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where α = ( ju . 1)π. The first term in the series (with ju = 1) is the unitary function that represents the rigid body motion which is kinematically admissible for this system. This is because the screw is attached to the base by the bearing, which is modeled as a lumped spring, as Fig. 2a shows. Although the axial rigid body motion
would not actually occur, it is advisable to introduce a unitary function to account for stiffness differences between the screw and the rigid bearing [11]. It is more obvious that the screw has a rigid body motion for the angular displacement; then, the displacement field to describe rotation in the screw can be represented by
where α = ( jθ . 1)π. The number of terms Nu and Nθ can be selected by studying the convergence of the model solutions.
2.2 Equations of motion based on energy and work formulation
A convenient way to find the equations of motion of a multibody system is using the energy and work formulation, which is more efficient and reliable than the Newton–Euler formulation based on momentum considerations. Using the defined variables, the kinetic energy can be computed as follows:
where the first and second terms represent the contributions from the mass of the carriage and the inertia of the rotor, respectively. The third term is the energy from the flexible coupling, for which an average speed between the angular velocity of the rotor and the angular velocity of the screw in x = 0 was considered. The fourth and fifth terms represent the kinetic energy from the distributed rotary inertia and the distributed linear inertia of the screw, respectively.
The potential energy stored in the elastic parts of the system can be computed according to where the first and second terms correspond to the potential energy in the rigid bearing and flexible coupling,respectively. Similarly, the third term corresponds to the potential energy stored in the ball nut, where δn is the axial deformation in the nut. Although the elastic deformation in the screw–nut interface produced by the normal contact force has axial and radial components [12], only the axial component influences the axial displacement field.
Therefore, the interface axial deformation can be expressed as
that denotes the difference between the absolute position of the carriage, uc(t), and the absolute position of the screw at the interface-point coordinate xc.Itis important to notice that Eq. 6 involves both axial and torsional displacements, showing a coupling between them, a fact that forbids each field to be treated separately. Finally, the fourth and fifth terms of Eq. 5 represent the potential energy stored in the continuous portion of the system, the screw, by torsional and axial displacements.
The net power input to the system results in
where the first term is the power input from the motor, the second term is the Coulomb friction dissipation in the ball nut due to the friction torque τf, and the third term represents the power required to move the carriage at the velocity u˙c against a disturbance force fc. Note that fc is a general variable to account for external forces acting on the carriage, which can include machining forces and Coulomb friction forces in guideways.
Finally, the power dissipation in the system due to viscous friction can be expressed as follows: The first two terms represent the power dissipation due to the viscoelastic behavior of the continuous portion, where γ.is the structural damping loss factor. The other four terms represent the power dissipation in rotor bearings, screw supporting bearings, ball nut, and guideways, respectively. Therefore, the coefficients cm, cb, cn,and cc are the viscous friction coefficients of these elements, as Fig. 2 shows.
Now the expressions of T, V, Pin,and Pdis are combined with the Ritz series to formulate the Lagrange procedure in order to find the equations of motion. Alternatively, the power balance methodology can be used in this particular case, giving the same results as those obtained by Lagrange formulation but with an