How delay equations arise in Engineering? G

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How delay equations arise in Engineering?Gábor
StépánDepartment of Applied MechanicsBudapest
University of Technology and Economics
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Contents

• Answer Delay equations arise in Engineering
• by the information system (of control), and
  by the contact of bodies.
• Linear stability subcritical Hopf bifurcations
• Robotic position and force control
• Balancing human and robotic
• Contact problems
• Shimmying wheels (of trucks and motorcycles)
• Machine tool vibrations

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Stability of linear RFDEs of n DoF systems

• Delayed mechanical systems include 2nd
  derivatives
• Autonomous systems
• Trial solution
• Characteristic roots Re ?j lt 0, j1,2, ?
  stability
• D-curves
• 
  ? stability

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Examples with 1 DoF, n 1
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Stability chart
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Delayed oscillators

• 
  vibration
  frequencies
• 
  
• 
  stability
  chart

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Delayed oscillator with damping

• 
  damping b01

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Non-autonomous linear RFDEs

• Time-periodic systems
• Trial solution
• Hills infinite dimensional determinant ?
• characteristic function ? characteristic roots
  ?
• Re ?j lt 0, j1,2, ? stability ? ??j?lt1, j1,2,
• for characteristic multipliers of
  fund. op. at T

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The delayed Mathieu equation

• 
  Harmonic balance
  ? Hills determinant

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The delayed Mathieu stability charts

• b0 (Strutt-Incze, 1928) e0 (Hsu-Bhatt,
  1966)
• e1

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Stability chart of delayed Mathieu

• 
  
• 
  Insperger,
• 
  Stépán (2002)

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Semi-discretization method introduction

• The approximating DDE is non-autonomous

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Introduction to SDM delayed oscillator

• 
  stability

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Delayed oscillator stability chart by SDM
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Full discretization - comparison

• Discretization also w.r.t. time derivatives
  slow
  convergence

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Introduction to SDM Mathieu equation
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Semi-discretization general case

• 
  Insperger, Stepan
  Int. J. of
  Numerical Methods
  in Engineering (2002)

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Examples test on delayed Mathieu
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Nonlinear RFDEs in Engineering

• Stability analysis of steady-states is followed
  bybifurcation analysis
• Hopf bifurcation self-excited vibrations
• Supercritical case easy to avoid vibrations by
  knowing the linear stability behaviour
• Subcritical case the unstable periodic solutions
  mean a limited domain of attraction for the
  desired steady-state behaviour cannot be
  predicted by linear stability analysis.

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Stickslip unstable periodic motion
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Unstable limit cycle ghost vibration