Nonlinear Reduced-Order Simulation Using Stress-Free and Pre-Stressed Modal Bases

Abstract

A study is undertaken to determine the advantages and disadvantages associated with application of stress-free and pre-stressed modal bases in a reduced-order finite-elementbased nonlinear simulation. A planar beam is chosen as an application example and its response due to combined thermal and random pressure loadings is examined. Combinations of two random pressure levels and two thermal conditions are investigated. The latter consists of an ambient temperature condition and an elevated temperature condition in the post-buckled regime. It is found that stress-free normal modes establish a broadly applicable modal basis yielding accurate results for all the loading regimes considered. In contrast, the range of applicability for a thermally pre-stressed modal basis is found to be limited. The behavior is explained by scrutinizing the coupling found in the linear stiffness and the effect this coupling has on the structural response characteristics under the range of loading conditions considered.

Improved Solution of Nonlinear Reduced-Order Models for Static and Dynamic Response Prediction

“…(20). Which for our two-mode example is ∇ ∇ ∇θ θ θ (q q q) = 2B 11 1 q 1 +B 12 1 q 2 +3A 111 1 q 1 q 1 +2A 112 1 q 1 q 2 +A 122 1 q 2 q 2 B 12 1 q 1 +2B 22 1 q 2 +A 112 1 q 1 q 1 +2A 122 1 q 1 q 2 +3A 222 1 q 2 q 2 2B 11 2 q 1 +B 12 2 q 2 +3A 111 2 q 1 q 1 +2A 112 2 q 1 q 2 +A 122 2 q 2 q 2 B 12 2 q 1 + 2B 22 2 q 2 +A 112 2 q 1 q 1 +2A 122 2 q 1 q 2 +3A 222 2 q 2 q 2 (A-4) which can be easily verified using Eq. (A-1) and the definition of the gradient of the two-mode model ∇ ∇ ∇θ θ θ (q q q) = ∂ θ θ θ ∂ q 1 ∂ θ θ θ ∂ q 2 .…”

“…For example, B 12 1 is assigned to two separate entries of the T 1 array (see Eq. (A-6)) and that B 11 1 is assigned twice into one location producing a factor of 2. The programming assignments for the θ θ θ 2 2 2 matrix are T 2(r, index( j, k), i) += A i jk r , (A-11)…”

“…The work was revisited with a study comparing the accuracy of cold and hot modes approaches on the same beam example. 9,11 The "hot" modes approach uses normal modes of the pre-stressed structure (at temperature) as the basis. Hot modes can be quite different from cold modes, especially if the change in temperature is large.…”