## 1 Introduction  
Rotating unbalance is a topic frequently mentioned in the analysis of the dynamics of rotary bodies (rotordynamics [1]). The rotating unbalance occurs due to an asymmetry of mass distribution (in some different regions of the rotary body, the center of mass is not placed on the axis of rotation). Centrifugal forces occur in these unbalanced regions of the rotary body. The resultant of these centrifugal forces is transmitted through the bearings to the structure where the rotary body is placed. In each bearing the resultant of centrifugal forces has two components at orthogonal directions. Each component acts as a harmonic excitation force against the structure, thus generating vibrations. For some specific appliances these vibrations are desirable (e. g. in vibration shakers used also as mechanical vibration exciters [2; 3], vibration alert systems in mobile phones, electronic vibrating bracelets [4], and haptic feedback devices with vibrations [5]). Generally, the vibrations due to the rotating body unbalances have highly undesirable effects (e.g., bad surface quality in the grinding process [6; 7], premature bearing destruction [8; 9]), and human body discomfort [10]). In order to measure [11] and to eliminate the unbalance of rotary bodies [12; 13; 14] (using additional balancing masses and additional inertia [15]), several special requirements must be met and specialized equipment must be used [16; 17]. Actually, one of the best ways to balance rotary unbalanced bodies is through the use of self-balancing systems [18; 19]. Sometimes two rotary unbalanced bodies having almost the same angular speed (or rotational frequency) which rotate in the same structure (e.g., in centerless grinding machines, [20]) produce a vibration beating phenomenon [10; 21; 22]. Each rotary unbalanced body generates a vibration. The addition of two vibrations, having slightly different frequencies, produces the aforementioned beating phenomenon. This is a resultant vibration with periodical variation of amplitude, with nodes (where the amplitude has a minimum value, the addition of the two vibrations produces destructive interference, 180 degrees out of phase between the constituents of the resultant vibration) and anti-nodes (a maximum amplitude, where the addition of the two vibrations produces constructive interference, with zero degrees shift of phase between constituents) [23]. Obviously the beating phenomenon in mechanics is not solely related to the vibrations produced by rotary unbalanced bodies, it also occurs when two vibration modes with almost the same modal frequency are excited [24; 25], and it occurs as well when a system vibrates simultaneously due to forced sinusoidal excitation close to a resonant frequency and due to a free response [26; 27; 28; 29]. Some specific appliances use the vibration beating phenomenon to monitor the condition of mechanical systems, e.g., monitoring the adhesion integrity of single lap joints [30], monitoring the structural integrity of helicopter rotor blades [31], and for seismic vibration testing [32]. A vibration beating mechanism in piezoelectric energy harvesting systems is proposed in [33]. In machine tools, in addition to a critical source of vibrations (self-excited vibrations in turning [34], milling [35] or grinding [36] processes), the vibrations produced by rotary unbalance (generated by tools [37], shafts [38] or work pieces [36]) and particularly the beating phenomenon [6] created by rotary unbalanced bodies, are important items. This paper proposes some approaches, in theoretical and experimental terms, to address the vibration beating phenomenon produced inside a Romanian lathe headstock SNA 360, by two inner unbalanced rotary shafts, rotating with very close angular speeds.