Hybrid modeling and analysis of structural dynamic of a ball screw feed drive system

Ball screw drives are widely used as the motion delivery mechanism due to their high stiffness and high accuracy. As the speed and precision requirements from machine tools increase, the effects of the structural flexibility of the drives on controller performance are becoming increasingly significant. The vibrations adversely affect the positioning accuracy and performance of the drive. The natural frequencies of a ball screw system vary as the nut moves along the screw. It is necessary to have an insight into the dynamic response of the rotating ball screw subjected to axially moving load during prototyping of the machine tool and controller [1]. Conventional dynamic modeling methods of the ball screw drives employ lumped parameter models [2-4], which cannot accurately characterize the high frequency mechanical resonance. More advanced models are obtained with the help of hybrid methods that consider torsional, axial, and possibly flexural vibrations. Varanasi et al. [5] and Whally et al. [6] captured the axial and torsional dynamics of a ball screw drive using beam formulations. In general, these models cannot accurately characterize the high frequency mechanical resonances. Frey et al. [7] proposed an approach aims at identifying the dominant effects of the ball screw and including them into a simple lumped mass model. Vicente et al. [8] studied the axial and torsional coupled vibration by Ritz serious method. Finite Element methods are often incorporated in modeling the drive systems. Zhou et al. [9] presented a model only considering the axial dynamics of the screw. However, neither model considers the flexural deformation. Okwudire et al. [10] proposed a model of ball screw using beam elements having axial, torsional, and lateral dynamics. This paper presents a hybrid model of a ball screw drive, in which the screw is considered as a Timoshenko’s beam having axial,torsional and flexural dynamics. The deformation of the screw is determined by assumed mode method. The method models the dynamic behavior of a ball screw considering the moving nut. The dynamic equations of a rotating screw subjected to the axially moving load with general boundary conditions are proposed.


Introduction
Ball screw drives are widely used as the motion delivery mechanism due to their high stiffness and high accuracy.As the speed and precision requirements from machine tools increase, the effects of the structural flexibility of the drives on controller performance are becoming increasingly significant.The vibrations adversely affect the positioning accuracy and performance of the drive.The natural frequencies of a ball screw system vary as the nut moves along the screw.It is necessary to have an insight into the dynamic response of the rotating ball screw subjected to axially moving load during prototyping of the machine tool and controller [1].
Conventional dynamic modeling methods of the ball screw drives employ lumped parameter models [2][3][4], which cannot accurately characterize the high frequency mechanical resonance.More advanced models are obtained with the help of hybrid methods that consider torsional, axial, and possibly flexural vibrations.Varanasi et al. [5] and Whally et al. [6] captured the axial and torsional dynamics of a ball screw drive using beam formulations.In general, these models cannot accurately characterize the high frequency mechanical resonances.Frey et al. [7] proposed an approach aims at identifying the dominant effects of the ball screw and including them into a simple lumped mass model.Vicente et al. [8] studied the axial and torsional coupled vibration by Ritz serious method.Finite Element methods are often incorporated in modeling the drive systems.Zhou et al. [9] presented a model only considering the axial dynamics of the screw.However, neither model considers the flexural deformation.Okwudire et al. [10] proposed a model of ball screw using beam elements having axial, torsional, and lateral dynamics.
This paper presents a hybrid model of a ball screw drive, in which the screw is considered as a Timoshenko's beam having axial，torsional and flexural dynamics.The deformation of the screw is determined by assumed mode method.The method models the dynamic behavior of a ball screw considering the moving nut.The dynamic equations of a rotating screw subjected to the axially moving load with general boundary conditions are proposed.

Hybrid modeling
A ball screw drive feed system proposed in this paper is shown in Fig. 1.Actuation is provided through a servomotor that is connected to the ball screw using a coupling.The screw is constrained to the machine base through a thrust bearing at the motor side.The vibration of a ball screw feed drive can be characterized by three different modes: axial, torsional and flexural as shown in Fig. 2. It should be notice that in order to keep the model complexity minimal the modeling of the nut dynamic, which have been successful modeled by Okwudire [10], is not considered in this article.The ball screw considered is an axisymmetric uniform shaft rotating about its longitudinal axis with a constant angular velocity , as shown in Fig. 2. The rotating screw is subjected to two forces P x , P y moving along the x-axis.P x , P y are applied in the directions of the X and Y axes.P x is the axial load.The mass of the ball in the nut is considered as concentrated mass expressed as P y .F is the preload.
The screw is characterized by density ρ, cross-section A, length L, polar moment of inertia J, Young's modulus E, Poisson's modulus G, and screw pitch l.
The relatively more rigid components are represented as lumped parameters connected by springs at joint interfaces.The lumped parameters are defined by the axial stiffness K b , the torsional stiffness K c of the coupling, the axial stiffness K n and radial stiffness K nr of the nut, the inertia of the motor J m and the coupling J c , the mass of the table m t .

Axial-torsional model
The continuous deformation of the ball screw can be represented by an axial displacement using a field func-tion U(x, t)，by an angular displacement using Θ(x, t).The u t (t) and θ m (t) each represents the displacement of the table and the angle of rotation of the motor.
where u i (x) and θ i (x) are shape functions that satisfy the corresponding boundary conditions;  i (t) and  i (t) are the respective modal displacements.
For the screw-nut interface, the preloaded nut serves as a special spring that couples different degrees of freedom, especially in the torsional and axial directions.
The axial deformation of the screw-nut interface  n can be expressed as: The kinetic energy T of the axial-torsional coupled system, due to the distributed inertia of the screw, the mass of the table, the inertia of the rotor and the energy from the flexible coupling, can be computed as: where the symbols ( . ) denotes differentiation with respect to time.The potential energy U, due to the potential energy stored in the elastic parts including the screw , coupling, screw nut interface and preload, is defined as: where the symbols " ′ " indicate differentiation with respect to x.
The energy can be expressed in matrix form: where the preload F can enhance the axial stiffness of the whole system.

Flexural model
Based on Timoshenko's beam theory, the deformed screw can be described by the transverse translations V(x, t) and W(x, t) in Y and Z directions and small rotations B(x, t) and (x, t) about the Y and Z, respectively.
to bending and a contribution V s ， W s due to transverse shear deformation.The relations among these variables are: Using the assumed mode method, V, W, , B can be expressed: where  i (x) and ψ i (x) are shape functions that satisfy the corresponding boundary conditions; v i (t), w i (t), p i (t) and q i (t) are the respective modal displacements.
The kinetic energy T f , due to the screw rotating at a constant speed : The kinetic energy T M , due to moving mass [11]: The potential energy U f ，due to shear deformation and elastic bending，the preload F and moving axial force P x is: where κ is the shear coefficient.Define: The energy can be expressed in matrix form as: The virtual work   . The equations described the rotating screw subjected to the axially moving load can be expressed as: where a time-dependent stiffness term generated by the moving axial force P x reduces (when compressive) or increases (when tensile) the overall stiffness of the ball screw drive system.

Shape functions for general boundary conditions
For axial-torsional model, the shape functions are For flexural model, three general geometric boundaries of a screw are introduced, including Clamped-Hinged (C-H), Clamped-Clamped (C-C), Hinged-Hinged (H-H).The corresponding shape functions are shown in Table 1.

Model verification
Eqs. ( 8) and ( 18) can be expressed as: where M, C and K are the inertia, damping, and stiffness matrices, respectively.The matrices are dependent of the time, and needs to be updated as the nut moving over the screw.The F includes the generalized forces and q is the generalized vector.Eq. ( 20) can be solved by a Runge-Kutta method to obtain the dynamic response of the rotating ball screw under the moving load.The vibration modes can also be obtained through the characteristic equation.It is observed that the first three modes present a favorable approximation using N = 4 terms in axial-torsional subsystem and n = 10 terms in flexural subsystem.In this section, measurements are made on a ball screw drive.The test bed shown in Fig. 3 moves a carriage of mass 50 kg.The parameters are listed in Table 2.The axial vibration is measured between an impact force(PCB/086C03) applied at the table (in the direction of motion of table) and axial acceleration measurements obtained from the table using accelerometer(PCB/356A32). The flexural vibration is measured using an accelerometer at each of 12 points on the screw.To investigate the variation as the table moves along the screw, each of the tests is measured at four distinct position of the table within its travel range.

Fig. 3 Ball screw drive setup
The predicted and measured resonance frequencies are listed in Table 3.It can be concluded that the predicted results have a good agreement with the measured ones.The deviation may due to the inaccuracies or ignorance in various stiffness and inertias.The axial and flexural mode shapes are shown in Fig. 4.

Numerical simulation and analysis
Table 5 gives the parameters used in simulation.Due to the table moves along the screw reducing the number of available measurement or excitation locations, it is impractical to measure the complex mode shapes of the ball screw assembly.Hence, the model presented here is instrumental in understanding the behavior of these modes.The ball screw feed drive is always subjected to a moving mass during the operation.The Fig. 5 illustrates the axial and torsional components of the mode shapes.The axial displacement of the table is described as a point value plotted at x = x t .Similarly, the torsional displacement of the motor rotor is plotted at x = 0.
In Mode 1, the angular component has the smallest value compared to the other three torsional components.In addition, the displacement of the table has the largest value, which indicates that the first mode is described as an axial mode with lightly coupled with the torsional mode.The mode shape is similar with the experimental results shown in Fig. 4, a.The axial deformation of screw increases from the thrust bearing to the table position, while the value of axial deformation remains constant from table position to the right end of the screw.
In Mode 2, the amplitude of the torsional compo-nent increases considerably, whereas the axial component decreases.The axial displacement of the table is very small.Fig. 5, b shows the neural position of the torsional mode is near the motor.The torsional displacement gradually increases from left to right along the ball screw and the maximum deflection happens at the free end.The motor has a negative displacement in this mode.It can be concluded that this is a torsional predominant mode.Finally, in Mode 3, the amplitude of the axial component shows the largest values, whereas the torsional component amplitude is also very large.There are two neutral positions in this torsional mode, one near the motor and the other between the table position and the free end.It is observed that in this mode exists a strong coupling between the axial and torsional deformation.
The variable table position almost has no influence on the torsional mode shapes, however, it has large influence on the axial mode shapes.It is obvious that the largest torsional deformation always appears at the right side of the ball screw.The mode shapes of the system provide a useful guideline for measurement.6 shows the first flexural mode shapes for variable table position.Comparison of the predicted and measured mode shapes (Fig. 4, b) indicates that the flexural model can accurately describe the mode shapes of the ball screw drive.As can be expected, the variable table position has a great influence on the flexural mode shapes.Due to the table is properly amounted to the rigid guideway, and the excitation on the flexural direction is considerably small, hence the flexural modes only have tiny influence on the feed motion controller and therefore will be neglected in the following discussion.However, the flexural nature frequency of the screw is the key point for determining the critical rotational speed.Axial distance x/L Fig. 6 Flexural mode shapes for l = 10 mm, m t = 400 kg ----: x t = 0.7L; -:x t = 0.5L; -.-.: x t = 0.4L

Analysis of vibration for variable operating conditions
From the controller design standpoint, it is of interest to know varies in the axial and torsional vibration according to variable operating conditions.Numerical results have been divided into three sections to investigate the effects of variable mass, screw pitch and table position, respectively.The simulation results are listed in Table 6.The first axial mode has a wide variation in resonance frequency as a function of the table position and table mass; it presents low sensitivity to the variable mass for high transmission ratio.The second mode seems almost independent of the table position and mass, which is due to the weaker stiffness coupling provided by the screw-nut interface in torsional direction.On the other hand, the third mode shows very low sensitive to the table mass, however, it presents a considerable shift for variable table position, due to its highly axial-torsional coupling.Due to the stiffness of the screw decreases with increasing effective length(from the fixed bearing location to the table position), the first mode frequency decreases steadily as the table moves away from the motor while the table's vibration amplitude follows the reverse pattern.Therefore, the first axial mode can be explained by a spring mass system: the higher the mass and the greater the distance between machine table and fixed bearing the smaller the value of the first eigenfrequency.The first axial mode, due to its sensitive to the table position and mass, plays an important role in controller design and performance simulation based on the feed drive model.
The interaction between axial and torsional modes in Eq. ( 2) is determined by the transmission ratio of the ball screw.As can be seen from Table 6, with the increasing screw pitch the first mode frequency reveals a general trend of increase while the second and third frequencies decrease insignificantly.It indicates that the axial-torsional coupling has a great influence on the first mode frequency.The effects of the axial-torsional coupling forbid the axial and torsional mode to be considered separately, especially in high transmission ratio.

Analysis of ball screw preload variation
The nominal life of the ball screw can be calculated by an equation using the values for the basic dynamic load rating C a and applied axial load F ma : where f w is the load factor, F ma , calculated by the overall applied external forces F and preload force F pr , is defined as: The stiffness value of the nut K n is obtained from the preload and elastic displacement under a preload of 10% of the basic dynamic loading rating C a .If the preload F pr is differs from 0.1C a , the stiffness K n can be expressed using the following equation [12]: 13 08 01 where K  is the rigidity value in the specification table.Due to friction, lubrication and machining condition the preload will gradually decrease, which reduces the life expectancy of the ball screw.In order to examine the effect of preload variation, the first three eigenfrequencies for variable preload are calculated in Table 7.It indicates that the first and third mode frequencies exhibit a large variation with the preload variation in the range of 0.5-10%, while the second mode frequency has no significant deviation.The first and third mode frequencies reveal a decreasing tread as the preload steps down.It can be concluded that the stiffness of the nut is of great importance on the axial mode.The simulated result has a good accordance with the experimental measurement in [4].Therefore, the preload level can be predicted by monitoring the measured vibration dynamics, and then the life expectancy and the running status of a ball screw can be estimated.

Servo-control model
The servo controller should compensate for the vibration of the system and be designed carefully for any variations in the dynamics of the system [13,14].The hybrid model of the ball screw drive can be implemented into servo control algorithms, and then the effectiveness of various servo control algorithms can be evaluated to achieve high bandwidth control during the draft design stage of a machine.Fig. 8 Frequency response for the rigid and hybrid model A typical servo control system is composed of the drive mechanism and the servo system, as shown in Fig. 7.In control engineering, the state space model for the mechanical model is preferred above other representations.There are a proportional controller Gp in the position loop and a proportional and integral controller Gv in the velocity loop.The velocity feedforward controller F is used to improve tracking performance.
The frequency response of hybrid and rigid model is shown in Fig. 8，it can be seen that the hybrid model captures the dynamics of the ball screw feed drives.
It is of interest to investigate the influence of the structural vibrations on the controller performance, particular the axial and torsional modes.Fig. 9 shows that due to the structural dynamics at 83 and 466 Hz, instead of obtaining a GM (Gain Margin) of 41.7 dB as indicated in rigid model, a GM of 3.67 dB is observed.In order to improve the performance of the controller, two common methods can be used: increasing the natural frequencies by the mechanical modification at the design stage and notch filtering of the structural modes of interest.

Conclusion
This paper has presented a hybrid dynamic model for a rotating ball screw drive subjected to axially moving load.The simulated results show a good accordance with the experimental data.Results of numerical simulations have been presented for various combinations of operation conditions.The proposed hybrid model can accurately capture the dynamic of the ball screw drive and be implemented into the design of the servo system.In order to get the complete structural dynamic of the ball screw drive this paper proposes a hybrid modeling method.The dynamic characteristic of a ball screw is analyzed by a continuous beam approach having axial, torsional and flexural dynamics.The dynamic equation of a rotating screw subjected to the axially moving load with general boundary conditions are obtained through Lagrangian approach.Compared with the experimental results, the model can accurately capture the structural dynamics of the ball screw drive.Influences of parameters, such as table mass, screw pitch, table position and preload on the structural dynamic of the ball screw drive are discussed.Finally, the hybrid model is integrated into a servo control system, which makes an integrated design of the servomechanisms possible.

Fig. 1
Fig. 1 Mechanical components of a ball screw feed drive

Fig. 2
Fig. 2 Dynamic structural model of a ball screw feed drive system mass moments of inertia of the shaft per unit length.

Fig. 4
Fig. 4 The measured mode shapes for the ball screw drive at x t = 1000 mm a) axial; b) flexural

Fig. 5
Fig.5Axial and torsional mode shapes for l = 10 mm, m t = 400 kg; △----: x t = 0.7L; ○-:x t = 0.5L; + -.-.: x t = 0.4L; a) axial ; b) torsional Fig.6shows the first flexural mode shapes for variable table position.Comparison of the predicted and measured mode shapes (Fig.4, b) indicates that the flexural model can accurately describe the mode shapes of the ball screw drive.As can be expected, the variable table position has a great influence on the flexural mode shapes.Due to the table is properly amounted to the rigid guideway, and the excitation on the flexural direction is considerably small, hence the flexural modes only have tiny influence on the feed motion controller and therefore will be neglected in the following discussion.However, the flexural nature frequency of the screw is the key point for determining the critical rotational speed.

Fig. 9
Fig. 9 Open loop bode diagram for position loop

Table 2
Parameters of the ball screw drive system

Table 3 Predicted
and measured resonance frequencies Position (x t , mm)

Table 5
Parameters of the ball screw drive system for simulation

Table 6
Simulation results of the hybrid model for variable mass, screw pitch and table position

Table 7
The first three frequencies shift for variable preload for x t = L / 2, m t = 400 kg