Glenn R. Heppler

Shaped Input Control of a System With Multiple Modes

This paper describes a method for limiting vibration in flexible systems that have more than one characteristic frequency and mode. It is only necessary to have knowledge of the component mode frequencies and damping ratios in order to be able to calculate the timing and magnitudes of the impulse sequence used in the shaping. Only two impulses, in the nonrobust case, or three impulses in a more robust case, are necessary regardless of the number of component frequencies. Simple tests are established to determine when this technique can be used and examples are presented.

Modeling impact on a one-link flexible robotic arm

A finite-element model for a single-link flexible robotic arm including the effects of beam damping, hub inertia and both Coulomb and viscous hub friction is derived. The initial conditions required to represent impact loading are determined, and the motion of the arm under impact loading is simulated. Simulation results are compared to experimental data. From the experimental results it is concluded that this model provides an accurate representation of the physical process. It also provides a vehicle for investigating the effects of parameters of the process model, in particular, frictional effects and load profiles. Impact loading has been effectively represented in terms of initial conditions and can be applied to a range of finite-element models. >

A Deformation Field for Euler–Bernoulli Beams with Applications to Flexible Multibody Dynamics

The deformation field commonly used for Euler–Bernoulli beamsin structural dynamics is investigated to determine its suitability foruse in flexible multibody dynamics. It is found that the traditionaldeformation field fails to produce an elastic rotation matrix that iscomplete to second-order in the deformation variables. A completesecond-order deformation field is proposed along with the equationsneeded to incorporate the beam model into a graph-theoretic formulationfor flexible multibody dynamics [1]. This beam modeland formulation have been implemented in a symbolic computer programcalled DynaFlex that can use Taylor, Chebyshev, or Legendrepolynomials as the basis functions in a Rayleigh–Ritz discretizationof the beams deformation variables. To demonstrate the effects of the proposed second-order deformationfield on the response of a flexible multibody system,two examples are presented.

Dynamics of a Flexible Cantilever Beam Carrying a Moving Mass

The motion of a flexible cantilever beam carrying a moving spring-mass system is investigated. The beam is assumed to be an Euler–Bernouli beam. The motion of the system is described by a set of two nonlinear coupled partial differential equations where the coupling terms have to be evaluated at the position of the mass. The nonlinearities arise due to the coupling between the mass and the beam. Due to the nonlinearities the system exhibits internal resonance which is investigated in this work. The equations of motion are solved numerically using the Rayleigh–Ritz method and an automatic ODE solver. An approximate solution using the perturbation method of multiple scales is also obtained.

Controlling the impact response of a one-link flexible robotic arm

A finite-element model of a single-link flexible robotic arm is derived, and the initial conditions required to represent impact loading are determined. An open-loop control system is designed using frequency-domain techniques to compute a desired hub torque profile. The motion of the arm after tip impact and including hub actuation is simulated. Simulation results are compared to experimental data. It is shown that the controller damps vibration caused by the impact and allows the steady-state tip deflection to be predetermined. >

A large-stroke electrostatic micro-actuator

Voltage-driven parallel-plate electrostatic actuators suffer from an operation range limit of 30% of the electrostatic gap; this has restrained their application in microelectromechanical systems. In this paper, the travel range of an electrostatic actuator made of a micro-cantilever beam above a fixed electrode is extended quasi-statically to 90% of the capacitor gap by introducing a voltage regulator (controller) circuit designed for low-frequency actuation. The voltage regulator reduces the actuator input voltage, and therefore the electrostatic force, as the beam approaches the fixed electrode so that balance is maintained between the mechanical restoring force and the electrostatic force. The low-frequency actuator also shows evidence of high-order superharmonic resonances that are observed here for the first time in electrostatic actuators.

Scaling laws for linear controllers of flexible link manipulators characterized by nondimensional groups

When constructing large robotic manipulators or space structures, it is advisable to begin with a small-scale prototype on which to perform the design, analysis, and debugging. To ensure that the results obtained on the scale-model apply directly to the actual manipulator, it is necessary that the prototype and the original robot are dynamically equivalent. This paper examines the single flexible link (SFL) manipulator. Dimensional analysis is used to identify the nondimensional groups for the SFL. These groups are present in the corresponding nondimensional equations of motion, which are also derived. To account for inherent manufacturing imprecision, tolerances are developed for the nondimensional groups. Scaling laws for continuous-time and discrete-time controllers are developed for dynamically equivalent SFL systems. These theoretical scaling laws are verified experimentally for an H/sub /spl infin// and a PD control strategy.

Analysis of a Chaotic Electrostatic Micro-Oscillator

The closed-loop dynamics of a chaotic electrostatic microbeam actuator are presented. The actuator was found to be an asymmetric two-well potential system with two distinct chaotic attractors: one of which occurs predominantly in the lower well and a second that visits a lower-well orbit and a two-well orbit. Bifurcation diagrams obtained by sweeping the ac voltage amplitudes and frequency are presented. Period doubling, reverse period doubling, and the one-well chaos through period doubling are observed in amplitude sweep. In frequency sweep, period doubling, one-well, and two-well chaos, superharmonic resonances and on and off chaotic oscillations are found. DOI: 10.1115/1.4002086

On the dynamic behaviour of a flexible beam carrying a moving mass

The behaviour of a system containing a mass traveling on a cantilever beam is considered. The mass is induced to move by an applied force as opposed to the case which has been considered in most literature where the position of the moving mass is assumed to be known and independent of the motion of the beam. Furthermore, the system to be discussed has the unique characteristic that the motions of the mass and the beam are coupled. The mathematical model of the system includes two coupled nonlinear integral/partial differential equations which are impossible to solve analytically and are difficult to solve numerically in their original form. As a remedy, the solution is discretized into space and time functions and the equations of motion are reduced to a set of ordinary differential equations. The shape function is chosen so that it satisfies the boundary conditions of the beam as well as the transient conditions imposed by the traveling mass. This choice of the shape function, which considers the mass-beam interaction, provides an improvement over the conventional method of using a simple cantilever beam mode shapes.The ordinary differential equations of motion using the ‘improved’ shaped functions, are solved numerically to obtain the dynamic behaviour of the system. The results illustrate the validity of the model, and demonstrate the advantages of the ‘improved’ model to the ‘un-improved’ equations.

Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations:

This paper is devoted to a review of the linear isotropic theory of micropolar elasticity and its development with a focus on the notation used to represent the micropolar elastic moduli and the experimental efforts taken to measure them. Notation, not only the selected symbols but also the approaches used for denoting the material elastic constants involved in the model, can play an important role in the micropolar elasticity theory especially in the context of investigating its relationship with the couple-stress and classical elasticity theories. Two categories of notation, one with coupled classical and micropolar elastic moduli and one with decoupled classical and micropolar elastic moduli, are examined and the consequences of each are addressed. The misleading nature of the former category is also discussed. Experimental investigations on the micropolar elasticity and material constants are also reviewed where one can note the questionable nature and limitations of the experimental results reported on the micropolar elasticity theory.