Structural Dynamics Laboratory

Engineering problem

Structural Dynamics Laboratory is a Python-based platform for analyzing structural response under time-dependent excitation, particularly earthquake ground motion. It combines matrix structural mechanics, finite-element assembly, modal analysis, numerical time integration, a reduced nonlinear shear-building model and engineering diagnostics.

Mathematical foundations

Dynamic equilibrium

The structural equations of motion are

$$M\ddot{\mathbf u}+C\dot{\mathbf u}+K\mathbf u=-M\mathbf r a_g(t),$$

where $M$, $C$ and $K$ are the mass, damping and stiffness matrices, $\mathbf u$ is the structural displacement vector, $\mathbf r$ maps ground motion to the degrees of freedom and $a_g(t)$ is ground acceleration.

Modal analysis

Natural frequencies and mode shapes follow the generalized eigenvalue problem

$$K\boldsymbol\phi=\omega^2M\boldsymbol\phi.$$

The modal workflow includes mass normalization, participation factors and effective modal mass so that individual modes can be interpreted in relation to the applied excitation.

Numerical time integration

Response histories use Newmark integration. The average-acceleration form uses

$$\beta=\frac14,\qquad\gamma=\frac12,$$

to update displacement, velocity and acceleration over discrete time steps.

Nonlinear structural response

The nonlinear module uses a bilinear elastoplastic lumped shear-building representation with yielding, post-yield stiffness, hysteresis and iterative equilibrium solution. This plasticity model applies to the shear-building system; it is not a general nonlinear frame-element solver.

Two-dimensional frame finite elements

Linear 2D beam-column elements support stiffness and mass assembly, static response, support reactions, element forces and modal analysis. The formulation is intended for transparent computational studies rather than design-code certification.

Implemented capabilities

• Floor displacement, velocity and acceleration histories
• Interstory drift and ground-acceleration response
• Natural frequencies, mode shapes and effective modal mass
• Elastic response spectra
• Static frame analysis, support reactions and element forces
• Equilibrium residuals and energy accounting
• Time-step studies and static benchmark comparisons

Numerical verification

Verification combines matrix and static benchmarks, equilibrium residual checks, energy accounting and time-step comparisons. These tests examine implementation correctness under controlled numerical conditions. They do not constitute physical validation or structural design-code certification.

Engineering applications

The laboratory supports seismic-response studies, modal and vibration analysis, finite-element implementation, numerical verification and the development of custom engineering software.

Technical resources

The verified live application uses the project’s custom domain.