•Design and analysis of a novel origami-inspired vibration isolator.•A three-mass body with multiple-degree-of-freedom motion is modelled.•Quasi-zero stiffness feature is achieved with compliant mechanisms.•Influence of structural weights are considered during dynamic behaviours.•Investigation of strong nonlinearity on isolation performance.•Nonlinear damping due to complex geometrical linkages is investigated. Origami-inspired structure has shown strong nonlinearity on its force response during morphing process between phases. When the origami-inspired structure is applied as vibration isolation system, the structural weight is rare to be considered and discussed in the modelling and analysis of vibration isolation system. The effects of the structural self-weight on the dynamic behaviour of the isolation system is not yet fully understood. Thus, this study aims to investigate the influence of the self-weight induced structural and damping nonlinearity on the dynamic performance of an origami-based vibration isolator. A three-mass body, which includes the payload mass, top facets’ mass and bottom facets’ mass, with multiple degree-of-freedom (DOF) motion is proposed to describe the vibration isolation system. First, a quasi-zero-stiffness feature is designed and its static performance is discussed for a set of specifically selected system parameters. Then, the equation of motion for such three-mass body with spring damping considered is derived by using the harmonic balance method (HBM) on its Lagrange's formulation, where the effects of strong nonlinearity on its dynamic performance can be investigated. The analytical expression is verified with the numerical solutions, which are obtained using the Newmark numerical integration method. The influences of each important system parameter on the dynamic nonlinearity are also discussed. It is expected that this study would provide valuable insights to the effects of structural self-weight in a quasi-zero-stiffness isolation system.