A Timoshenko beam reduced order model for shape tracking with a slender mechanism

Sensing linear viscoelastic constitutive parameters with a Timoshenko beam on a multi-layer foundation: Modeling and simulation

“…This class of numerical optimization tools has been used in [51,52] to estimate time dependent material parameters, and in [53][54][55] to solve inverse problems in linear and non-linear vibrating systems. Predictions of this sensor model are in good agreement with published results, suggesting that the system can be used in a versatile way as an autonomous agent operating in a generic environment, and simultaneously as a sensor that could inform the action of the system itself [56], or that could be used to monitor the environment. The modeling work done in this study opens can be implemented in engineering systems applied to environmental monitoring and health applications, in which we envision the system to be used to estimate material properties of living tissues, that can be correlated to the diagnosis of classes of diseases.…”

“…The deformable body along with the distributed system of coupling elements is inspired by a class of organisms that move on unstructured substrates by adapting the shape of their bodies to non-zero curvatures of the substrates (shape morphing) and by forward locomoting through a traveling wave-like motion transmitted by the legs in contact with the substrate. These features have been used in [57,56] to model and simulate shape morphing and forward locomotion. Here, these ideas are extended to exploit the system's features as a sensor, in which case the shape morphing parameters are observable (measured) and suitable characteristics of the substrate are reconstructed based on the model presented below.…”

“…This reproduces the salient characteristics of the bodies of millipedes [58][59][60] that specifically characterizes their peculiar shape morphing coupled with forward locomotion. In order to retain the independent relative rotational motion between two contiguous elements, it is natural to adapt a Timoshenko beam model that is a one dimensional continuum (axis) with a local Euclidean structure describing the state of cross section [56]. We consider the variation of the curvature of the substrate to be small, and the mechanism to be locally parallel to the substrate, which implies that deformations are small [61].…”

“…Basis functions w and w are obtained by solving the vector eigenvalues problem associated with a planar Timoshenko beam with free end boundary conditions, see [56] and the Appendix for details. Mode shapes are normalized with respect to the maximum amplitude.…”

“…Simulations do not require the specification of the scaling time s; however, the choice of this parameter dictates the density and the length of the device once the shear modulus is given. Simulation scenarios are always set with measuring data set for the vertical and rotational deflection of the system with a 3 ¼ 10, meaning that the stiffness of the coupling elements is ten times larger than the shear stiffness of the body so that the body of the sensor morphs according to the shape of the substrate, see [56]. The undisturbed depth of the substrate is set to d ¼ 1.…”