Tailoring of Nonlinear Waves by Purposeful Introduction of Defects in Periodic Mechanical Metamaterials Phase Space Analysis USNCTAM 2022 Mohammed A Mohammed PhD student Dynamical Systems Lab ID: 1014851
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1. Piyush Grover, Assistant ProfessorTailoring of Nonlinear Waves by Purposeful Introduction of Defects in Periodic Mechanical Metamaterials: Phase Space AnalysisUSNC/TAM 2022Mohammed A. Mohammed, Ph.D. studentDynamical Systems LabMechanical and Materials EngineeringUniversity of Nebraska-Lincoln, NE, USA
2. 1D chain of bistable elements: Transition wavesOnsite bistable spring/mass elements (geometric nonlinearity)Linear intersite springsSupports propagation of nonlinear transition waves for all inputs above certain amplitude thresholdImpulseInputNadkarni et. al., PRE 2014Hwang and Arrieta, Smart Material Structures 2022, SPIE 2019, Scientific Reports 2018, PRE 2018
3. Introduction of a localized defectLocalized inhomogeneity in the onsite spring stiffnessGives rise to localized oscillations (breather) near the defect Energy transfer between the travelling wave and the local oscillationsComplicated (chaotic) dynamics due to this interaction* Figotin and Klein: Journal of statistical Physics 1997, Fei et al., PRA 1992, Zhou et al., PRE 2017, Dauxois and Peyrard, 2006
4. Defect locationDisplacementPositionWave transmission, capture and reflection
5. Input-output velocity relation for the system with defectIncoming velocityOutgoing velocityReflected wavesTransmitted wavesCaptured waves
6. Full-order Discrete Model and Continuum ApproximationNondimensionalized N-DOF model :Continuum approximation (nonlinear wave equation)with nonlinear potentialfor masses , * Nadkarni et. al., PRE 2014
7. Continuum approximations for the system with and without defectWithout Defect :Transition (travelling) wave : moves with fixed speed below cDispersion relation (linearizing about 0 solution ):With Defect :A time-periodic (linearized) mode localized in space appears : Breather Breather (temporal) frequency lies in the bandgap oscillatoryBandgapPhonon spectrum, q = wavenumber
8. Low-order model of wave-breather interaction : ModesTwo-DOF low-order model with fixed shapesDOFS: Solitary wave center position X(t) and amplitude of the breather a(t) solitary waveIgnoring energy transfer To phonons due to discreteness effects (present in discrete model only)To phonons due to wave-breather interaction (present in discrete and continuum models)To the other bound mode (present in discrete and continuum models)+ Breather0
9. Low-order model : Method of collective coordinatesStandard method to obtain finite dim. governing equations from Lagrangian densityExtremize the Lagrangian among the fields represented by the ansatz K.E.Linear springP.E.Nonlinear springP.E.A conservative 2-DOFsystem: Coupled oscillator* Dauxois and Peyrard, 2006
10. Low-order model prediction : Input-Output RelationIncoming velocityOutgoing velocity
11. Reflected wavesCaptured wavesPosition of Wave-frontLow-order model prediction : Trajectory of the solitary wave Transmitted wavesInteraction with the breather
12. Wave transmission, capture and reflection : Reduced order model
13. Low-order model : Phase space analysisStep 1: Switch to Hamiltonian approachStep 2: Split the Hamiltonian in a specific way* and introduce an artificial coupling parameter The breather dynamics are uncoupled from the solitary wave: Fully integrable systemThe actual system is recoveredFour-dimensional nonlinear Hamiltonian system-0.2-0.100.20.1Heteroclinic orbit1DOF Nonlinear oscillatorParticle movingin the potential well0* Goodman et al., Physica D 2002
14. Phase space analysis: The solitary wave and breather dynamics are fully-coupled in 4D: cannot study the dynamics in separate 2D phase plotsFor small , the periodic orbits ( ) persist and can be used to define a Poincare mapNear-integrable system Action-angle transformationPoincare map P: Fix the total energy levelFix the phase of breather (say )We obtain a discrete-time 2D dynamical system on the surface Dynamics are constrained to 3D hypersurface
15. Unstable Manifold of periodic orbit at - Stable Manifold of periodic orbit at + PeriodicOrbit at -PeriodicOrbit at +*Koon et al., Chaos 2000
16. Phase space analysis for Stable and unstable manifolds of periodic orbits at infinities intersect transversallyTransport is possible between the three regions : Lobe Dynamics*Wiggins, Chaotic Transport in Dynamical Systems, Springer 2013
17. Transmitted Wave: Lobe Dynamics
18. Reflected Wave: Lobe Dynamics
19. What determines the threshold for wave transmission across the defect ?Range ofallowableincomingwave speedsFlow with fixed total energy in breather + solitary waveZero incomingwave speed,all energy in breatherMax. incoming wave speed,no energy in breatherGap >0 above critical energyCritical energy= energy of the incoming wave with (and no energy in breather)Trajectory in thegap avoids the lobes
20. Ongoing workImproved reduced-order models by incorporating energy leakage to phonons as a (nonlinear) damping termExperimental validation using 3D printed models Modulation of breather energy in open-loop or feedback manner to tailor nonlinear wavesExtensions to 2D latticesPapers/Preprints/VideosDynamical Systems Lab: engineering.unl.edu/dslEmail: Piyush.grover@unl.edu
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22. with nonlinear potentialfor masses , Dimensional: