Representation of Long Elastic Ropes in Automatic Control Systems
Abstract Generated abstract
The paper addresses how long flexible elastic ropes should be represented as dynamic elements in automatic regulation systems, with emphasis on vertically suspended weighted ropes whose tension is controlled by changing rope length. It derives the static tension, elongation relationship and shows that the static transfer coefficient varies inversely with rope length, then formulates longitudinal rope vibrations with damping as a distributed parameter problem and obtains transfer functions for control and disturbance actions. A simplified rational approximation is proposed for practical calculation, and comparison with an experimental step response for a short test rope shows close agreement, supporting the use of the derived expressions in control system design.
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REPRESENTATION OF LONG ELASTIC ROPES IN AUTOMATIC CONTROL SYSTEMS
V. A. Beinarovich, A. G. Borodich, A. D. Kozhukhovskii,
S. A. Kopanev, L. S. Udut
(Presented by the scientific and technical seminar of the NII AEM)
In a number of industrial and special installations that have automatic regulation and control systems, long flexible elastic ropes are used; these ropes are objects of regulation or are included in the regulation system as one of the elements of the object of regulation (the working machine). Such installations include mine hoists, aerial ropeways, towing cable devices, extra-long-wave antenna devices, rope log-haulers, and others. When designing automatic regulation systems (ARS) containing long flexible elastic ropes, it becomes necessary to represent the latter in the form of dynamic elements included in a definite way in the structural diagram of the closed-loop ARS. The present article considers the problem of representing a long flexible elastic weighted vertically arranged rope as a dynamic element of an ARS.
Consider the case in which, with the upper end of the rope fixed, the rope-tension force is regulated by changing the length of the rope by winding or unwinding part of the rope on the drum of a winch. In the problem under consideration (Fig. 1, a), a flexible homogeneous vertically arranged rope of diameter \(d\) and length \(l\) is acted upon by two tensile forces applied to the upper \((T_{\mathrm{в}})\) and lower \((T_{\mathrm{н}})\) ends of the rope. In the steady state (in statics) it is obvious that at the upper point of the rope the tensile force \(T_{\mathrm{в}}\) is equal to the sum of the force \(T_{\mathrm{н}}\) and the weight of the rope \(Q\). Thus, for a vertical homogeneous flexible rope with constant diameter and known weight \(Q\), the static tensile forces at any point depend uniquely on its distance from the end of the rope and on the magnitude of the tensile force \(T_{\mathrm{н}}\) at the lower point.
As the static characteristics of the rope as an element of the regulation system we shall take the dependence of the increment of the tensile force \(T_{\mathrm{н}}\) at the lower point on the elongation of the rope, with the upper end of the rope fixed. By Hooke’s law [1], the total elongation of the rope is determined by
\[ \Delta l=\frac{\Delta T_{\mathrm{н}} l}{EF}+\frac{\gamma l^{2}}{2E} =\Delta l_{T}+\Delta l_{Q}. \tag{1} \]
where \(E\) is the modulus of elasticity of the rope;
\(F\) is the cross-sectional area of the rope;
\(\gamma\) is the specific weight of the rope;
\(\Delta l_T\) is the elastic deformation of the rope under the action of the force \(T_{\mathrm{н}}\);
\(\Delta l_Q\) is the elongation of the rope under the action of the rope’s own weight.
It follows from (1) that the static transfer coefficient depends nonlinearly on the length of the rope and is determined by the expression
\[ K_{\mathrm{op}}=\frac{\Delta T_{\mathrm{н}}}{\Delta l_T}=\frac{EF}{l}. \tag{2} \]
Thus, in systems for regulating the tensile force by changing the length of the rope with the upper end fixed, the static coefficient varies inversely with the rope length, and as the length \(l\) decreases to zero the value \(K_{\mathrm{op}}\) tends to infinity. This should be taken into account when developing automatic control systems for rope tension by means of winches, using structures that ensure the dynamic stability of the automatic control system for an arbitrarily large transfer coefficient of the controlled object. Figure 1, b shows the dependence of the static transfer coefficient for a steel rope of diameter 4.7 mm and linear weight 87.7 kg/km over the range of length variation from 1 to 4 km.
Fig. 1.
To determine the dynamic characteristics of a long elastic vertically arranged rope, we shall use the theory of oscillatory systems with distributed parameters, considering the natural longitudinal vibrations of the rope under the condition that the frictional force causing damping of the vibrations is proportional to the displacement velocity of an infinitesimal rope element. The equation of free longitudinal vibrations of the rope, taking account of vibration damping, has the form [2, 3]
\[ m\frac{\partial^2 u}{\partial t^2}+k\frac{\partial u}{\partial t}-EF\frac{\partial^2 u}{\partial x^2}=0, \tag{3} \]
where \(m\) is the mass of the rope;
\(k\) is the damping coefficient;
\(u\) is the absolute elongation of the rope at the considered length \(x\);
\(x\) is the length of the rope under consideration, measured from the upper point (Fig. 1, a), taken as the origin of coordinates \((0 \le x \le l)\);
\(t\) is the current time.
The problem of determining the dynamic properties of the rope as an element of an automatic-control system reduces to finding the time function of the tensile force for a specified time function of the change in the rope length.
With zero initial conditions, the boundary conditions for equation (3) are written as [7]:
\[ \left. \begin{array}{l} EF \left(\dfrac{\partial u}{\partial x}\right)\bigg|_{x=l}=T_{\mathrm{н}}(t),\\[6pt] u(x,t)\big|_{x=0}=0 . \end{array} \right\} \tag{4} \]
Taking account of the zero initial conditions, the oscillation equation (3) in operator form is written as [7]:
\[ \frac{\partial^2 U(x,p)}{\partial x^2} - \frac{p}{a^2}(p+2h)U(x,p)=0, \tag{5} \]
where \(p\) is the Laplace operator;
\(U\) is the operational image of the elongation;
\[ 2h=\frac{k}{m}; \qquad a^2=\frac{EF}{m}. \]
Equation (5) has the following solution in general form [7]:
\[ U(x,p)=C_1 \operatorname{ch}(x\sqrt{b})+C_2 \operatorname{sh}(x\sqrt{b}), \tag{6} \]
where
\[ b=\frac{p}{a^2}(p+2h). \]
Using the boundary conditions (4), we determine the constants of integration in equation (6):
\[ C_1=0; \qquad C_2=\frac{T_{\mathrm{н}}(p)}{EF\sqrt{b}\operatorname{ch}(l\sqrt{b})}, \tag{7} \]
Taking (7) into account, solution (6) is written as
\[ U(x,p)= \frac{T_{\mathrm{н}}(p)\operatorname{sh}(x\sqrt{b})} {EF\sqrt{b}\operatorname{ch}(l\sqrt{b})}. \tag{8} \]
For \(x=l\), the function \(U(x,p)\) will represent the elongation of the rope \(\Delta l(p)\). Taking \(\Delta l(p)\) as the input control signal, we obtain the transfer function of an elastic rope of length \(x=l\) with respect to the control action in the form
\[ W_y(p)=\frac{\Delta T_{\mathrm{н}}(p)}{\Delta l(p)} = \frac{EF\sqrt{b}\operatorname{ch}(l\sqrt{b})} {\operatorname{sh}(l\sqrt{b})}. \tag{9} \]
To simplify the calculations in expression (9), the hyperbolic functions may be expanded in power series [7] and, taking account of the first three terms of the expansion, represented in the form
\[ \operatorname{ch}\sqrt{bl^2}\simeq 1+\frac{bl^2}{2}+\frac{b^2l^4}{24}; \tag{10} \]
\[ \operatorname{sh}\sqrt{bl^2}\simeq \sqrt{bl^2}\left(1+\frac{bl^2}{6}+\frac{b^2l^4}{120}\right). \tag{11} \]
Then the transfer function (9), taking (10) and (11) into account, assumes the simplified expression
\[ W_{yy}(p)=5\cdot K_{\mathrm{op}}\, \frac{l^{4}p^{4}+4hl^{4}p^{3}+(12l^{2}a^{2}+4h^{2}l^{4})p^{2}+24hl^{2}a^{2}p+24a^{4}} {l^{4}p^{4}+4hl^{4}p^{3}+(20l^{2}a^{2}+4h^{2}l^{4})p^{2}+40hl^{2}a^{2}p+120a^{4}} . \tag{12} \]
The dynamic properties of a long elastic rope in transmitting disturbances in the form of an increment of the tension force \(\Delta T_{\mathrm{в}}\) at the upper point are evaluated on any segment of the rope of length \(0\le x\le l\) by the ratio of the increment of the tension force \(\Delta T\) in the section under consideration, located at a distance \(x\) from the end of the rope, to the increment of the force \(\Delta T_{\mathrm{в}}\) at the upper point of attachment of the rope. Between the relative elongation of the rope \(\dfrac{\partial u(x,t)}{\partial x}\) and the tension \(T(x,t)\) there is the relation [4]:
\[ T(x,t)=EF\,\frac{\partial u(x,t)}{\partial x}, \tag{13} \]
where \(EF\) is the longitudinal stiffness of the rope.
Then, taking into account expressions (8) and (13), the transfer function of an elastic rope of length \(x=l\) with respect to the disturbing action takes the following form:
\[ W_b(p)=\frac{\Delta T_{\mathrm{н}}(p)}{\Delta T_{\mathrm{в}}(p)} =\frac{1}{\operatorname{ch}\left(\sqrt{lb}\right)} . \tag{14} \]
Verification of the calculated expressions obtained was carried out on an experimental setup with a rope of diameter 8 mm, length 3.75 m, linear weight 0.07 kg/m, Young’s modulus 78 kg/cm\(^2\), damping coefficient 0.0364 sec, and transfer coefficient 10.4 kg/m. The calculation of the transient process of the change in tension force under a stepwise elongation of the rope by the amount \(\Delta l_T=0.04\) m was performed using the simplified expression (12) for the transfer function of the rope, which for the specified parameters has the following form:
\[ W_{yy}(p)=52\, \frac{p^{4}+2{,}72p^{3}+127{,}8p^{2}+172{,}5p+2668{,}1} {p^{4}+2{,}72p^{3}+213p^{2}+287{,}5p+13340{,}5}, \tag{15} \]
Figure 2 shows the curves of the transient process obtained experimentally (by oscillograph recording)—solid curve 1—and by calculation, according to expression (15)—dashed curve 2.
Fig. 2.
To solve equation (15), an analog computer was used according to the circuit presented in Fig. 3.
Comparison of the calculated and experimental curves of the transient process shows their close agreement (the phase discrepancy is caused by the inaccuracy in determining the damping coefficient of the rope), which in-
Fig. 3.
shows the practical suitability of the obtained computational expressions for the design of various control systems containing flexible, elastic, heavy, vertically arranged ropes.
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