Two-to-one internal resonance in microscanners

Mohammed Daqaq, Eihab M. Abdel-Rahman, Ali H. Nayfeh

Research output: Contribution to journalArticle

Abstract

To realize large scanning angles, torsional microscanners are normally excited at their natural frequencies. Usually, a bias DC voltage is also applied to scan around a desired nonzero tilt angle. As a result, a deep understanding of the mirror's response to a DC-shifted primary resonance excitation is imperative. Along these lines, we use the method of multiple scales to obtain a second-order nonlinear approximate analytical solution of the mirror steady-state response. We show that the response of the mirror exhibits a softening-type behavior that increases as the magnitude of the DC component increases. For a given mirror, we can also identify a DC voltage range wherein the mirror exhibits a two-to-one internal resonance between the first two modes; that is, ω 2≈2ω 1. To analyze the mirror behavior within that range, we first treat the case where the excitation frequency is near the first-mode frequency; that is, Ω≈ω 1. Then we treat the case where the excitation frequency is near the second-mode frequency; that is, Ω≈ω 2. We analyze the stability of the response and compare the analytical results to numerical solutions obtained via long-time integration of the equations of motion. We show that, due to the internal resonance, the mirror exhibits complex dynamic behavior characterized by aperiodic responses to primary resonance excitations. This behavior results in undesirable oscillations that are detrimental to the mirror performance, namely bringing the target point in and out of focus and resulting in distorted images.

Original languageEnglish (US)
Pages (from-to)231-251
Number of pages21
JournalNonlinear Dynamics
Volume57
Issue number1-2
DOIs
StatePublished - Jul 1 2009

Fingerprint

Internal Resonance
Mirror
Mirrors
Excitation
Primary Resonance
Voltage
Angle
Method of multiple Scales
Softening
Electric potential
Complex Dynamics
Tilt
Time Integration
Natural Frequency
Range of data
Dynamic Behavior
Equations of motion
Scanning
Natural frequencies
Equations of Motion

Keywords

  • Method of multiple scales
  • Microscanner
  • Nonlinear interactions

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering

Cite this

Two-to-one internal resonance in microscanners. / Daqaq, Mohammed; Abdel-Rahman, Eihab M.; Nayfeh, Ali H.

In: Nonlinear Dynamics, Vol. 57, No. 1-2, 01.07.2009, p. 231-251.

Research output: Contribution to journalArticle

Daqaq, Mohammed ; Abdel-Rahman, Eihab M. ; Nayfeh, Ali H. / Two-to-one internal resonance in microscanners. In: Nonlinear Dynamics. 2009 ; Vol. 57, No. 1-2. pp. 231-251.
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