Rotors instability is one of the most serious problems of modern engineering. There are many approaches for solution of the problem which is especially important for high-speed machinery. A known model of piece-wise linear system of oscillator supported by bearings having clearances is chosen for analytical consideration of a rotor motion and its stability. Since the procedure of prediction of the dynamical stability of investigated motions near harmonic oscillations is based on analysis of the first derivative of restoring force it is convenient to represent the derivative as expansion of Fourier series. Two cases of symmetry and asymmetry of acting external force are studied and compared. Approximate equations for computing and plotting maps of instability zones are found. The existence of a difference in principle between both considered cases is resulted from different types of expansions of Fourier series. Hence, it follows that if in a symmetric case the main (first) instability zone of the motion, supposed close to harmonic, coincides with the region of the primary resonance, asymmetry of restoring force shifts the first instability zone to the right direction.