Quadratic mode shape components from linear finite element analysis

Points on a vibrating structure move along curved paths rather than straight lines; however, this is largely ignored in modal analysis. Applications where the curved path of motion cannot be ignored include beamlike structures in rotating systems, e.g., helicopter rotor blades, compressor and turbine blades, and even robot arms. In most aeroelastic applications the curvature of the motion is of no consequence. The flutter analysis of T-tails is one notable exception due to the steady-state trim load on the horizontal stabilizer. Modal basis buckling analyses can also be performed when taking the curved path of motion into account. The effective application of quadratic mode shape components to capture the essential kinematics has been shown by several researchers. The usual method of computing the quadratic mode shape components for general structures employs multiple nonlinear static analyses for each component. It is shown here how the quadratic mode shape components for general structures can be obtained using linear static analysis. The derivation is based on energy principles. Only one linear static load case is required for each quadratic component. The method is illustrated for truss structures and applied to nonlinear static analyses of a linear and a geometrically nonlinear structure. The modal method results are compared to finite element nonlinear static analysis results. The proposed method for calculating quadratic mode shape components produces credible results and offers several advantages over the earlier method, viz., the use of linear analysis instead of nonlinear analysis, fewer load cases per quadratic mode shape component, and user-independenc