Observations of modal interactions in resonantly forced beam-mass structures

We present a collection of experimental results on the influence of modal interactions (i.e., internal or autoparametric resonances) on the nonlinear response of flexible metallic and composite structures subjected to a range of resonant excitations. The experimental results are provided in the form of frequency spectra, Poincaré sections, pseudo-phase planes, dimension calculations, and response curves. Experimental observations of transitions from periodic to chaotically modulated motions are also presented. We also discuss relevant analytical results. The current study is also relevant to other internally resonant structural systems.     

A unified nonlinear formulation for plate and shell theories

A general geometrically exact nonlinear theory for the dynamics of laminated plates and shells under-going large-rotation and small-strain vibrations in three-dimensional space is presented. The theory fully accounts for geometric nonlinearities by using the new concepts of local displacements and local engineering stress and strain measures, a new interpretation and manipulation of the virtual local rotations, an exact coordinate transformation, and the extended Hamilton principle. Moreover, the model accounts for shear coupling effects, continuity of interlaminar shear stresses, free shear-stress conditions on the bonding surfaces, and extensionality. Because the only differences among different plates and shells are the initial curvatures of the coordinates used in the modeling and all possible initial curvatures are included in the formulation, the theory is valid for any plate or shell geometry and contains most of the existing nonlinear and shear-deformable plate and shell theories as special cases. Five fully nonlinear partial-differential equations and corresponding boundary and corner conditions are obtained, which describe the extension-extension-bending-shear-shear vibrations of general laminated two-dimensional structures and display linear elastic and nonlinear geometric coupling among all motions. Moreover, the energy and Newtonian formulations are completely correlated in the theory.     

Design and Modeling for Chatter Control

Boring bars for single-point turning on a lathe are particularly susceptible to chatter and have been the subject of numerous studies. Chatter is, in general, caused by instability. Clearly, the cutting process can be limited to regions of known stable operation. However, this severely constrains the machine-tool operation and causes a decrease in productivity. The more aggressive approach is to attack the stability problem directly through application of vibration control. Here, we demonstrate a new biaxial vibration control system (VPI Smart Tool) for boring bars. We present the experimentally determined modal properties of the VPI Smart Tool and demonstrate how these properties may be used to develop models suitable for chatter stability analysis, simulation, and development of feedback compensation. A phenomenological chatter model that captures much of the rich dynamic character observed during experiments is presented. We introduce the notion that the mean cutting force changes direction as the width of cut increases due to the finite nose radius of the tool. This phenomenon is used to explain the progression from chatter that is dominated by motions normal to the machined surface at small widths of cut to chatter that is dominated by motions tangential to the machined surface at large widths of cut. We show experimental evidence to support our assertion that a biaxial actuation scheme is necessary to combat the tendency of the tool to chatter in both directions. We then present some preliminary theoretical results concerning the persistence of subcritical instability as we expand consideration to high-speed machining.     

Input-shaping control of nonlinear MEMS

We develop a new technique for preshaping input commands to control microelectromechanical systems (MEMS). In general, MEMS are excited using an electrostatic field which is a nonlinear function of the states and the input voltage. Due to the nonlinearity, the frequency of the device response to a step input depends on the input magnitude. Therefore, traditional shaping techniques which are based on linear theory fail to provide good performance over the whole input range. The technique we propose combines the equations describing the static response of the device, an energy balance argument, and an approximate nonlinear analytical solution of the device response to preshape the voltage commands. As an example, we consider set-point stabilization of an electrostatically actuated torsional micromirror. The shaped commands are applied to drive the micromirror to a desired tilt angle with zero residual vibrations. Simulations show that fast mirror switching operation with almost zero overshoot can be realized using this technique. The proposed methodology accounts for the energy of the significant higher modes and can be used to shape input commands applied to other nonlinear micro- and macro-systems.     

Nonlinear dynamics of a resonant gas sensor

We develop a mathematical model for a resonant gas sensor made up of an microplate electrostatically actuated and attached to the end of a cantilever microbeam. The model considers the microbeam as a continuous medium, the plate as a rigid body, and the electrostatic force as a nonlinear function of the displacement and the voltage applied underneath the microplate. We derive closed-form solutions to the static and eigenvalue problems associated with the microsystem. The Galerkin method is used to discretize the distributed-parameter model and, thus, approximate it by a set of nonlinear ordinary-differential equations that describe the microsystem dynamics. By comparing the exact solution to that associated with the reduced-order model, we show that using the first mode shape alone is sufficient to approximate the static behavior. We employ the Finite Difference Method (FDM) to discretize the orbits of motion and solve the resulting nonlinear algebraic equations for the limit cycles. The stability of these cycles is determined by combining the FDM discretization with Floquet theory. We investigate the basin of attraction of bounded motion for two cases: unforced and damped, and forced and damped systems. In order to detect the lower limit of the forcing at which homoclinic points appear, we conduct a Melnikov analysis. We show the presence of a homoclinic point for a loading case and hence entanglement of the stable and unstable manifolds and non-smoothness of the boundary of the basin of attraction of bounded motion.     

Finite-amplitude longitudinal waves in non-uniform bars

An analysis is presented of the non-linear propagation of longitudinal waves along a bar whose geometrical and material properties vary slowly along its length, taking into account the effects of a small viscosity. A first-order uniform expansion for small but finite amplitudes is obtained by using the method of multiple scales. The analytical solution shows the effects of non-linearity, heterogeneity, and dissipation on the distortion of the waves and the formation of shock waves. The present results show that if the speed of sound or the cross-sectional area of the rod decreases in the direction of propagation, the wave amplitude and the rate of progress of finite-amplitude distortion increase as the wave propagates.     

Enhancement of power harvesting from piezoaeroelastic systems

We investigate the effects of varying the eccentricity between the gravity axis and the elastic axis on the level of energy harvested from a piezoaeroelastic energy harvester consisting of a pitching and plunging rigid airfoil supported by nonlinear springs. The normal form of the dynamics of the harvester near the Hopf bifurcation is used to determine the critical nonlinear coefficients of the springs and maximize the harvested power for different eccentricities. Two configurations are evaluated in terms of the power generated from limit cycle oscillations and a range of operating wind speeds. The impact of the load resistance on the harvested power is also assessed.     

Normal form representation of the aeroelastic response of the Goland wing

We follow two approaches to derive the normal form that represents the aeroelastic response of the Goland wing. Such a form constitutes an effective tool to model the main physical behaviors of aeroelastic systems and, as such, can be used for developing a phenomenological reduced-order model. In the first approach, an approximation of the wing’s response near the Hopf bifurcation is constructed by directly applying the method of multiple scales to the two coupled partial-differential equations of motion. In the second approach, we apply the same method to a Galerkin discretized model that is based on the mode shapes of a cantilever beam. The perturbation results from both approaches are verified by comparison with results from numerical integration of the discretized equations.     

Effect of the aerodynamic-induced parametric excitation on the longitudinal stability of hovering MAVs/insects

Longitudinal flight dynamics of hovering micro-air-vehicles and insects is considered. The natural oscillatory flapping motion of the wing leads to time-periodic stability derivatives (aerodynamic loads due to perturbation in the body-motion variables). Hence these terms play the role of parametric excitation on the flight dynamics. The main objective of this work is to assess the effects of these aerodynamic-induced parametric excitation terms that are neglected by the averaging analysis. The method of multiple scales is used to determine a second-order uniform expansion for the response of the time-periodic system at hand. The proposed approach is applied to the hovering flight dynamics of five insects that cover a wide range of operating frequency ratios to assess the applicability of the averaging analysis.     

A Nonlinear Vibration Absorber for Flexible Structures

An approach for implementing an active nonlinear vibration absorber for flexible structures is presented. The technique exploits the saturation phenomenon exhibited by multidegree-of-freedom systems with quadratic nonlinearities possessing two-to-one autoparametric resonances. The strategy consists of introducing second-order controllers and coupling each of them with the plant through a sensor and an actuator, where both the feedback and control signals are quadratic. Once the structure is forced near its resonances, the oscillatory response is suppressed through the saturation phenomenon. We present theoretical and experimental results of the application of the proposed vibration absorber. The structure consists of a cantilever beam, the feedback signal is generated by a strain gage, and the actuation is achieved through piezoceramic patches. The equations of motion are developed and analyzed through perturbation techniques and numerical simulation. Then, the strategy is tested by assembling the controllers in electronic components and suppressing the vibrations of the first and second modes of two beams.