Chaotic Inertial Energy Systems Using Railgun Pulses and Nonlinear Spring Architectures
Creators: name: Adam, McEvoy
Publisher: name: Zenodo
Year: 2025
Subjects: railgun
dampening
chaos
spring

description: Title:

Chaotic Inertial Energy Systems Using Railgun Pulses and Nonlinear Spring Architectures



Abstract:

This simulation framework explores three chaotic mechanical systems—pulsed linear railgun actuators, single nonlinear spring oscillators, and coupled asymmetric spring systems—as candidates for inertial energy generation and modulation. Each system produces non-harmonic acceleration signatures, which can be harvested and stabilized through a predictive regulation framework. The models are evaluated for their kinetic energy output, instantaneous acceleration, residual force error relative to Earth gravity, and energy harvesting efficiency. A double pendulum model is included as a baseline reference. All systems are run in parallel, and their results are plotted comparatively to assist in real-world applicability assessment for energy capture, damping stabilization, or onboard power systems in dynamic platforms.



System Descriptions and Mathematical Basis:

1. Double Pendulum (Baseline Reference)

The double pendulum exhibits chaotic motion due to its non-integrable equations of motion. Its Lagrangian mechanics are modeled with coupled second-order differential equations derived from Newtonian dynamics. The ODEs used are:

d2θ1dt2,d2θ2dt2=functions of θ1,θ2,θ˙1,θ˙2\frac{d^2\theta_1}{dt^2}, \frac{d^2\theta_2}{dt^2} = \text{functions of } \theta_1, \theta_2, \dot{\theta}_1, \dot{\theta}_2dt2d2θ1,dt2d2θ2=functions of θ1,θ2,θ˙1,θ˙2

Kinetic energy is calculated as:

KE=12M1(L1θ˙1)2+12M2(L2θ˙2)2KE = \frac{1}{2} M_1 (L_1 \dot{\theta}_1)^2 + \frac{1}{2} M_2 (L_2 \dot{\theta}_2)^2KE=21M1(L1θ˙1)2+21M2(L2θ˙2)2

2. Railgun Burst System

This model assumes discrete high-force electromagnetic pulses applied at periodic intervals. The governing second-order ODE:

d2xdt2=F(t)m,where F(t)={200,if mod(t⋅10,5)=00,otherwise\frac{d^2x}{dt^2} = \frac{F(t)}{m}, \quad \text{where } F(t) = \begin{cases} 200, & \text{if } \text{mod}(t \cdot 10, 5) = 0 \\ 0, & \text{otherwise} \end{cases}dt2d2x=mF(t),where F(t)={200,0,if mod(t⋅10,5)=0otherwise

This system produces sharp inertial spikes ideal for short bursts of high kinetic energy, but lacks continuous stabilization.

3. Single Spring Oscillator (Driven Nonlinear System)

This model includes linear spring dynamics with a sinusoidal external driving force:

d2xdt2=−kx−cdxdt+Asin⁡(ωt)\frac{d^2x}{dt^2} = -kx - c\frac{dx}{dt} + A\sin(\omega t)dt2d2x=−kx−cdtdx+Asin(ωt)

Where:





k=10k = 10k=10 is the spring constant,




c=0.5c = 0.5c=0.5 is the damping coefficient,




A=10A = 10A=10, ω=5\omega = 5ω=5 for external forcing.



It shows phase-locked but nonlinear oscillations which produce harvestable energy in low-frequency domains.

4. Coupled Spring Oscillator (Asymmetric System)

Two masses are connected by springs with unequal stiffness and coupled interaction:

{d2x1dt2=−k1x1+c(x2−x1)d2x2dt2=−k2x2+c(x1−x2)\begin{cases} \frac{d^2x_1}{dt^2} = -k_1 x_1 + c(x_2 - x_1) \\ \frac{d^2x_2}{dt^2} = -k_2 x_2 + c(x_1 - x_2) \end{cases}{dt2d2x1=−k1x1+c(x2−x1)dt2d2x2=−k2x2+c(x1−x2)

Where k1=10k_1 = 10k1=10, k2=12k_2 = 12k2=12, and c=3c = 3c=3. This model yields more unpredictable yet continuous inertial profiles due to constructive and destructive interference between the coupled oscillators.



Harvesting and Stabilization Framework:

For all models, the kinetic energy is harvested using:

Eharvested=η⋅KE(t),η=0.85E_{\text{harvested}} = \eta \cdot KE(t), \quad \eta = 0.85Eharvested=η⋅KE(t),η=0.85

Residual force relative to Earth gravity is calculated as:

Fresidual=M⋅g−M⋅a(t)F_{\text{residual}} = M \cdot g - M \cdot a(t)Fresidual=M⋅g−M⋅a(t)

Efficiency is measured by the instantaneous ratio:

Efficiency(t)=Eharvested(t)KE(t)\text{Efficiency}(t) = \frac{E_{\text{harvested}}(t)}{KE(t)}Efficiency(t)=KE(t)Eharvested(t)

Comparison charts include:





Acceleration over time




Kinetic energy profiles




Residual force (error from 1g)




Harvest efficiency





Conclusion:

The results demonstrate that chaotic systems—when driven or pulsed appropriately—can produce non-linear, rich acceleration curves that allow for meaningful energy harvesting. Each model offers unique tradeoffs:





Railgun: High-intensity, low-continuity energy.




Single Spring: Low-cost, oscillatory energy profile.




Coupled Spring: Complex dynamics with high entropy, useful for stochastic energy harvesters.




Double Pendulum: A well-characterized chaotic benchmark.



This simulation platform is extensible to other physical domains and systems, including magnetic repulsion, tensegrity frames, and biological actuators. The general principle remains: chaos, when anticipated and modulated, becomes a viable source of order and energy.