This work examines the transient nonlinear oscillations of complex composite systems under mechanical pressure utilizing Reddy’s theory of higher-order shear deformation plates. The analysis entails the formulation of nonlinear equations that regulate the stress and displacement fields inside the composite structure. The dynamic behavior is defined by using Hamilton’s idea. The transient response of the composite system under various loading conditions is determined by deriving analytical solutions. The work offers a full knowledge of the dynamic behavior of advanced composite structures by using Reddy’s higher-order shear deformation plate theory, which takes into account the effects of transverse shear deformation and rotating inertia. The intricate interplay of material properties, shape, and external force is elucidated by the nonlinearity inherent in the equations. This provides valuable information on the temporary vibrations that occur in the composite system. The work employs rigorous mathematical analysis and closed-form solutions to clarify the intricate transient dynamics of modern composite structures, providing insights into how they react to mechanical loads over time. The results enhance the overall comprehension of composite material behavior and provide useful insights for the technical design and optimization of composite structures.