Source-linked AI summary
Enhancing Energy Harvesting in Harmonically Forced Oscillators via Regenerative Impulsive Braking
Nilay Kant
TL;DR
To test whether regenerative impulsive braking can augment passive harvesting, the paper derives the braking-induced transient and an exact finite-horizon energy-gain expression. Its analysis finds zero-crossing optimal braking over an infinite horizon and finite-horizon gains that can exceed the infinite-horizon value.
Problem
The paper asks whether regenerative braking can augment passive harvesting, where prior impulsive damping work lacked a closed-form characterization of incremental gain and optimal braking conditions.
Source: §I · p. 1Method
The authors derive the braking-induced transient analytically and use it to obtain exact harvested-energy gains over finite and infinite post-braking horizons.
Source: §I · p. 1, §III · p. 2, A. Finite-Horizon Harvested-Energy Gain · p. 3Results
Finite-horizon gain can exceed its infinite-horizon value; for an infinite horizon, optimal braking occurs at excitation zero crossings, with complete braking maximizing achievable positive gain.
Source: §I · p. 1, A. Insights from Analytical Results · pp. 4–5Takeaways & Limitations
Regenerative braking can add energy extraction beyond the energy recovered during braking by reshaping the transient and subsequent passive harvesting.
Source: §VI · p. 6
Abstract
from arXiv · showhide
This paper investigates whether regenerative impulsive braking can enhance passive energy harvesting in harmonically forced oscillators. An impulsive braking event directly extracts kinetic energy while simultaneously inducing a transient that alters subsequent passive harvesting. The analysis shows that suitably timed braking can produce a net energy gain over passive harvesting alone. The braking-induced transient is derived analytically and used to obtain an exact expression for the harvested-energy gain over an arbitrary finite post-braking horizon, together with conditions for positive gain and the optimal braking instant. The corresponding infinite-horizon analysis shows that optimal braking occurs at zero crossings of the harmonic excitation, with complete braking maximizing the gain. Numerical results further show that the finite-horizon harvested-energy gain can exceed its infinite-horizon value. Simulations using short-duration high-gain braking closely match analysis, supporting potential of the proposed approach for energy-harvesting applications.
I. Introduction
The paper studies regenerative impulsive braking as a semi-active addition to passive harvesting, addressing the lack of closed-form harvested-energy gains and optimal braking conditions. It derives finite- and infinite-horizon results, including cases where finite-horizon gain exceeds the infinite-horizon value.
I. Introduction
- Vibration energy harvesting spans applications from ocean waves and transportation to infrastructure, human motion, and low-power devices.Passive, active, and semi-active methods differ in their dissipation or actuation strategies; semi-active methods modulate system quantities using comparatively low-power actuation.Source: §I · p. 1
- The analysis derives exact finite- and infinite-horizon harvested-energy gains, conditions for positive gain, and optimal braking conditions.The finite-horizon expression applies over an arbitrary post-braking horizon without requiring the transient to be negligible.Source: §I · p. 1
- For an infinite horizon, optimal braking occurs at zero crossings of the external force, and complete braking maximizes gain whenever positive gain is achievable.Source: §I · p. 1
II. Problem Formulation
The paper formulates a harmonically forced underdamped linear oscillator whose damper harvests energy passively. It compares energy extracted after regenerative impulsive braking with passive damping alone over a fixed post-braking horizon.
II. Problem Formulation
- The model is a harmonically forced underdamped linear oscillator with mass, damping, and stiffness parameters, and a passive damper that harvests energy from the mass's motion.Source: §II · pp. 1–2
- Before braking, the oscillator follows its periodic steady-state displacement and velocity under the harmonic excitation.Source: §II · p. 2
- At a chosen time, the ideal regenerative brake instantaneously reduces velocity without changing position, retaining a fraction α of the pre-braking velocity.Source: §II · p. 2
- The objective is to compare braking and passive-only energy extraction over a fixed post-braking horizon, derive the net gain, and find conditions for positive gain and optimal braking time.The brake directly harvests the reduction in the mass's kinetic energy.Source: §II · p. 2
III. Braking-Induced Transient Response
Because braking leaves the harmonic forcing unchanged, its effect is an additional transient superposed on the oscillator's steady-state response. The transient follows homogeneous dynamics from the braking-induced change in velocity and is used to quantify subsequent harvested-energy change.
III. Braking-Induced Transient Response
- After braking, the unchanged harmonic forcing means the response differs from steady state only by an added braking-induced transient.The post-braking displacement is written as the steady-state displacement plus the transient response.Source: §III · p. 2
- Subtracting the steady-state equation cancels the common forcing, so the transient obeys a homogeneous system with initial conditions set by unchanged position and reduced velocity.Source: §III · p. 2
- The braking-induced transient is solved using the oscillator's natural frequencies and decay rate, and the resulting post-braking velocity is used to quantify harvested-energy change.Source: §III · p. 2
A. Finite-Horizon Harvested-Energy Gain
The section derives an exact expression for the harvested-energy gain over any finite post-braking horizon and identifies when braking is beneficial. It also characterizes the braking instant that maximizes that gain.
A. Finite-Horizon Harvested-Energy Gain
- The exact finite-horizon gain compares regenerative braking with passive damping over [t0, t0 + T] and does not require neglecting the braking-induced transient.The common passive contribution cancels, leaving the incremental gain due to braking and its transient.Source: A. Finite-Horizon Harvested-Energy Gain · p. 3
- Braking provides a net gain when P_T cos^2 ψ + Q_T sin ψ cos ψ > 0.The gain also depends on the decay rate, excitation frequency, and damped natural frequency.Source: A. Finite-Horizon Harvested-Energy Gain · p. 3
- The gain formula follows by adding the direct braking contribution to the change in harvested energy caused by the transient.The derivation substitutes the transient response and braking energy into the energy comparison.Source: A. Finite-Horizon Harvested-Energy Gain · p. 3
- For a given T, the maximizing braking instant satisfies 2ψ − γ_T = 2nπ, with the maximum given by the corresponding expression.When R_T = 0, the net gain is zero; otherwise, the optimal instant depends on T, the braking ratio α, and the oscillator-response terms P_T and Q_T.Source: A. Finite-Horizon Harvested-Energy Gain · p. 3
B. Infinite-Horizon Harvested-Energy Gain
The section takes the finite-horizon gain to its infinite-horizon limit and derives conditions for positive gain and optimal braking. The optimum occurs at excitation-force zero crossings, with complete braking maximizing the gain when the stated phase condition holds.
B. Infinite-Horizon Harvested-Energy Gain
- Theorem 2 gives the infinite-horizon harvested-energy gain as the limit of the finite-horizon gain.The limit describes the net effect after the induced transient has decayed and the oscillator has returned to periodic steady state.Source: B. Infinite-Horizon Harvested-Energy… · p. 4
- For any fixed braking ratio 0 ≤ α < 1, net infinite-horizon gain ΔW∞ > 0 occurs if and only if the stated criterion is satisfied, though its form is not supplied here.At ω = ω_n, ΔW∞ ≤ 0 for every braking instant, so regenerative impulsive braking cannot provide a positive infinite-horizon energy gain.Source: B. Infinite-Horizon Harvested-Energy… · p. 4
- Optimal braking instants correspond to zero crossings of the external harmonic force, and α⋆ = 0 maximizes gain when cos φ ≠ 0.The overall maximum is given by the corresponding expression.Source: B. Infinite-Horizon Harvested-Energy… · p. 4
A. Insights from Analytical Results
Finite-horizon harvested-energy gain and the optimal braking phase vary with the post-braking horizon; at intermediate horizons, the gain can exceed its infinite-horizon value. The optimal gain also depends on braking ratio, with stronger braking producing larger maximum gain.
A. Insights from Analytical Results
- For short horizons, both harvested-energy gain and optimal braking phase depend strongly on Tn; as horizons grow, the optimal phase approaches ωt0 = nπ.The finite-horizon curves approach the infinite-horizon gain as Tn increases.Source: A. Insights from Analytical Results · p. 4, Figure 2 · p. 5
- At Tn = 3, the harvested-energy gain and its maximum exceed their infinite-horizon values, while convergence with increasing horizon is nonmonotonic.The plots compare normalized gain across braking phase and braking ratio; the latter panel optimizes the braking instant for each horizon and ratio.Source: A. Insights from Analytical Results · pp. 4–5, Figure 2 · p. 5
- The gain’s rate can change sign as the braking-induced transient evolves, so increasing the horizon can first increase and later reduce accumulated gain.The gain eventually converges to its infinite-horizon value as the transient decays.Source: A. Insights from Analytical Results · pp. 4–5
- The maximum gain vanishes at α = 1, increases with stronger braking, and approaches a linear dependence on (1−α) as Tn increases.At short horizons, the dependence on α is nonlinear because braking strength affects both directly extracted energy and the subsequent transient contribution.Source: A. Insights from Analytical Results · p. 5
B. Simulation with Approximate Impulsive Braking
A short-duration high-gain feedback brake approximates the ideal complete-braking case. The simulation reproduces both immediate regenerative extraction and the post-braking transient’s effect on passive harvesting.
B. Simulation with Approximate Impulsive Braking
- Braking reduced velocity from 0.1869 m/s to 0.0100 m/s in about 0.02 s, giving αeff = 0.0535 and closely approximating ideal complete braking.The feedback brake was activated at the optimal phase for Tn = 3 and α = 0, then deactivated at | ˙x| ≤ 0.01 m/s.Source: B. Simulation with Approximate… · p. 5, Figure 3 · p. 5
- After braking, the system evolves passively from the modified state, producing a transient that gradually approaches periodic steady-state.The passive and braked responses coincide before braking because the oscillator is already in periodic steady-state.Source: B. Simulation with Approximate… · p. 5, Figure 3 · p. 5
- The braked cumulative-energy curve remains above the passive curve after braking, reflecting transient-modified passive harvesting in addition to direct brake extraction.The simulation therefore reproduces both energy-extraction mechanisms of the ideal impulsive model.Source: B. Simulation with Approximate… · p. 5, Figure 3 · p. 5
VI. Conclusion
The paper derives finite-horizon harvested-energy gain for regenerative impulsive braking and identifies conditions for benefit and optimal timing. It concludes that the gain can exceed its infinite-horizon value at intermediate horizons, while simulations support approximate realization using short-duration high-gain braking.
VI. Conclusion
- An analytically derived braking-induced transient yields an exact harvested-energy-gain expression over arbitrary finite post-braking horizons, with conditions for positive gain and optimal braking instant.For infinite horizons, optimal braking occurs at external-force zero crossings, and complete braking maximizes gain whenever positive gain is achievable.Source: §VI · p. 6
- Finite-horizon gain need not approach its infinite-horizon value monotonically and can exceed it at intermediate horizons.The gain’s largest value can occur before the braking-induced transient has completely decayed.Source: §VI · p. 6
- Short-duration high-gain braking simulations closely reproduce the ideal impulsive model’s predicted mechanism, supporting approximate practical braking actions.The braking-induced transient changes subsequent passive harvesting in addition to directly extracting energy.Source: §VI · p. 6
- Future work will examine repeated braking and hybrid steady-state dynamics to maximize average harvested power through event timing and braking strength.The framework is also slated for extension to general periodic forcing, active harvesting, and uncertain nonlinear multidegree-of-freedom oscillators.Source: §VI · p. 6