In this paper, we study a kinematic friction model of a single degree-of-freedom oscillating system. This model has the following dimensionless equation of motion:  
y+2ζy+y=-μky
in which the kinematic friction is described by a quadratic function of the velocity as the following:  
μky=μ0+μ2y2sgny+μ1y
The sign function preserves the asymmetry of the friction force. The model captures most of the friction behavior that has been observed experimentally using a recent apparatus known as “dynamic oscillating tribometer”. This experimental setup is based on the non-linear free dynamic response of this kind of oscillator. However, this technique is able to carefully determine, with no need for any force transducer, the velocity-independent and velocity-dependent friction coefficients, μ0 and μ1 respectively, for a linear description of the kinematic friction, μk, around y′ = 0 where:  
μky=μ0sgny+μ1y
In this context, the principal aim is to investigate numerically and analytically the effect of the additional quadratic term. To analyze the free dynamic response and their corresponding envelops, different numerical methods are performed. New characteristics of the envelops are discussed in detail with respect to the form of the kinematic friction coefficient. This allows a better comprehension of the results observed experimentally.
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