Automatic Device with Self-Braking for Fixation Objects in Executive Links of Technological Equipment

The most common typical structure of a mechanical CM that provides reliable maintenance of the clamped state of the object of fixation without external energy consumption based on the principle of geometric locking of the mechanism. This is most often realized through the use of a lever mechanism and an input link containing conical and cylindrical surfaces (Fig. 1). When the point of interaction between a large arm of a lever reaches the cylindrical part of the specially-shaped link, the mechanism becomes geometrically locked. Such a structure provides a constant amount of movement of CM elements, and what is essential are the clamping elements of a chuck. As a result, the CM system's stress state values will vary if the diameter of the clamping object deviates from the nominal values for which the CM is configured.

Fig. 1.

Mechanical CM’s actuator with geometric locking: a) functioning scheme in combination with a lever chuck (in the “clamped” state); b) general view of the mechanism (in the “unlocked” state)

This places significant limitations on the clamping of, uncalibrated objects. In addition, the presence of radially moving drive elements worsens the conditions for implementing and maintaining high-precision balancing of the spindle assembly. This disadvantage is common to all CM with geometric locking.

A new alternative structure of the electromechanical actuator of the CM has been designed based on the recommendations of previous research and structure-scheme synthesis [2]. The actuator with the new proposed structure is expected to have improved operating characteristics and is adapted to operate autonomously as well as at high rotational frequencies for use as part of a spindle assembly. The design of the proposed CM is devoid of most of the disadvantages of the hydraulic and mechanical prototypes and is protected by Patent 95323 (Ukraine). The CM drive is mounted at the rear end of spindle 1 (Fig. 2). Its body in the form of sleeve 3 is held against rotation relative to spindle 1 by key 2. Part of the external cylindrical surface of sleeve 3 is provided with an external threaded surface on which rotor 5 and nut 6 that restricts its stroke are screwed. On sleeve 3 there is also axially movable bushing 4 which has the possibility of simultaneous force interaction with cylinder 7 and rotor 5 via thrust bearings 18 (fixed with nut 19) and cylinder 17 which is rigidly connected to rotor 5. In the rear end of sleeve 3, there is installed drive bush 8 which can move in the axial direction and is capable of interacting simultaneously with nut 9 placed on drawbar 10 in the form of tube and cylinder 7 through a set of elastic elements 11. The preload of elastic elements 11 is provided by the nut 12 mounted on drive bush 8. The right end of drawbar tube 10 is connected to a clamping chuck (not shown in Fig. 2). Electric windings 13 are located on rotor 5 and can interact with electromagnetic windings 14 located on stator 15, which is rigidly connected to the spindle body 16.

Fig. 2.

Design of the new electromechanical CM actuator

The device presented (Fig. 2) works as follows. To perform the clamping operation, windings 14 of stator 15 are supplied with electricity. At the same time, a magnetic field is created around windings 14, which interacts with windings 13 of rotor 5 and gives them torque. As windings 13 are rigidly connected to rotor 5, this causes the rotational and translational movement of rotor 5 relative to body 3. The axial force from rotor 5 is transmitted through a set of structural elements to the end of drawbar 10, which is connected to the clamping chuck at the other end (not shown). The clamping process continues until the required clamping force is reached, which depends only on the parameters of the electric current supplied to stator windings 14 and is stable. This means that it does not depend on the radial dimensional deviation of the workpiece. The rotation-translation movement of rotor 5 and the clamping process stop when the required clamping force is reached, thus increasing the force opposing the axial movement. A moment of completion of the clamping (rotor stop) is reflected in the characteristics (pattern of change) of the electric current of winding 14. At the end of the clamping operation, the electric current supplied to stator windings 14 stops and the clamping force is maintained by the self-braking of the threaded gear between rotor 5 and sleeve 3. In order to initiate the process of releasing a workpiece, a current with characteristics is supplied to windings 14, which determine the movement of rotor 5 in the opposite direction to the clamping process, until it stops at nut 6, located on body 3. The workpiece releases under the elastic forces of the collet petals (not shown).

The design of the proposed clamping mechanism possesses the following advantages: no need for mechanical contact with the elements of a spindle unit for power supply; a few movable and complex elements and no radially movable elements improves the conditions of balancing a spindle unit; an absence of subsystems for the conversion and transfer of energy to the clamping driver (electrical energy is fed directly). In order to compensate for the loss of clamping force when machining at high frequencies of rotation, the presented clamping drive allows for changing the amount of clamping force by supplying power to the stator windings with the appropriate characteristics.

For calculation parameters of the proposed CM the special calculation scheme (Fig. 3) is created.

Fig. 3.

The scheme for calculating CM parameters

For the calculation, the maximum output axial force that can be produced by the driver of CM is S = 20 kN. The thread screw transmission “rotor-spindle” is self-braking and has the diameter d₂ = 85 mm and the pitch p = 2 mm. In order to reduce wear of the thread transmission it is reasonable to choose antifriction pair materials of the inner threaded surface (in the rotor) – cast iron and the outside thread (on the spindle) – steel.

Verification of the thread diameter by wear resistance condition is done with the formula: (1) d2≥S/π⋅ψH⋅ψh⋅σCM {d_2} \ge \sqrt {S/\left( {\pi \cdot {\psi _H} \cdot {\psi _h} \cdot \left[ {{\sigma _{CM}}} \right]} \right)} where [σ_см] = 4…6 MPa – permissible tension values of crumpling for selected materials; ψ_h = 0.75 – coefficient of height of the thread; ψ_H = 1.2 – length coefficient of internal threaded surface of the rotor. Accordingly to (1) 85 mm>20000/π⋅1.2⋅0.75⋅5=38 mm 85\,{\rm{mm}} > \sqrt {20000/\left( {\pi \cdot 1.2 \cdot 0.75 \cdot 5} \right)} = 38\,{\rm{mm}} which means the wear resistance is provided more than a double reserve that allows the use of less durable and lightweight materials for the inside of the threaded rotor surface.

The minimum length of the length of the threaded surface of the rotor H that is needed to provide calculated values of wear resistance, H = ψ_H · d₂ = 1.2 · 38 ≈ 46mm. The value of pitch of a thread also was selected based on provided a self-braking ψ < ϕ_p, where ψ – an angle of rise of a thread turn; φ_p – the angle of friction in the thread. To greased, polished and fitted surfaces threaded friction a coefficient accept is f_p = 0.07. The coefficient of friction f_RP in the screw gear takes into account the influence of an angle thread profile and is determined from the f_RР = f_p/cosγ_p, where for a thrust thread γ_p = 3° – the angle of slope of the profile line of the thread. Considering the small value of the angle γ_p (cosγ_p = 0.999) in the thrust thread, equality f_RР ≈ f_p is accepted, which defines the angle of friction ϕ_p = arctgf ≈ 4°. The helix angle of the thread ψ = arctg[p/(π · d₂)] ≈ 0.43°. Thus, with the selected thread pitch, a self-locking reserve (0.43° << 4°) is provided in the screw drive, which is necessary to prevent unscrewing in the event of alternating loads, vibrations and the occurrence of radial elastic deformations.

Axial force S, which is transferred from the rotor through the drawbartube to the collet, is counterbalanced by reactions of the spindle and clamping forces. To calculate parameters of the screw gear, the forces acting on the surface of the thread are conditionally replaced with concentrated force that is normal to the surface of the thread profile. Taking into account the small angle of the thread profile γ_p, we count it they are coaxial with the axis of rotation and with S. Tangential force F_t which must be applied to the rotor on its radius to obtain the required amount of axial force S is determined from: (2) Ft=S⋅tg(ψ+ϕp). {F_t} = S \cdot tg(\psi + {\phi _p}).

Torque generated by a tangential force: (3) Tp=0.5d2Ft. {T_p} = 0.5{d_2}{F_t}. From (3) T_p = 0.5 · 0.085 ⋅ 20000 ⋅ tg4.43° = 66Nm.

From expressions (1) and (2) the expression for finding the axial force S generated at the output of the actuator is obtain: (4) S=Tp/0.5⋅d2⋅tg(ψ+ϕp). S = {T_p}/0.5 \cdot {d_2} \cdot tg(\psi + {\phi _p}).

Forces in the collet clamping chuck can be calculated from the scheme (Fig. 3). In the calculation, the stiffness of the collet petals is neglected. The equation of a balance of powers which operate on the collet in conditional planes according to Fig. 3: in horizontal plane, (5) FTP⋅cos α+R⋅sin α−S=0; {F_{TP}} \cdot cos\,\alpha + R \cdot sin\,\alpha - S = 0; in vertical plane, (6) FTP⋅sin α−R⋅cos α+TΣ+Fω=0, {F_{TP}} \cdot sin\,\alpha - R \cdot cos\,\alpha + {T_\Sigma } + {F_\omega } = 0, where Т_∑ – total radial clamping force; R – normal reaction that acts from the spindle cone; F_TP – a force of friction on the cone of the collet and spindle; α – half of an angle of a cone of a collet; F_ω – centrifugal force of inertia that acts on the petals of the collet chuck during its rotation around the axis. Taking into account F_ω = m ⋅ _ω² ⋅ r; F_TP = R ⋅ f and f = tgϕ the value S can be derived from (5) and (6): (7) S=TΣ+mω2r⋅tg(α+ϕ), S = \left( {{T_\Sigma } + m{\omega ^2}r} \right) \cdot tg(\alpha + \phi ), where m – the mass of the collet petals; r – the distance from the axis of rotation to the centre of weight of the collet petals; f – the coefficient of friction.

As a prototype electric motor for the drive of the proposed CM a three-phase asynchronous motor with a squirrel-cage rotor is chosen as the most common type. Initial information for the calculation includes the previously calculated torque on the rotor T_p = 66 Nm, which is required to produce the maximum force S = 20 kN, and the maximum frequency of rotation of the rotor approximately 6000 rpm. Utilizing the kinetic energy gathered by the rotor during the free run can help to produce part of the force S. The relationship between the maximal torque that can be generated by an electric motor and its nominal value is limited by its resistance to overheating (properties of insulation materials, means of cooling conditions) and displays the motor's overload capacity. The overload capacity of modern motors reaches up to three times, and there are features of the electric motor, which is part of the CM drive, that contribute to increasing the overload capacity of the motor: –

a special positive feature of the proposed design of the CM drive is the possibility of controlling its operation and, as a result, the final stage of clamping during spindle rotation, which is assumed to be 5000 rpm;

Taking everything into consideration, it was agreed that the electric motor could overload about two times. Thus, the calculation of electric motor parameters was conducted with the condition of providing a nominal torque during work in long-duration mode M₁ = 35 Nm.

Nominal mechanical power P₁ of electric motor is determined from: (8) P1=0.1046⋅M1⋅n1. {P_1} = 0.1046 \cdot {M_1} \cdot {n_1}.

For the case of the operation mode of the proposed CM, which involves its functioning (changing the value of the clamping force) during the spindle rotation, the rotational speed n₁ will be determined from n₁ = n_p – n as relative frequency of rotation of the rotor, where n_p – the frequency of rotation of the rotor, n – the frequency of rotation of the spindle. According to (8), the motor power that should be used to drive the proposed СM: (9) P1=0.1046⋅35⋅6000−5000≈3.7kW. {P_1} = 0.1046 \cdot 35 \cdot \left( {6000 - 5000} \right) \approx 3.7{\rm{kW}}{\rm{.}}

Defined power is produced by the CM motor during the relatively short clamping time. After the clamping process is complete, the motor rotor rotates with the spindle due to the torque from the spindle drive. This rotation of the CM motor rotor contributes to its cooling for a period of time that is at least several times longer than the time of its operation under load. At the same time, the tension state of the system, which provides the clamping force, is ensured by self-braking. This load mode, which involves short-term operation and a long cooling period, provides improved opportunities for using motor loads above the rated values.

The first stage of work of the proposed CM is characterised by the action of forces of dissipative (not potential) character since no tension is created in the system. The proposed CM design can be considered a system with one degree of freedom. The position of other elements of the clamping mechanism of the proposed design depends on the angle of rotation of the rotor Ω_p, and is considered a generalised coordinate in this study. To conduct studies on the relationship between the movement of CM links in the general form is advisable to use Lagrange's Equation of second kind. With respect to the generalised coordinate Ω_p, the divergent equation of motion will be written: (10) ddt∂E∂Ω•p−∂E∂Ωp=Qr, {d \over {dt}}\left( {{{\partial E} \over {\partial {{\mathop \Omega \limits^ \bullet }_{\rm{p}}}}}} \right) - {{\partial E} \over {\partial {\Omega _{\rm{p}}}}} = {Q_r}, where E – kinetic energy of the clamping mechanism, Ωp• \mathop {{\Omega _{\rm{p}}}}\limits^ \bullet – the first derivative of the angle of rotation of the rotor, which is its angular velocity, Q_r – generalised force in the form of a reduced torque is equal to the main torque of external forces relative to the rotor rotation axis, t – the time during which the action takes place.

During the operation of the SM mechanism, its kinetic energy E does not depend on the angle of rotation of the rotor Ω_p, so the partial derivative of the kinetic energy E by the angle of rotation of the rotor ∂E∂Ωp=0 {{\partial E} \over {\partial {\Omega _{\rm{p}}}}} = 0 , and therefore (10) takes shape: (11) ddt∂E∂Ω•p=Qr. {d \over {dt}}\left( {{{\partial E} \over {\partial {{\mathop \Omega \limits^ \bullet }_{\rm{p}}}}}} \right) = {Q_r}.

The kinetic energy E of the proposed CM can be defined as the sum of the kinetic energies of its links. In general, it can be expressed as: (12) E=JpΩp2•2+m1V122+m2V222, E = {J_{\rm{p}}}{{\mathop {\Omega _{\rm{p}}^2}\limits^ \bullet } \over 2} + {m_1}{{V_1^2} \over 2} + {m_2}{{V_2^2} \over 2}, where J_p is the moment of inertia of the rotor about the axis of rotation; m₁ – total mass of elements moving in the axial direction with a speed of V₁: masses of the rotor, draw bar (in the form of a pipe), collet and workpiece in the form of a bar (it is assumed that the bar moves with the collet at the moment of clamping); m₂ – total mass of elements moving in the radial direction (clamping elements) with a speed of V₂.

Taking into account the characteristics of kinematic transmission, movements of links and theit masses can be reduced to the rotor. The kinetic energy of the CM links is expressed through the reduced moment of inertia as: (13) E=JrΩp2•2, E = {J_{\rm{r}}}{{\mathop {\Omega _{\rm{p}}^2}\limits^ \bullet } \over 2}, where J_r – is the moment of inertia of links of the proposed CM reduced to its rotor.

The partial derivative of the expression of the kinetic energy E at the generalised velocity Ωp• \mathop {{\Omega _{\rm{p}}}}\limits^ \bullet is as follows ∂E∂Ω•p=Ω•pJr {{\partial E} \over {\partial {{\mathop \Omega \limits^ \bullet }_{\rm{p}}}}} = {\mathop \Omega \limits^ \bullet _{\rm{p}}}{J_r} . When time t changes, only the angle of rotation Ω_p changes, the time derivative of the partial derivative of the previous expression will be equal to: (14) ddt∂E∂Ω•p=Ω••pJr. \frac{d}{{dt}}\left( {\frac{{\partial E}}{{\partial \mathop \Omega \limits^ \bullet _{\text{p}} }}} \right) = \mathop \Omega \limits^{ \bullet \bullet } _{\text{p}} J_r .

Taking into account (11), expression (14) becomes: (15) Ω••pJr=Qr. {\mathop \Omega \limits^{ \bullet \bullet } _{\rm{p}}}{J_r} = {Q_r}.

The generalised force (torque) Q_r can be derived from the equality of the elementary work of the forces acting in CM. That is, it is necessary to consider the virtual work of active forces acting in the mechanism at small displacements Δ: (16) ΔΩpQr=ΔΩpTp+ΔΩpTsf+Δx∑i=1nFiT+ΔyTΣ, \Delta {\Omega _{\rm{p}}}{Q_r} = \Delta {\Omega _{\rm{p}}}{T_{\rm{p}}} + \Delta {\Omega _{\rm{p}}}T_s^f + \Delta x\mathop \sum \nolimits_{i = 1}^n F_i^T + \Delta y{T_\Sigma }, where T_p – torque developed by the rotor of the CM motor during the clamping process, Tsf T_s^f – the resistance torque generated by the friction forces in the rotor screw gear, ΔΩ_p – elementary rotor rotation angle, Δx – elementary movements in friction pairs, FiT F_i^T – the friction force that occurs in the friction pair of the i-th link, Δy – elementary movement of the clamping elements, T_Σ – the amount of clamping force.

The motions Δx and Δy are functions of ΔΩ_p and can be determined in accordance with the characteristics of the CM kinematic chain and in particular gear ratios for a particular mechanism. After substituting Δx=f(Ω_p) and Δy=f(Ω_p) and performing the transformations (16) can be presented in the following form: (17) Qr=ξ1+£1Ωp, {Q_r} = {{\rm{\xi }}_1} + {\pounds_1}{\Omega _{\rm{p}}}, where the values ξ₁ and £₁ does not contain variables and is a function of the geometrical parameters of the mechanism links and the forces of their interaction.

When substituting (17) into (15), the equation is formed as: (18) Ω••pJr=ξ1+£1Ωp. {\mathop \Omega \limits^{ \bullet \bullet } _{\rm{p}}}{J_r} = {{\rm{\xi }}_1} + {\pounds_1}{\Omega _{\rm{p}}}.

After the notation ξ=ξ1Jr {\rm{\xi }} = {{{{\rm{\xi }}_1}} \over {{J_r}}} аnd £=£1Jr \pounds = {{{\pounds_1}} \over {{J_r}}} has been introduced in (18): (19) Ω••p=ξ+£Ωp. {\mathop \Omega \limits^{ \bullet \bullet } _{\rm{p}}} = {\rm{\xi }} + \pounds{\Omega _{\rm{p}}}.

Since dΩpdt=ωp {{d{\Omega _{\rm{p}}}} \over {dt}} = {\omega _{\rm{p}}} as result it can be derived d2Ωpdt2=dωpdt=dωpdΩp⋅dΩpdt=dωpdΩp=ωp {{{d^2}{\Omega _{\rm{p}}}} \over {d{t^2}}} = {{d{{\rm{\omega }}_{\rm{p}}}} \over {dt}} = {{d{{\rm{\omega }}_{\rm{p}}}} \over {d{\Omega _{\rm{p}}}}} \cdot {{d{\Omega _{\rm{p}}}} \over {dt}} = {{d{{\rm{\omega }}_{\rm{p}}}} \over {d\Omega {\rm{p}}}} = {{\rm{\omega }}_{\rm{p}}} . The consequence is dωpdΩpωp=ξ+£Ωp {{d{{\rm{\omega }}_{\rm{p}}}} \over {d{\Omega _{\rm{p}}}}}{{\rm{\omega }}_{\rm{p}}} = {\rm{\xi }} + \pounds{\Omega _{\rm{p}}} ; ∫ ω_p dω_p = ∫(ξ + £Ω_p) dΩ_p and ωp22=ξΩp+£Ωp22+X1 {{{\rm{\omega }}_{\rm{p}}^2} \over 2} = {\rm{\xi }}{\Omega _{\rm{p}}} + \pounds{{\Omega _{\rm{p}}^2} \over {\rm{2}}} + {X_1} . The integration constant X₁ for the initial conditions of the CM functionind is when Ω_p = 0 and the rotor is motionless in relation to the spindle so their relative speed ω_p = 0, value X₁=0. As a result, dΩpdt=£Ωp2+2ξΩpor∫dΩp£Ωp2+2ξΩp=∫dt+X2/,∫dΩp£Ωp2+2ξΩp=1£ln2£Ωp+2ξ+22ξ£Ωp+£2Ωp2. \eqalign{ & {{d{\Omega _{\rm{p}}}} \over {dt}} = \sqrt {\pounds\Omega _{\rm{p}}^2 + 2{\rm{\xi }}{\Omega _{\rm{p}}}} {\rm{or}}\int {{{d{\Omega _{\rm{p}}}} \over {\sqrt {\pounds\Omega _{\rm{p}}^2 + 2{\rm{\xi }}{\Omega _{\rm{p}}}} }}} = \int {dt + X_2^/} , \cr & \int {{{d{\Omega _{\rm{p}}}} \over {\sqrt {\pounds\Omega _{\rm{p}}^2 + 2{\rm{\xi }}{\Omega _{\rm{p}}}} }} = {1 \over {\sqrt \pounds }}ln } \left[ {2\pounds{\Omega _{\rm{p}}} + 2{\rm{\xi + 2}}\sqrt {2{\rm{\xi }}\pounds{\Omega _{\rm{p}}} + {\pounds^2}\Omega _{\rm{p}}^2} } \right]. \cr}

The solving the equation can be presented in the form, ln[2£Ωp+2ξ+22ξ£Ωp+£2Ωp2]=£⋅t+X2. ln [2\pounds{\Omega _{\rm{p}}} + 2{\rm{\xi + 2}}\sqrt {{\rm{2\xi }}\pounds{\Omega _p} + {\pounds^2}\Omega _{\rm{p}}^2} ] = \sqrt \pounds \cdot t + {X_2}.

Under the initial conditions when t = 0 and Ω_p = 0 the value ln 2 ξ = X₂, after substitution into the equation: (20) £ξΩp+2£ξΩp+£2ξ2Ωp2=e£⋅t−1,Ωp=e£⋅t−122£ξe£⋅t. \eqalign{ & {\pounds \over {\rm{\xi }}}{\Omega _{\rm{p}}} + \sqrt {2{\pounds \over {\rm{\xi }}}{\Omega _{\rm{p}}} + {{{\pounds^2}} \over {{{\rm{\xi }}^2}}}\Omega _{\rm{p}}^2} = {e^{\sqrt \pounds \cdot t}} - 1, \cr & {\Omega _{\rm{p}}} = {{{{\left( {{e^{\sqrt \pounds \cdot t}} - 1} \right)}^2}} \over {2{\pounds \over {\rm{\xi }}}{e^{\sqrt \pounds \cdot t}}}}. \cr}

Formula (20) describes the dependence of the generalised coordinate Ω_p on time. The positive or negative values £ in (20) can indicate the type of change of Ω_p. That is, it contributes to preliminarily determine certain characteristics of the rotational motion of the CM rotor of the proposed type.