Chaos Inertial Dampening and Field based Impact nullification
Creators: name: Adam, McEvoy
Publisher: name: Zenodo
Year: 2025
Subjects: safety
impact
chaos
air vehicle safety

description: Overview

This simulation models a chaotic double pendulum system enhanced with an inertial dampening feedback mechanism. The purpose is to explore how predictive force injection during chaotic oscillations can amplify kinetic energy output while maintaining dynamic stability. By leveraging the natural energy of chaotic systems, the controller introduces minimal input energy at high-leverage moments—analogous to a well-timed push on a swing—resulting in disproportionately high output energy.

The system highlights how bounded, non-conservative injection paired with chaos-aware prediction algorithms can lead to high-efficiency energy harvesting or stability in mechanically volatile environments.



System Architecture

The core of the system is a classic double pendulum, simulated using nonlinear differential equations. It is augmented with an adaptive force feedback controller that monitors angular velocity and kinetic energy to decide when to inject a small counterforce into the platform.

1. Double Pendulum Equations of Motion

The state vector is y=[θ1,z1,θ2,z2]y = [\theta_1, z_1, \theta_2, z_2]y=[θ1,z1,θ2,z2], where θ\thetaθ are angles and zzz are angular velocities. The coupled second-order nonlinear ODEs governing the system are derived from Lagrangian mechanics:

δ=θ2−θ1den1=(M1+M2)L1−M2L1cos⁡2(δ)den2=L2L1⋅den1z˙1=M2L1z12sin⁡(δ)cos⁡(δ)+M2gsin⁡(θ2)cos⁡(δ)+M2L2z22sin⁡(δ)−(M1+M2)gsin⁡(θ1)den1z˙2=−L2z22sin⁡(δ)cos⁡(δ)+(M1+M2)gsin⁡(θ1)cos⁡(δ)−(M1+M2)L1z12sin⁡(δ)−(M1+M2)gsin⁡(θ2)den2\begin{aligned} \delta &= \theta_2 - \theta_1 \\ \text{den}_1 &= (M_1 + M_2)L_1 - M_2 L_1 \cos^2(\delta) \\ \text{den}_2 &= \frac{L_2}{L_1} \cdot \text{den}_1 \\ \dot{z}_1 &= \frac{M_2 L_1 z_1^2 \sin(\delta) \cos(\delta) + M_2 g \sin(\theta_2) \cos(\delta) + M_2 L_2 z_2^2 \sin(\delta) - (M_1 + M_2) g \sin(\theta_1)}{\text{den}_1} \\ \dot{z}_2 &= \frac{-L_2 z_2^2 \sin(\delta) \cos(\delta) + (M_1 + M_2) g \sin(\theta_1) \cos(\delta) - (M_1 + M_2) L_1 z_1^2 \sin(\delta) - (M_1 + M_2) g \sin(\theta_2)}{\text{den}_2} \end{aligned}δden1den2z˙1z˙2=θ2−θ1=(M1+M2)L1−M2L1cos2(δ)=L1L2⋅den1=den1M2L1z12sin(δ)cos(δ)+M2gsin(θ2)cos(δ)+M2L2z22sin(δ)−(M1+M2)gsin(θ1)=den2−L2z22sin(δ)cos(δ)+(M1+M2)gsin(θ1)cos(δ)−(M1+M2)L1z12sin(δ)−(M1+M2)gsin(θ2)



2. Kinetic Energy Modeling

The instantaneous total kinetic energy of the system is given by:

KEtotal=12M1(L1z1)2+12M2[(L1z1)2+(L2z2)2+2L1L2z1z2cos⁡(θ1−θ2)]\text{KE}_{\text{total}} = \frac{1}{2} M_1 (L_1 z_1)^2 + \frac{1}{2} M_2 \left[ (L_1 z_1)^2 + (L_2 z_2)^2 + 2 L_1 L_2 z_1 z_2 \cos(\theta_1 - \theta_2) \right]KEtotal=21M1(L1z1)2+21M2[(L1z1)2+(L2z2)2+2L1L2z1z2cos(θ1−θ2)]

This value is logged over time as the harvestable energy output.



3. Predictive Control and Force Injection

The control algorithm monitors angular velocities and kinetic energy. If either exceeds a threshold, a counterforce is triggered:

trigger=∣θ˙1∣>8or∣θ˙2∣>8orKEtotal>150 J\text{trigger} = |\dot{\theta}_1| > 8 \quad \text{or} \quad |\dot{\theta}_2| > 8 \quad \text{or} \quad \text{KE}_{\text{total}} > 150 \, \text{J}trigger=∣θ˙1∣>8or∣θ˙2∣>8orKEtotal>150J

The counterforce is computed as:

Finject=min⁡(0.5⋅(∣θ˙1∣+∣θ˙2∣),80)F_{\text{inject}} = \min \left( 0.5 \cdot (|\dot{\theta}_1| + |\dot{\theta}_2|), 80 \right)Finject=min(0.5⋅(∣θ˙1∣+∣θ˙2∣),80)

This force is subject to a control latency (e.g., 20 ms), which simulates real-world actuator delay.



4. Energy Calculations and Efficiency

Energy input and output are tracked continuously:





Energy Input:
Einput=∣Finject⋅a⋅Δt∣E_{\text{input}} = |F_{\text{inject}} \cdot a \cdot \Delta t|Einput=∣Finject⋅a⋅Δt∣



Energy Output:Derived from kinetic energy as shown above.




Energy Error:
Eerror=Einput−EoutputE_{\text{error}} = E_{\text{input}} - E_{\text{output}}Eerror=Einput−Eoutput



Dampening Efficiency:
η=1−∣EerrorEinput∣(if Einput>0)\eta = 1 - \left| \frac{E_{\text{error}}}{E_{\text{input}}} \right| \quad \text{(if } E_{\text{input}} > 0\text{)}η=1−EinputEerror(if Einput>0)



Residual Force (for evaluating uncancelled dynamic load):
R=∣platform acceleration−g∣R = |\text{platform acceleration} - g|R=∣platform acceleration−g∣




Interpretation of Results





A single energy input spike (~5 J) at the beginning initiated oscillations.




The system consistently produced energy outputs in the 15–25 J range, showing a clear amplification effect.




Residual force and acceleration returned to near steady-state following counterforce application, confirming the effectiveness of inertial dampening.




The dampening efficiency remained high after initialization, and the jerk prediction logic successfully prevented unbounded motion while preserving energy oscillation.





Applications

This system has potential real-world applications in:





Vibration harvesting from chaotic mechanical environments.




Passive stabilization for satellites or spacecraft during uncontrolled motion.




Control systems in robotics where chaotic motion must be harnessed, not eliminated.




Educational or research tools for chaos theory and nonlinear system control.