2026/07/24 by Mateus Ferraz, Daniel Longo, Marcus Varanis
paper · doi:10.1088/1361-6404/ae9015
Abstract Multi-degree-of-freedom (MDOF) vibration problems are commonly introduced through the second-order matrix equation M q¨ + C q˙ + Kq = F(t), an approach that is effective but often leaves the underlying energy structure implicit. In this tutorial, we present a Hamiltonian-first workflow for quadratic MDOF systems with coupled translational-rotational motion. Starting from compact expressions for the kinetic and potential energies, we construct the canonical momenta and an explicit Hamiltonian, derive Hamilton's equations for the conservative core, and incorporate linear dissipation and external actuation as nonconservative generalized forces while preserving a transparent energy-based interpretation.

As a worked example, we consider a rigid regular cube with six degrees of freedom supported by spring-damper elements at its bottom vertices. The central step is a geometry-driven assembly in which each element elongation is written as δkα=Bkαq, leading to K=Σk,α kkα BkαT Bkα and C=Σk,α ckα BkαT Bkα. This construction connects the canonical formulation to the standard mass-damping-stiffness (MCK) form and to familiar modal concepts, including a physical interpretation of the modal degeneracy induced by the geometric symmetry of the support layout. To probe the system dynamically, we employ a chirp-to-harmonic excitation together with a continuous wavelet transform (CWT), used as a pedagogical time-frequency modal scanner to make resonance crossings visible in a single run. In the conservative limit, we compare a fixed-step fourth-order Runge-Kutta (RK4) integrator with a second-order leapfrog, or Störmer-Verlet, scheme using the Hamiltonian H(q(t),p(t)) as an energy diagnostic, showing that the structure-preserving method suppresses the secular energy drift exhibited by a general-purpose explicit integrator at the same time step. A fully documented reference implementation, including both integrators, is provided to support classroom use and independent replication.