Computational technique to free vibration response in a multi-degree of freedom parametric system

•This paper provides a new idea to solve free vibration response of MDOF parametric system with a special trigonometric series, and gives a closed-form solution based on an equivalent dynamic model.•Its practicability and availability are verified by a real application.•The principal oscillation frequencies and combined frequencies can be accurately computed with known parametric excitation frequency.•The time periodic stiffness will affect principal oscillation frequencies and modal, and splitting modals corresponding to combined harmonic resonance will arise in the response.