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Cover
Title Page
Preface
Chapter 1: Background and Introduction
1. 1.1 Examples of Bistable Structures and Systems
2. 1.2 Characteristics of Bistable Structural Dynamics
3. 1.3 The Exploitation of Bistable Structural Dynamics
4. 1.4 Outline of This Book
5. References
Chapter 2: Mathematical Modeling and Analysis of Bistable Structural Dynamics
1. 2.1 A Linear Oscillator
2. 2.2 Stability
3. 2.3 A Monostable Nonlinear Oscillator
4. 2.4 A Bistable Oscillator
5. 2.5 Analytical Methods for Steady‐state Dynamics
6. 2.6 Bifurcations of Bistable Systems
7. 2.7 Multiple Degrees‐of‐Freedom Systems
8. 2.8 An Electromechanical Bistable System
9. 2.9 Summary
10. References
Chapter 3: Vibration Control
1. 3.1 Topic Review
2. 3.2 High and Adaptable Damping Using Bistable Snap‐through Dynamics
3. 3.3 Isolating Structures Under Large Amplitude Excitations Through Activation of Low Amplitude Interwell Dynamics: Criteria for Excitation‐induced Stability
4. 3.4 Exploiting Excitation‐induced Stability for Dual‐stage Vibration Isolation
5. 3.5 Dynamic Stabilization of a Vibration Suspension Platform Attached to an Excited Host Structure
6. 3.6 Snap‐through Dynamics for Vibration Absorption
7. References
Chapter 4: Vibration Energy Harvesting
1. 4.1 Topic Review
2. 4.2 Effective and Straightforward Design Guidelines for High Performance Operations
3. 4.3 Understanding Superharmonic Energy Diffusion in Bistable Energy Harvesters
4. 4.4 Optimal and Robust Energy Harvesting from Realistic Stochastic Excitations Using the Dynamics of Structures Designed Near the Elastic Stability Limit
5. 4.5 Amplifying the Snap‐through Dynamics of a Bistable Energy Harvester Using an Appended Linear Oscillator
6. 4.6 A Linear Dynamic Magnifier Approach to Bistable Energy Harvesting
7. References
Chapter 5: Sensing and Detection
1. 5.1 Topic Review
2. 5.2 Detecting Changes in Structures by Harnessing the Dynamics of Bistable Circuits
3. 5.3 Improving Damage Identification Robustness to Noise and Damping Using an Integrated Bistable and Adaptive Piezoelectric Circuit
4. 5.4 Passive Microscale Mass Detection and Progressive Quantification by Exploiting the Bifurcations and Resonant Dynamics of a Two DOF Bistable Sensor
5. References
Chapter 6: Emerging Themes and Future Directions
1. 6.1 Emerging Themes
2. 6.2 Challenges and Future Outlooks
3. References
Index
End User License Agreement

List of Illustrations

Chapter 01
1. Figure 1.1 Schematic of a bistable system composed of springs, mass, and frame. Here which makes the central configuration an unstable position of static equilibrium.
2. Figure 1.2 Dependence of (a) spring force and (b) stored potential energy on the displacement position of the inertial mass.
3. Figure 1.3 (a) Damping module of buckling beams arranged in series and parallel within a housing frame for energy attenuation between the ends of the module. (b) Bistable device for vibration isolation of suspended top bearing mass. (c) Cantilevered beam with piezoelectric PVDF patches and magnetic attraction induced bistability for energy harvesting. (d) Bistable circuitry attached to host beam structure via piezoelectric transducer for sensing structural change.
4. Figure 1.4 (a) For an excitation frequency at which both the low and high intrawell dynamics coexist, the time series of the displacement responses are presented. The thick dotted lines indicate the statically‐stable equilibria. (b) The displacement magnitude spectral responses of the low and high amplitude intrawell dynamics are shown as computed from the time series data plotted in (a). The energy diffusion to other harmonics of the excitation frequency is apparent and distinct between the response forms. In (a,b), the parameters are (p, ω, γ, β) = (0.1, 1.1, 0.09,1), with initial conditions = (1.0041, 0.0961) for the low amplitude oscillations and (2.2858, –0.9302) for the high amplitude dynamics.
5. Figure 1.5 (a) For an excitation frequency at which both the low and high interwell dynamics coexist, the time series of the displacement responses is presented. (b) The displacement magnitude spectral responses of the low and high amplitude interwell dynamics are shown as computed from the time series data plotted in (a). The energy diffusion to other harmonics of the excitation frequency is apparent and distinct between the response forms. In (a, b), the parameters are (p, ω, γ, β) = (8, 2.5, 0.09, 1), with initial conditions (–1.9297, –0.0944) for the low amplitude interwell responses and = (–2.9443, –0.1791) for the high amplitude interwell vibrations.
6. Figure 1.6 Depending on the excitation level, the single‐periodic, steady‐state response of a bistable system may exhibit non‐unique dynamic forms. (a) A high amplitude interwell and both low and high amplitude intrawell responses may occur depending on the excitation frequency. Here, the parameters are (p, γ, β) = (0.18,0.09,0.7). The corresponding linear system stationary response computed using the counterpart equation . (b) A greater excitation level than that used in (a) may induce low or high interwell responses or an intrawell response. Here, the parameters are (p, γ, β) = (8, 0.09, 1). (c) Responses of softening and hardening Duffing oscillators contrasted against the corresponding linear system. Here, the parameters are (p, γ, β) = (0.18, 0.09, 0.7). The thin gray curves indicate unstable response forms.
7. Figure 1.7 The transient responses of bistable systems may exhibit high sensitivity to initial conditions. In this example, the initial conditions are =(0.292721, –0.3) for the response settling to the normalized displacement stable equilibria position of –1 (black curve), while the small change of initial conditions to = (0.292722, –0.3) causes the transient dynamics to tend to the stable equilibrium of normalized displacement +1 (gray curve). Here, the parameters are (γ, β) = (0.09, 1). The thick dotted lines indicate the statically‐stable equilibria.
8. Figure 1.8 (a) An harmonically‐excited bistable system may lead to aperiodic dynamics. This characteristic is also strongly sensitive to initial conditions. Thus, after only a small number of excitation periods, two nearly identical initial conditions show great divergence in the transient, aperiodic snap‐through dynamics of the bistable systems. (b) After 500 excitation periods, the aperiodic responses bear little resemblance. Here, the parameters are (p, ω, γ, β)=(0.37, 1.29, 0.1, 1), with initial conditions = (0.292707, –0.3) leading to the responses plotted in the black curves and with initial conditions = (0.292706, –0.3) leading to the responses plotted in the gray curves. The thick dotted lines indicate the statically‐stable equilibria. The normalized sampling frequency in the 4th‐order Runge–Kutta numerical integration is 32.
9. Figure 1.9 Depending on the excitation frequency, the single‐periodic, steady‐state response of a bistable system may exhibit non‐unique dynamic forms. In the following examples, changes in normalized excitation frequency ω lead to variation in the amplitude and existence of the low and high amplitude intrawell dynamics and the high amplitude interwell vibrations of a bistable system, as functions of changing normalized excitation level p. In contrast, only the amplitude of the counterpart linear system responses is seen to vary. Results computed for (a) a normalized excitation frequency ω = 0.8, (b) ω = 1.0, and (c) ω = 1.2. Here, the parameters are (γ, β) = (0.09, 0.7). The corresponding linear system stationary response computed using the counterpart equation .
10. Figure 1.10 Examples of the frequency response function (frf) dependence of a bistable system as compared to a linear system. The frf is the displacement amplitude with respect to the excitation level. The linear system frf (dash‐dot curve) is independent of excitation level. For bistable systems, the interwell response frfs (solid curves) reduce in amplitude but extend to higher frequencies as the excitation level increases. The high amplitude intrawell frfs (dashed curves) reduce in amplitude and frequency range as the excitation level increases, while the low amplitude intrawell dynamics (dotted curves) primarily reduce in frequency bandwidth without notable change in frf level. The light to dark curves for the bistable system are determined using p from 0.17, 0.22, to 0.27. Here, the parameters are (γ, β) = (0.09, 1). The linear system frf is computed using damping factor γ = 0.09.
11. Figure 1.11 The stochastic resonance phenomenon occurs when combined periodic and stochastic excitations, which are individually insufficient to activate snap‐through dynamics of a bistable system, lead to large amplitude interwell dynamics which closely follow the excitation period of the periodic excitations. Here, the parameters are (p, ω, γ, β) = (0.2, 0.05, 0.5, 1) for the cases involving the periodic excitations. The normalized, additive white noise variance is 0.04. All cases use initial conditions = (0, 0).
12. Figure 1.12 (a) Although bistable systems may be excited at a frequency ω, the steady‐state dynamics may occur primarily at integer multiples or fractions of the excitation frequency. Two cases are shown for which the system primarily vibrates at a frequency of 2ω (gray curve) or ω/3 (black curve. (b) The displacement magnitude spectral responses of the bistable system dynamics, as computed from the time series data plotted in (a), show clearly the unique spectral concentrations of the input excitation energies. In (a,b), for the dynamics tending to the frequency of 2ω, the parameters are (p, ω, γ, β) = (0.2, 1.3, 0.0001, 1), with initial conditions = (–3.2987, 1.5132); while for the dynamics tending to the frequency ω/3, the parameters are (p, ω, γ, β) = (0.2, 3, 0.01, 1) with initial conditions = (–0.7507, 1.4841).
Chapter 02
1. Figure 2.1 General schematics of several base‐excited systems. (a) An oscillator with arbitrary spring restoring force . (b) The fundamental mode response of a simply‐supported beam with axial load. (c) The fundamental mode response of a ferromagnetic beam tip in the presence of magnetic fields. (d) A bistable composite plate mid‐point response.
2. Figure 2.2 (a) Spring restoring force profiles for linear, monostable hardening Duffing, and bistable oscillators, and (b) corresponding potential energies.
3. Figure 2.3 Example free response (a) time series and (b) phase portraits, of a linear oscillator for various initial conditions and ζ = 0.05 and rad s^–1.
4. Figure 2.4 Monostable nonlinear oscillator free response (a) time series and (b) phase portrait, for m = 1 kg, d = π N s m^–1, k₁ = (2π5)² N m^–1, k₃ = 100(2π5)² N m^–1.
5. Figure 2.5 (a) Example free response displacement time series of bistable system. (b) Basin of attraction map with single example case of phase portrait free response overlaid as light dashed curve. β = 1, γ = 0.1.
6. Figure 2.6 Basin of attraction maps for an excited bistable system where (γ, β, p, ω) = (0.1, 1, 0.2, 0.8). Plot (b) is a close‐up view of (a) indicated by the bounded region in (a). Plot (c) is a close‐up view of (b) indicated by the bounded region in (b).
7. Figure 2.7 (a) Illustration of the point sampling strategy to generate a Poincaré map. (b) Poincaré map for excited bistable system with parameters. (γ, β, p, ω) = (0.1, 1, 0.37, 1.29).
8. Figure 2.8 (a) Intrawell oscillation displacement amplitude, and (b) phase. (c) Comparison of directly simulated results (dashed curves) against those predicted by the method of multiple scales (solid curves) for . In all results γ = 0.1, β = 1, and ε = 0.1.
9. Figure 2.9 Analytical and numerical predictions of steady‐state intra‐ and interwell bistable system dynamics as nonlinearity strength β increases from (a) 0.01 to (b) 0.1 to (c) 1, as a function of normalized excitation frequency ω. (d) Predictions of dependence on excitation level p. Numerical results plotted as unfilled circles. Notable bifurcations denoted by crosses and labels.
10. Figure 2.10 Schematic of electromechanical bistable system having piezoelectric transduction in parallel with restoring and damping forces. The corresponding circuit schematic, where the resulting transduced voltage is loaded by a resistance R, is shown on the right.
Chapter 03
1. Figure 3.1 (a–c) Bilinear bistable spring properties where the force–displacement relationship is described by a discontinuous pair of linear functions. (d–f) Bistable spring properties where the force–displacement relationship is defined by a smooth cubic polynomial.
2. Figure 3.2 Schematic of the mass–spring‐damper bistable oscillator excited by a ground motion z(t). The pivoting points are shown as black dots.
3. Figure 3.3 (a1–e1) Plots of the displacements of the bistable device as functions of time. Thick dotted lines indicate the stable equilibria positions. (a2–e2) Phase portraits of displacement and velocity. Circle points indicate the stable equilibria. All results computed for excitation frequency Hz. Excitation amplitude Z is (a) 1 mm, (b) 6 mm, (c) 7 mm, (d) 13 mm, and (e) 15 mm.
4. Figure 3.4 Average power dissipated by the bistable device damper. The plots (a–e) here correspond to the counterpart results (a–e) in Figure 3.3. Long‐dash lines are ±5% of the final value, short‐dash lines are ±2% of the final value.
5. Figure 3.5 (a1–c1) Phase portraits and (a2–c2) computations of average power dissipated for bistable device parameters that induce aperiodic motions. Long‐dash lines are ±5% of the final value, short‐dash lines are ±2% of the final value. The results show that average power dissipation of aperiodic dynamics does converge given a long enough window of the integration used in Eq. (3.7). (a) kg, kN m^–1, Hz, mm. (b) kg, kN m^–1, Hz, mm. (c) kg, kN m^–1, Hz, mm.
6. Figure 3.6 Average power dissipated as a function of excitation amplitude at a frequency ω/2π of (a) 6, (b) 9, and (c) 12 Hz. Results for 20 different initial conditions are shown as individual stars whereas the circles indicate the mean power dissipation for a given excitation amplitude. The shadings indicate (white) intrawell oscillations, (light gray) multiharmonic or aperiodic motions, (medium gray) snap‐through or aperiodic behaviors, and (dark gray) snap‐through or EIS dynamics. To distinguish aperiodic or EIS from the snap‐through dynamics, the aperiodic or EIS motions yield much smaller values of average power dissipation.
7. Figure 3.7 Bistable device composed using a frame excited by shaker platform motions in the direction indicated by the double‐arrow and a sliding bearing mass which moves with respect to the frame according to the pivot point attachments of the coil spring. A dashpot damper dissipates the relative motion between the bearing mass and frame.
8. Figure 3.8 (a1–d1) Plots of the measured displacements of the bistable device as functions of time. Thick dotted lines indicate the stable equilibria positions. (a2–d2) Phase portraits of displacement and velocity. Circle points indicate the stable equilibria. All results measured for excitation frequency Hz. Excitation amplitude Z is (a) 1.0, (b) 6.8, (c) 8.1, (d) 11.6 mm.
9. Figure 3.9 Experimentally determined average power dissipation as the excitation amplitude of shaker motions is varied at 6 Hz. The cross (square) points indicate the average power measures for an increasing (decreasing) amplitude of excitation. The shadings indicate: white, intrawell oscillations; medium gray, multiharmonic or aperiodic motions; and dark gray, snap‐through or aperiodic behaviors.
10. Figure 3.10 Schematic of stable and unstable domains of Mathieu’s equation, where the stable dynamics occur in the hatched regions. B₁, B₂, and B₃ are the first three transition curves on the plane.
11. Figure 3.11 (a) Steady‐state response amplitude. Black solid (dashed) lines are stable interwell (intrawell) responses whose stability is determined via Jacobian analysis. Dotted lines are the first three approximated transition curves, S₁, S₂, and S₃. Circles (crosses) correspond to interwell (intrawell) response amplitudes computed via direct numerical integration. Gray lines are analytically predicted unstable responses. Criteria predict stable interwell response regions between S₁ and S₂ or above S₃ (as indicated by arrows). Two interwell responses A and B coexist at and the corresponding (b) phase portraits, (c) time series and (d) frequency spectra are presented, with black and gray representing snap‐through and EIS dynamics, respectively. Filled circles or dashed lines indicate the positions of the stable equilibria in (b) and (c), respectively.
12. Figure 3.12 (a) EIS response amplitude as a function of damping and frequency with system parameter . Increasing lightness of the contour represents increasing response amplitude. Dashed lines correspond to estimated frequency boundaries using simplified form of Eqs (3.18) and (3.19). (b) and (c) are two analytically predicted, representative responses with damping and , respectively. Solid black (gray) lines indicate interwell (intrawell) responses and dashed gray lines are unstable responses.
13. Figure 3.13 (a) Basin of attraction map for system parameters , , where unshaded, gray, and black shaded areas indicate snap‐through, EIS, and period‐3 harmonic interwell responses, respectively. (b) Steady‐state time domain responses where is the excitation period, and (c) corresponding phase portraits for different initial conditions with damping . In (b) and (c) thick dashed lines and filled circles, respectively, represent the two statically‐stable equilibria positions.
14. Figure 3.14 (a) EIS response amplitude as a function of excitation level and frequency with system parameter . Increasing lightness of the contour represents increasing response amplitude. Dashed lines correspond to estimated frequency boundaries using simplified form of Eqs (3.18) and (3.19). (b) and (c) are two analytically predicted, representative responses with excitation level and , respectively. Solid black (gray) lines indicate interwell (intrawell) responses and dashed gray lines are unstable responses.
15. Figure 3.15 Schematic and photograph of the experimental system.
16. Figure 3.16 (a) Experimentally determined phase portraits with dashed black, solid gray, and black curves representing intrawell, EIS, and snap‐through dynamics, respectively. Filled circles represent two stable equilibrium positions of the bistable oscillator. (b) Experimentally determined displacement frf of the bistable oscillator as a function of shaker input frequency and amplitude. Circles and triangles are experimentally determined frequency boundaries of stable EIS region.
17. Figure 3.17 Schematic of the bistable dual‐stage isolator and the harmonic exciting forces.
18. Figure 3.18 Force transmissibility comparison of a bistable dual‐stage isolator with an equivalent linear dual‐stage isolator and a linear single‐stage isolator: (a) force transmissibility; (b) stroke motion amplitudes of the first stage.
19. Figure 3.19 Force transmissibility of the valley response as a function of the bistable stage stroke amplitude.
20. Figure 3.20 Force transmissibility comparison of the bistable and linear dual‐stage isolators corresponding to variation in frequency tuning ratio where .
21. Figure 3.21 Force transmissibility comparison of the bistable and linear dual‐stage isolators corresponding to variation in mass ratio where .
22. Figure 3.22 Force transmissibility comparison of the bistable and linear dual‐stage isolators corresponding to variation in the loss factor of the first stage where .
23. Figure 3.23 Numerically predicted responses of the bistable dual‐stage isolator at ω = 0.8 subjected to two different initial conditions : the top row, (−1, 0, 0, 0); the bottom row: (0, 0, 0, 0). From left to right: (a) and (d), time series plots of the transmitted force p_t; (b) and (e), phase plots of the bistable stage motion x and velocity x′; (c) and (f), FFT plots of the bistable stage motion x.
24. Figure 3.24 Basin of attraction map at . The initial condition ranges of (x₀, y₀) are both from –3 to 3. Single‐periodic valley response, black‐shaded area. Multiharmonic response with ω/3 component, unshaded area.
25. Figure 3.25 Percentage likelihood of the single‐periodic valley responses in and single‐periodic intrawell responses in for three cases of the bistable stage loss factor γ₁.
26. Figure 3.26 Schematic of a prototype of the bistable dual‐stage isolator configured by cantilevered beams and mutually repulsive magnets (a) and bistable stage restoring spring forces F of the prototype and analytical model with respect to the bistable stage’s stroke X_t.
27. Figure 3.27 Force transmissibility of the force P_t to the excitation force P as a function of normalized excitation frequency ω.
28. Figure 3.28 Experimental set‐up: (a) schematic and (b) photograph, to validate the valley response of a bistable dual‐stage isolator prototype.
29. Figure 3.29 RMS transmissibility as a function of normalized bistable stage stroke.
30. Figure 3.30 Schematic of the two DOF coupled system where a bistable suspension is attached to a flexible host structure.
31. Figure 3.31 Analytically predicted transmissibility |T | of the two DOF coupled systems with the bistable and linear suspensions in bandwidth .
32. Figure 3.32 (a) Transmissibility |T| and (b) normalized suspension relative amplitudes r₁ of the two DOF coupled systems with the bistable and linear suspensions.
33. Figure 3.33 (a) Transmissibility contour map of the two DOF coupled system with the bistable suspension and (b) the transmissibility maximum |T | corresponding to variation of normalized excitation level p₀.
34. Figure 3.34 (a) Photograph and (b) schematic of experimental set‐up.
35. Figure 3.35 (a) Experimental and (b) analytical transmissibility |T | corresponding to excitation level variation.
36. Figure 3.36 (a) Experimental and (b) analytical transmissibility |T | corresponding to variation of bistable suspension damping.
37. Figure 3.37 (a) Experimental and (b) analytical transmissibility |T | corresponding to suspended mass variation.
38. Figure 3.38 Excited host linear structure with bistable device attachment.
39. Figure 3.39 System dynamics as tuning ratio f varies. (a) Bistable attachment displacement amplitude and (b) phase lag, (c) host structure displacement amplitude.
40. Figure 3.40 (a) Force amplitude and (b) force phase as applied by attachment. (c) Total force amplitude applied to the host structure for different tuning frequency ratios f.
41. Figure 3.41 System dynamics as mass ratio μ varies. (a) Host displacement amplitude, (b) total force applied to host structure.
42. Figure 3.42 System dynamics as damping constant γ₁ varies. (a) Host structure displacement amplitude, (b) total force amplitude on host structure, (c) bistable attachment force amplitude and (d) phase.
43. Figure 3.43 Experimental system of primary host structure and bistable attachment.
44. Figure 3.44 Relative acceleration frf magnitude of host system with and without the bistable attachment for several base acceleration RMS levels: (a) 1.07, (b) 1.35, and (c) 1.60 m s^–2.
45. Figure 3.45 Time series of forces as computed via measured data. Forces plotted correspond to the following dynamic phenomena: (a) Constructive snap‐through at 2.62 Hz. (b) Destructive snap‐through during near‐stabilization of unstable equilibrium at 5.25 Hz (OOP). (c) Destructive snap‐through at 5.75 Hz (OOP). (d) P2 snap‐through at 5.75 Hz.
Chapter 04
1. Figure 4.1 Common design strategies to induce bistability into vibration energy harvesting systems. (a) Two magnets repulse each other. (b) A ferromagnetic substructure is attracted to a pair of frame‐mounted magnets. (c) Geometric influences or axial compressive forces are exerted on the energy harvesting structure. In all sub‐figures, the lightly shaded bistable energy harvester represents the other statically stable equilibrium configuration. The piezoelectric layers are one representative example of including electromechanical transduction.
2. Figure 4.2 Analytical predictions of (a) normalized displacement amplitude transfer function and (b) displacement phase lag in degrees relative to the harmonic excitation. Parameters are .
3. Figure 4.3 (a) Schematic and (b) photograph of experimental set‐up.
4. Figure 4.4 Measurements of (a) normalized displacement amplitude transfer function and (b) displacement phase lag in degrees relative to the harmonic excitation.
5. Figure 4.5 Comparison of critical normalized excitation frequency and amplitudes which activate snap‐through dynamics for high power generation. Criteria A (dashed) and B (dash‐dot) and Melnikov's equation (solid) are labeled, denoting the boundaries described in the text. Parameters (a) and (b) . Light, medium, and dark gray shadings, respectively, refer to simulation results for which 10, 50, and 90% of the numerical trials resulted in steady‐state snap‐through vibrations of the bistable harvester.
6. Figure 4.6 Bistable energy harvester driven by base excitation having either piezoelectric or electromagnetic transduction mechanisms. The respective harvesting circuits of the transduction mechanisms are shown on the right.
7. Figure 4.7 Comparison of 1‐term and 2‐term harmonic balance predictions for an excitation amplitude , showing those responses proportional to the excitation frequency ω and three times the excitation frequency 3ω. Excitation frequency dependence of the (a) displacement amplitudes r_1,3 and (b) average powers P_1,3.
8. Figure 4.8 Excitation frequency dependence of the (a) displacement amplitudes r_1,3 and (b) average powers P_1,3 for an excitation amplitude .
9. Figure 4.9 Zoom on Figure 4.8 in the excitation frequency bandwidth of significant activation of superharmonic dynamics.
10. Figure 4.10 Excitation frequency dependence of the (a) displacement amplitudes r_1,3 and (b) average powers P_1,3 for an excitation amplitude .
11. Figure 4.11 Excitation amplitude dependence of (a) displacement amplitudes r_1,3 and (b) average harvested powers P_1,3. .
12. Figure 4.12 Experimental set‐up. Bistable ferromagnetic cantilever with piezoelectric PVDF patches, attached to shaker platform, with PVDF electrode leads connected to external resistance (not shown).
13. Figure 4.13 Experimentally measured bistable harvester responses. Left‐hand column (1), average power across the load resistance showing spectral components of fundamental ω and superharmonic 3ω in consequence to excitation at frequency ω. Right‐hand column (2), piezoelectric voltage amplitude during excitation frequency sweep. Using base acceleration levels of (a) 2.53, (b) 3.01, (c) 3.48, (d) 3.94, and (e) 4.39 m s^–2.
14. Figure 4.14 Nonlinear energy harvester with piezoelectric and electromagnetic induction mechanisms and corresponding circuitry.
15. Figure 4.15 Measured natural frequencies of beam as equal magnet spacing d is adjusted. Inset shows experimental platform schematic.
16. Figure 4.16 Piezoelectric cantilever free tip velocity vibrations due to initial displacement imposed on beam tip. Here, the magnet spacing is d = 15 mm, near point of essential nonlinearity.
17. Figure 4.17 Energy harvester mean‐square velocity normalized to noise correlation maxima. Part (a,b) shows the analytical results and simulated data (filled data points). Part (c,d) shows the experimental results with legends indicating mean‐square noise input acceleration values.
18. Figure 4.18 Influence of linear stiffness term κ on harvester mean‐square velocity normalized to excitation level.
19. Figure 4.19 Analytically‐predicted influence of bias α on harvester mean‐square velocity normalized to excitation level.
20. Figure 4.20 (a) Experimental set‐up for evaluating influence of gravitational bias. (b,c,d) Experimental harvester mean‐square voltage responses normalized to data for inclination ° using increasing noise excitation levels from (b) to (d). Legends indicate mean‐square base acceleration levels for each test.
21. Figure 4.21 Schematic of base‐excited, inductive bistable energy harvester to which is attached a linear oscillator. The corresponding external harvesting circuit is shown on the right.
22. Figure 4.22 Analytically predicted responses of the bistable harvester as the mass ratio μ and excitation frequency ω are varied, corresponding to the snap‐through dynamic regime. (a) Fundamental displacement amplitude r₁ and (b) phase lag φ₁ (degrees). (c) Average power density determined from .
23. Figure 4.23 Analytically predicted responses of the auxiliary linear harvester as the mass ratio μ and excitation frequency ω are varied, corresponding to the snap‐through dynamic regime of the harvesting system. (a) Fundamental displacement amplitude r₂ and (b) phase lag φ₂ (degrees).
24. Figure 4.24 Analytically predicted responses of the bistable harvester as the frequency ratio f and excitation frequency ω are varied, corresponding to the snap‐through dynamic regime. (a) Fundamental displacement amplitude r₁ and (b) average power density .
25. Figure 4.25 Comparison of numerically‐ and analytically‐predicted snap‐through (a) displacement amplitudes of the bistable harvester r_1,3 and (b) average powers P_1,3, considering contributions from the fundamental and third‐order superharmonic. The auxiliary linear oscillator mass and frequency ratios are and , respectively.
26. Figure 4.26 Experimental set‐up of bistable device coupled to auxiliary linear oscillator.
27. Figure 4.27 Snap‐through acceleration frf amplitude of the bistable device with and without the linear oscillator attachment.
28. Figure 4.28 Snap‐through due to base acceleration excitation at 4 Hz. (a) Acceleration autospectra and (b) time series for the dynamic regime dominated by motions proportional to the frequency of excitation.
29. Figure 4.29 Snap‐through due to base acceleration excitation at 4 Hz. (a) Acceleration autospectra and (b) time series for the dynamic regime dominated by motions proportional to the third‐order superharmonic.
30. Figure 4.30 Comparison of linear and bistable energy harvesting system architectures which leverage auxiliary linear oscillators (left) compared to linear dynamic magnifier stages (right). The stage linearity is denoted by the text on the respective inertial mass. The components labeled by EH are the energy harvesting transduction mechanisms.
31. Figure 4.31 Base‐excited linear oscillator with attached bistable energy harvester having electromagnetic transduction mechanism, with corresponding resistive load power harvesting circuit shown on the right right.
32. Figure 4.32 Snap‐through dynamics of the bistable harvester as a function of excitation frequency ω, for a variety of mass ratios μ. Shown are the bistable harvester (a) displacement amplitude r₁; (b) phase lag ϕ₁; and (c) average power density P₁.
33. Figure 4.33 Snap‐through dynamics of the bistable harvester as a function of excitation frequency ω, for a variety of frequency ratios f. Shown are the bistable harvester (a) displacement amplitude r₁ and (b) average power density P₁.
34. Figure 4.34 Snap‐through dynamics of the bistable harvester as a function of excitation frequency ω, for a variety of electromechanical coupling coefficients θ. Shown are the bistable harvester (a) displacement amplitude r₁ and (b) average power density P₁.
35. Figure 4.35 Snap‐through dynamics of the bistable harvester in the coupled system as a function of excitation frequency ω, for a variety of linear dynamic magnifier stage damping factors γ₂.
36. Figure 4.36 Snap‐through dynamics of the undamped coupled bistable energy harvesting system with linear magnifier stage as a function of excitation frequency ω for a variety of mass ratios μ. The dashed lines are the respective computations of the limiting frequency .
37. Figure 4.37 Photograph of coupled linear dynamic magnifier and bistable device system.
38. Figure 4.38 Relative acceleration frf magnitude of the bistable oscillator as a function of the base excitation frequency, for a variety of damping factors.
39. Figure 4.39 Snap‐through dynamics of the bistable oscillator of the coupled system as functions of the excitation frequency for a variety of attached bistable device mass ratios. Shown in (a) is the bistable harvester displacement amplitude r₁ predicted analytically in comparison to the corresponding (b) measurements of relative acceleration frf of the bistable device.
Chapter 05
1. Figure 5.1 Distinction between out‐of‐plane and in‐plane buckling of bistable microstructures, and illustration of the pull‐in phenomenon.
2. Figure 5.2 Configuration of the sensing platform. In experiments, a cantilevered beam is excited at its base. A piezoelectric patch attached near the cantilever clamped end provides a transduced voltage to a bistable circuit.
3. Figure 5.3 Bifurcations in the normalized displacement amplitude r response, as a function of excitation level p. Results are computed using the Eq. (3) and (4) in [90]. In (a) and (b) . . Arrows indicate the trajectories followed in consequence to increasing and decreasing level of excitation.
4. Figure 5.4 Voltage function of the bistable circuit.
5. Figure 5.5 Theoretical and experimental responses of the cantilever beam tip velocity and the normalized bistable circuit output voltage. The base acceleration on the beam is (a) 0.438 and (b) 2.17 m s^–2.
6. Figure 5.6 Experimentally measured bistable circuit output voltage frf magnitude. Lines of constant frf phase lag in degrees are plotted as the thin solid curves. The thin dashed curve represents the approximate transition region from the intra‐ to the interwell dynamic regimes.
7. Figure 5.7 (a) Damping ratio and change in natural frequency of the monitored structure as mass is added to the host monitored structure: cantilever beam. (b) Output voltage autospectra normalized to the case of no added mass. The letter locations correspond to similar response features as exhibited in the bifurcation curve presented in Figure 5.3a.
8. Figure 5.8 (a) Bistable circuit input voltage level. (b) Representative output voltage dynamic regime triggering as observed from the bistable circuit.
9. Figure 5.9 Illustration of monitored structure with piezoelectric transducer and adaptive circuit.
10. Figure 5.10 (a) Configuration of the cantilever beam integrated with bistable and adaptive piezoelectric circuitry. Damage is simulated as a 15% stiffness loss on the second beam element. (b) Results of the damage identification model.
11. Figure 5.11 (a) Frequency shifts determined by 40 runs of BB method each having random initial conditions for each inductance. (b) Damage identification using the mean results from the 40 resonance frequency shifts.
12. Figure 5.12 Example of (a) noise‐free integrated system resonance and (b) that with 32 dB SNR additive noise.
13. Figure 5.13 Frequency shifts determined by DPD or BB approaches and their standard deviations as signal‐to‐noise ratio increases. Values of standard deviations for each case of SNR and detection method are provided in the figure alongside the data points.
14. Figure 5.14 (a) Damage identification comparison for structure with mild damping and with varying degrees of low level additive noise. (b) RMSD of damage identification.
15. Figure 5.15 (a) Damage identification comparison for structure with increased damping and for varying degrees of low level additive noise. (b) RMSD of damage identification.
16. Figure 5.16 Types of dynamic bifurcations in a bistable system. SN: saddle‐node. PD: period‐doubling. PF: pitchfork. Responses predicted using a constant and one‐term Fourier series expansion as the steady‐state solution to the equation of motion . Parameters (p, γ, β) are (a) (0.06, 0.09, 0.7), (b) (0.25, 0.09, 0.7), and (c) (4.5, 0.09, 0.7).
17. Figure 5.17 Schematic of coupled linear–bistable micromechanical sensor architecture. (a) Realization via host microcantilever and inset buckled cantilever. (b) Schematic showing coordinate convention used in model. (c) Lumped parameter representation of (b).
18. Figure 5.18 (a) Fundamental mode displacement response transfer function amplitude, H, presented as a function of the excitation frequency and increasing fundamental modal mass. This amplitude is equivalent to the excitation level which is passed to the bistable inclusion of the system so as to induce the bifurcation featured in (b). The numbers in (a,b) are correlated between phenomena happening with respect to modal dynamics and those with respect to bifurcation dynamics of the bistable inclusion.
19. Figure 5.19 Proof‐of‐concept experimental sensor architecture. (a) Host beam with inset beam, showing fabrication details prior to magnet attachment. (b) System cantilevered from shaker for testing. (c) and (d) Buckled states of inset bistable beam due to magnetic repulsion.
20. Figure 5.20 Slow frequency sweep into the SN bifurcation. (a) Experiment: host beam velocity. (b) Simulation: host beam velocity. (c) Simulation: bistable inclusion displacement. Inset images denote zoomed views of bifurcation activation. The reduced‐darkness portions of the curves in the simulated results (b,c), for times approximately greater than 92 s, are responses which are predicted to occur after the activation of the jump but which were not measured experimentally because the trial was finished.
21. Figure 5.21 Exemplary model simulation of sequential bifurcation activation due to mass accumulation upon host structure. Sensor sub‐system (a) response displacements and (b) velocities.
22. Figure 5.22 (a) Host beam frf due to low‐level white noise input denoting natural frequencies and the corresponding frequencies at which the SN bifurcation is triggered for cases of mass addition. (b) Measured frf magnitudes for all cases of added mass ratio and triggering frequencies.
23. Figure 5.23 (a) Experimentally measured frf around the fundamental mode as added mass increases (light to dark). (b) Comparable simulated frf responses due to stochastic excitation. (c) Measured and numerically simulated fundamental mode natural frequencies as mass increases, with linear fits and residual norms (in inset).
24. Figure 5.24 Dependence on additive noise excitation contribution for (a) controlled excitation frequency sweeps into the bifurcation and (b) passive, mass accumulation sweeps into the bifurcation (mr = mass ratio).
25. Figure 5.25 (a) Measured adsorbed mr as a function of constant excitation frequency and level. Mass ratio accumulation rate for all trials is 0.001. Data points are mean values across 40 simulated trials, bars are first standard deviations. (b) Measured adsorbed mr as a function of mr adsorption rate and excitation level. Data points are mean values across 5 sets of excitation frequency each using 40 simulated trials, bars are first standard deviations.
26. Figure 5.26 Ratio of host oscillator velocity² variance during energetic response period to before/after as a function of mass ratio adsorption rate and excitation level. Data points are mean values across 5 sets of excitation frequency each using 40 simulated trials.
27. Figure 5.27 Schematic of coupled linear–bistable sensor architecture employing in‐plane dynamics of electrostatically‐actuated, host comb drive shuttle oscillator and inset curved beam.
Chapter 06
1. Figure 6.1 (a) Before and (b) after collapse of the cellular metamaterial internal topology between two states of equilibrium for the same load condition.
2. Figure 6.2 Axial compliance integrated into a compressed nonlinear energy harvesting beam system. .
3. Figure 6.3 Bistable waterbomb origami showing the two stable states of the rotation (a) 'in' and (b) 'out'.

Harnessing Bistable Structural Dynamics

For Vibration Control, Energy Harvesting and Sensing

Preface