A Simplified Direct Method for Calculating the Steady-State Motion of Mechanisms and Its Comparison with Other Methods
Abstract Generated abstract
A simplified direct energy method is developed for calculating the steady-state angular velocity variation of one-degree-of-freedom mechanisms under position-dependent forces, with a prescribed mean speed. Starting from the kinetic energy balance, the authors derive first and corrected approximations for velocity deviations, give an estimate of the limiting relative error, and connect the same formulation to approximate and refined flywheel moment of inertia calculations. A comparison with several established graphical, energy, and successive-approximation methods on a difficult example indicates that the improved simplified method attains accuracy comparable to the best alternatives while retaining moderate labor input and direct applicability to flywheel design.
Full Text
A Simplified Direct Method for Calculating the Steady-State Motion of Mechanisms and Its Comparison with Other Methods
Yu. Ya. Kovylyn, N. D. Grebeneva
We shall consider the motion of mechanisms with one degree of freedom under the action of forces that depend only on the position of the driving link. Then the increment of the kinetic energy of the mechanism during the displacement of the driving link from the initial position, determined by the coordinate \(\varphi=\varphi_0\), to the current position corresponding to the coordinate \(\varphi=\varphi_i\) \((i=0,1,2,3,\ldots,z)\), is determined by the well-known [1] equation
\[ \frac{I_i\omega_i^2}{2}-\frac{I_0\omega_0^2}{2}=A_i . \tag{1} \]
Here \(\omega_0,\ \omega_i\) are the values of the angular velocity of the driving link in its initial and current positions;
\(I_0,\ I_i\) are the values of the moment of inertia of the mechanism, reduced to the driving link, in the same positions;
\[ A_i=\int_{\varphi_0}^{\varphi_i} M(\varphi)\,d\varphi \]
is the work of the moment of the forces \(M(\varphi)\), reduced to the driving link, during the displacement \(\varphi_i-\varphi_0\) of this link.
Let \(I_i=I(\varphi_i)\) and \(A_i=A(\varphi_i)\) be specified over the entire period \(\varphi_z-\varphi_0\) of variation of the angular velocity of rotation of the driving link.
Moreover,
\[ A_z\equiv\int_{\varphi_0}^{\varphi_z} M(\varphi)\,d\varphi=0 \]
in accordance with the known [1] periodicity conditions.
It is required to determine the law of variation of the angular velocity of rotation of the driving link in the regime of steady-state motion with a specified mean angular velocity equal to \(\omega_{\mathrm{av}}\).
This problem, often encountered in applications, has no exact solution.
S. N. Kozhevnikov [2] gives the following approximate solution. Specifying a series of arbitrary values of the coefficient of nonuniformity of motion of the driving link \(\delta',\ \delta'',\ \delta''',\ldots\), one finds the corresponding required values of the flywheel moment of inertia reduced to this link, \(I_m',\ I_m'',\ I_m''',\ldots\), using the well-known graphical Wittenbauer method in combination with the assumption customary for this method
\[ \omega_{\mathrm{av}}\simeq\frac{\omega_{\max}+\omega_{\min}}{2}. \tag{2} \]
Then, in the coordinates \((\delta, I_m)\), the curve of the found dependence \(I_m = I_m(\delta)\) is plotted, and its intersection is found with the straight line drawn at the level \(I_m\) actually present in the mechanism. In this way the value of the coefficient \(\delta\) is determined.
The subsequent operations for determining the dependence \(\omega = \omega(\varphi)\) are carried out in full accordance with the above-mentioned graphical method of Wittenbauer.
The described procedure has not become widely used because of a number of significant shortcomings, the chief of which are as follows:
a) assumption (2) is in many cases too crude, especially when applied to mechanisms with a relatively light flywheel and with an “impulsive” character of the variation of the applied forces;
b) the Wittenbauer method underlying the solution permits only a graphical implementation, in connection with which it is impossible to use numerical methods that allow efficient application of computers (apart from the preparation of the initial data);
c) the origin of coordinates of the Wittenbauer curve \(A = A(I)\) usually lies far outside the limits of the drawing. Therefore a sufficiently accurate determination of the angular coefficients of the straight lines that must be drawn through the indicated point in order to determine the velocity of the driving link becomes impossible.
The shortcomings listed are entirely predetermined by the internal properties of the Wittenbauer method, which, one may say, is slowly but surely leaving the scene, giving way to methods that allow a numerical solution. Among these, first of all, should be included the method developed by E. M. Gut’yar [3], which is free of the shortcomings noted in items b and c. However, the application of this method, instead of the Wittenbauer method, within the framework of the plan of analysis of steady-state motion set forth above according to [2], does not eliminate the need to use assumption (2) and, moreover, sharply increases the labor involved.
Another well-known energy method, based on the ideas of N. I. Mertsalov [4] and K. E. Rehrich [5] and developed in the work of M. I. Bat’ [6], in contrast to the method described, while having low labor intensity, makes it possible to solve the stated problem without relying on assumption (2). However, in certain cases this method may give a considerable error [7], which, while remaining within the framework of this method, it does not seem possible to reduce.
A more accurate, but also considerably more labor-intensive, method is given in the work of V. A. Yudin and E. E. Kol’man-Ivanov [8]. But this method is not directly connected with the solution of the problem of determining the moment of inertia of the flywheel, which is its essential shortcoming.
In the case where high requirements are not imposed on the accuracy of the solution, the dependence \(\omega(\varphi)\) can be determined approximately directly from equation (1), if, instead of the unknown quantity \(\omega_0\), the prescribed value \(\omega_{\mathrm{av}}\) is substituted there. This simplest solution of the problem is given in the work of A. M. Antoviliya [9]. An important advantage of the method is also its low labor intensity.
By using equation (1) directly, it would be possible to find a solution of the problem as accurate as desired if the method of successive approximations were applied. Within the framework of this method, the solution by the procedure of [9] will be only the first approximation. To obtain the solution in the second approximation, it is necessary first to find the average velocity and the deviation of the instantaneous velocity from the average value found for the initial position of the mechanism from the results of the first approximation, and then to repeat the calculation by equation (1), now taking the value \(\omega_0\) equal to ...
the sum of the specified mean velocity and the deviation just found. Higher approximations are constructed analogously.
Good results are obtained already in the second approximation. In this case the labor involved is still comparatively small. But, like the method of V. A. Yudin and E. E. Kolman-Ivanov, this method also is not directly connected with the calculation of the flywheel.
Let us consider a method that combines the simplicity of rough methods with the high accuracy of the method of successive approximations. The effectiveness of the method is increased by the availability of an estimate of the limiting error and by its suitability for determining the moment of inertia of the flywheel.
From (1) we find
\[ \omega_i=\sqrt{\frac{I_0\omega_0^2+2A_i}{I_i}} . \tag{3} \]
Transform the expression under the radical in (3), adding and subtracting the quantity \(I_i\omega_{\mathrm{cp}}^2\) and representing in the numerator
\[ I_i=I_{\mathrm{m}}+I_{\mathrm{link},\,i}, \tag{4} \]
where \(I_{\mathrm{link},\,i}\) is the reduced moment of inertia of the mechanism links without the flywheel:
\[ \omega_{\mathrm{cp}}^2+ 2\, \frac{ \dfrac{I_0\omega_0^2}{2} -\dfrac{I_{\mathrm{m}}\omega_{\mathrm{cp}}^2}{2} -\dfrac{I_{\mathrm{link},\,i}\omega_{\mathrm{cp}}^2}{2} +A_i }{I_i} = \omega_{\mathrm{cp}}^2+ 2\,\frac{L_i-L^*}{I_i}. \]
Here the following notation has been introduced:
\[ L_i=A_i-\frac{I_{\mathrm{link},\,i}\omega_{\mathrm{cp}}^2}{2}, \]
\[ L^*=\frac{I_{\mathrm{m}}\omega_{\mathrm{cp}}^2}{2} -\frac{I_0\omega_0^2}{2} =\mathrm{idem}. \]
Now rewrite expression (3) in the form
\[ \omega_i=\omega_{\mathrm{cp}} \sqrt{1+2\,\frac{L_i-L^*}{I_i\omega_{\mathrm{cp}}^2}} . \tag{5} \]
Represent the right-hand side of (5) as the result of expansion in a binomial series,
\[ \omega_i=\omega_{\mathrm{cp}} \left[ 1+\frac{L_i-L^*}{I_i\omega_{\mathrm{cp}}^2} -\frac{1}{2} \left( \frac{L_i-L^*}{I_i\omega_{\mathrm{cp}}^2} \right)^2 +\ldots \right], \]
whence, putting \(\omega_i=\omega_{\mathrm{cp}}+\Delta\omega_i\), we obtain
\[ \Delta\omega_i= \frac{L_i-L^*}{I_i\omega_{\mathrm{cp}}} -\frac{1}{2\omega_{\mathrm{cp}}} \left( \frac{L_i-L^*}{I_i\omega_{\mathrm{cp}}} \right)^2 +\ldots \tag{6} \]
Since in cases of practical significance the second term under the radical in (5) is, in absolute value, considerably less than unity, the series written above converges very rapidly. Therefore, for practical calculations it is sufficient to restrict oneself to the terms written out. Denoting the quantity equal to the first term in (6) by \(\Delta\omega_i\), we express the second term through this quantity.
Then we obtain
\[ \Delta\omega_i\approx \frac{L_i-L^*}{I_i\omega_{\mathrm{cp}}} -\frac{1}{2}\cdot \frac{(\Delta\omega_i)^2}{\omega_{\mathrm{cp}}}. \tag{7} \]
Next, taking the mean conditional velocity to be equal to its mean planimetric value (with respect to the angle \(\varphi\)), from the periodicity condition
\[ \int_{\varphi_0}^{\varphi_z} \Delta\omega(\varphi)\,d\varphi \simeq \frac{\varphi_z-\varphi_0}{z}\sum \Delta\omega_i=0 \]
and taking (7) into account, we find
\[ L^*= \frac{\sum L_i/I_i-\dfrac{1}{2}\sum(\Delta\omega_i^{\mathrm{I}})^2} {\sum 1/I_i}. \tag{8} \]
Here and below the summation is over the index \(i\), which assumes the values \(i=0,1,2,\ldots,z-1\).
Substituting the value \(L^*\) from (8) into (7), after simple transformations, instead of the system (7)–(8) we obtain the new system
\[ \Delta\omega_i= \frac{L_i-L^{**}}{I_i\omega_{\mathrm{av}}} + \frac{1}{2\omega_{\mathrm{av}}} \left[ \frac{\sum(\Delta\omega_i^{\mathrm{I}})^2}{I_i\sum 1/I_i} - (\Delta\omega_i^{\mathrm{I}})^2 \right], \tag{9} \]
\[ L^{**}= \frac{\sum L_i/I_i}{\sum 1/I_i}. \tag{10} \]
Since \(|\Delta\omega|_{\max}\ll\omega_{\mathrm{av}}\), it is evident that the deviations \(\Delta\omega_i\) of the angular velocity are due mainly to variation of the first term on the right-hand side of (9), which differs little from the first term in (6), whose value is denoted by \(\Delta\omega_i^{\mathrm{I}}\). Therefore one may approximately take
\[ \Delta\omega_i^{\mathrm{I}}\simeq \frac{L_i-L^{**}}{I_i\omega_{\mathrm{av}}}. \tag{11} \]
The second term on the right-hand side of (9) (denote it by \(\Delta\omega_i^{\mathrm{II}}\)) plays the role of a small correction and is equal to
\[ \Delta\omega_i^{\mathrm{II}}= \frac{1}{2\omega_{\mathrm{av}}} \left[ \frac{\sum(\Delta\omega_i^{\mathrm{I}})^2}{I_i\sum 1/I_i} - (\Delta\omega_i^{\mathrm{I}})^2 \right]. \tag{12} \]
Finally, in accordance with (9), (11), and (12), we have
\[ \Delta\omega_i\simeq \Delta\omega_i^{\mathrm{I}}+\Delta\omega_i^{\mathrm{II}}. \tag{13} \]
A detailed analysis shows that the limiting relative error of the quantity \(\Delta\omega_i\) when it is determined from (9)–(13) is no greater than
\[ \frac{1}{2}\left(\frac{\Delta\omega}{\omega_{\mathrm{av}}}\right)^2_{\max}, \]
which is always admissible.
Let us consider the inverse problem—the calculation of the flywheel.
Let \(A=A(\varphi)\), \(I_{\mathrm{red}}=I_{\mathrm{red}}(\varphi)\), \(\omega_{\mathrm{av}}\), and the admissible value of the coefficient of nonuniformity of rotation of the reduction link \([\delta]\) be given.
It is required to determine the moment of inertia of the flywheel \(I_{\mathrm{m}}\), which ensures the prescribed uniformity of motion.
Putting in (11)
\[ I_i\simeq I_{\mathrm{m}}+I_{\mathrm{red,av}}, \]
where
\[ I_{\mathrm{red,av}}= \frac{I_{\mathrm{red,max}}+I_{\mathrm{red,min}}}{2}, \]
we obtain
\[ \Delta\omega_i\simeq \frac{L_i-L^{**}}{(I_{\mathrm{m}}+I_{\mathrm{red,av}})\omega_{\mathrm{av}}}. \]
Since \(L^{**}\), \(I_{\mathrm{cp}}\), and \(\omega_{\mathrm{cp}}\) are constant, then
\[ \Delta \omega_{\max} \approx \frac{L_{\max}-L^{**}}{(I_{\mathrm{m}}+I_{\mathrm{eq.cp}})\omega_{\mathrm{cp}}}, \]
\[ \Delta \omega_{\min} \approx \frac{L_{\min}-L^{**}}{(I_{\mathrm{m}}+I_{\mathrm{eq.cp}})\omega_{\mathrm{cp}}}. \]
The coefficient of nonuniformity of rotation, as is known, is equal to
\[ \delta=\frac{\Delta\omega_{\max}-\Delta\omega_{\min}}{\omega_{\mathrm{cp}}}. \]
Substituting here the approximate values found for \(\Delta\omega_{\max}\) and \(\Delta\omega_{\min}\), and setting \(\delta=[\delta]\), we obtain:
\[ I_{\mathrm{m}} \approx \frac{L_{\max}-L_{\min}}{[\delta]\omega_{\mathrm{cp}}^{2}} -I_{\mathrm{eq.cp}}. \tag{14} \]
This result coincides with the solution obtained by the method of N. I. Mertsalov—K. E. Rerikh, refined in the work of Yu. Ya. Kovylin [7].
According to the foregoing, the limiting relative error of the value \(I_{\mathrm{cp}}\) determined from (14) is of the order of
\[ \frac{\Delta I_{\max}}{I_{\mathrm{cp}}}+ \frac{|\Delta\omega|_{\max}}{2\omega_{\mathrm{cp}}} \]
and, as a rule, is not of substantial importance, since the quantity \(\delta\) is not strictly standardized.
| Method | Greatest relative error, % | Mean-square error, % of the mean-square deviation | Error in the range of angular velocity, % of its exact value | Relative labor input, % | Relation to flywheel calculation | Possibility of continuing iteration |
|---|---|---|---|---|---|---|
| Simplified direct method. First approximation — formula (11) | 6,7 | 5,2 | 5,2 | 60 | yes | yes |
| Simplified direct method. Improved first approximation — formula (13) | 1,0 | 0,7 | 0,8 | 100 | yes | yes |
| Method of V. A. Yudin and E. E. Kolman-Ivanov | 2,1 | 5,8 | 0,00 | 120 | no | yes |
| Method of N. I. Mertsalov — K. E. Rerikh — M. I. Bat | 16,4 | 14,6 | 13,1 | 40 | yes | no |
| Method of successive approximations. First approximation | 13,7 | 22,5 | 0,5 | 55 | no | yes |
| Method of successive approximations. Second approximation | 1,0 | 1,6 | 0,05 | 105 | no | yes |
On the other hand, in order to carry out refined calculations of the mechanism parts for strength, durability, etc., sufficiently accurate information is needed on the actual course of the curve \(\Delta \omega(\varphi)\). For this purpose one may use (9)–(13), substituting into them \(I_i\) according to (4), where instead of \(I_m\) one should use its approximate value just found.
This calculation makes it possible to refine the character of the dependence \(\Delta \omega(\varphi)\) over the entire interval \(\varphi_z - \varphi_0\) and to determine the actual value of \(\delta\).
Comparing it with the permissible value, one can refine the required magnitude of the flywheel moment of inertia
\[ I_{m.\,\mathrm{ref}} \approx \frac{\delta}{[\delta]}\left(I_m + I_{\mathrm{red.\,av}}\right) - I_{\mathrm{red.\,av}} . \]
Here \(I_m\) is the approximate value according to (14).
To compare the method considered with other energy methods known from the literature, a calculation was carried out for a specific problem that represents a fairly difficult case for all methods: a high degree of nonuniformity of motion, \(\delta = 0.202\), and a significant variable component in the reduced moment of inertia,
\[ \frac{2\left(I_{\max} - I_{\min}\right)}{I_{\max} + I_{\min}} = 0.478 . \]
The results of the calculations are summarized in the table.
The comparative estimates demonstrate the advantage of the method set forth above, whose accuracy is at the level of the most accurate methods, with an average labor intensity and with a direct connection to the flywheel calculation.
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