Modeling the Kinematics of Planar Mechanisms of Class IV on Analog Computers
Abstract Generated abstract
This paper addresses the kinematic analysis and synthesis of complex planar mechanisms, focusing on class IV mechanisms and the limitations of graphical and analytical calculation methods. It presents an analog computer modeling approach based on decomposing a mechanism into interconnected oblique triangles, using implicit-function equations to determine link positions, angles, and coordinate transformations. The worked formulation for a class IV mechanism shows how four-bar substructures can be coupled with tracking amplification, and supports the broader claim that planar mechanisms of classes II, III, and IV can be modeled on analog computers, with digital computers reserved for final precise calculations.
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MODELING THE KINEMATICS OF PLANAR MECHANISMS OF CLASS IV ON ANALOG COMPUTERS
A. G. Kokin
(Presented by the scientific seminar of the VL Research Institute of AEM at TPI)
With the modern development of machine building, when complex mechanisms of classes II, III, and IV are being used more and more widely, the problem of calculating and designing such mechanisms is of considerable importance. Graphical and analytical methods of calculation no longer satisfy designers. They are being replaced by machine methods for the analysis and synthesis of mechanisms [3, 4], which make it possible to use the most modern analog and digital computers. Graphical methods, while highly visual and simple, prove to be insufficiently accurate, and the degree of such accuracy is unknown. Moreover, in calculating complex mechanisms they are labor-intensive. Analytical methods, although more accurate, are insufficiently visual; therefore, during calculations it is difficult to monitor the correctness of the results obtained. They are also labor-intensive [5].
In the design and calculation of complex mechanisms, the advantages of machine methods, in comparison with analytical and graphical methods, increase considerably. Machine methods for the analysis and synthesis of mechanisms make it possible to reduce many times over the labor expended on investigating mechanisms, to increase the accuracy of calculation, and to increase the number of mechanisms studied. In addition, they ensure visibility of the solution and the possibility of monitoring by the designer.
Obtaining high accuracy in the calculation of mechanisms and selecting the corresponding mechanism scheme when using machine methods is achieved through the use of an analog-digital complex.
Each of the mechanisms has kinematic, dynamic, and other qualities specific to it; therefore, the most advantageous mechanism scheme can be selected only on the basis of a sufficiently complete comparison of different variants of mechanisms that perform one and the same functional purpose. All this is most consistent with the use of analog computers, since modern analog machines have a speed in periodic-operation mode of up to 1000 solutions per sec, i.e., in other words, a cam speed of 1000 rev/sec can be achieved without too great a phase lag. The use of an automatic optimizer makes it possible to select the mechanism scheme quickly and to determine approximately the kinematic dimensions of the mechanism links. The final precise calculation in the kinematic and dynamic analysis and synthesis of mechanisms is carried out on a digital computer.
The separation of the functions of search and calculation of the required mechanism determines the role of each of the machines—the analog machine as a search machine and the digital machine as a calculating machine. It follows from this that, for the analysis and synthesis of the kinematics of planar mechanisms, it is necessary to consider the possibilities of investigating any planar mechanisms on an analog computer. Until now we have considered the modeling of the kinematics of planar mechanisms of classes II and III on analog computing machines. Let us consider the modeling on an analog computer of the kinematics of planar mechanisms of class IV. Mechanisms that include groups no higher than class IV of the second order are called class IV mechanisms (according to the classification of I. I. Artobolevsky) [1] (Fig. 1).
Fig. 1. Diagram of a class IV mechanism
Let us consider modeling the kinematics of such a mechanism by the method of solving oblique triangles on an analog computer. In this example we shall show the universality of the method and its possibilities in modeling the kinematics of complex mechanisms. Its essence is that the mechanism is geometrically divided into a certain number of interconnected oblique triangles, the solution of which makes it possible to determine the position of all links of the mechanism. In the given mechanism, by introducing the fictitious link \(AB\), we obtain the oblique triangle \(OAB\). The fictitious link \(AB\)—a side of triangle \(OAB\)—is determined by the law of cosines:
\[ AB^2 = OA^2 + OB^2 - 2OA \cdot OB \sin \varphi . \tag{1} \]
When solving on an analog computer by the method of implicit functions [2], the expression is written in the form:
\[ AB^2 - OA^2 - OB^2 + 2OA \cdot OB \sin \varphi = -\frac{AB}{\mu}, \tag{2} \]
\(\mu\) is the gain factor of the following amplifier.
\[ \mu > 1 \cdot 10^6 . \]
From the equations of the projections of \(OAB\) on the coordinate axes we find \(\sin \psi\), \(\cos \psi\), which determine the position of \(AB\) in the coordinate system:
\[ OA \cos \varphi - AB \cos \psi = 0, \]
\[ OA \sin \varphi + AB \sin \psi - OB = 0. \tag{3} \]
\[ OA\cos\varphi - AB\cos\psi = -\frac{\cos\psi}{\mu}, \]
\[ OA\sin\varphi + AB\sin\psi - OB = -\frac{\sin\psi}{\mu}. \tag{4} \]
Let us transfer the origin of coordinates to point \(A\) so that the abscissa axis coincides with \(AB\), and consider the group \(ACDBFE\). In this group one can distinguish two quadrilaterals, \(ACDB\) and \(AEFB\). We divide them respectively into oblique triangles \(ACB\), \(CDB\), \(AEB\), \(EFB\), and solve, taking into account that the angles \(\varphi'_1\) and \(\varphi'_2\) are related to each other.
For the quadrilateral \(ACDB\) we find the length of the fictitious link \(BC\) as the side of the oblique triangle \(ACB\):
\[ BC^2 - AC^2 - AB^2 + 2AC\cdot AB\sin\varphi'_1 = -\frac{BC}{\mu}. \tag{5} \]
From the projection equations for \(ACB\) onto the coordinate axes \(x' y'\) we find \(\sin\psi'\), \(\cos\psi'\), which determine the position of \(CB\) in the new coordinate system:
\[ AC\cos\varphi'_1 - BC\cos\psi' = -\frac{\cos\psi'}{\mu}, \]
\[ AC\sin\varphi'_1 + BC\sin\psi' - AB = -\frac{\sin\psi'}{\mu}. \tag{6} \]
The angle \(\alpha\), determining the position of \(CD\) in the coordinate system \(x' y'\), as the exterior supplementary angle of triangle \(CNK\), is equal to:
\[ \alpha = 180^\circ - (\gamma + \psi'). \tag{7} \]
The unknown angle \(\gamma\) is determined from triangle \(CDB\) by the law of cosines:
\[ DB^2 - CD^2 - CB^2 + 2CD\cdot CB\cos\gamma = -\frac{\gamma}{\mu}. \tag{8} \]
The position of \(CD\) is determined by \(\sin\alpha\), \(\cos\alpha\) according to the formula for finding the functions of the sum of two angles \(\gamma\) and \(\psi'\):
\[ \cos\alpha = \cos[180^\circ - (\gamma + \psi')] = -\cos\gamma\cos\psi' + \sin\gamma\sin\psi', \]
\[ \sin\alpha = \sin[180^\circ - (\gamma + \psi')] = \sin\gamma\cos\psi' + \cos\gamma\cdot\sin\psi'. \tag{9} \]
From the closure equations of the contour \(ACDB\) we find \(\sin\xi_1\), \(\cos\xi_1\), determining the position of the link \(DB\):
\[ AC\cos\varphi'_1 + CD\cos\alpha - DB\cos\xi_1 = -\frac{\cos\xi_1}{\mu}, \]
\[ AC\sin\varphi'_1 + CD\sin\alpha - DB\sin\xi_1 = -\frac{\sin\xi_1}{\mu}. \tag{10} \]
We do the same for the quadrilateral \(AEFB\), taking into account that
\[ \varphi'_1 + \varphi'_2 + \angle CAE = 180^\circ,\quad \xi_1 + \xi_2 + \angle DBF = 180^\circ. \tag{11} \]
The angles \(CAE\) and \(DBE\) are known and connect both quadrilaterals into a single system. In order to fulfill equality (11), it is necessary to include a tracking amplifier. The tracked difference is fed to the formation of the initial angle \(\varphi'\). It remains now to connect the new system
Fig. 2. Block diagram for modeling the kinematics of a class IV mechanism.
Visible inscriptions in the diagram:
- “Scheme for modeling the kinematics of the four-bar linkage \(AEFB\)”
- “Scheme for modeling the kinematics of the four-bar linkage \(ACDB\)”
coordinates with the old one. The coordinate axes are related by the formulas of analytic geometry:
\[ \begin{aligned} x &= x' \cos \psi + y' \sin \psi + OA \sin \varphi,\\ y &= x' \sin \psi - y' \cos \psi + OA \cos \varphi. \end{aligned} \tag{12} \]
The block diagram for modeling the kinematics of a class IV mechanism is given in Fig. 2. It consists of a driving analog generator for \(\sin \varphi\), \(\cos \varphi\), two identical schemes for modeling the kinematics of a four-bar linkage, and blocks for forming, by the method of implicit functions, the angles \(\varphi_1'\), \(\varphi_2'\).
In conclusion, it may be stated that modeling the kinematics of planar mechanisms of classes II, III, and IV of any complexity is possible on an analog computer when the method of oblique-angle triangles is applied. The combined use of analog and digital computers will make it possible to carry out the kinematic analysis and synthesis of planar mechanisms accurately and rapidly.
REFERENCES
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- M. G. Vintizenko, A. G. Kokin. Method of analog modeling of planar hinged mechanisms (with rotation of the driven link through \(360^\circ\)). Proceedings of TPI, vol. 187 (in press).
- A. G. Kokin. Modeling of planar mechanisms of the third class on analog computers. Proceedings of TPI, vol. 203 (in press).
- V. Zinoviev. Analytical Methods for the Calculation of Planar Mechanisms. Gostekhizdat, 1949.