Explicit sensitivity analysis of spectral submanifolds of mechanical systems

Model reduction via spectral submanifolds (SSMs) has displayed benefits such as the facilitation of nonlinear analysis and significant speed-up gains. One needs the sensitivity of the SSM-based model reduction to carry over these benefits to the settings of optimal design, modal updating, and uncertainty quantification of high-dimensional nonlinear mechanical systems. Here, we construct explicit third-order, SSM-based model reduction for general mechanical systems. We further derive the explicit sensitivity of the third-order SSM-based reduction using direct and adjoint methods. We demonstrate the effectiveness of the derived explicit sensitivity via a few examples with increasing complexity. We also show that the obtained sensitivity can be used to effectively construct perturbed SSMs, backbone curves, and forced response curves.