Modeling the modal shift in ATSSS prototypes

Modeling the modal shift in ATSSS prototypes involves a comparative eigenvalue analysis between a core-only baseline (Model A) and the integrated helical system (Model B). The methodology focuses on quantifying the increase in the fundamental torsional frequency ($\omega_1$) driven by the parallel stiffness contribution of the helical exoskeleton.

1. Analytical Formulation

The natural frequency is governed by the standard SDOF torsional relationship: $$\omega = \sqrt{\frac{K_\theta}{I_\theta}}$$ Where:

• $I_\theta$: Polar mass moment of inertia of the structure.
• $K_\theta$: Total torsional stiffness.

For the ATSSS prototype, the total stiffness is the sum of the core and helical contributions: $K_{\theta,total} = K_{\theta,c} + K_{\theta,h}$.

2. Helical Stiffness Projection

The primary mechanism for the modal shift is the "re-vectoring" of axial stiffness into torsional resistance. The helical stiffness contribution ($K_{\theta,h}$) is derived by projecting the axial stiffness of the exoskeleton members ($k_a = EA/L_h$) through the system geometry: $$K_{\theta,h} = \sum_i \frac{E_i A_i}{L_{h,i}} R_i^2 \cos^2\alpha_i$$ Where:

• $R$: Structural radius (lever arm).
• $\alpha$: Helix angle relative to the horizontal, defined by $\tan\alpha = p / (2\pi R)$.
• $\cos^2\alpha$: The geometric efficiency factor capturing how effectively axial stiffness resists rotational twist.

3. Numerical Implementation

In a 300m (75-story) prototype, the model parameters typically involve a polar mass moment of $I_\theta = 6.0 \times 10^8 \text{ kg·m}^2$ and a core stiffness of $K_c = 4.0 \times 10^9 \text{ N·m/rad}$. The addition of the helix ($K_h \approx 3.0 \times 10^9 \text{ N·m/rad}$) shifts the frequency from $\omega_A \approx 2.24 \text{ rad/s}$ to $\omega_B \approx 2.96 \text{ rad/s}$, representing a ~32% migration.

The following Python-based eigenvalue routine illustrates the modeling approach used in the numerical validation framework:

import numpy as np
from scipy.linalg import eig

# Prototype Parameters
I_theta = 6.0e8  # Polar mass moment of inertia (kg·m^2)
K_core = 4.0e9   # Core torsional stiffness (N·m/rad)
EA = 3.0e9       # Helix axial stiffness (N)
R = 25.0         # Structural radius (m)
alpha = np.deg2rad(35.0) # Helix angle
L_ref = 50.0     # Reference length

# 1. Calculate Helical Stiffness Contribution
K_helix = (EA * (R**2) * (np.cos(alpha)**2)) / L_ref

# 2. Define Stiffness for Models
K_A = K_core             # Baseline
K_B = K_core + K_helix   # ATSSS

# 3. Eigenvalue Analysis (Natural Frequencies)
omega_A = np.sqrt(K_A / I_theta)
omega_B = np.sqrt(K_B / I_theta)

frequency_shift_pct = ((omega_B - omega_A) / omega_A) * 100

print(f"Baseline Frequency: {omega_A:.3f} rad/s")
print(f"ATSSS Frequency: {omega_B:.3f} rad/s")
print(f"Modal Shift: {frequency_shift_pct:.2f}%")

4. Spectral Significance

The objective of modeling this shift is to move the fundamental frequency away from the dominant energy bands of site-specific wind torque spectra (e.g., Kaimal or Davenport). The "Modal Migration Control" principle states that the benefit of the shift is contingent upon the migrated frequency ($\omega_B$) encountering lower power spectral density (PSD) in the wind forcing. This ensures that the structural stiffening is coupled with a reduction in dynamic resonance.