Linear harmonic response

Analysis methodROUTE-DEPENDENT · frequency sweep is model-specific

A complex displacement amplitude is solved independently at each declared excitation frequency.

01 Model and scope

Assuming sinusoidal steady-state motion converts time derivatives into factors of iω. The dynamic stiffness K − ω²M + iωC is solved at each requested frequency; amplitude and phase both matter, especially near resonance.

Route assumptions and calculation checks

Complex residual, amplitude, and phase are more informative than peak-only comparison.
PETSc is not enabled for the current complex harmonic route.
Frequency grid, damping, shell condensation, and load assumptions remain part of the evidence.

Useful foundations and prerequisites

Linear static Units, signs and error measures

02 Essential equations and symbols

(K - \omega{}^2M + i\omega{}C) \hat{u}(\omega{}) = f̂(\omega{})

Definitions and conventions

$\omega{}$: Excitation angular frequency.
û, f̂: Complex displacement and load amplitudes.
$C$: Damping matrix, including the declared Rayleigh model where used.
$i$: Imaginary unit.

01Complex excitation

02Z(ω) = K − ω²M + iωC

03Amplitude + phase

04Residual by frequency

Linear harmonic response · method mapEach excitation frequency gives a complex solve whose amplitude and phase are both retained.

4 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Assume a steady sinusoidal load and response

Represent load and displacement by complex amplitudes multiplying e^{iωt}. Differentiating in time becomes multiplication by iω, so acceleration contributes −ω² and velocity contributes iω. This turns the linear differential system into one complex algebraic system per frequency.

\mathbf u(t)=\Re\{\widehat{\mathbf u}e^{i\omega t}\},\qquad \mathbf f(t)=\Re\{\widehat{\mathbf f}e^{i\omega t}\}

2 · Assemble the complex dynamic stiffness

The frequency-dependent operator combines elastic stiffness, inertial softening, and damping. Solve it independently at each requested frequency unless the implementation proves a valid reuse. Complex values retain both amplitude and phase; using magnitudes alone can conceal sign and phase errors.

\left(\mathbf K-\omega^2\mathbf M+i\omega\mathbf C\right)\widehat{\mathbf u}(\omega)=\widehat{\mathbf f}(\omega)

3 · Read the single-degree-of-freedom oracle

For an oscillator, the complex response is force divided by dynamic stiffness. The denominator becomes small near the undamped natural frequency; damping bounds the peak and introduces a phase shift. This closed-form case checks the algebra and response convention.

\widehat u(\omega)=\frac{F_0}{k-m\omega^2+i c\omega},\qquad |\widehat u|=\frac{|F_0|}{\sqrt{(k-m\omega^2)^2+(c\omega)^2}}

4 · Audit the complex response and scope

Report the complex residual, amplitude, phase, frequency grid, damping model, and any shell condensation or recovery. The current complex route does not enable PETSc. Each published MITC4 comparison belongs to its named mesh, loading, and output definition; it is not a general harmonic validation.

r(\omega)=\frac{\|\mathbf Z(\omega)\widehat{\mathbf u}-\widehat{\mathbf f}\|_2}{\max(\|\widehat{\mathbf f}\|_2,1)}

04 Algorithm and implementation

Read the linked versioned source for the actual assembly and solver behavior. The derivation above explains the mathematics and is not a verbatim source excerpt.

Harmonic implementation

05 Verification and limitations

ROUTE-DEPENDENT · frequency sweep is model-specific

MITC4 evidence includes modal-frequency, broadband, damping-sensitivity, and NAFEMS 13H cases.

The formulation and its explanatory checks do not by themselves establish a qualified application domain. Consult the capability matrix and the applicable recorded case.

06 Sources and related sheets

Harmonic implementation Run input Method demonstrations

Continue the reasoning

Newmark average-acceleration method Sparse generalized eigenproblem

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