference commands to closed-loop systems has been proposed for a range of aerospace applications. In this paper, we revisit the design of the CG and describe approaches to its implementation based directly on a bilevel (inner loop + outer loop) optimization problem in the formulation of CG. Such approaches do not require offline construction and storage of constraint admissible sets nor the use of online trajectory prediction, and hence can be beneficial in situations when the reference command is high-dimensional and constraints are nonlinear and change with time or are reconfigured online. In the case of linear systems with linear constraints, or in the case of nonlinear systems with linear constraints for which a quadratic Lyapunov function is available, the inner loop optimization problem is explicitly solvable and the bilevel optimization reduces to a single stage optimization. In other cases, a reformulation of the bilevel optimization problem into a mathematical programming problem with equilibrium constraints (MPEC) can be used to arrive at a single stage optimization problem. By applying the bilevel optimization-based CG to the classical low thrust orbital transfer problem, in which the dynamics are represented by Gauss-Variational Equations (GVEs) and the nominal controller is of Lyapunov type, we demonstrate that constraints, such as on the radius of periapsis to avoid planetary collision, on the osculating orbit eccentricity and on the thrust magnitude can be handled. Furthermore, in this case the parameters of the Lyapunov function can be simultaneously optimized online resulting in faster maneuvers.more » « less
