Linear Versus Nonlinear Response of a Cantilevered Beam Under Harmonic Base Excitation: Theory and Experiment

A computational and experimental study of a uniform cantilever beam with a tip mass under base excitation was performed. The beam was excited at various levels of base displacement to provoke tip displacements greater than 15% of the beam length. Damping and yield stress of the beam were both considered. It was found that a large tip displacement causes nonlinear inertial (NLI) and structural (NLS) effects to arise. Each of the structural and inertial nonlinearities has an opposite effect on the resulting resonance frequency, which are nearly mutually canceling. The result was that resonant frequency calculated using the full nonlinear (FNL) model was essentially equal to the value calculated by linear (LIN) theory, and the tip displacement amplitude varied only modestly from the LIN value. It was also observed that the damping in this system is likely nonlinear, and depends on tip displacement amplitude. A theoretical model for fluid damping is suggested. Initial investigation shows encouraging agreement between the theoretical fluid damping and the measured values.

Nonlinear Response of CNT Cantilever Based Nano-Resonators

This paper presents nonlinear dynamic response of carbon nanotube (CNT) cantilevers incorporating electrostatic forces and van der Waals interactions between the CNT and the conducting plane. The CNT cantilever models including geometric and inertial nonlinearities for predicting unexpected phenomena when the deflection of the CNT is increased. As a result, the CNT cantilever shows complex dynamic responses due to the applied voltage. At the low voltages, the cantilever has only linear response at fundamental resonance except the superharmonic response due to the harmonic excitation of electrostatic field. The secondary resonance response branches off through period-doubling (PD) bifurcation, and becomes softened through saddle-node (SN) bifurcation when the applied voltage is increased. After SN bifurcation, the lower branch of the solution near resonance loses its stability and becomes unstable. This theoretical finding can help the prediction of complex response of the nano-resonators.

Modelling and Control of a Closed Chain Flexible Linkage

The objective of this paper is to describe and numerically test a control scheme for a flexible four-bar linkage. The dynamic response of the mechanism is reproduced by means of an accurate finite element model accounting for geometric and inertial nonlinearities. A reduced number of measured variables is selected to control both rigid-body motion and vibration separately without implementing an estimator of the state of the system. Rigid-body motion control is performed by means of a PID-like regulator while proportional controllers are employed to damp link oscillations. Appropriate devices are proposed to avoid coupling effects among variables. Numerical results demonstrate the effectiveness of the control when it operates at different sampling times.

Comparison and Validation of Dynamics Simulation Models for a Structurally Flexible Manipulator

This paper presents a series of experimental results obtained with a 2-DOF flexible-link direct-drive manipulator. First, we conduct a frequency analysis by comparing experimental natural frequencies with those predicted by the finite element model. Then, the time responses from four dynamics models are compared with each other and with the experiment. It is demonstrated that higher order nonlinearities are less important for slow maneuvers by close agreement between all four simulation models. For fast maneuvers, the two simpler models fail to predict a physically meaningful response. Good agreement with experimental results is attained with a model which accounts for all inertial nonlinearities. It is also shown that inclusion of damping in the dynamics models has a significant impact on their performance, as well as improving the correlation with experimental data.

A Parametric Study of the Modified Model Reference Adaptive Control Scheme

The validity of the claim by many studies that the damping and stiffness forces can be ignored when designing a model reference adaptive controller, is examined. For a simple plant, the sensitivity of the closed loop system to the inertial, damping, and stiffness nonlinearities are investigated, through a simulation analysis. It is shown that the closed loop system is sensitive to the changes in the inertial nonlinearities, and relatively insensitive to variations in the damping and stiffness forces. This supports the assumption made in many previous studies.

Simulation of Flexible-Link Manipulators With Inertial and Geometric Nonlinearities

Several important issues relevant to modeling of flexible-link robotic manipulators are addressed in this paper. First, we examine the question of which inertial nonlinearities should be included in the equations of motion for purposes of simulation. A complete model incorporating all inertial terms that couple rigid-body and elastic motions is presented along with a rational scheme for classifying them. Second, the issue of geometric nonlinearities is discussed. These are terms whose origin is the geometrically nonlinear theory of elasticity, as well as the terms arising from the interbody coupling due to the elastic deformation at the link tip. Accordingly, a general way of incorporating the well-known geometric stiffening effect is presented along with several schemes for treating the elastic kinematics at the joint interconnections. In addition, the question of basis function selection for spatial discretization of the elastic displacements is also addressed. The finite element method and an eigenfunction expansion techniques are presented and compared. All issues are examined numerically in the context of a simple beam example and the Space Shuttle Remote Manipulator System. Unlike a single-link system, the results for the latter show that all terms are required for accurate simulation of faster maneuvers. Hence, the conclusions of the paper are contrary to some of the previous findings on the validity of various models for dynamics simulation of flexible-body systems.