Dynamic Modelling of Resonance Behavior in Four Cylinder Engines Mounted on Viscoelastic Foundation

An integrated nonlinear dynamic model was developed to investigate resonance in a four-cylinder engine mounted on a viscoelastic foundation. A coupled lumped-parameter formulation captures vertical and torsional responses under unbalanced inertial forces, combustion torque, and stochastic base excitation. Time-domain simulations show that at low rotational speeds the vertical displacement reaches transient amplitudes before converging to periodic oscillations, whereas higher excitation speeds reduce steady-state amplitudes. Torsional motion exhibits initial angles near 0.05 rad that decay below 0.01 rad in steady state, with further reduction at higher speeds. Frequency-domain analysis indicates that vibration energy is concentrated in engine-order harmonics between approximately 8 and 50 Hz, while components above 60 Hz are strongly attenuated, yielding a dynamic range exceeding 50 dB. Finite element modal analysis identifies the first four structural modes between 18 Hz and 666 Hz, revealing an increasingly dominant overall translational mode and a localized directional behavior at higher frequencies. A high-dimensional kernel density spectrogram integrates modal and spectral features to map resonance regions. Results indicate that increasing rotational excitation enhances inertial stiffening, systematically reduces displacement amplitudes, and preserves bounded periodic dynamics without instability.