Sensitivity analysis of the non-linear dynamic response of beams using space-time perturbation modes

The focus of the present work is directed towards the development of an effective reduced basis technique for calculating the sensitivity of the non-linear dynamic structural response of mechanical systems with respect to variations in the design variables.The proposed methodology is formulated within the context of a mixed space-time finite element method, which naturally allows the treatment of initial and boundary value problems. The time dependency of the solutions is implied in the assumed space-time modal shapes, and hence the partial differential equations of motion are directly reduced to a set of non-linear simultaneous equations of a purely algebraic nature.The independent field variables are approximated in terms of perturbations modes or path derivatives with respect to a load control parameter. These modes, extracting information about the kinematic and dynamic behavior of the structural system through the higher order derivatives of the strain and kinetic energies, are appropriate bases for non-linear dynamic problems. The sensitivity derivatives of the field variables are then approximated using a combination of perturbation modes and of their sensitivity derivatives.The resulting computational procedure offers high potential for the effective and numerically efficient sensitivity analysis of dynamic systems exhibiting periodic-in-time response. The proposed methodology is illustrated addressing non-linear beam problems subjected to harmonic loading and the results obtained are compared with those of a full finite-element model.Nomenclature (O, I _i), (is1, 2, 3) Inertial frame of origin O - (P, s _i), (is1, 2, 3) Local frame in the undeformed configuration - (Q, s _i ^*), (is1, 2, 3) Local frame in the deformed configuration - t Time - l Abscissa along the beam reference line - L Beam length - (·)s(·)/t Partial derivative with respect to time - (·)s(·)/l Partial derivative with respect to space - u Position vector of the beam reference line - r Rotation parameters - ds(u, r) Generalized displacement vector - R(r) Rotation tensor associated with r - (r) Tensor defined in equations (4) and (5) - s· Finite rotation vector - as(a _s, a _v) Conformal rotation vector - Angular velocity - ws(u, ) Generalized velocity vector - k Curvature - e Generalized strains - ps(h, l) Generalized momenta - fs(s, m) Generalized sectional stress resultants - f _es(S _e, m _e) Applied external loads - M Inertia tensor