Modeling and optimization of dynamic performances of large-scale lead screws whirl milling with multi-point variable constraints

Journal of Materials Processing Tech. 276 (2020) 116392

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Modeling and optimization of dynamic performances of large-scale lead screws whirl milling with multi-point variable constraints Yulin Wanga, Chen Yina, Li Lib, Wenbin Zhaa, Xiaonan Pua, Yan Wangc, Jianxiu Wangd, Yan Hee,

T ⁎

a

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China Beijing Machine Tool Research Institute CO., LTD, Beijing, 100102, China c Department of Computing, Mathematics and Engineering, University of Brighton, Brighton, BN2 4GJ, UK d Hanjiang Machine Tool Co, Ltd, Hanzhong, 723003, China e State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China b

A R T I C LE I N FO

A B S T R A C T

Associate Editor: E. Budak

The complex nonlinear dynamic responses of the whirl milling system, caused by discontinuous cutting forces, could considerably affect the machining quality of large-scale lead screws (LLS). Fixtures including floating supports and holding devices are frequently utilized to improve the dynamic performance of the system, however leading to the Multiple-Point Variable Constraints (MPVC) of LLS workpiece during machining. It makes the dynamic performances of whirl milling complicated and hard to predict. In this paper, the dynamic model of whirl milling process is established considering MPVC induced by floating supports and holding devices, and the dynamic performances of large-scale lead screws are effectively calculated based on the logical assumption of boundary conditions of LLS workpiece and accurate representation of the three-dimensional cutting forces. Experiments of whirl milling for the eight-meter LLS workpiece are performed to validate the proposed dynamic model, and the results calculated from the proposed model are in good agreement with the experimental results. Further, the effects of floating supports are investigated to optimize the dynamic performances of whirl milling. The results show that the proposed dynamic model is feasible and effective for the modelling and optimization of dynamic performance of whirl milling processes.

Keywords: Dynamic model Whirl milling Large-scale lead screws Multi-point variable constraints Dynamic responses

1. Introduction Lead screws are widely used in industrial fields such as machine tools and aerospace because of their efficiency in transmission of rotational motion to linear motion. It enables a driving system reaches high degrees of precision, accuracy, and reliability. As manufacturing equipment in these fields are moving towards large-size and heavyduty, large-scale lead screws (LLS) (axial length > 6 m, ratio of length to diameter > 30) that are accurate and reliable under high speeds and varying loads are greatly demanded. Whirl milling, a novel and efficient machining process, is often employed for the machining of LLS. It is a typical discontinuous machining process that the workpiece is encompassed by a tool holder mounted with cutters. The workpiece and tool holder have the same direction of rotation, but the workpiece rotates slowly, e.g. around 3–30 rpm while the tool holder rotates at a high speed of 300–3000 rpm (Mohan and Shunmugam, 2007). Moreover, the tool holder is eccentrically set to the workpiece to form cutting motions, removing the material from the workpiece in radius

⁎

direction, meanwhile it feeds at a constant feed speed along the axis of workpiece to produce a helical forming. Compared with traditional screw machining processes such as grinding, the smooth and tangential cutting motions of whirl milling processes not only minimize the cutting forces, but also allow a high material removal rate. Thus, whirl milling process could produce better quality LLS in a feasible and efficient manner. In addition, whirl milling generates large amount of thin and short chip carrying away majority of cutting heat, thus reducing thermal deformation. It makes the whirling processes environment-friendly as cutting fluid are rarely used. These advantages make the whirl milling a popular manufacturing process for LLS. However, due to the ordinal cutting methods of multiple cutters, workpieces need to bear the periodic variable cutting forces and intermittent shocks in the whirl milling. It makes the dynamic responses of machining system hard to predict, leading to varying surface quality of LLS. Therefore, researches on the modelling of dynamic performances of whirl milling is significant and necessary. The manufacturing of LLS is similar to that for typical slender shaft

Corresponding author at: State Key Laboratory of Mechanical Transmission, Chongqing University, School of Mechanical Engineering, Chongqing, 400030, China. E-mail address: [email protected] (Y. He).

https://doi.org/10.1016/j.jmatprotec.2019.116392 Received 22 January 2019; Received in revised form 18 August 2019; Accepted 9 September 2019 Available online 10 September 2019 0924-0136/ © 2019 Elsevier B.V. All rights reserved.

Journal of Materials Processing Tech. 276 (2020) 116392

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Nomenclature LLS MPVC ω fx, U V, W B, Γ N a, b, c Tε Uε ρ A

Large-scale lead screws Multiple-point variable constraints Relative angular velocity between the cutters and LLS workpiece fy, fz cutting force component in x, y, z direction Axial deflection Transverse deflection in Y, Z direction Rotational deflection in Y, Z direction Number of mode shape The generalized coordinates with respect to displacements Kinetic energy of the machining system Potential energy of the machining system Mass density of the LLS workpiece Cross-sectional area of the LLS workpiece

l mb E I k vf, i, k wf , i k vb, j, k wb , j

Length of the LLS workpiece Mass of the holding device Young’s modulus, Area moment of inertia of LLS workpiece Stiffness of the i-th floating support in Y, Z direction Stiffness of the j-th holding device in Y, Z directions

x i f , x jb

Position of the i-th floating support and the j-th holding device Damping of the i-th floating support in Y, Z directions Damping of the j-th holding device in Y, Z directions Total number of the floating supports Rotation angular velocity of the LLS workpiece Dissipated energy caused by the damping Gravity effect of the LLS workpiece The uniform load of LLS gravity

c vf, i, c wf , i c vb, i, c wf , i k Ω D WG qw

processes of LLS have its unique characteristics which should be taken into account for the dynamic modelling. Basically, a shaft with a ratio of length to diameter greater than 25 is referred as the slender shaft, while this ratio of the LLS workpiece is usually greater than 30. Due to the bigger large length to diameter ratio, a LLS workpiece has a poorer bending stiffness than the typical slender shaft. In this case, two kinds of auxiliary devices, holding devices and floating supports, are utilized to reduce cutting deformation and vibration of the LLS workpiece during machining. In other words, the constraints on workpiece are not only depended on the clamping constraints at the both ends, but also affected by the holding devices and floating supports. Thus, the varying constraints are employed in modelling of the whirl milling to represent the clamping, floating supports and holding devices. These constraints are defined as ‘Multi-Point Variable Constraints’ (MPVC) in this paper. However, the dynamic responses of the machining system under MPVC are far more complex and difficult to model compared to that for traditional modelling of large scale shaft. As reported by Zhang (2015); Zhang et al. (2018) and Zhang et al. (2015), the previous published literatures on screw dynamic modeling were mainly concentrated on the LLS during its service. FEM methods used by Wang et al. (2015) and Altintas et al. (2005) and lumped mass method employed by Zaeh et al. (2004) are the two universal methods employed to establish the dynamic model. However, little researches have been carried out regarding the dynamic modelling for manufacturing process of screws. Zanger et al. (2017) derived the cutting tool profile for whirl milling and designed workpiece geometry based on process parameters. The coupling relationship between tool and workpiece was investigated. However, the dynamic performances of the machining process were not studied. Araujo et al. (2015) analyzed the relationship between cutting forces and the thread roughness which was machined by thread milling. Song and Zuo (2014) investigated the cutting force and chip morphology in whirl milling, and utilized the equivalent cutting volume method to validate the proposed dynamic model. Although the cutting forces were investigated through experiments by Araujo et al. (2015) and Song and Zuo (2014), the interactions between cutting forces and dynamic responses of the machining system were not further analyzed. Wang et al. (2014) established a dynamic model about whirl milling processes to study the dynamic response with the employment of Generalized Polynomial method. Nevertheless, the effects of workpiece rotation and gravity, axial machining force, the MPVC representing the clamping and supporting conditions are ignored in the dynamic modelling. In fact, the MPVC directly affect the dynamic response of machining system. The workpiece vibration, induced by the travelling cutting force, has been recognized as the main influencing factor of machining quality in whirl milling processes. Especially, for whirl milling, the vibrations at cutting point have a great impact on the cutter-workpiece engagement and cutting force, leading to machining

as both have large length to diameter ratios. Regarding the modelling of dynamic responses of machining systems of the large-scale shaft, Shiau et al. (2009) analyzed the dynamic responses of a spinning multi-span shaft which were acted by axially moving loads. Both the intermediate supports and boundaries were taken into consideration. Runge–Kutta method was employed to solve the numerical results of the dynamic model. The effects of spans between ends, rotation speed, velocity of moving loads on the deflections of the shaft are further investigated. Hsu et al. (2014) studied the deflections of a rotating shaft with double cutting forces and time-varying mass were considered, and the shaft were treated as Timoshenko beam which boundary conditions are assumed as clamped-hinged supports. Mohammad-Abadi and Daneshmehr (2015) investigated the vibration of a composite laminated beams based on Timoshenko and Euler–Bernoulli beam theory, with three boundary conditions: hinged-hinged, clamped-clamped and clamped-hinged are considered. Moreover, a rotating Timoshenko beam subjected moving forces was studied with the same three boundary conditions by Shiau et al. (2006). Huang and Yang (2009) studied the dynamic responses of workpieces that were subjected to moving forces from the cutters of the lathe. Both axial and transverse forces were considered to simulate these moving forces. Rayleigh beam was chosen for the workpiece. The results indicated that the axial moving force is one of the major factors causing instability in workpiece machining. Tabejieu et al. (2016) studied the effects of distribution and speed of moving loads on the vibration amplitude of the Rayleigh beam. The results revealed that a Rayleigh beam vibrates the least when it was subjected to uniformly distributed moving loads upon the structure length. In these literatures, the moving forces are generally assumed as a function related to the feed rate, and the boundary conditions representing the supporting, clamping constraints are mainly set as fixed end, linear spring or hinged end, etc. Especially, the machining processes of the large-scale shaft are modelled as a rotating flexible beam under the action of the moving loads. Despite different beam theories including Rayleigh beam and Euler Bernoulli beam were considered, some researches also revealed that Rayleigh beam model is more accurate and reliable in certain frequencies. For instance, Piccolroaz et al. (2017) suggested that Euler-Bernoulli beams and Rayleigh beams have a similar dynamic response at low frequencies, but Rayleigh beams demonstrated better dynamic properties at higher frequencies. Zhu and Chung (2016) modeled a rotating beam based on two different beam theories: Rayleigh beam theory and Euler-Bernoulli beam theory. The distinctions on natural frequencies and dynamic responses were analyzed. The results showed that the Rayleigh beam theory were more suitable and reliable in the modelling of a rotating beam because more accurate features could be extracted in the aspect of dynamic responses. Although similarities exist between whirl milling process of LLS and the machining processes for large-scale shaft, the whirl milling 2

Journal of Materials Processing Tech. 276 (2020) 116392

Y. Wang, et al.

(1) The cross section is assumed to be a circle and the effect of the thread on the diameter of the LLS workpiece is omitted; (2) A zero lead angle is assumed (the lead angle is 1.82°); (3) The rotary inertia of the LLS workpiece is omitted due to the large length to diameter ratio; (4) The LLS workpiece is assumed to vibrate freely during the cutting interval of adjacent cutters;

errors which are resulted from the deviations between the true cutting depth and nominal values. This paper firstly establishes a dynamic model of LLS workpiece under the action of three-dimensional moving forces, while the global assumed mode method is employed to derive the system dynamic equations. Although the FEM combined with structural dynamic modification method can be applied for the model of whirl milling process, vast computer memory and considerable computational time are required because the calculation of FEM is based on large dimension of system matrices. On the other hand, due to the mode shape functions used by the proposed method are selected as a series of polynomial functions, computational cost can be efficiently saved through the proposed method. Moreover, the varying constraints caused by auxiliary devices are represented as MPVC, and the modified transformation matrix is used to solve the boundary constraints of the LLS workpiece, which also contributes to achieve low computational cost. The whirl milling experiments of an eight-meter LLS are performed to verify the proposed model. Additionally, the changes of vibration responses along the length of workpiece under the travelling machining forces and MPVC are then revealed, and the optimization analyses regarding the number and layout of floating supports are also carried out. The rest of the paper is organized as follows: The modeling method of whirl milling system and derivation of the dynamic equation are illustrated in Section 2. Section 3 presents the verifications of the proposed dynamic model based on experimental and simulation results. In Section 4, optimizations of the dynamic responses are carried out based on the proposed dynamic model. Finally, Section 5 is the conclusions of this paper.

Fig. 1 shows the configuration of the whirl milling system with the above hypothesis. The chuck-center clamping is simplified to the fixedhinged end respectively. Floating supports as well as holding devices are assumed to linear spring constraints to represent MPVC. Two coordinates are adopted to describe the motion of whirl milling system: a fixed coordinates X-Y-Z and a rotating coordinates x-y-z. The X axis is collinear with the x axe along the axial direction of the LLS workpiece, and the Z axes is along the tangential direction in horizontal plane while Y axes is along the radial direction in vertical plane. The rotating coordinates about the x axis have a rotating speed of ω , leading to a rotation difference ω⋅t . fx , f y and fz denote the three components of the moving force acting on the LLS workpiece in X, Y and Z directions respectively. The deflection of the LLS workpiece care functions of position along the rotating axis x and time t. Thus, U(x, t) denotes axial deflection in X direction. The deflection in a cross-section of LLS workpiece are constitutive of two translations: V(x, t), W(x, t), and two rotations B(x, t), Γ (x, t). The function of the translations V(x, t), W(x, t) are:

2. Dynamic modelling of LLS workpiece under whirl milling

⎧ B (x , t ) = − ∂W (x , t ) ⎪ ∂x ⎨ ∂V (x , t ) ⎪ Γ (x , t ) = ∂x ⎩

2.1. Establishment of dynamic model of LLS workpiece under whirl milling

Based on modal shape function and generalized modal polynomial method, the associated deflections can be expressed as, N

(1)

N

⎧ i−1 ⎪ U (x , t ) = ∑ ϕi (x ) ci (t ) = ∑ x ci (t ) i=1 i=1 ⎪ N N ⎪ W (x , t ) = ∑ ϕn (x ) bn (t ) = ∑ x m − 1ci (t ) ⎨ n=1 i=1 ⎪ N N ⎪ n−1 ⎪V (x , t ) = ∑ ϕm (x ) am (t ) = ∑ x ci (t ) 1 1 m i = = ⎩

In whirl milling, holding devices are mounted on both sides of the whirling head to stabilize the processing LLS workpiece, feeding together with the whirling head along the axial direction of the LLS workpiece. Floating supports which equally distribute along the length of LLS workpiece are used to support the processing workpiece. The position of floating supports is fixed, yet they have two working status in whirl milling: engagements, e.g. the floating supports are in contact with the LLS workpiece to reduce deformation, and non-engagement, namely, the floating supports lose contact with the LLS workpiece to avoid interference with the moving holding devices. The constraints on workpiece are affected by holding devices, multiple floating supports and chuck-center clamping simultaneously. According to Shiau et al. (2009), the machining system of the LLS workpiece can be simplified as rotating Rayleigh beam subjected to three-dimensional moving forces. Following assumptions are made in the establishment of dynamic model:

(2)

N

⎧ ⎪ B (x , t ) = ∑ ϕn′ (x ) bn (t ) ⎪ n=1 N ⎨ ⎪ Γ (x , t ) = ∑ ϕm′ (x ) am (t ) ⎪ m=1 ⎩

(3)

Where am(t), bn(t) and ci(t) are generalized coordinates varying with time, and N denotes the number of mode shape functions. The polynomial functions are employed to describe the mode shape functions

Fig. 1. The configuration of the simplified LLS whirl milling system. 3

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ϕi (x ) , ϕn (x ) and ϕm (x ) as follows:

l

∫ ∫

ϕ (x ) = x i − 1, i = 1, 2, ⋯, N ⎧ ⎪ i ϕ (x ) = x n − 1, n = 1, 2, ⋯, N ⎨ m ⎪ ϕn (x ) = x m − 1, m = 1, 2, ⋯, N ⎩

⎧ A (m , n) = ρAϕm ϕn dx 0 ⎪ l ⎪ B (m , n) = ρIϕm′ ϕn′ dx ⎪ 0 ⎪ 2 ⎪ C (m , n) = ∑ mb ϕm ϕn ⎪ ⎪ j=1 x = x bj ⎪ ⎪ 2 k ⎪ DW (m , n) = + ∑ Cwb , j ϕm ϕn ∑ Cwf ,i ϕm ϕn ⎪ j=1 i=1 x = xi f ⎪ ⎪ 2 k ⎪ f + ∑ Cvb, j ϕm ϕn ⎪ DV (m , n) = ∑ Cv, i ϕm ϕn ⎪ j=1 i=1 x = xi f ⎪ l ⎪ E (m , n) = EIϕm′ ′ ϕn′ ′ dx 0 ⎨ l ⎪ F (m , n) = f y ϕm′ ϕn′ dx ⎪ 0 ⎪ l ⎪ G (m , n) = kGAϕ′m ϕ′n dx 0 ⎪ l ⎪ H (m , n) = 2Ω ρIϕm′ ϕn′ dx ⎪ 0 ⎪ 2 k ⎪ f + ∑ k vb, j ϕm ϕn ⎪ KV (m , n) = ∑ k v, i ϕm ϕn ⎪ j=1 i=1 x = xi f ⎪ ⎪ k 2 ⎪ KW (m , n) = ∑ k f ϕ ϕ + ∑ k wb , j ϕm ϕn w, i m n ⎪ j=1 i=1 x = xi f ⎪ ⎪ l ⎪ EA (m , n) = EAϕm′ ϕn′ dx 0 ⎩

(4)

The equations regarding to the potential and kinetic energy of the system are required for the established of dynamic model. Taking gyroscopic effect as well as the motions in rotational and translational direction into consideration, the kinetic energy and potential energy can be expressed as,

1 2

Tε =

∫0

+ 2Ω

Uε =

1 ρA (V˙ 2 + W˙ 2 + U˙ 2) dx + 2

l

∫0

l

˙ ) dx + ρI (BΓ

j=1

+

1 2

∫0

+

1 2

∑ (k vf,i V 2 + k wf ,i W 2)| x=xif

l

∫0

l

ρI (B˙ 2 + Γ˙ 2) dx + ρIlΩ2

(5)

EI (V ″2 + W ″2) dx

kGA (V ′2 + W ″2) dx −

1 2

∫0

k

+

i=1

l

1 2

f y (V ′2 + W ′2) dx 2

∑ (k vb,j V 2 + k vb,j W 2)| x=x bj j=1

(6) Where ρ denotes the mass density, l is the length of the LLS workpiece and A denotes its cross-sectional area, mb is the mass of the holding devices, I is the area moment of inertia of LLS workpiece, E denotes the Young’s modulus. k vf, i and k wf , i respectively denote the stiffness of the ith floating support in Y, Z directions. k vb, j and k wb , j denote the stiffness of the j-th holding device in Y, Z directions respectively. k is the number of the floating supports. x i f and x jb denote the position of the i-th floating support and the j-th holding device in the fixed coordinates system X–Y–Z. Ω is the rotation angular velocity of the LLS workpiece. “.” is the derivative of ‘t’. ‘'’ is the derivative of ‘x’. The dissipated energy induced by the damping is expressed as,

D=

1 2

k

∑ (Cvf,i V˙ 2 + Cwf ,i W˙ 2)| x=xif i=1

+

1 2

j=1

∫0

l

qw V (x , t ) =

∫0

l

∑ ϕm (x ) am (t ) m=1

[aT ,

bT ,

(11)

cT ],

FT

[QaT ,

QbT ,

(12)

As the MPVC caused by floating supports and holding devices have been analyzed by the assumption of linear spring constraints, the chuckcenter clamping simplified to the fixed-hinged is investigated in this part. Namely, the boundary conditions of the dynamic model are used to represent the clamp. The geometric boundary conditions can be represented as

(7)

⎧ v (x w , t ) = 0 ⎧ w (x w , t ) = 0 ⎧ u (x c , t ) = 0 , v (x c , t ) = 0 , w (x c , t ) = 0 ⎨ ⎩u′ (x c , t ) = 0 ⎨ v′ (x c , t ) = 0 ⎨ w′ (x c , t ) = 0 ⎩ ⎩

(8)

(13)

Where, xc and xw denote the position of the chuck and the center in the fixed coordinates system X–Y–Z. Then the transformation matrix for geometric constraints is introduced,

a4 a ⎡ 1⎤ T ⎡ ⎤ ⎢ a2 ⎥ = ⎡ a1 ⎤ ⎢ a5 ⎥ = Ta [ad ] ⎢⋯⎥ ⎣ I ⎦⎢⋯⎥ ⎣ aN ⎦ ⎣ aN ⎦

(9)

Combining the Eqs. (4)–(8), the matrix form of dynamic equations are arranged as,

0 0 ⎤ ⎧ a¨ ⎫ ⎡ DV H 0 ⎤ ⎧ a˙ ⎫ ⎡A + B + C 0 0 ⎥ b¨ + ⎢−H DW 0 ⎥ b˙ A+B+C ⎢ ⎨ ⎬ ⎬ ⎢ 0 0⎦ ⎨ 0 0 A + C⎥ ⎣ ⎦ ⎩ c¨ ⎭ ⎣ 0 ⎩ c˙ ⎭ Qa 0 0 ⎤ a ⎡ E + (G − F ) + KV ⎧ ⎫ ⎧ ⎫ +⎢ 0 E + (G − F ) + KW 0 ⎥ b = Qb ⎨c ⎬ ⎨ ⎬ ⎢ 0 0 EA ⎥ ⎣ ⎦ ⎩ ⎭ ⎩ Qc ⎭

qT

2.2. The boundary and loading conditions of the dynamic model

Where qw is the uniform load of LLS gravity. By Lagrange principle,

d ⎛ ∂T ⎞ ∂T ∂U ∂D ⎜ ⎟ − + + =Q dt ⎝ ∂q˙ ⎠ ∂q ∂q ∂q˙

FT

[M ] q¨ + [C ] q˙ + [K ] q = [Fˆ ]

N

qw

x = x bj

= = QcT ], and are defined as Then where aT = [a1, a2 , ⋯, aN ], bT = [b1, b2 , ⋯, bN ], cT = [c1, c2, ⋯, cN ], QaT = f y [ϕ1, ϕ2 , ⋯, ϕN ], QbT = fz [ϕ1, ϕ2 , ⋯, ϕN ], QcT = fx [ϕ1, ϕ2 , ⋯, ϕN ], For simplicity, Eq. (9) can be expressed as,

qT

Where c vf, i and c wf , i respectively denote the damping of the i-th floating support in Y, Z directions. c vb, i and c wf , i denote the damping of the j-th holding device in Y, Z directions respectively. Gravity effect can be expressed as,

WG =

x = x bj

∫

2

∑ (Cvb,j V˙ 2 + Cwb ,j W˙ 2)| x=x bj

x = x bj

∫ ∫ ∫ ∫

2

∫0

EAU′2dx +

l

∑ mb (V˙ 2 + W˙ 2 + U˙ 2)| x=x bj

1 2

l

1 2

1 2

∫0

x = x bj

⎡ b4 ⎤ ⎡ b1 ⎤ ⎢ b2 ⎥ = ⎡Tb1 ⎤ ⎢ b5 ⎥ = T [b ] b d ⎢⋯⎥ ⎣ I ⎦⎢⋯⎥ ⎢ bN ⎥ ⎢ bN ⎥ ⎣ ⎦ ⎣ ⎦ c3 c ⎡ ⎤ ⎡ 1⎤ T ⎢ c2 ⎥ = ⎡ c1 ⎤ ⎢ c4 ⎥ = Tc [cd] ⋯ ⋯ ⎢ ⎥ ⎣ I ⎦⎢ ⎥ ⎣ cN ⎦ ⎣ cN ⎦

(10)

Where, 4

(14)

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3 4 x wN − 1 ⎡ xw xw ⋯ ⎤ ⎢ x3 x4 ⋯ ⎥ xcN − 1 c c ⎢ ⎥ ⎢3xc2 4xc3 ⋯ (N − 1) xcN − 2 ⎥ ⎣ ⎦

−1

3 4 x wN − 1 ⎡ xw xw ⋯ ⎤ ⎢ x3 x4 ⋯ ⎥ xcN − 1 c c ⎢ ⎥ − 2 3 2 N ⎢3xc 4xc ⋯ (N − 1) xc ⎥ ⎣ ⎦

2 ⎡1 x w x w ⎤ Ta1 = −⎢ 1 x c xc2 ⎥ ⎥ ⎢ ⎣ 0 1 2x c ⎦ 2 ⎡1 x w x w ⎤ Tb1 = −⎢ 1 x c xc2 ⎥ ⎥ ⎢ ⎣ 0 1 2x c ⎦

2 3 xcN − 1 1 x c ⎤− 1 ⎡ xc xc ⋯ ⎤ Tc1 = −⎡ ⎢ 2 0 1 ⎣ ⎦ ⎣ 2x c 3xc ⋯ (N − 1) xcN − 2 ⎥ ⎦

The uniform layout is also the conventional distribution method of floating supports in whirl milling. Fig. 3(b) is the tool holder which is embed in whirling head during machining, and cutters mounted in the tool holder are shown in the Fig. 3(c). In whirl milling, the high-speed rotating tool holder feeds along the axial direction drives the cutters to generate cutting motions. Fig. 3(d) shows the machining process of whirl milling, and it can be seen that the holding devices tightly contact with the workpiece. The moving holding devices and the fixed floating supports are shown in the Fig. 3(e) and Fig. 3(f) respectively. Since the effective thread length of the LLS workpiece is 7m, the corresponding coordinates of the thread in fixed coordinates system X-Y-Z are 0.5m -7.5m along X direction, which is also the actual machining part of the LLS workpiece. Hence, the position of three uniformly distributed floating supports are x1f = 2m , x 2f = 4m , x 3f = 6m . The detailed machining parameters are given in Table 1. As mentioned before, the vibrations at cutting point have a great impact on the cutter-workpiece engagement, and reducing the vibrations of the cutting point could improve the machining quality of the LLS workpiece in overall length. In this case, the vibrations at the cutting point are investigated to give a comprehensive view of the effects caused by the complex whirl milling system on workpiece quality. However, due to the cutting position is inside the whirling head and the workpiece stays rotating during machining, the acceleration sensors cannot be directly mounted on the workpiece to measure the workpiece vibrations. In addition, since tool holder travels along the axial direction together with the cutting tool in relation to workpiece position, the relative position between workpiece and cutting position is changing. As a result, mounting the acceleration sensors on the workpiece could lead to unstable sensor signals. In this case, the indirect measurement method, which is proved as an effective method to measure the cutting vibrations by Lu et al. (2018) and Cao et al. (2013), is employed to obtain the vibrations of the cutting position. As shown in the Fig. 4, the sensors are mounted in both right (positon1) and left (position 2) holding device to indirectly measure the vibration responses of the workpiece at cutting position. As well known, the cutting forces model developed by Altintaş and Budak (1995) is accurate method to obtain the cutting forces theoretically. However, due to that the thin and varying chip thickness in whirl milling is significantly affected by the intermittent cuttings, the theoretical cutting forces calculated by the cutting forces model may have significant error from the real ones. Therefore, the cutting forces served as the input of the simulations are obtained by experiments. In order to obtain the three-dimensional cutting forces during machining process, a tool embedded with three-axis force sensors (Kistler9602A3) are specially designed. The detailed force signal collection devices shown in the Fig. 5. Fig. 5(a) shows the installation of the specially designed jig holding the force sensors, the structure of which is shown in the Fig. 5(b). Fig. 5(c) is the slip ring, which rotates with the tool holder and wire out the force sensor cables during machining. The diagram of force signal collection mechanism is shown in the Fig. 5(d).

(15)

The generalized coordinates of the system can be transformed into,

ad Ta 0 0 ad a ⎡ ⎤ = ⎡ 0 T 0 ⎤⎡b ⎤ = T ⎡b ⎤ ⎥⎢ d ⎥ b ⎢ d⎥ ⎢b⎥ ⎢ ⎥ ⎣ cd ⎥ ⎢ ⎣c ⎦ ⎢ ⎣ cd ⎥ ⎦ ⎦ ⎣ 0 0 Tc ⎦ ⎢

(16)

Where, T is the transformation matrix with the boundary conditions. Denoting qdT = [adT , bdT , cdT ], substituting Eq. (15) into Eq. (11) to obtain the final system dynamic model with associated geometric constraints,

[T T ][M ][T ] q¨d + [T T ][C ][T ] q˙ d + [T T ][K ][T ] qd = [T T ][Fˆ ]

(17)

Due to the fact that cutting forces are the main contributing factor for deformation and vibration of the LLS workpiece, an accurate representation of the three-dimensional cutting forces in whirl milling is the key to obtain accurate solution results. In order to increase the accuracy of solutions of the developed dynamic model, the three-dimensional cutting forces set as the input of the proposed model are obtained from experiments. Namely, the amplitude and period of the three-dimensional cutting force are obtained by collecting cutting force data and vibration data. The detail will be introduced in Section 2. The central difference method and direct integration method are often used to solve dynamic models. But for large and complex models, the direct integration method shows better performances of computational accuracy and efficiency, noted by (Ding et al., 2018). Wilson-θ methods, Runge-Kutta methods, Linear-acceleration methods and Newmark methods are the four main methods of direct integration method. In this paper, the Runge-Kutta method is chosen to solve the proposed dynamic model in simulation. In summary, the calculation procedure of the dynamic model is shown in Fig. 2. 3. Experiment verification 3.1. Experiment setup In this section, the whirl milling experiments of an eight-meter LLS are performed to validate the proposed model. Both the cutting forces and workpiece vibrations are measured during machining process. As shown in Fig. 3(a), the experiments are conducted on the Eight-meter CNC Whirl Milling Machine with uniform layout, which means all the floating supports equally distribute along the axial direction of the LLS.

Fig. 2. Solution flowchart of the dynamic model. 5

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Fig. 3. Experiment setup. Table 1 Machining parameters. Parameter type

Parameter items

LLS workpiece parameters

Material GCr15 Chamfer 0.15 × 25°

Cutter parameters

Hardness (HRC) 62 Clearance angle 5°

Diameter (mm) 99 Rake angle −7°

Length (mm) 800 Number of cutters 6

Lead(mm) 10 Speed (rev/min) 476.1

Speed (rev/min) 3.08 Eccentricity (mm) 20.44

3.2. Model of cutting force

The force signals are collected from one of the six cutting tools during machining. The specifications of force and acceleration sensors are given in Table 2, and the sensor signals are simultaneously collected by the Prosig8020 data acquisition system online.

As mentioned earlier, the cutting forces served as the input of the simulations are obtained by experiments. Fig. 6 shows experimental results of cutting forces and vibrations. The measured cutting force

Fig. 4. The measurements of vibration signal. 6

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Fig. 5. The cutting force signal collection devices. Table 2 Sensor parameters. Force sensor

Fx / Fy Measurement range −0.5˜0.5KN

Fz Measurement range −1˜1KN

Fx / Fy Sensitivity 10mV/N

Acceleration sensor

Measurement range 500g

Frequency range 0.5Hz˜6.5kHz

Sensitivity 10mV/g

Fz Sensitivity 5 mV/N

Fig. 6. Experimental results of cutting force and LLS workpiece vibration. (a)Cutting force signals in X direction; (b)Cutting force signals in Y direction; (c)Cutting force signals in Z direction; (d)Acceleration signals in X direction;(e) Acceleration signals in Y direction; (f) Acceleration signals in Z direction.

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constraint, floating supports 1, 2, 3 engaged), while the simulation time is 0.22s and 1s respectively. Fig. 8(c) and (d) are the simulation results with cutting position at x = 3.729m (the floating support 1, 3 engaged, and 2 disengaged). The simulation results obviously indicated that the vibration responses present a periodic trend with variable amplitude, and the most significant vibration responses appear in Z direction, which is in good agreement with the experimental results. Moreover, the simulation results indicate that the vibration amplitudes at x = 3.729m are 2.01 × 10−5m and 4.35 × 10−5m in Y and Z direction respectively, which are larger than that at x = 2.974m with vibration amplitudes of 1.86 × 10−5m and 3.74 × 10−5m in Y and Z direction respectively. The difference between these two conditions is that the floating support 2 disengages with the LLS workpiece at x = 3.729m, which suggests that the vibration responses of LLS workpiece are significantly affected by the engage status of floating supports. Both the simulation and experimental results demonstrate that the minimum vibration amplitude appears in the X direction. Thus, it is omitted in the subsequent analyses, while the vibration responses in Y and Z direction are further discussed. Fig. 9 and Table 3 show the experimental results of vibration responses at different cutting points. It can be concluded from Fig. 9 that the vibration responses in Y direction and Z direction have similar trend, though more significant in Z direction. Therefore, the experimental results of the vibration responses in the Z direction are used to verify the dynamic model. Fig. 10 shows the comparison between the simulation and the experimental vibration data. The errors at different cutting points are shown in Table 4. The results indicate that the simulation and experimental vibration responses have similar trend along the Z direction. Furthermore, the errors of the vibration amplitude shown in the Table 4 are in the range of 1.6% to 8.4%, which are within acceptable limits. These suggested that the simulation and experimental results are in good agreement. Moreover, considering that the natural frequency is the basic dynamical parameter in an oscillatory system, and the natural frequencies of the LLS workpiece can be obtained by solving the eigenvalues problem of Eq. (17), the comparisons of the natural frequencies between the simulation and experiments is made to further validate the efficiency of the proposed dynamical model. In this case, a series of offline model experiments are carried out on the LLS workpiece under four different constraint conditions: three floating supports with no holding devices (Situation Ⅰ), no floating supports with two holding devices (Situation Ⅱ), two floating supports with two holding devices (Situation Ⅲ) and three floating supports with two holding devices (Situation Ⅳ). The first three natural frequencies of the LLS workpiece obtained by the offline model experiments and the presented dynamical model are given in the Table 5. The percentage error for all situations are below 10%, which show that the theoretical results are in good agreement with the experimental results. By the prediction of the proposed dynamic model, the optimization of the whirl milling is conducted in the next section.

signals shown in Fig. 6(a–c) indicate that the cutting forces are discontinuous and has periodic variable amplitude in three directions. Because only the force signals of one cutter are collected, the cutting period of one cutter is about 0.12 s which is the distance of adjacent peak values. Therefore, the cutting interval of adjacent cutters is about 0.02s because 6 cutters are set in the tool holder. Due to the fact that the width of each cutting force wave is determined by the effective signals shown in the Fig. 6. Moreover, the maximum cutting force is about 150N in Z direction (tangential direction), while the cutting forces are less than 60N in both X (axial direction) and Y (radial direction) directions. Due to the experimental results of vibrations show that the vibration amplitudes at position 1 are larger than that at position 2, the signals measured at position 1 are used to verify the proposed model. Fig. 6(d–f) show the experimental vibration signals of the left holding device in three directions. It can be seen that there obviously exists a periodical variation trend in three directions. Although only the vibration signals of one cutter are collected, indeed, each of the six cutters will cause vibration on the LLS workpiece during machining. Thus, there are 6 extrema of vibration signals in each period. The acceleration signals also indicate the interval time of the adjacent amplitude extremum is about 0.02s, corresponding to the cutting interval of adjacent cutters. The results coincide with the conclusion inferred by force signals. Furthermore, the acceleration signals show that the maximum vibration appears in the Z direction with an amplitude of 3.4g (m/s2), which is much larger than these in both X and Y directions. Based on these experimental results, the simplified three-dimensional cutting forces are assumed as a combination of discontinuous periodical triangle waves shown in the Fig. 7. The period of the triangle waves is about 0.022s, corresponding the cutting period of real cutting forces. The width of the triangle waves is referred to the effective duration of real cutting forces, which is 0.002s in each period. The amplitude of the triangle waves is set as 58N, -58N and 152N in X, Y and Z direction respectively. The simplified three-dimensional cutting forces are served as the input of the dynamic model during simulations. 3.3. Model verification The dynamic model is verified in this section based on the comparison between simulation and experimental results of workpiece vibrations. Due to the solutions of the dynamic equation are vibration displacements of LLS workpiece, the acceleration signals measured by the data acquisition system are transformed to the vibration displacements in the data acquisition software. The simulation parameters, such as rotational speed and radius of LLS workpiece, number of cutters and so on, are set the same as that in experiments. As mentioned earlier, the vibrations at the cutting point have great impacts on the machining quality of LLS workpiece. Thus, the dynamic response of cutting points is used for the verification. Fig. 8 shows the simulation results of workpiece vibrations at different cutting position. As mentioned before, the period of system excitation is 0.022s. Fig. 8 (a) and (b) are the simulation results with cutting position at x = 2.974m (Normal

Fig. 7. System excitation of simplified three-dimensional cutting forces. 8

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Fig. 8. Simulation results of workpiece vibrations at different cutting position (x = 2.974m and x = 3.729m).

Fig. 9. The vibration responses at different cutting points.

Fig. 10. The comparison between the simulation and experimental results.

4. Optimization of whirl milling based on the multi-points variable constraints

the effects of multi-points variable constraints are investigated through the proposed dynamic model. More specifically, the number, location and contact status of floating supports are the influencing factors on constraints of the LLS workpiece. The contact status of floating supports varies depending on the position of the holding devices that are feeding along the axial direction in whirl milling. In this section, the number and location of floating supports are manually changed in simulation to compute the optimal dynamic performance.

As mentioned earlier, the holding devices and floating supports are auxiliary devices employed to decrease the workpiece vibrations in whirl milling. The simulation and experimental results in section 2 have suggested that the constraints representing the holding devices and floating supports applied to LLS workpiece have significant impacts on its dynamic performance. Indeed, smaller vibration amplitude of LLS workpiece are beneficial to machining quality. Therefore, this section investigates the effects of holding devices and floating supports to achieve optimal dynamic performance of the LLS workpiece. Namely,

Table 3 The vibration responses at different cutting points. Cutting points location (m) 0.745 1.821 2.974 3.729 4.834 5.775 7.123

Constraint conditions Normal constraint Floating Support 1 disengaged Normal constraint Floating Support 2 disengaged Normal constraint Floating Support 3 disengaged Normal constraint

Vibration amplitude in Y direction(m) −5

1.77 × 10 2.10 × 10−5 1.86 × 10−5 2.01 × 10−5 1.66 × 10−5 2.02 × 10−5 1.59 × 10-5

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Vibration amplitude in Z direction(m) 3.34 × 10−5 4.51 × 10−5 3.74 × 10−5 4.35 × 10−5 3.84 × 10−5 4.18 × 10−5 3.25 × 10−5

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Table 4 The comparison between the experimental results and the simulation values. Location of cutting point /m 0.745 1.821 2.974 3.729 4.834 5.775 7.123

Constrain conditions

Experimental vibration amplitude in Z direction /m

Normal constraint Floating Support 1 disengaged Normal constraint Floating Support 2 disengaged Normal constraint Floating Support 3 disengaged Normal constraint

−5

−5

3.06 × 10 4.41 × 10−5 3.55 × 10−5 4.10 × 10−5 3.78 × 10−5 4.03 × 10−5 3.03 × 10−5

3.34 × 10 4.51 × 10−5 3.74 × 10−5 4.35 × 10−5 3.84 × 10−5 4.18 × 10−5 3.25 × 10−5

1 2 3

Situation Ⅱ

Situation Ⅲ

Situation Ⅳ

Error (%)

11.0|11.2 18.8|19.2 28.7|30.1

10.3|10.1 28.8|27.9 55.9|53.4

13.5|14.2 26.4|28.4 52.2|56.6

20.2|21.5 45.5|49.2 64.5|70.8

1.8|1.9|5.2|6.4 2.1|3.1|7.6|8.1 4.9|4.4|8.4|9.8

Error 8.40% 2.20% 5.10% 5.70% 1.60% 3.80% 6.80%

disengages, reaching a level similar to that without floating supports. With the number of floating supports increases, the average vibration amplitude in overall length tends to decrease, and for each case the vibration amplitude increases near the location where the floating support disengages. When the number of floating supports is 5, the average vibration amplitude in overall length is 2.34 × 10−5m, which reduce 0.77 × 10-6m compare to that of 4 floating supports. However, the reduction of average vibration amplitude is 1.50 × 10-6m when the number of floating supports changes from 3 to 4. It suggests that the number of floating supports has little impacts on the vibration responses of LLS if sufficient floating supports are utilized in whirl milling. Considering the slight increases of maximum vibration amplitudes shown in Fig. 12, the optimal number of the uniformly distributed floating supports should be 4 in theory. Nevertheless, 3 floating supports may be another feasible option. Indeed, the problems such as system complexity and the economic cost may be encountered with the increase in number of floating supports. Moreover, when the number of floating supports is changed from 4 to 3, the LLS workpiece vibration is almost equivalent while total floating supports keep contact with LLS workpiece. Therefore, in eight-meter LLS machining under uniform layout, 3 floating supports are considered an another optimal option which is more advised to apply in practical production.

Table 5 The natural frequencies of the LLS workpiece with different constraint conditions obtained by experiments | simulation (Unit: Hz). Situation Ⅰ

Simulation vibration amplitude in Z direction /m

4.1. The effects of floating supports on dynamic responses of the LLS workpiece The simulation results shown in the Fig. 11 reveal that the contact status of the floating supports obviously affects the LLS workpiece vibration. Meanwhile, the LLS workpiece vibration amplitude decreases with the increases in the number of floating supports leading to higher bending stiffness of the machining system. However, too many floating supports may lead to redundant constrains and high economic cost. Thus, it is necessary to investigate the effects of contact status and numbers of floating supports on LLS workpiece vibration. In order to map the real cutting process and study the effects of contact status, the dynamic responses of LLS workpiece is studied with the cutting forces moving along the length of LLS. It’s worth to note that a floating support will disengage with the LLS workpiece to avoid collision when the travelling holding device is close to the floating support. Fig. 11 shows the simulation results of vibration amplitude in both Y and Z directions. The vibration responses reach peak level at the points A, C and E where the floating supports start to disengage with the LLS workpiece, however decrease quickly after the points B, D and F where floating supports are re-engaged with the LLS workpiece. Thus, the simulation results in Fig. 11 verify that the contact status change of the floating supports will directly affect the dynamic responses of LLS workpiece. In addition to the contact status of the floating supports, the number of floating supports also affect the dynamic responses of the LLS workpiece. Six conditions are analyzed: the number of floating supports utilized in whirl milling changes from 0 to 5. The uniform layout of the floating supports is assumed for each case. Fig. 12 shows the results of the maximum vibration amplitudes in both Z and Y directions along the length of LLS workpiece. The results indicate that the LLS workpiece vibration is easily affected by the number of floating supports, while the vibration amplitude decreases with the increase of the number of floating supports in general. However, the trend of vibration amplitude shows a slightly increase when the number of floating supports changes from 4 to 5, indicating that the increase in floating supports does not necessarily reduce the vibration amplitude. Therefore, the vibration responses of LLS workpiece in the Z direction are further analyzed in order to study the resultant effects of the number and contact status of the floating supports. Fig. 13 shows the results in overall length of LLS workpiece with floating supports number changes from 0 to 5. When the number of floating support is 1, the vibration amplitude reduces significantly when the floating support is engaged with the LLS workpiece, yet it increases significantly when the floating support

4.2. Optimal analysis of the floating supports location Since chuck clamping is applied at one end and center clamping at the other end, different boundary conditions applied at the two ends of the LLS workpiece in whirl milling. As shown in the Fig. 13, the vibration responses are asymmetric near two ends. In general, the vibration responses near the end constrained by the center (location x = 0(m)) are more severe than that near the end held by chuck (location x = 8(m)). Hence, the non-uniform layout (the floating supports are distributed unequally along the length of the LLS) shall be investigated. In this section, the optimal dynamic responses of the LLS workpiece under the non-uniform layout is studied in the case of three

Fig. 11. Dynamic response in overall length. 10

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Table 6 The coordinates and the vibration responses with different generations. Generations

x1 (m)

x2(m)

x3(m)

Tangential vibration amplitude at the point of x = 4 (m)

10 20 30 40 50

1.537 1.569 1.589 1.641 1.546

3.589 3.572 3.62 3.624 3.608

5.539 5.603 5.582 5.651 5.531

3.29 × 10−5 3.31 × 10−5 3.28 × 10−5 3.29 × 10−5 3.27 × 10−5

x ⎡ 1⎤ ⎡1 ⎤ ⎡3⎤ − 1 1 0 ⎤ b = ⎡ 0.5⎤ Where x = ⎢ x2 ⎥, A = ⎡ , , L = ⎢3⎥, U = ⎢ 5 ⎥, and 0.5 0 − 1 1 ⎣ ⎦ ⎣ ⎦ ⎣5 ⎦ ⎣7⎦ ⎣ x3 ⎦ L, U represent the range of the independent location variables of the floating supports. The computational accuracy of the GA is directly affected by the its generations. In theory, the computational accuracy of the algorithm improves with the increase in the number of generations, however after the calculation results are converged, any further increase in the number of generation will lead to little change in results. In this paper, the generations are respectively set as 10, 20, 30, 40 and 50. The calculated coordinates of the three floating supports and the tangential vibration amplitude are shown in Table 6. Fig. 14 shows the trend of the best and mean fitness of the objective function with 40 and 50 generations. It can be seen from Fig. 14 that the best fitness and the mean fitness are converged at the 50th iterations because they are approximated to each other. Moreover, both the best fitness and the mean fitness change a little from the 40th iterations to the 50th iterations. Therefore, the coordinates of the 50th iterations shown in the Table 6 are set as the optimal location of the three floating supports. The locations of the three floating supports which are set as x1f = 2m , x 2f = 4m , x 3f = 6m for the uniform layout of floating supports are changed to x1f = 1.546m , x 2f = 3.608m , x 3f = 5.531m after the GA optimization. The vibration responses along the length of LLS workpiece before and after optimization are given in Fig. 15. The results show that the vibration responses before optimization are more severe than that after the optimization. The vibration amplitude decreases the most at the location of xmax = 6.053m, decreasing from 4 × 10−5m to 2.5 × 10−5m in Z direction. Meanwhile, the location of minimum amplitude reduction appears at xmin = 3.526m, which vibration amplitude decreases from 3.9 × 10−5m to 3.6 × 10−5m in Z direction. The average vibration amplitude reduction is about 22.2% along the length of LLS workpiece. It is proved that the optimization of floating supports location effectively reduces the dynamic responses in overall length of the LLS workpiece. Thus, machining quality of LLS workpiece can be improved with the distribution of non-uniform layout.

Fig. 12. The effects of the floating support number.

Fig. 13. Dynamic responses in overall length with different number of floating supports.

floating supports utilized in eight-meter LLS machining. Genetic Algorithm(GA) is used in this paper to solve the optimal location of floating supports. Compared to other traditional algorithms, GA is more efficient in solving problems which the objective function is discontinuous, stochastic, or highly nonlinear. It can be concluded from Fig. 13 that the vibration amplitude at x = 4m is relatively larger along length of LLS workpiece. Therefore, the tangential (Z direction) vibration amplitude at the location of x = 4m is set as the objective function, and the locations of floating supports are the independent variables in GA. Assume the coordinates of the three floating supports are x1, x2, x3 respectively, thus, the location matrix of the three floating supports is expressed as x = [x1 x2 x3]. For the case study, the length of the holding devices is about 0.25m, the theoretical spacing of floating supports should be greater than 0.5m to avoid collision with the holding devices since there are two holding devices located on both sides of the travelling whirling cutting head. Hence, the constraint conditions representing the floating supports can be expressed as,

5. Conclusion This paper is focused on the dynamic modeling method and the optimization of whirl milling processes. Multi-point variable constraints imposed by floating supports and holding devices, three-dimensional force, workpiece rotation and gravity are considered in the dynamic modeling. The numerical results calculated by the proposed dynamic model are compared with those obtained by experimental tests carried out on the Eight-meter CNC Whirl Milling Machine. Furthermore, the optimal analyses of dynamic responses are conducted base on the proposed model. The effects of contact status, the number and layout of the floating supports are further analyzed. The following conclusions are achieved:

⎧ 1 < x1 < 3 x − x2 > 0.5 3 < x2 < 5 , and ⎧ 3 ⎨ ⎨ ⎩ x2 − x1 > 0.5 ⎩5 < x3 < 7 The matrix presenting of the constraint conditions of the floating supports can be expressed as

(1) Experimental verifications indicate that the proposed dynamic model is feasible and effective for predicting the dynamic responses in whirl milling, with good agreements exist between the measured and calculated results at different cutting points.

{ L Ax< x>
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Fig. 14. (a) Fitness curves with 40 generations, (b) Fitness curves with 50 generations.

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Fig. 15. Vibration responses of the LLS workpiece before and after the GA optimization.

(2) The fact that dynamic responses of LLS workpiece is related to the contact status and number of floating supports is analytically confirmed. When a floating support disengages with the LLS, the dynamic response is much more severe than that all the floating supports are in contact with the LLS workpiece. The average vibration amplitude tends to decrease with the increase of the number floating supports, but change a little if enough floating supports are set. Numerical results indicate that the optimal number of floating supports in the eight-meter LLS machining is 4, yet 3 floating supports are advised considering practical factors. (3) Compared with the traditional distribution method of uniform layout, the optimal technology of non-uniform layout effectively reduces the dynamic responses of LLS workpiece in overall length. Based on this manageable method, whirl milling of typical LLS workpiece (axial length > 6 m, ratio of length to diameter > 30) are suggested to be optimized in practical production. The optimal location of floating supports can be efficiently calculated by Genetic Algorithm.

Acknowledgements The authors would like to thank the support from the National Key R&D Program of China (Grant No. 2018YFB2002200), the National Natural Science Foundation of China (Grant No.51575072), the Fundamental Research Funds for the Central Universities, China (No. 2018CDQYJX033).

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