Visual representation

Figure 1: High-level architectural overview of the simulation engine.

Technical Code Analysis

MachineDyna V.02 ( 2026-ongoing)
After a successful prototype, the biggest challenge lies in shaping the entire software into a form that is easier and faster to use, more effective, more modular, and equipped with a better graphical user interface—this time using Dear PyGui. It has turned out that this approach significantly complicates the programming itself, but such software could possess combinatorial capabilities that are logistically unsurpassed by standard approaches (in a practical design sense).

Visual representation

Figure 1: High-level architectural overview of the simulation engine.

Elastic suspension solver prototype

MachineDyna V.01 ( Sept. 2025 )
This project addresses the problem of simulating the dynamic behavior of a rigid body with six degrees of freedom (6DOF) under time-varying forces, such as unbalanced rotational excitation. The goal is to accurately capture the coupled translational and rotational motion through mass, damping, and stiffness matrices, while accounting for the spatial distribution of mounts relative to the center of gravity. By applying advanced numerical methods like the Newmark-Beta and Runge-Kutta 4 schemes, the model ensures stable and reliable time integration of the equations of motion. Additionally, 3D visualization is used to validate the physical setup and reduce input errors, enabling effective vibration analysis, structural optimization, and prediction of system response under realistic operating conditions.

Key Strengths

Modularity: Decoupled logic using specialized modules allows for isolated physical component testing.
Numerical Stability: Dual-solver implementation (Newmark-Beta & RK4) acts as an excellent internal verification tool.
Accurate 3D Physics: Proper use of orthonormal bases for unbalance modeling ensures realistic 3D spatial rotation.

Engineering Code Analysis

This code turned out to be serious piece of engineering work. It isn't just a script; it is a structured 6DOF (Six Degrees of Freedom) Rigid Body Dynamics Simulator utilizing two numerical methods.

1. Features

Modularity: Using custom modules (_calc, _visual, _solve, _class) enables scalability and isolated testing.
Numerical Robustness: Implementation of both Newmark-Beta and RK4 methods allows internal validation of results.
Unbalance Physics: Proper 3D rotational excitation using an orthonormal basis instead of simplified 2D force.
Visual Integration: Use of PyVista reduces human error in coordinate setup.

2. Mathematical problem statement

Algorithm follows the classic Linear Time-Invariant (LTI) system:

$$ [M]\ddot{x} + [C]\dot{x} + [K]x = F(t) $$

The derivation of stiffness and damping matrices relative to the Center of Gravity (CoG) correctly introduces coupling between translational and rotational motion.

3. Possible Improvement

Newmark-Beta Optimization:
Precompute the decomposition (LU or inverse) of $ M_{eff} $ outside the loop to significantly improve performance.
Error Handling:
Simplify try-except blocks by defining default values instead of duplicating logic.
Time Step Validation:
Add a warning condition:
$$ f_s < 10 \cdot f_{max} $$
to ensure RK4 stability.

Future work might be adding non-linear elements (e.g., progressive elastomer stiffness).

4. Visual Identity

ASCII headers and structured comments improve readability and maintain an "engineering-grade" style appreciated in simulation environments.

Overall

Prototype turned out to be technically sound, demonstrating strong understanding of both programming and vibration mechanics.

Figure 1: Prototype structure

Results

Figure 2: CoG displacements calculated based on forced response of 11.000 tons machine (m),both green and black line, two numeric methods match perfectly for this case.

Figure 2: CoG velocities (m/s)

Figure 2: CoG accelerations calculated (m/s2)

1. Solver I/O parameters

	MACH
	├─ Static mass: 11000.0
	├─ Machine_L: 5.0
	├─ Machine_B: 3.0
	├─ Machine_H: 3.65
	├─ Machine_Xs: 0.37
	├─ Machine_Ys: 0.32
	└─ Machine_Zs: 2.4

	DYN
	├─ FORCE_dyn.Fx: 0
	├─ FORCE_dyn.Fy: 0
	├─ FORCE_dyn.Fz: 1000
	├─ X-koordinata: 1.8
	├─ Y-koordinata: 0
	└─ Z-koordinata: 1.5

	FLO
	├─ Floor_L: 10.0
	├─ Floor_B: 15.0
	└─ Floor_H: 0.35

	MNT
	├─ 1: Lx=0.3, Ly=0.3
	├─ 2: Lx=0.4, Ly=0.5
	├─ 3: Lx=0.4, Ly=0.4
	└─ 4: Lx=0.45, Ly=0.55

	calc_MNT
	├─ 1: Lx=0.3, Ly=0.3, AREA_1=0.09, x1=-2.5, y1=-1.5, D=0.15, z1=0.0, xc=-2.35, yc=-1.35
	├─ 2: Lx=0.4, Ly=0.5, AREA_2=0.2, x2=-2.5, y2=1.5, D=0.15, z2=0.0, xc=-2.3, yc=1.25
	├─ 3: Lx=0.4, Ly=0.4, AREA_3=0.16, x3=2.5, y3=-1.5, D=0.15, z3=0.0, xc=2.3, yc=-1.3
	└─ 4: Lx=0.45, Ly=0.55, AREA_4=0.2475, x4=2.5, y4=1.5, D=0.15, z4=0.0, xc=2.275, yc=1.225

	calc_STATIC
	├─ m_tot[kg]: 11000.0
	├─ A_tot[m2]: 0.698
	├─ s_tot[N/mm2]: 0.155
	├─ s_tot[N/m2]: 154709.677
	├─ Fs_tot[N]: 107910.0
	├─ Fs_tot->[kN]: 107.91
	├─ L1 [kN]: 13.924
	├─ L2 [kN]: 30.942
	├─ L3 [kN]: 24.754
	└─ L4 [kN]: 38.291

	calc_DYN
	├─ fx[Hz]: 16
	├─ fy[Hz]: 16
	├─ fz[Hz]: 21
	├─ zeta_X: 0.1
	├─ zeta_Y: 0.1
	├─ zeta_Z: 0.15
	├─ K1: kx1=14344674.022, ky1=14344674.022, kz1=24710942.359
	├─ K2: kx2=31877053.383, ky2=31877053.383, kz2=54913205.241
	├─ K3: kx3=25501642.706, ky3=25501642.706, kz3=43930564.193
	├─ K4: kx4=39447853.561, ky4=39447853.561, kz4=67955091.486
	├─ C1: cx1=28537.822, cy1=28537.822, cz1=56183.837
	├─ C2: cx2=63417.383, cy2=63417.383, cz2=124852.972
	├─ C3: cx3=50733.906, cy3=50733.906, cz3=99882.378
	└─ C4: cx4=78479.011, cy4=78479.011, cz4=154505.553

	calc_EXCITE
	├─ t_start: 0
	├─ t_finish: 5
	├─ iteracija: 4096
	├─ fs: 819.2
	├─ Centrifugal Force: True
	├─ f_start: 1
	├─ f_finish: 10
	├─ C: x=0.0, y=0.0, z=3.1025
	├─ m_c: 2.0
	├─ r_c: 0.25
	├─ rot_axis: [1.0, 0.0, 0.0]
	├─ e1: [0.0, -1.0, 0.0]
	├─ e2: [0.0, 0.0, -1.0]
	├─ Aksijalne sile: True
	├─ fo: 0
	├─ fend: 5
	├─ Aksijalna sila: True
	├─ Fx: 0
	├─ Fy: 0
	├─ Fz: 1000
	├─ magnituda: 1000.0
	├─ jedinicni: [0.0, 0.0, 1.0]
	└─ K: x=1.8, y=0, z=1.5

	calc_MODAL
	├─ 1: frequency=8.114, active_dofs=y, θx
	├─ 2: frequency=11.798, active_dofs=x
	├─ 3: frequency=21.0, active_dofs=z
	├─ 4: frequency=23.456, active_dofs=θz
	├─ 5: frequency=33.055, active_dofs=x, y, θx, θy
	└─ 6: frequency=34.492, active_dofs=x, y, θx, θy