Some Copying Errors of Single-Coordinate Hydraulic Tracer Systems for Automating Lathes
Abstract Generated abstract
This article analyzes copying errors in single-coordinate hydraulic tracer systems used to automate lathes, focusing on transient copying modes under aperiodic stability. It formulates the error as a function of hydraulic cylinder force and copying speed, then linearizes the system’s static stiffness and velocity characteristics to derive differential equations for a hydraulic slide copying a transition from a cylindrical to a conical surface. The resulting equations provide a basis for calculating axial and radial profile errors introduced by the tracer system during transient operation. The paper also notes that more realistic treatment of nonlinear characteristics, dead zones, and falling friction behavior would require nonlinear solution methods such as analog computation or approximate graphical analytical techniques.
Full Text
Some Copying Errors of Single-Coordinate Hydraulic Tracer Systems for Automating Lathes
I. A. Nemirovskii
(Presented by the Department of Mechanical-Engineering Technology, Tomsk Polytechnic Institute)
Single-coordinate hydraulic tracer systems are widely used for automating metal-cutting machine tools. Such systems are used, in particular, in the hydraulic carriages GS-1, UP-240, KST-1, and others.
An important indicator characterizing the quality of a hydraulic tracer system is the copying error.
This article considers the copying errors introduced by the tracking hydraulic system in transient copying modes, under the condition of aperiodic stability of the system.
With all other conditions being equal, the copying error caused by the mismatch \(h\) in the spool unit of the hydraulic tracking system is a function of the copying speed \(V_k\) and the force \(P_{dv}\) developed by the hydraulic cylinder:
\[ h = h(P_{dv}, V_k), \tag{1} \]
where \(h\) is the displacement of the tracking spool from the neutral position, at which \(P_{dv} = 0\) and \(V_k = 0\).
The traction force on the piston \(P_{dv}\) depends on the forces being overcome: cutting forces, dry and viscous friction forces, inertial forces, and weight forces (if there are units moved by the hydraulic system in the vertical direction).
The magnitude and direction of the copying speed \(V_k\) developed by the system depend on the profile being copied and on the speed of the command feed.
Consequently, \(P_{dv}\) and \(V_k\) in expression (1) are independent variables.
Hence
\[ dh = \frac{\partial h}{\partial P_{dv}} \cdot dP_{dv} + \frac{\partial h}{\partial V_k} \, dV_k . \tag{2} \]
The differential equation (2) includes the static characteristics of the system ...
\[ \frac{\partial P_{\partial v}}{\partial h}=E, \tag{3} \]
where \(E\) is the stiffness of the hydraulic system, measured in kg/mm;
\[ \frac{\partial V_k}{\partial h}=C, \tag{3a} \]
where \(C\) is the velocity gain coefficient of the system, \(1/\text{sec}\).
The static characteristics \(E\) and \(C\) of real systems are nonlinear [4, 5, 6]. It has been customary [1, 2, 3] to linearize these characteristics (Fig. 1), which naturally leads to certain errors in the calculations.
Fig. 1a.
Fig. 1b.
Let us take
\[ E=E_0=\text{const}, \]
\[ C=C_0=\text{const}. \]
The tractive force developed by the piston of the hydraulic copying device is equal to
\[ P_{\partial v}=P_0+M\frac{dV_k}{dt}, \tag{4} \]
where \(P_{\partial v}\) is the force developed by the hydraulic system;
\(P_0\) is the sum of the cutting forces and dry friction, assumed constant (we restrict the motion of the working member to one direction, which corresponds to copying turning of stepped shafts with increasing diameter dimensions, with aperiodic stability of the system; the friction forces are assumed not to depend on the speed of motion);
\(M\) is the reduced mass of the moving units;
\(V_k\) is the copying-motion speed.
Differentiating expression (4), we obtain
\[ dP_{\partial v}=M\frac{d^2V_k}{dt^2}\,dt. \tag{5} \]
Substituting (3), (3a), and (5) into (2), we obtain
\[ dh=\frac{1}{E_0}M\frac{d^2V_k}{dt^2}\,dt+\frac{1}{C_0}\,dV_k. \tag{6} \]
After integration,
\[ h=\frac{M}{E_0}\frac{dV_k}{dt}+\frac{V_k}{C_0}+\text{const}. \tag{7} \]
For \(V_k=0\) and \(\dfrac{dV_k}{dt}=0\), we have \(h=\dfrac{P_0}{E_0}\).
Thus,
\[ h=\frac{M}{E_0}\frac{dV_k}{dt}+\frac{V_k}{C_0}+\frac{P_0}{E_0}, \tag{8} \]
or else
\[ V_k=C_0\left[h-\frac{M}{E_0}\frac{dV_k}{dt}-\frac{P_0}{E_0}\right]. \tag{9} \]
Let us consider the specific case of copying, by a hydraulic copying slide, the transition from a cylindrical to a conical surface (Fig. 2). Let the angle of inclination of the guides of the hydraulic slide to the axis of the part be \(\beta^\circ\), and the angle between the generatrix of the cone being copied and the axis of the part be \(\alpha^\circ\).
Fig. 2.
Line I in Fig. 2 represents the profile of the template, and line II the profile of the part copied by the hydraulic system. Point \(N\) determines the position of the feeler of the hydraulic slide at some instant of time \(t\), and point \(A\) the corresponding position of the cutter forming the profile on the part.
Let us denote:
\(s\)—the longitudinal feed of the slide, in mm/rev;
\(n\)—the number of revolutions of the part, in rev/min;
\(V_{\mathrm{pr}}\)—the longitudinal feed rate, in mm/sec,
\[ V_{np}=n\cdot s\cdot \frac{1}{60}, \]
\(x\) and \(y\) are the current coordinates of the feeler corresponding to the template profile
\[ x=V_{np}\cdot t\left[1-\frac{\sin\alpha\cos\beta}{\sin(\alpha+\beta)}\right], \tag{10} \]
\[ y=x\cdot \operatorname{tg}\alpha, \tag{11} \]
\(x_p\) and \(y_p\) are the current coordinates of the cutter;
\(i\) is the transmission ratio of the lever mechanism transmitting motion from the feeler to the spool valve; \(i\geq 1\);
\(h_{\sh}\) is the displacement of the copying feeler from the neutral position.
The displacement \(\delta x\) along the \(x\)-axis of the machined profile relative to the template profile can be calculated as the difference \((x-x_p)\) for various \((y=y_p)\), i.e.,
\[ \delta x=(x-x_p)_{y=y_p}. \tag{12} \]
Similarly, the displacement \(\delta y\) along the \(y\)-axis is
\[ \delta y=(y-y_p)_{x=x_p}. \tag{13} \]
The relation \(x=f(y)\) is specified by the template profile, and \(x=f_1(t)\) by the machining regime.
To determine \(\delta x\) and \(\delta y\), it is necessary to know
\[ x_p=\varphi(y)\quad \text{and}\quad x_p=\varphi_1(t). \]
Let us consider the position of the cutter and the feeler at time \(t\). According to Fig. 2,
\[ h_{\sh}=\frac{AK\sin\alpha}{\sin(\alpha+\beta)};\qquad AK=x_p-\overline{KM};\quad \overline{KM}=y_p\operatorname{ctg}\alpha. \]
Hence
\[ h_{\sh}=(x_p-y_p\operatorname{ctg}\alpha)\frac{\sin\alpha}{\sin(\alpha+\beta)}. \tag{14} \]
Substituting (14) into formula (9) and taking into account that \(V_y=V_k\sin\beta\), and \(h=h_{\sh}\cdot i\), we obtain
\[ V_y=C_0\left[ i(x_p\sin\alpha-y_p\cos\alpha)\frac{\sin\beta}{\sin(\alpha+\beta)} -\frac{M}{E_0}\frac{dV_y}{dt} -\frac{P_0}{E_0} \right]. \tag{15} \]
According to Fig. 2,
\[ x_p=\overline{OB}-\overline{EB}=V_{np}t-y_k\operatorname{ctg}\beta. \tag{16} \]
Substituting (16) into (15), after elementary transformations we obtain
\[ \frac{M}{E_0}\frac{d^2y_p}{dt^2} +\frac{1}{C_0}\frac{dy_p}{dt} +i y_p = -\frac{V_{np}i}{\operatorname{ctg}\alpha+\operatorname{ctg}\beta}\cdot t +\frac{P_0}{E_0}\sin\beta. \tag{17} \]
Similarly (taking into account that \(V_{x_p}=V_k\cos\beta\)),
\[ -\frac{1}{\sin\beta}\frac{M}{E_0}\frac{d^2x_p}{dt^2} +\frac{1}{C_0}\frac{dx_p}{dt} +\frac{i}{\sin\beta}x_p = \frac{V_{np}\,i\,\operatorname{ctg}\beta}{\operatorname{ctg}\alpha+\operatorname{ctg}\beta}\cdot t +\frac{P_0}{E_0}\operatorname{ctg}\beta. \tag{18} \]
The solution of differential equations (17) and (18) presents no difficulty [8, 9] and is not given here.
The derivation of the comparatively simple equations (17) and (18), which, together with expressions (12) and (13), make it possible to calculate the axial $\delta_x$ and radial $\delta_y$ errors introduced by the system in the transient copying regime, became possible as a result of linearizing the static characteristics of the system and considering only those systems that are aperiodically stable. In doing so, it was not taken into account that the friction force has a “falling” characteristic.
If, however, one considers oscillatory-stable systems while taking into account the nonlinearity of the static characteristics, dead zones, and discontinuities of the “falling” characteristic of the friction forces, then the solution of the resulting nonlinear differential equations becomes possible either by means of an electronic analog installation or by approximate graphical-analytical methods, for example, by the method of A. V. Basharin [10].
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