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Models accounting for lateral deformation, in addition to the axial and torsional, were presented by Zaeh et al. [9] and Okwudire and Altintas [10]. The main difference between both models is that the latter is able to capture the coupling between axial–torsional and lateral vibrations whereas the former only captures the coupling between the axial and torsional. For both models, a detailed screw–nut interface is proposed, which implies that a significant number of parameters must be known.
From this review, it can be seen a need for a model with capabilities for virtual machine design as well as for controller selection and tuning. The model must
capture the dynamics of the first resonant modes with parameters that are generally available from component manufacturers or easily estimated. Particularly, it is important for modern control strategies to study the degree of coupling between the main modes as a function of system parameters and operating conditions. In this way, this work proposes a model based on a combination of concentrated and distributed parameters. After that, the mode shapes are obtained and analyzed for two typical transmission ratios, and the variations of the mode frequencies are studied for different carriage positions and different moving masses. Finally, the results are analyzed drawing conclusions about the system behavior, which can be helpful as a first guideline for control process design and system identification.
2 Servomechanism model
The most common feed drive for precision positioning consists of a ball screw, assembled to the machine base by rotary bearings, which is driven by an electric servomotor through a flexible coupling, as Fig. 1 shows. The ball nut is attached to the carriage that is constrained to move axially on linear bearings and guideways.
The schematic model considered here is presented in Fig. 2, in which the screw is considered as a continuous system, whereas the remaining elements are assumed
in the lumped form. In these conditions, the screw can be considered as a straight bar with three fundamental types of deformations: axial deformation, by traction or compression; angular deformation, by torsion; and flexural deformation. Flexure is discarded, assuming the screw is suitably mounted in the servomechanism and then minimizing buckling due to non-concentric forces produced by misalignments.
In this way, the continuous deformation can be represented by an axial displacement using a field function u(x, t) and by an angular displacement using θ(x, t). This continuous portion is characterized by mass density ρ, cross section A, moment of inertia Jt, length L, modulus of Young E, modulus of Poisson G, and screw lead l (also cited as transmission ratio).
Fig. 1 Ball screw feed system
Fig. 2 Schematic of the axial (a) and rotational (b) decoupled models
The elements assumed in the lumped form are the rotor of the electric motor with moment of inertia Jm, the flexible coupling with moment of inertia Ja and stiffness ka, the rigid bearing with stiffness kb, the nut with stiffness kn, and the carriage with mass mc.
2.1 Selection of the basis functions The deformation in a continuous general system can be represented by a displacement field u(x, t) that is a function of time and spatial coordinates. The Ritz series method [11], also known as the method of assumed modes, uses a series expansion to describe the displacement field as follows:
where ψj(x), known as basis functions, represent the displacement field as a function of the x coordinate and the coefficients qj(t) represent the instantaneous
contribution of ψj(x) over the displacement field. The basis functions must fulfill certain conditions to obtain a valid formulation of the Ritz series. All basis functions must be continuous, linearly independent, and must satisfy the geometric boundary conditions [11]. In this way, a suitable axial field equation can be constructed using cosine basis functions as follows: