Monitoring of structural deformations/vibrations superposed upon finite pre-deformations

The signal of a continuously distributed strain-type sensor is proportional to the weighted integral of the local strains. Choosing the weight functions as a solution to a particular auxiliary problem of statics, we design a sensor, whose signal in the geometrically linear setting equals to a desired kinematic entity. In the present paper we study the design of such sensors in a geometrically nonlinear regime. In particular, we first discuss the implications of the type of the Lagrangian strain tensor, which is integrated by the sensor, on the problem of the design of the sensor. Then, we formulate the method of designing the sensor for the measurement of a kinematic entity in configurations, which remain in vicinity of a known largely pre-deformed state. The application of the method to the dynamical behavior of a straight beam under the action of a large bending moment is considered as an example problem.