WAVE PROPAGATION AND BEHAVIOR IN NANOSCALE BISTABLE SYSTEMS
WAVE PROPAGATION AND BEHAVIOR IN NANOSCALE BISTABLE SYSTEMS Bistable Mechanisms Compliant Mechanisms
BACKGROUND PAPER AND RESEARCH 2
Band gap transmission in periodic bistable elements, Frazier et Kochmann Mechanical Bistability • • Formed by tilted beam springs, which alter the energy potential for a lateral driving force. There are three different equilibrium positions that develop: o Stable equilibrium of the unperturbed system. o Unstable equilibrium during the transition between stable states. o Stable equilibrium, which is a mirror image of the initial undisturbed system. Mechanical bistability, with three different equilibrium positions.
Band gap transmission in periodic bistable elements, Frazier et Kochmann Mechanical Bistability • • The system achieves a stable configuration at 0 and 2 L, which both reference the initial geometric setup of the system. o This would be the initial configuration and the mirror image of the switch. At the midpoint, L, the system reaches an unstable equilibrium point, where it will force to one side of the potential well or the other. The potential energy well for the bistable system, along with the initial geometric configuration of the system.
Band gap transmission in periodic bistable elements, Frazier et Kochmann Chain of Bistable Elements • Individual bistable elements can be combined to form a chain of bistable elements, which are referred to as snapthrough switches. • The combination of bistable elements leads to unique properties of the entire system, especially once a driving frequency and amplitude are introduced. • The nonlinear response of one element drives the unique properties of the system as a whole. The complete system, which is a periodic chain of bistable elements connected by linear springs and driven by an input frequency/amplitude.
Band gap transmission in periodic bistable elements, Frazier et Kochmann Linear Wave Propagation • From the simple unit cell, a governing characteristic equation was developed to theoretically evaluate the response of the periodic bistable chain to an input frequency. • The key parameters that determine the response of the system are as follows: o Linear spring, k 1 o Beam spring, k 2 o Mass of the bead, m o Driving frequency, ω o Propagation constant, γ o Geometric ratio, d Characteristic equation governing the nonlinear response of the periodic bistable chain. The real and imaginary components of the normalized wavenumber, where γ is the propagation constant.
Band gap transmission in periodic bistable elements, Frazier et Kochmann Frequency Band Diagram • Based on the initial parameters of the system, a frequency band diagram can be produced, solely from theoretical equation. • • • For low input frequencies below a lower threshold value of 1. 386, the input signal does not pass through the system. For high input frequencies above an upper threshold value of 6. 474, the input signal also does not pass through the system. For intermediate input frequencies between the upper and lower limit, transmission occurs where the signal does pass through the system. Emergence of an upper and lower bandgap for the mechanical bistable chain.
Band gap transmission in periodic bistable elements, Frazier et Kochmann Amplitude Analysis in the Lower Bandgap • The displacement, energy, and snapthrough events for the first 50 sites in the periodic chain are analyzed. • As the driving amplitude is increased in the lower bandgap region, three separate responses occur. o No transmission where the wave diminishes throughout the system for lower amplitudes. o Weak transmission, where the network becomes more transparent as wave propagation begins to occur. o Supratransmission, where a strong nonlinear response begins to occur with high values of snapthrough. Amplitude response of the periodic bistable chain in the lower bandgap region (a low driving input frequency).
MOLECULAR DYNAMICS (MD) 9
Computational Nanodynamics Why this system? Why choose MD? • The system acts as a bandpass filter, which at a smaller scale has applications in nanoelectronics and signal transfer/amplification (overall signal processing). • Unlike at the macroscopic scale, in MD the behavior of the system can be analyzed while taking pressure, volume, and temperature into account. • • A nanoscale system is not simply a miniaturization. There are fundamental differences in the behaviors at this scale. We can gauge the validity of the approximate theories and the experimental data. Through simulation and visualizations, we can understand the underlying mechanisms of nanoscale phenomena. We can develop analytical explanations that would not be possible with just experimental data.
NANOSCALE BISTABLE SYSTEMS 11
Nanoscale Bistable System Creation and Parameters • First goal was to replicate the macroscale bistable system in MD, and then begin to compare the bandgap and amplitude response in the paper. • Second goal was to determine tunability factors of the system as a bandpass filter. • • The parameters utilized in both the paper and the current system are as follows: o k 1 = 10 o k 2 = 1 o m=1 o d = ∂/2 L The length (L), lattice spacing (a), and vertical offset (∂), were chosen such that each unit cell was sufficiently spaced out. x ∂/2 L a k 1 k 2
Nanoscale Bistable System Creation
Nanoscale Bistable System Creation Molecular Dynamics (MD) Setup • For all non-temperature simulations, the system is run using the NVE ensemble. • For all temperature simulations, the system is run using the NVT ensemble. • Here a constant integration updates the positions and velocities of atoms for each timestep, where: o N is the number of particles o V is the volume o E is the energy o T is the temperature Also called the microcanonical ensemble, every state has the same energy and the same likelihood of being in one state or another. • Also called the canonical ensemble, the probability of a certain configuration will depend on the energy of that configuration. • This performs time integration on Nose. Hoover style non-Hamiltonian equations of motion (extra DOF for heat bath): •
Bandgap Frequency Analysis Frequency (ω) is varied: • ω = 0. 2 : 7 • Lowerthreshold = 1. 387 • Upperthreshold = 6. 474 Summary: • The displacement of a single node, the final center bead in the simulation, can be reduced using RMSD to measure the wave transmission. • This indicates that the wave propagated through the system.
Amplitude Analysis System Parameters Input sinusoidal wave: • Frequency (ω) is fixed • Amplitude (U) varied x ∂/2 Snapshot Analysis: • Region I: no transmission • Region II: weak transmission • Region III: supratransmission ∂/2 L a k 1 k 2
Amplitude Analysis Amplitude (U) is varied: • U = 0. 05 : 10 • Normalized: (0 -2 L) • Uthreshold ≅ 1. 62 Summary: • Three separate behaviors occur for this frequency in the lower bandgap region: § No transmission, U = 0. 5 § Weak transmission, U = 1. 61 § Suptratransmission, U = 1. 62
Amplitude Analysis Amplitude (U) is varied: • U = 0. 05 : 10 • Normalized: (0 -2 L) • Uthreshold ≅ 1. 62 Summary: • The first transition occurs at about 1. 24, where the first snapthrough event after the first bead occurs. • The transition to supratransmission occurs around 1. 62, when both the energy profile and snapthrough count dramatically increase.
Amplitude Analysis Amplitude (U) is varied: • U = 0. 05 : 10 • Normalized: (0 -2 L) • Uthreshold ≅ 1. 62 Summary: • The RMSD data for the last point indicates that the transition behavior across the three different regions persists throughout the entire system.
Snapthrough Summary Amplitude (U) is varied: • U = 0. 05 : 10 • Normalized: (0 -2 L) Frequency (ω) is varied: • ω = 0. 2 : 7 Summary: • The MAX Ein for each driving frequency is shown on the right, which reflects the profile that is obtained in the initial bandgap results. A peak in both energy and snapthrough occurs at ω = 5.
Snapthrough Summary Amplitude (U) is varied: • U = 0. 05 : 10 • Normalized: (0 -2 L) Frequency (ω) is varied: • ω = 0. 2 : 7 Summary: • In the lower bandgap region at high amplitudes the wave can still permeate throughout the system, whereas above the upper bandgap this no longer remains possible—even at extremely high amplitudes.
Snapthrough Summary • • On the right, the driving amplitude at which snapthrough first occurs is plotted along with the threshold amplitudes for supratransmission. Three separate responses occur across each region of the initial bandgap analysis: o Lower bandgap: the wave can propagate as the amplitude increases to high values, and the wave propagates throughout the entire system. o Transmission region: the wave will transmit and propagate for lower driving amplitudes. o Upper bandgap: the wave will not propagate throughout the entire system despite snapthrough occurrences.
Theoretical System, k 2 Constant Parameters defined by the paper: • k 1 = 10 • k 2 = 1* • d = 0. 2 Summary: • Following theoretical equation outlined in the paper, we can now define a transmission region that is dependent on the ratio of k 1/k 2, with maintaining k 2 constant*.
k 2 Constant Simulation Parameters defined by the paper: • k 1 = 10 • k 2 = 1* • d = 0. 2 Summary: • Like theoretical profile, in simulation the lower bandgap limit remains constant as the ratio is changed, whereas the upper limit transitions higher with an increasing ratio.
Theoretical System, k 1 constant Parameters defined by the paper: • k 1 = 10* • k 2 = 1 • d = 0. 2 Summary: • Following theoretical equation outlined in the paper, we can now define a transmission region that is dependent on the ratio of k 1/k 2, with maintaining k 1 constant*.
k 1 Constant Simulation Parameters defined by the paper: • k 1 = 10* • k 2 = 1 • d = 0. 2 Summary: • Like theoretical profile, in simulation the lower bandgap limit increases as the ratio decreases, and the upper limit also rapidly increases for low ratios (k 1/k 2 < 1).
Theoretical System, Springs Constant Parameters defined by the paper: • k 1 = 10 • k 2 = 1 • d = 0. 2* • • When d is varied from the parameter in the paper, the switch can be thought of as straightening out vertically or flattening out horizontally. Either way, this parameter does not have an affect on the overall limits regardless.
Bandgap Temperature Frequency (w) is varied: • U = 0. 2 : 7 Temperature (T) is varied: • T = 100: 50: 750 Summary • System is first equilibrated by incrementally raising the temperature over an initialization period. • The overall profile remains the same, and temperature does not alter the upper or lower limit a significant amount.
Bandgap Temperature Frequency (w) is varied: • U = 0. 2 : 7 Temperature (T) is varied: • T = 5 : 100 Summary • Here, theoretical limits are compared to the simulated limits, which are determined by the largest increase/decrease indices of the RMSD profiles.
Temperature Amplitude (U) is varied: • U = 0. 25 : 10 Temperature (T) is varied: • T = 25 : 50 : 750 Summary: • Here, the RMS displacement is provided for the temperature range. The overall profile remains similar to the initial results and relatively unaffected by the temperature. • The supratransmission transition does not occur at a significantly lower amplitude for any of the temperatures.
Temperature Amplitude (U) is varied: • U = 0. 25 : 10 Temperature (T) is varied: • T = 25 : 50 : 750 Summary: • Here, the threshold amplitudes have been determined for the entire temperature range and compared to the expected threshold value. • There is no temperature trend despite a slight lowering in the threshold value below T = 100. • Damping occurs for both the energy and displacement profiles, meaning the supratransmission likely is not occurring. It was defined as a total snap count above 100.
Amplitude Temperature Max Energy obtained for each temperature Max Total Snapthrough obtained for each temperature
Conclusions Summary Future Work • • Created a bistable switch at the nanoscale, which behaves similarly to the experimental counterpart: o Bandpass filter o Nonlinear snapthrough Snapthrough response throughout the entire frequency spectrum: • Snapthrough thresholds • Nonlinear capabilities Lack of a distinct temperature effect on both the bandgap and nonlinear response. • • Analyzing the system response to changes in geometry along with changes to the overall length of the periodic bistable chain. Another extension to this research could be exploring the mechanical diode, which could be achieved by a system with two different linear spring constants. Additional exploration into temperature could also provide more clarity to our results (smaller step-size or a more efficient way of determining nonlinear snapthrough occurrences).
THANK YOU QUESTIONS 34
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