Two-Port Dynamic Stiffness Model for Flexure Displacement Amplifier

This project will implement the general two-port dynamic stiffness model for three bridge-type flexure displacement amplifiers (Rhombic, Parallel, Aligned) as described in Ling (2019).

I selected Mojo for education purposes, to get some idea about the language and its ecosystem. For now this file is an AI-generated plan with some manual edits for myself, with some basic structure to guide the implementation.

Modules

1. model.mojo

This file will contain the physics and mathematics.

• Structs: To hold geometric parameters ($h, L, \theta, etc.$) and material properties ($E, \rho$).
• Functions:
  • get_beam_stiffness(params, freq): Calculates $D_e(\omega)$ (Eq 1, 2).
  • transform_matrix(local_matrix, angle): Rotates matrices (Eq 3, 5).
  • get_transfer_matrix(stiffness_matrix): Converts stiffness to transfer matrix (Eq 11).
  • condense_limb( ... ): Combines multiple beam elements into one equivalent stiffness matrix (Eq 13).
  • assemble_system( ... ): Constructs the final system matrix (Eq 15).
  • solve_...(): Functions to find determinants (frequencies) and inverse (static/dynamic response).

2. main.mojo

This file drives the simulation.

• Define specific dimensions for the Rhombic/Parallel/Aligned amplifiers.
• Call model functions to build the system.
• Iterate through frequencies to find roots (natural frequencies).
• Solve for steady-state response at $\omega=0$ (static amplification/stiffness).

Implementation Plan

Step 1: Data Structures

Goal: Create a way to pass parameters around easily.

• Define a struct for GeometricParameters (Beam dimensions $L, h, b$, Angle $\theta$).
• Define a struct or class for MaterialProperties (Young's modulus E, density rho).

Step 2: Beam Dynamic Stiffness Matrix

Goal: Implement Equation (1) and (2).

• Create a function calc_beam_stiffness(geom, mat, omega) returning a 6x6 Matrix.
• Key Point: The matrix elements ($\delta_1 .. \delta_8$) depend on frequency dependent parameters $\alpha$ and $\beta$.
• Tip: Mojo's SIMD or Tensor types can be useful for matrices, or use a simple 2D array/pointer wrapper if external linear algebra libs are not yet available.

Step 3: Coordinate Transformation

Goal: Implement Equation (3) and (5).

• Implement the Rotation Matrix $R_i$ (Eq 5).
• Create a function to apply $R^T D_e R$.

Step 4: Transfer Matrix & Condensation

Goal: Implement simplified limb logic (Eq 11, 12, 13).

• Implement conversion from Stiffness $D$ to Transfer $T$ (Eq 11).
• Implement matrix multiplication for $T_{total} = T_4 T_3 T_2 T_1$.
• Convert back from $T_{total}$ to equivalent Stiffness $D_{eq}$ (Eq 13).

Step 5: System Assembly

Goal: Build the final matrix (Eq 15).

• Combine the condensed limb matrices into the global system matrix.
• Note the structure of Equation 15 – it's a sparse block matrix assembly.

Step 6: Solvers

Goal: Extract results.

• Statics ($\omega=0$):
  • Solve $F = K X$. Invert the matrix (or use Gaussian elimination).
  • Calculate Amplification Ratio $R$, Input Stiffness $K_{in}$ (Eq 18).
• Dynamics ($\omega > 0$):
  • Natural Frequency: Find $\omega$ where $\det(D(\omega)) = 0$. You can use a root-finding algorithm (e.g., Bisection or Newton-Raphson) on the determinant function.

Verification

We will verify the implementation using the Rhombic Amplifier case from Section 3.1 of the paper.

• Parameters: $h=1mm, L=15mm, \theta=10^\circ$, etc.
• Expected Result: Compare against Figure 4 curves.