Delayed control of an elastic beam

Abstract

Stabilization of longitudinal vibrations along an elastic beam with delayed feedback is analyzed. The beam is modeled as a series of equal masses connected by springs and dashpots. The degree of freedom (DoF) of the model is increased-first 1, then 3 and 9 DoF-with stability charts in terms of the gain and the time delay presented in each case for various levels of damping. With zero damping, for 9 DoF no stability region remains at all. Yet at the continuum case, when the partial delay differential equation is examined with zero damping, discrete stable intervals of gain are found at specific delay values. These intervals largely match the stability charts of the higher DoF models.

Exact stability chart of an elastic beam subjected to delayed feedback

“…Consequently, undamped continuum systems with delayed boundaries can be unstable for any irrational ratios of the arising time delays in the system. Still, a few zeromeasure (in engineering terms, zero-probability) stability domains were identified in some numerical examples, while these tiny stable domains could not be obtained through finite DoF approximations [9]. Similar numerical examples were presented also in section 1.2.5 of [25] related to the continuity properties of the spectrum with respect to the time delays.…”

“…For the system governed by (15), numerical results in [9] show that τ values that are odd integer multiples of T also result in exponential stability for certain gain parameters K.…”

“…In both cases, they used low DoF approximations since the PDE models with delayed boundary conditions were too complex for analytical approaches. A basic microphone feedback system was also modelled as a chain of rigid blocks with elastic springs and time delay in the control loop in [9], which again leads to a finite DoF approximation of a wave equation with delayed boundary conditions. The results showed discrepancies in convergence as the DoF was increased.…”

The Microphone Feedback Analogy for Chatter in Machining

“…The time delay, , was first calculated as the inverse of the predicted spindle speed vector from (4). The time delay was then related to the distance between the speaker and microphone using (6), where V is the velocity of sound in dry air at 20 deg C (343 m/s). Consider…”

“…Many additional publications are available: research in [2] provides a comprehensive review and [3,4] are two available textbooks. The purpose of this paper is to demonstrate the analogy between the squeal frequency experienced when audio feedback causes instability in public address systems [5,6] and the chatter frequency exhibited by machining operations. A simple microphone/speaker setup is described and experimental results are quantitatively and qualitatively compared to machining stability predictions.…”