J. Cent. South Univ. (2013) 20: 2359−2365 DOI: 10.1007/s11771­013­1744­z

Model­based parameter identification of comprehensive friction behaviors for giant forging press

LI Yi­bo(李毅波), PAN Qing(潘晴), HUANG Ming­hui (黄明辉)

State Key Laboratory of High Performance Complex Manufacturing (Central South University), Changsha 410083, China

© Central South University Press and Springer­Verlag Berlin Heidelberg 2013

Abstract: A new experimental apparatus was set up to investigate the actual friction characteristics on the basis of speed control of the serve system. A modified friction model was proposed due to real time varying deformation resistance. The approach to identify the parameters of comprehensive friction behaviors based on the modified model was proposed and applied to the forging press. The impacts on parameters which the external load had were also investigated. The results show that friction force decreases with velocity in the low velocity regime whereas the friction force increases with the velocity in the high velocity regime under no external load. It is also shown that the Coulomb friction force, the maximum static friction force and the vicious friction coefficient change linearly with the external load taking the velocity at which the magnitude of the steady state friction force becomes minimum as the critical velocity.

Key words: friction; forging press; modified model; LuGre model; parameter identification

1 Introduction

The giant forging press is currently used in a wide range of applications in national defense, infrastructure construction, aero transportation and energy equipment [1−2]. The velocity control precision of the moving beam of the hydraulic forging press will have a direct impact on the quality of processing and productivity. Successful application of the isothermal forging process demands low strain rate, namely, the moving beam must be driven in an ultra low velocity [3].

However, the velocity of the moving beam presents the phenomenon of instability and fluctuation when it is driven in such ultra low velocities. Nonlinear friction, deteriorating the tracking performance critically, is the leading factor of oscillating and crawling [4].

To deal with the friction existing in the mechanical control system, model based friction compensation methods have been proposed. Modeling friction and identification of its parameters are the first steps towards effective friction compensation in mechanical control systems [5]. Hence, the effectiveness of friction compensation depends heavily on the accuracy and simplicity of the friction model utilized [6−9]. There are many friction mathematical models been utilized and contributed to describe friction characteristics [10−11].

CANUDAS et al presented LuGre model to describe most of the friction behaviors such as presliding displacement, Stribek effect and stick­slip motion [12−15]. YANADA et al [16−17] and HIDEKI and SEKIKAWA [18] have modified the LuGre model by taking lubricant film dynamics into consideration and have shown that the modified one can simulate the friction behaviors with a good accuracy.

The other key issue is the identification approach applied to friction models. In recent decades, a great number of methodologies are developed to identify friction characteristic parameters. In Ref. [13], a two­step off­line method to estimate the nominal friction parameters associated with the LuGre model was developed. In Ref. [19], the corresponding parameters are estimated using a single pair of displacement (excitation)­friction (response) signals based on the LuGre and Maxwell models. A new open­loop (off­line) identification approach to determine the friction parameters in the joints of robotic manipulators is presented in Ref. [20]. In Ref. [21], parameters were identified by applying fast Fourier transformation (FFT) algorithm to analyze the frequency characteristics. In Ref. [22], particle swarm optimization algorithm was proposed to be applied to the parameters identification. A frequency domain identification method was suggested in Ref. [23].

Foundation item: Project(51005251) supported by the National Natural Science Foundation of China; Project(2011CB706802) supported by the National Basic Research Development Program of China (973 Program)

Received date: 2012−05−23; Accepted date: 2012−09−29 Corresponding author: PAN Qing, PhD Candidate; Tel: +86−15111023006; E­mail: panqing0905@csu.edu.cn

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However, previous studies focused on the identification of velocity dependent friction model [24−25]. Working condition changes such as load uncertainty, lubrication condition and temperature variation impose friction parametric uncertainties on the hydraulic control system inevitably.

The purpose of the current work is to modify the LuGre model by incorporating time­varying external load into the factors imposed on the friction model. Also, a simple approach to identify the parameters of the modified friction model is applied to the forging press. How the parameters of this friction model can be influenced by the external load is shown as well.

2 Experiments

2.1 Isothermal forging Isothermal forging is an advanced hot working

process technology. In an isothermal forging process, the dies are heated to the same temperature as the workpiece in a low strain rate, allowing near­net shape configurations to be formed. Thus, it is possible to make full use of raw material and minimize postforging machining. Isothermal forging requires that forging press should be driven in ultra low velocity steadily, thus, it improves the microstructure and capability differences of forgings caused by internal deformation inhomogeneity. On the other hand, the heating system must be established in order to keep the mould temperature the same with that of the metal. The temperature was kept almost constant in this experiment. One of isothermal forging process rules is given in Table 1. And the forging press equipment and forged piece are also shown in Fig. 1 and Fig. 2, respectively.

Table 1 Isothermal forging process rules Step Reduction/mm Time/s Velocity/(mm∙s −1 ) 1 2 3 4

6024 4 2

50 240 80

1 000

1.200 0.100 0.050 0.002

2.2 Experimental apparatus and methods The apparatus used in this work is shown in Fig. 3.

Three hydraulic cylinders were fixed vertically on a frame in the upper position of moving beam and the pistons were connected to the moving beam. Similarly, four hydraulic cylinders were fixed in the lower position of the frame and the pistons were connected to the moving beam; the moving beam moved along the guide way vertically. Pressure sensors were used to measure major cylinders pressure p1 and that of supporting cylinders p2; the moving beam displacement was measured using displacement sensors mounted on the

Fig. 1 Forging press equipment

Fig. 2 Forged piece

frame as shown in Fig. 3 (velocity and acceleration can be obtained by derivation). The motion of the moving beam was controlled by hydraulic system directly. Signals from the sensors were read into a computer through A/D (analog­to­digital) converters and signals from the computer were supplied to the servo­valve via D/A (digital­to­analog) converters. Experimental data, including displacement, velocity and pressures, were recorded on the computer via a data acquisition card.

The comprehensive friction force Fr was obtained based on the equation of motion of moving beam and hydraulic cylinder pistons as follows:

2

r 1 1 σ 2 2 2 d d x F p A mg F p A m t

= + − − − (1)

where m is the mass of moving beam and hydraulic cylinder pistons, g is the acceleration of gravity, A1 and A2 are the piston areas of major and supporting cylinders respectively and Fσ is the deformation resistance. The load force is the function of temperature, forging rate and degree of deformation for a typical material, i.e., it can be described by

( ) σ , , F f T µ γ = (2)

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1−Major cylinders; 2−Moving beam; 3−Supporting cylinders; 4−Upper mould; 5−Lower mould; 6−Forging workpiece; 7−Displacement sensor; 8, 9−Pressure sensor; 10−Tank; 11−Guideway; 12−Lower beam

Fig. 3 Experimental device and sketch map: (a) Front view; (b) Side view

where μ is forging rate corresponding to velocity of moving beam, T is deformation temperature, and γ stands for deformation extent. Apparently, there is only one variable parameter since temperature and forging rate were kept almost constant during every stage in the period of this work [26].

3 Friction models

3.1 Comprehensive friction analysis Seal structure between the piston and the cylinder is

presented in Fig. 4. The frictional condition at the moving beam/guideway interface is shown in Fig. 5. The gap between the moving beam and the guideway is full of lubrication oil. Since the guiding device is equipped with lubrication mechanisms which can guarantee the guiding work in good lubrication. Hence, comprehensive friction force is described as r m k F F F = + (3)

where Fm stands for friction between piston seal and the cylinder, Fk is the friction between the moving beam and guideway.

Fig. 4 Seal structure between piston and cylinder

Fig. 5 Interface between moving beam and guideway

3.2 LuGre friction model Visualizing two contacting surfaces with a number

of asperities as two rigid bodies that make contact through elastic bristles [13−14], the LuGre model (see Fig. 6) captures the Stribeck effect, therefore, it is possible to get an access to describe stick­slip motion.

Fig. 6 Bristles model

The LuGre model is given by

r 0 1 2 d d z F z v t

σ σ σ = + + (4)

( ) 0 d d

v z v z t g v

σ = − (5)

where z is the average deflection of the bristles, ν is the relative velocity between the two surfaces in contact, σ0

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is the stiffness of bristles, σ1 is the micro damping coefficient, σ2 stands for the vicious friction coefficient, and Fr is the comprehensive friction force, g(v) models the function to describe the Stribeck effect, a common form of g(v) can be given by

( ) ( ) 2 s ( / )

c s c e v v g v F F F − = + − (6)

where Fc is the Coulomb friction force, Fs stands for the maximum static friction force, vs characteristic Stribeck velocity, and the parameter vs determines how quickly g(v) approaches Fc. Since only the extending stroke period was investigated, the absolute value sign in Eq. (2) is not necessary.

For steady­state friction behaviors, substituting dz/dt=0 into Eqs. (1) and (2) and friction force Fr yields

( ) rs 2 F g v v σ = + (7)

Inserting Eq. (3) into Eq. (4), the steady­state friction force can be expressed as

( ) 2 s ( / )

rs c s c 2 e v v F F F F v σ − = + − + (8)

3.3 Modified model As set forth, the load force varies with the

displacement of the moving beam (shown in Fig. 3). Experimental results show that the parameters, Fc, Fs and σ2 of the LuGre were affected linearly by the external load. Accordingly, the Coulomb friction force Fc and the maximum static friction force Fs can be expressed as

c c0 1 F F k F σ = + (9)

s s0 2 F F k F σ = + (10)

2 20 3 k F σ σ σ = + (11)

where Fσ is deformation resistance, Fc0, Fs0 and σ20 are Coulomb friction force, maximum static friction force and vicious friction coefficient identified respectively in case of no load (i.e, Fσ=0), and k1, k2, k3 are modified coefficients.

For the steady state, the relationship of modified coefficients k1, k2 and k3 is given by the following inequalities:

( ) ( ) ( ) ( )

2 1 20 3 s0 c0 c

2 1 20 3 s0 c0 c

0 k k k F F v v k k k F F v v

σ σ

− < − < < − ≥ − >

(12)

where vc is the velocity at which the magnitude of the steady state friction force becomes minimum, and vc0 stands for the velocity corresponding to the minimum friction force in no­loading condition. And vc can be obtained by solving the following nonlinear equation:

( ) 2 s ( / )

s c 2 s 2 e v v v F F v σ − − = (13)

4 Parameter identification

Experiments were conducted on the basis of speed control of the servo system. With reference to Eq. (1) when the moving beam moves at a series of constant velocities, the value of the friction force can be obtained.

The identification procedure is divided into two steps.

Firstly, in a constant velocity experiment under no external load (Fσ=0), the moving beam was driven at series of velocities 1 ( 1, 2, , ), N

i i v i N =

= L and the value of the friction force Fr(vi) can be computed by Eq. (1). Hence, the friction−velocity map is conducted (Fig. 7).

Fig. 7 Steady­state friction behaviors (no­loading)

Define the identified parameters as

[ ] c0 s0 s 20 , , , F F v σ = Φ (14)

Define the identification error as

r i r ( , ) ( ) ( , ) i i e v F v F v = − )

Φ Φ (15)

where Fr(Φ, vi) is the modeled friction force determined by Eq. (5).

The objective function is defined as

( ) 2

1 ,

N

i i

L e v =

= ∑ Φ (16)

Then, the parameters Fc0, Fs0, vs and σ20 are estimated by minimizing the objective function, and all of them are given in Table 2.

Table 2 Static parameters Parameter Value

Fs0/kN 377.39

Fc0/kN 203.23

vs/(mm∙s −1 ) 0.076

σ20/ (kN∙s∙mm −1 ) 19.77

vc/(mm∙s −1 ) 0.19

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Secondly, several dedicated experiments under external load with a range of constant velocities were conducted. The linear relations between the friction force measured and deformation force at different velocities were obtained.

Define the modified parameters vector as

[ ] T 1 2 3 , , k k k = K (17)

The modified coefficients 1 2 3 , , k k k are identified based on Eq. (15)

1 1 1 1 1

2 2 2 2 2

3 3 3 3 3

1 1 1

v k F v k F v k F

σ

σ

σ

ξ ξ ξ ξ ξ ξ

− − + −

1 1 c0 r 1

2 2 s0 c0 r 2

3 3 20 r 3

1 ( ) 1 ( ) 1 ( )

v F F v v F F F v v F v

ξ ξ ξ σ

′ ′ − = ′

(18)

where 2

s ( / ) e ( 1, 2, 3 ) i v v i i ξ − = = L , Fr(vi) is the value of

the friction force measured under external load, Fσi=Fσ(xi,vi), stands for deformation resistance corresponding to the displacement and velocity of the moving beam.

Thus, the modified coefficients k1, k2 and k3 are

achieved successfully (given in Table 3).

Table 3Modified coefficients Parameter 0<v<vc ν>νc

k1 −0.87 103.95

k2 0.33 102.88

k3/(s∙mm −1 ) 0.04 319.91

5 Results and discussion

Figure 7 shows the relations between the friction force measured and the velocity of the moving beam in the case of no­loading, which is known as the Sribeck curve. As shown in Fig. 7, for the velocity changes from 0 to νc, friction force decreases with the increase of velocity, whereas friction force increases with the growing of velocity when the velocity of the moving beam is larger than νc. Tables 2 and 3 give the identified static parameters of the modified model. Those parameters were identified by using the approach introduced in Section 4.

The relations between the friction force measured and deformation force in different velocities are shown in Fig. 8. Friction force decreases linearly with the external

Fig. 8 Relations between Fr and external load: (a) ν=1.2 mm/s; (b) ν=0.1 mm/s; (c) ν=0.05 mm/s; (d) ν=0.002 mm/s

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load in the case that velocity is 1.2 mm/s and 0.1 mm/s, respectively, whereas the friction force increases linearly with the external load when the velocities are 0.05 mm/s and 0.002 mm/s, respectively. The results in Fig. 8 indicate that parameters Fc, Fs and σ2 change linearly with the external load, which agree with the experimental results shown in Fig. 9.

Fig. 9 Relations between Fc, Fs, σ2 and external load: (a) 0<ν<νc; (b) ν>νc

As can be seen in Fig. 9, the magnitudes of Fc, Fs and σ2 are influenced linearly by external load variation. The affection is small when the velocity is in the range of 0 to νc, however, magnitudes of Fc, Fs and σ2 change rapidly in the case that the velocity is larger than νc. It seems that the difference results from sharply increasing pressure in the cylinder chambers which is significantly affected by the external load. The growing pressure in the chambers indicates that the normal force between the piston seals and contacting surfaces increases intensely. Therefore, parameters Fc, Fs and σ2 change with external load roughly.

Figure 10 shows the experimental result obtained in the condition that has been described in Section 2. It is demonstrated that a relatively large friction force is

observed after the dwelling. Also, the characteristics that the friction force changes with deformation force are shown in Fig. 10.

Fig. 10 Friction force variation

6 Conclusions

1) In the case of no­loading, the friction force decreases with velocity in the low velocity regime; the friction force increases with the velocity in the high velocity regime.

2) Coulomb friction force, the maximum static friction force and the vicious friction coefficient change linearly with the external load taking νc as the critical velocity. In detail, these three parameters, namely, Fc, Fs and σ2 are affected slightly by the external load (deformation resistance) in the low velocity regime, whereas these parameters are influenced evidently by the external load in the high velocity regime.

(3) The velocity and displacement of the moving beam is nonlinearly related to the comprehensive friction force. Also, the investigation of the effect of compensation based on the modified model is remained to be studied.

Nomenclature

m Mass of the moving beam and cylinder pistons x Displacement of the moving beam Fr Friction force Fσ Deformation resistance Ai Piston area (i=1, 2) pi Pressure (i=1, 2) g Acceleration of gravity μ Deforming rate T Temperature γ Deformation extent ν Velocity σ0 Stiffness of bristles

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σ1 Micro damping coefficient σ2 Vicious friction coefficient σ20 Vicious friction coefficient for no­load period νs Stribeck velocity Frs Steady­state friction force Fc Coulomb friction force Fc0 Coulomb friction force for no­load period Fs Maximum static friction force Fs0 Maximum static friction force for no­load period Z Average deflection of bristles ki Modified coefficient (i=1, 2, 3) νc Velocity at minimum steady­state friction force Fk Friction between the moving beam and guideway Fm Friction between piston seals and contacting

surfaces

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(Edited by HE Yun­bin)