Abstract In this paper the vibration of a mass–spring oscillator with strong quadratic nonlinearity and one degree of freedom is analyzed. The both, strong and hard, springs are considered. The restoring force in the spring which is the function of the quadratic deformation has to satisfy the condition of antisymmetry. The mathematical model is an ordinary second order differential equation where the quadratic nonlinear term changes the sign. The quantitative and the qualitative analysis of the equation is done. The exact analytical solution is obtained. It depends on the Jacobi elliptic function.