W. Steve Shepard

Comparative analysis of actuator concepts for active gear pair vibration control

Abstract Four actuation concepts for the active suppression of gearbox housing mesh frequency vibrations due to transmission error excitation from the gear pair system are modelled and compared by computing the required actuation forces and amplifier power spectra. The proposed designs studied consist of (1) active inertial actuators positioned tangentially on the gear body to produce a pair of reactive force and moment, (2) semi-active gear–shaft torsional coupling to provide tuned vibration isolation and suppression, (3) active bearing vibration control to reduce vibration transmissibility, and (4) active shaft transverse vibration control to suppress/tune gearbox casing or shaft response. Numerical simulations that incorporate a transmission error term as the primary excitation are performed using a finite element model of the geared rotor system (dynamic plant) constructed from beam and lumped mass/stiffness elements. Several key comparison criteria including the required actuation effort, control robustness and implementation cost are examined, and the advantages and disadvantages of each concept are discussed. Based on the simulated data, the active shaft transverse vibration control scheme is identified as the most suitable approach for this application.

Estimation of structural wave numbers from spatially sparse response measurements.

A method is presented for estimating the complex wave numbers and amplitudes of waves that propagate in damped structures, such as beams, plates, and shells. The analytical basis of the method is a wave field that approximates response measurements in an aperture where no excitations are applied. At each frequency, the method iteratively adjusts wave numbers to best approximate response measurements, using wave numbers at neighboring frequencies as initial estimates in the search. In comparison to existing methods, the method generally requires far fewer measurement locations and does not require evenly spaced locations. The number of locations required by the method scales with the number of waves that propagate in the structure, whereas the number of locations required by existing methods scales with the minimum wavelength. In addition, the method allows convenient inclusion of the analytic relationships between wave numbers that exist for flexural vibrations of beams and plates. Advantages of the method are illustrated by an example in which a beam is excited by a transverse force at one end. Using analytic data and experimental measurements, the method produces a wave field that matches response measurements to within 1 percent. One interesting feature of the new method is that, when applied to analytic data, it supplies more robust wave number estimates using responses at unevenly spaced locations.

Modeling active vibration control of a geared rotor system

This paper analyzes the practicality of active internal shaft transverse vibration control using a piezoelectric stack actuator for reducing external gearbox housing structure response due to transmission error excitation from a gear pair. The proposed adaptive controller that is designed specifically for tackling mesh frequency vibrations is based on an enhanced filtered-x least mean square algorithm with frequency estimation to synthesize the required reference signal. The vital system secondary path characteristic that relates the actuation force to the housing response of interest is identified in offline or online mode using the additive random noise technique. Numerical studies employing practical design parameters in order to provide a realistic assessment of the proposed approach are performed for a geared rotor system (dynamic plant) that is represented as finite and lumped elements. The simulations reveal very promising vibration control results, which will be utilized to guide future experimental studies.

Experimental active vibration control of gear mesh harmonics in a power recirculation gearbox system using a piezoelectric stack actuator

An experimental study of an active shaft transverse vibration control system for suppressing gear mesh vibratory response due to transmission error excitation in a high power density gearbox is presented. The proposed active control concept employs a piezoelectric stack actuator to deliver the control force through a secondary bearing. A versatile test stand that includes a closed-loop, power recirculating, dual-gearbox set-up capable of high load transfer is specially designed for this work. The underlying controller for computing the actuation signal is based on a modified filtered-x LMS algorithm with a robust frequency estimation technique. In order to avoid the common out-of-band overshoot problem, an integrated adaptive linear enhancer is also applied. Both single mesh frequency and multi-harmonic control cases are examined to evaluate the performance of the active control system. Additionally, the impact of the adaptive linear enhancer order as well as the controller adaptation step size on active control performance is evaluated. The experiments performed show more than 10 dB reduction in housing vibrations at certain targeted mesh harmonics over a range of operating speeds.

Active vibration control of a plate-like structure with discontinuous boundary conditions

This research involves the application of methods to actively control the vibration of a plate-like structure with discontinuous boundary conditions. The research is motivated by the need to control vibrations on rack shelves in use on the International Space Station (ISS). Vibration of the rack shelves can adversely affect experiments being performed on those shelves. In this work, control of a rack shelf similar to those in use on the ISS is examined. Piezoelectric actuators bonded to the shelf structure are proposed as a method for controlling rack shelf vibrations. A two-dimensional asymmetric piezoelectric actuator model is first developed. The Ritz expansion method is then employed to derive the equations of motion for the combined piezoelectric actuators and rack shelf system with discontinuous boundary conditions. Model parameters from the analytical solution are used in conjunction with experimentally obtained parameters to develop a control model for the active structure. The control model is then used, together with a linear quadratic approach, to develop two different control strategies: collocated output feedback control and modal control. Results from an experimental evaluation of the two control approaches are presented. Based on the experimental results, the two control strategies are shown to be effective in controlling the first several modes of the rack shelf system at frequencies below 800 Hz.

Direct Hybrid Adaptive Control of Gear Pair Vibration

A direct hybrid adaptive approach based on the Lyapunov stability theorem is proposed for performing active vibration control of a rotational gear pair subject to multiple-harmonic, transmission error disturbances. The analysis applies a reduced single-degree-of-freedom gear pair model of the elastic mesh mode with time-varying tooth mesh stiffness. It is assumed that the resultant actuation force for suppressing the rotational vibration of the gear pair can be directly applied along the tooth contact line-of-action by employing a set of suitably configured inertial actuators. The proposed controller simultaneously adapts both the feedback and feed-forward gains, and only requires knowledge of the fundamental gear mesh frequency that is given by the product of the instantaneous gear rotational speed and the number of gear teeth. The analysis indicates that the proposed controller is insensitive to the gear mesh frequency estimation errors, and the resulting vibration control is more effective than those provided by the adaptive notch filter and filtered-x LMS algorithms. The control theory also incorporates dynamic normalization and leakage enhancements in order to optimize performance and improve robustness. Finally, the salient features are demonstrated in several numerical examples.

Sensitivity of structural acoustic response to attachment feature scales

When implementing a discrete computational method to predict the vibro-acoustic behavior of a larger structure one must decide how much effort and refinement to expend on modeling smaller attached structural components. In representing these attachments, or features, it is important to know what scales are important with regard to a valid prediction of the system response. To aid in the study of the impact of feature scales, sensitivity relationships for important system variables are developed in this paper. The sensitivity analysis will, for one, consider the change in the radiated power with respect to changes in the scale of an attached mass. These relations are developed analytically and do not require finite difference methods or eigenvalue derivatives. It is shown that computing the sensitivity of the radiated acoustic power only requires evaluating one additional term as all of the other terms are found during the normal course of modeling the system. Furthermore, the terms that include fluid-load...

Experimental analysis of an active vibration control system for gearboxes

An internal active vibration control system for suppressing gearbox housing vibrations due to gear transmission error excitation is developed and evaluated experimentally. The approach is based on an active shaft transverse vibration control concept that resides within the gearbox. The system contains a piezoelectric stack actuator for applying control forces to the shaft via a secondary rolling element bearing. An enhanced filtered-x least mean squares control algorithm with frequency estimation capability is employed to generate the appropriate actuation signal. The experimental results show up to about 20 dB reduction in the vibration levels of the first two mesh harmonics at the control positions on the housing structure for a range of operating speeds. The experimental studies also indicate that, under certain narrow operating conditions, the vibrations at some non-control positions may be amplified slightly due possibly to the effect of unmodeled dynamics. In spite of this limitation, the proposed active vibration control approach is quite promising for reducing gear vibration and ultimately gear noise.

Actuator design and experimental validation for active gearbox vibration control

A straightforward approach that combines a linear piezoelectric stack actuator model and the dynamics of the host system is proposed for predicting the required control voltage and power levels of an actuator targeted for active gearbox vibration control applications. Even though piezoelectric actuators are widely applied, their electrical requirements for particular active control systems are seldom addressed directly. The proposed theory is validated by applying the experimental results obtained from a closed-loop, power recirculating gearbox set-up equipped with an active shaft transverse vibration control system. The proposed methodology provides a formal framework for designing actuators and amplifiers without the usual costly trial and error process.

Reducing the Impact of Measurement Errors When Reconstructing Spatial Dynamic Forces

Inferring external spatially distributed dynamic forces from measured structural responses is necessary when direct measurement of these forces is not possible. The finite difference method and the modal method have been previously developed for reconstructing these forces. However, the accuracy of these methods is often hindered due to the amplification of measurement errors in the computation process. In order to analyze these error amplification effects by using the singular value decomposition approach, the mathematic expressions for these two force reconstruction methods are first transformed into a certain linear system of equations. Then, a regularization method, the Tikhonov method, is applied to increase computational stability. In order to achieve a good regularized result, the L-curve method is used in conjunction with the Tikhonov method. The effectiveness in reducing the influence of the measurement errors when applying the regularization method to the finite difference method and the modal method is investigated analytically and numerically. It is found that when the regularization method is appropriately applied, reliable computational results for the reconstructed force can be achieved.