Theory

This section describes the theoretical foundation of the Complex Exponential Algorithm (CEA) for time-domain modal estimation.

Mathematical Background

Free Vibration Response

For a multi-degree-of-freedom linear system with viscous damping, the free vibration response can be expressed as a sum of decaying exponentials:

\[y(t) = \sum_{k=1}^{n} A_k e^{\lambda_k t}\]

where:

• \(A_k\) are the modal participation factors (complex)

• \(\lambda_k\) are the modal frequencies (complex)

• \(n\) is the number of modes

Modal Frequencies

The modal frequencies are complex:

\[\lambda_k = -\zeta_k \omega_{n,k} \pm i\omega_{d,k}\]

where:

• \(\omega_{n,k}\) is the natural frequency of mode \(k\)

• \(\zeta_k\) is the damping ratio (fraction of critical damping)

• \(\omega_{d,k} = \omega_{n,k}\sqrt{1-\zeta_k^2}\) is the damped frequency

For real-valued responses, the modal frequencies occur in complex conjugate pairs.

Complex Exponential Algorithm

The CEA extracts modal parameters through the following steps:

Step 1: Toeplitz Matrix (Equation 7)

Build a Toeplitz matrix from the response time series:

\[\begin{split}Y = \begin{bmatrix} y_1 & y_2 & \cdots & y_{2n} \\ y_2 & y_3 & \cdots & y_{2n+1} \\ \vdots & \vdots & \ddots & \vdots \\ y_{N-2n} & y_{N-2n+1} & \cdots & y_{N-1} \end{bmatrix}\end{split}\]

where \(N\) is the total number of samples and \(n\) is the number of modes.

Step 2: Polynomial Coefficients (Equation 11)

Solve the least squares problem:

\[Y \boldsymbol{\alpha} = -\mathbf{y}_{target}\]

where \(\mathbf{y}_{target} = [y_{2n+1}, y_{2n+2}, \ldots, y_N]^T\) and \(\boldsymbol{\alpha} = [\alpha_0, \alpha_1, \ldots, \alpha_{2n-1}]^T\) are the characteristic polynomial coefficients.

Step 3: System Poles (Equation 8)

Find the roots of the characteristic polynomial:

\[\prod_{k=1}^{2n} (z - z_k) = z^{2n} + \alpha_{2n-1}z^{2n-1} + \cdots + \alpha_1 z + \alpha_0 = 0\]

The roots \(z_k\) are the system poles in the discrete-time domain.

Step 4: Modal Frequencies (Equations 9-10)

Convert discrete-time poles to continuous-time modal frequencies:

\[\lambda_k = \frac{1}{\Delta t} \ln(z_k)\]

where \(\Delta t\) is the sampling interval.

In Hz:

\[f_k = \frac{1}{2\pi\Delta t} \ln(z_k)\]

Step 5: Vandermonde Matrix (Equation 12)

Build the modal matrix:

\[\begin{split}\Lambda_{1,N} = \begin{bmatrix} e^{\lambda_1 t_1} & e^{\lambda_1 t_2} & \cdots & e^{\lambda_1 t_N} \\ e^{\lambda_2 t_1} & e^{\lambda_2 t_2} & \cdots & e^{\lambda_2 t_N} \\ \vdots & \vdots & \ddots & \vdots \\ e^{\lambda_n t_1} & e^{\lambda_n t_2} & \cdots & e^{\lambda_n t_N} \end{bmatrix}\end{split}\]

Step 6: Mode Shapes (Equation 13)

Solve for the modal participation factors:

\[\mathbf{A} = \Lambda_{1,N}^{-1} \mathbf{y}\]

where \(\mathbf{A} = [A_1, A_2, \ldots, A_n]^T\).

Since \(\Lambda\) is typically not square, the pseudo-inverse is used:

\[\mathbf{A} = (\Lambda^T \Lambda)^{-1} \Lambda^T \mathbf{y}\]

Physical Interpretation

Natural Frequency

The natural frequency \(f_k\) is:

\[f_k = \frac{|\lambda_k|}{2\pi}\]

in Hz, or:

\[\omega_{n,k} = |\lambda_k|\]

in rad/s.

Damping Ratio

The damping ratio is:

\[\zeta_k = -\frac{\text{Re}(\lambda_k)}{|\lambda_k|}\]

A positive damping ratio indicates stable (decaying) modes.

Damped Frequency

The damped frequency is:

\[\omega_{d,k} = \text{Im}(\lambda_k)\]

or in Hz:

\[f_{d,k} = \frac{\omega_{d,k}}{2\pi}\]

Assumptions and Limitations

Assumptions

1. Linear system: The structure behaves linearly

2. Viscous damping: Damping is proportional to velocity

3. Free vibration: No external forcing during measurement

4. Observable modes: All modes of interest are excited

5. Stationary: System properties don’t change during measurement

Limitations

1. Noise sensitivity: CEA is sensitive to measurement noise

2. Model order: Requires knowing or estimating the number of modes

3. Closely spaced modes: May have difficulty distinguishing modes with similar frequencies

4. Non-proportional damping: Assumes classical (proportional) damping for real mode shapes

Practical Considerations

Sampling Requirements

• Nyquist criterion: Sample at least 2x the highest frequency of interest

• Recommended: 10x the highest frequency for good accuracy

• Duration: Record at least 3-5 complete cycles of the lowest frequency

Model Order Selection

The model order \(n\) must be chosen appropriately:

• Too low: Misses important modes

• Too high: Introduces spurious (noise-driven) poles

Consider using stabilization diagrams or information criteria for order selection.

Noise Mitigation

For noisy data:

• Use longer time series

• Apply appropriate filters before analysis

• Average multiple measurements if possible

• Validate results against physical expectations

Eigensystem Realization Algorithm

The ERA extracts modal parameters from impulse response data by constructing a minimal state-space realization.

Mathematical Background

For a linear time-invariant system, the impulse response can be expressed as:

\[y_k = C A^{k-1} B\]

where:

• \(A\) is the state matrix

• \(B\) is the input matrix

• \(C\) is the output matrix

• \(k\) is the discrete time index

The modal parameters are extracted from the eigenvalues and eigenvectors of the state matrix \(A\).

ERA Algorithm Steps

Step 1: Hankel Matrix (Equation 29)

Build a Hankel matrix from the impulse response:

\[\begin{split}H(k) = \begin{bmatrix} y_k & y_{k+1} & \cdots & y_{k+s-1} \\ y_{k+1} & y_{k+2} & \cdots & y_{k+s} \\ \vdots & \vdots & \ddots & \vdots \\ y_{k+r-1} & y_{k+r} & \cdots & y_{k+r+s-2} \end{bmatrix}\end{split}\]

where \(r\) and \(s\) determine the size of the Hankel matrix.

Step 2: Singular Value Decomposition (Equation 30)

Perform SVD on the Hankel matrix:

\[H(0) = U \Sigma V^T\]

where \(U\) and \(V\) are orthogonal matrices and \(\Sigma\) contains the singular values.

Step 3: Reduced-Order Projection (Equation 31-32)

Project onto the dominant subspace:

\[U_n = U[:, :n], \quad \Sigma_n = \Sigma[:n, :n], \quad V_n = V[:, :n]\]

where \(n\) is the model order (typically \(2 \times\) number of modes).

Step 4: State-Space Realization (Equation 33-35)

Recover the system matrices:

\[A = \Sigma_n^{-1/2} U_n^T H(1) V_n \Sigma_n^{-1/2}\]

\[B = \Sigma_n^{1/2} V_n^T e_1\]

\[C = e_1^T U_n \Sigma_n^{1/2}\]

where \(e_1\) is the first unit vector.

Step 5: Modal Parameter Extraction (Equations 27-28)

Extract modal parameters from eigenvalues of \(A\):

\[\mu_k = \frac{\ln(\lambda_k)}{\Delta t}\]

where \(\lambda_k\) are the eigenvalues of \(A\) and \(\Delta t\) is the sampling interval.

Natural frequencies (Hz):

\[f_k = \frac{|\mu_k|}{2\pi}\]

Damping ratios:

\[\zeta_k = -\frac{\text{Re}(\mu_k)}{|\mu_k|}\]

ERA vs CEA Comparison

Feature	CEA	ERA
Input data	Free decay response	Impulse response
Matrix formulation	Toeplitz + polynomial roots	Hankel + SVD
Noise robustness	Moderate	Good (SVD filtering)
Model order	Must specify exactly	Flexible via SVD truncation
Computational cost	Lower	Higher (SVD)
Multiple outputs	Single output	Multiple outputs native

References

The implementation is based on:

Fahey, S. O’F., & Pratt, J. (1998). Time domain modal estimation techniques. Experimental Techniques, 22(6), 45-49.

Juang, J.-N., & Pappa, R. S. (1985). An eigensystem realization algorithm for modal parameter identification and model reduction. Journal of Guidance, Control, and Dynamics, 8(5), 620-627.

See also classical modal analysis literature for additional context and extensions.