Study of Steady-State Motion of Mechanisms by the Method of Dynamic Work with Torques Depending on the Position and Velocity of the Reduced Link
Abstract Generated abstract
This paper develops a dynamic work method for calculating the steady-state motion of mechanisms when the reduced torque depends on both the position and angular velocity of the reduced link. Starting from the periodicity condition and the kinetic energy equation, it derives approximate formulas for the mean angular velocity, the velocity deviation over a cycle, and the resulting coefficient of nonuniformity, while accounting for variable reduced inertia and an initially estimated flywheel inertia. The proposed procedure reduces repeated computation after flywheel correction because the dynamic work function depends only on position and can be reused in successive approximations. The method is presented as an extension of earlier dynamic work approaches for position-dependent torques and is indicated to be adaptable to more general torque functions.
Full Text
Proceedings of the Tomsk Polytechnic Institute Named after S. M. Kirov, Holder of the Order of the October Revolution and the Order of the Red Banner of Labor
Vol. 235
1973
Study of the Steady-State Motion of Mechanisms by the Method of Dynamic Work for Torques Depending on the Position and Velocity of the Reduced Link
Yu. Ya. Kovylyn
The problem stated in the title of the article arises because, after an approximate preliminary determination of the flywheel moment of inertia by some approximate method*), it is necessary to calculate the course of the curve \(\omega(\varphi)\) and establish how much the obtained value of the coefficient of nonuniformity of motion of the reduced link differs from the prescribed one. If the difference is large, then the adopted approximate value of the flywheel moment of inertia should be increased or decreased [1]. After this, the calculation for determining the law of motion is repeated at least once more in order to make sure that the corrected value of the flywheel moment of inertia is sufficient to ensure motion of the reduced link with the prescribed coefficient of nonuniformity. At the same time, all information about the course of the curve \(\omega(\varphi)\) is obtained that is needed for carrying out refined calculations of mechanism parts for strength, durability, etc.
The dependence \(\omega(\varphi)\) in steady-state motion can in principle be found with the aid of known general methods [1], [2]. However, their practical application to the solution of this problem is associated with a number of inconveniences. First, the initial conditions in calculating steady-state motion are chosen only very approximately. As a result, to obtain a solution corresponding to the steady-state regime, one has to investigate the motion of the reduced link over an angle of rotation larger than that for one cycle [2]. Second, after each change in the magnitude of the flywheel moment of inertia, practically the entire volume of computations must be repeated anew. As a result, the total amount of work, especially when the reduced moment of inertia of the mechanism is variable, becomes rather large.
The method of dynamic work presented below gives a very simple and compact solution of the entire complex of problems in the dynamics of steady-state motion of mechanisms, if the prescribed forces depend on the position and velocity of the reduced link.
Let the moment of the forces reduced to the reduced link be specified by the function \(M = M^{\mathrm{I}}(\varphi) + M^{\mathrm{II}}(\omega)\), and let the moment of inertia of the links (without the flywheel), reduced to the same link, be specified by the function \(I_{\mathrm{red}} = I_{\mathrm{red}}(\varphi)\). Here \(\varphi\) and \(\omega\) are the current values of the angle determining the position of the reduced link and of the angular velocity of its rotation,
*) There is no exact method for determining the flywheel moment of inertia when the forces depend on displacements and velocities.
Let, further, \([\delta]\) be the prescribed permissible value of the coefficient of nonuniformity of motion of the reduced link. Suppose that, guided by this number, by some method it has been possible to find the approximate value of the moment of inertia of the flywheel \(I_m\) installed on the reduced link. It is now necessary to calculate the shape of the curve \(\omega(\varphi)\) and determine the actual (computed) value of the coefficient \(\delta\) of nonuniformity of its motion.
We shall represent the sought angular velocity as
\[ \omega = \omega_{\mathrm{cp}} + \Delta \omega, \tag{1} \]
where \(\Delta \omega\) is the deviation of the instantaneous angular velocity \(\omega\) of rotation of the reduced link from its mean planimetric (with respect to the angle \(\varphi\)) value \(\omega_{\mathrm{cp}}\).
The quantity \(\omega_{\mathrm{cp}}\) must satisfy the known condition of periodicity
\[ \int_{\varphi_0}^{\varphi_0+\Phi} M(\varphi, \omega_{\mathrm{cp}}+\Delta \omega)\, d\varphi = 0 . \tag{2} \]
Here \(\Phi\) is the angle of rotation of the reduced link corresponding to one period of steady-state motion.
Substituting (1) into (2), we find
\[ M_{\mathrm{cp}}^{\mathrm{I}}+\frac{1}{\Phi} \int_{\varphi_0}^{\varphi_0+\Phi} M^{\mathrm{II}}(\omega_{\mathrm{cp}}+\Delta \omega)\, d\varphi = 0, \tag{3} \]
where \(M_{\mathrm{cp}}^{\mathrm{I}}\) is the mean, over the period \(\Phi\), value of the moment \(M^{\mathrm{I}}(\varphi)\), equal to
\[ M_{\mathrm{cp}}^{\mathrm{I}}=\frac{1}{\Phi} \int_{\varphi_0}^{\varphi_0+\Phi} M^{\mathrm{I}}(\varphi)\, d\varphi . \tag{4} \]
In the case, sufficiently often encountered in practice [1], when the characteristic \(M^{\mathrm{II}}(\omega)\) is expressed by a linear function of the velocity \(\omega\), equation (3) can be integrated completely. Thus the exact value of the mean planimetric velocity is determined.
For a nonlinear form of the function \(M^{\mathrm{II}}(\omega)\), the quantity \(\omega_{\mathrm{cp}}\) can be found only approximately. It is not difficult to show that if, on the working portion of the curve \(M^{\mathrm{II}}(\omega)\), lying between the ordinates \(\omega=\omega_{\min}\) and \(\omega=\omega_{\max}\), the derivative \(\dfrac{dM^{\mathrm{II}}}{d\omega}\) does not change sign (and this is characteristic of all modern machines), then the mean velocity of steady-state motion, in a first approximation, can be determined as
\[ \omega_{\mathrm{cp}}\approx \omega_{\mathrm{e}} \left[ 1+0.35[\delta]\, \frac{ M^{\mathrm{II}}(\omega')+M^{\mathrm{II}}(\omega'')+2M_{\mathrm{cp}}^{\mathrm{I}} }{ M^{\mathrm{II}}(\omega')-M^{\mathrm{II}}(\omega'') } \right]. \tag{5} \]
Here \(\omega_{\mathrm{e}}\) is the so-called mean energy angular velocity, determined by the equation
\[ M_{\mathrm{cp}}^{\mathrm{I}}+M^{\mathrm{II}}(\omega_{\mathrm{e}})=0; \]
\[ \omega'=\omega_{\mathrm{e}}(1-0.35[\delta]); \]
\[ \omega''=\omega_{\mathrm{e}}(1+0.35[\delta]). \]
To determine the dependence \(\Delta\omega(\varphi)\), divide the angle \(\Phi\) into \(z\) identical small intervals \(\Delta\varphi\) and write the kinetic-energy equation
\[ \frac{I_k\omega_k^2}{2}-\frac{I_0\omega_0^2}{2}=A_k \qquad (k=1,2,\ldots,z). \tag{6} \]
Here \(\omega_0\) and \(\omega_k\) are the values of the angular velocity of the reduced link in the initial (zero) and \(k\)-th positions;
\(I_0\) and \(I_k\) are the values of the reduced moment of inertia of the mechanism (with allowance for the flywheel) in the same positions;
\(A_k\) is the work of the reduced moment of forces \(M\) over the displacement of the reduced link equal to \(\varphi_k-\varphi_0\).
Since in the regime of steady-state motion even the greatest deviations of the velocity \(|\Delta\omega_{\max}|\) are considerably smaller than \(\omega_{\mathrm{av}}\), with sufficient accuracy we have
\[ \omega^2 \simeq \omega_{\mathrm{av}}^2+2\omega_{\mathrm{av}}\Delta\omega . \tag{7} \]
On the same basis, the curvilinear working segment of the given dependence \(M^{\mathrm{II}}(\omega)\) may be approximately replaced by a straight line passing through the point with coordinates \(\omega=\omega_{\mathrm{av}}\) and \(M=-M_{\mathrm{av}}^{\mathrm{I}}\). The equation of this straight line is
\[ M_*^{\mathrm{II}}=-M_{\mathrm{av}}^{\mathrm{I}}+C\Delta\omega, \]
where \(C\) is an angular coefficient, which may be determined as
\[ C=\left(\frac{dM^{\mathrm{II}}(\omega)}{d\omega}\right)_{\omega=\omega_{\mathrm{av}}}, \tag{8} \]
or
\[ C\simeq \frac{M^{\mathrm{II}}(\omega'')-M^{\mathrm{II}}(\omega')}{0.7[\delta]\omega_{\mathrm{av}}}. \tag{8'} \]
Here
\[ \omega'=\omega_{\mathrm{av}}(1-0.35[\delta]); \qquad \omega''=\omega_{\mathrm{av}}(1+0.35[\delta]). \]
Then the work \(A_k\), performed by the reduced moment of forces, will be equal to
\[ A_k=\int_{\varphi_0}^{\varphi_k} M^{\mathrm{I}}(\varphi)\,d\varphi -kM_{\mathrm{av}}^{\mathrm{I}}\Delta\varphi +C\int_{\varphi_0}^{\varphi_k}\Delta\omega\,d\varphi . \]
Applying the trapezoidal rule to compute the last integral, we find:
\[ A_k \simeq \int_{\varphi_0}^{\varphi_k} M^{\mathrm{I}}(\varphi)\,d\varphi -kM_{\mathrm{av}}^{\mathrm{I}}\Delta\varphi +C\Delta\varphi \left( \sum_{i=0}^{k-1}\Delta\omega_i -\frac{\Delta\omega_0}{2} +\frac{\Delta\omega_k}{2} \right). \tag{9} \]
\[ (k=1,2,\ldots,z). \]
Finally, substituting (7) and (9) into (6), after simple transformations we ultimately obtain
\[ \Delta\omega_k \simeq \frac{ L_k-L^{**}+C\Delta\varphi\displaystyle\sum_{i=0}^{k-1}\Delta\omega_i }{ I_k\omega_{\mathrm{av}}-\dfrac{C\Delta\varphi}{2} }, \tag{10} \]
where, to shorten the notation, the following designations have been introduced:
\[ L_k= \int_{\varphi_0}^{\varphi_k} M^{\mathrm{I}}\,d\varphi -kM_{\mathrm{av}}^{\mathrm{I}}\Delta\varphi -\frac{I_{33.k}\omega_{\mathrm{av}}^2}{2}; \tag{11} \]
\[ L^{**}=-\frac{I_{\mathrm{red}.0}\omega_{\mathrm{av}}^{2}}{2} -\Delta\omega_0\left(I_0\omega_{\mathrm{av}}-\frac{C\Delta\varphi}{2}\right). \tag{12} \]
The function \(L\) (it is convenient to call it the dynamic work) depends only on \(\varphi\). Therefore, for all \(\varphi_k\) (from \(\varphi_0\) to \(\varphi_z\)) its values can easily be computed directly from the initial data, taking into account the previously found velocity \(\omega_{\mathrm{av}}\).
By \(L^{**}\) is denoted a certain constant quantity (12), dependent on the as yet unknown deviation \(\Delta\omega_0\), which must be specified. In the first approximation the quantity \(\Delta\omega_0\) may be determined as
\[ \Delta\omega_0 \approx -\frac{0.5 I_{\mathrm{red}.0}\omega_{\mathrm{av}}^2+L_{\mathrm{av}}} {I_{\mathrm{av}}\omega_{\mathrm{av}}}, \tag{13} \]
where
\[ L_{\mathrm{av}}=\frac{1}{z}\sum_{k=1}^{z}L_k; \]
\[ I_{\mathrm{av}}=I_m+I_{\mathrm{red}.\mathrm{av}};\qquad I_{\mathrm{red}.\mathrm{av}}=0.5\left(I_{\mathrm{red}.\min}+I_{\mathrm{red}.\max}\right). \]
Formula (13) can be obtained from (10), if in the latter one puts \(C=0\) and \(I_k=I_{\mathrm{av}}\).
Since the value \(\Delta\omega_0\) computed in this way is not exactly equal to the sought deviation, this will inevitably lead to the fact that the value \(\Delta\omega_z\) of the velocity deviation at the end of the considered period of steady-state motion will not be exactly equal to the adopted value \(\Delta\omega_0\). This is also promoted to some extent by the accumulation of error in the sum appearing in the numerator of (10*).
If the discrepancy \(\Delta\omega_z-\Delta\omega_0\) is large, this indicates a significant mismatch in the approximate value of \(\Delta\omega_0\). In such a case the calculation should be repeated with the aid of the same formulas (10)–(12), now taking \(\Delta\omega_0\) equal to \(\Delta\omega_z\). If necessary, the calculations may be continued until the difference \(\Delta\omega_z-\Delta\omega_0\) decreases to an admissible value. These refining calculations require a minimal expenditure of labor, since the values of the dynamic work \(L\) found in the first approximation and the denominator in formula (10) remain the same in the subsequent approximations. However, as numerous examples of calculating steady-state motion by the method presented show, the need to repeat the calculations arises infrequently.
The remaining discrepancy may be distributed according to a linear law over all values \(\Delta\omega_k\) within the period. Taking the quantity \(\Delta\omega_z\) to be closer to the true velocity deviation than \(\Delta\omega_0\), we finally obtain
\[ \Delta\omega_k^{T} =\Delta\omega_k+\frac{\Delta\omega_z-\Delta\omega_0}{z}(z-k), \tag{14} \]
\[ (k=0,1,2,\ldots,z), \]
where \(\Delta\omega_0\), \(\Delta\omega_k\), and \(\Delta\omega_z\) denote the velocity deviations corresponding to the last approximation.
Having investigated the periodic dependence \(\Delta\omega(\varphi)\) found, we determine the coefficient of nonuniformity of motion of the reduced link by the formula
\[ \delta=\frac{\Delta\omega_{\max}-\Delta\omega_{\min}} {\omega_{\mathrm{av}}}. \]
* The same consequences will also occur when using any other calculation method.
If the value \(\hat{\delta}\) differs only slightly from the prescribed value \([\delta]\), then the approximate value \(I_M\) found may be regarded as satisfying the stated conditions. If, however, as a result of the check calculation the difference between \(\hat{\delta}\) and \([\delta]\) proves greater than the permissible one, the moment of inertia of the flywheel should be refined. In doing so, one may assume that the known [1] hyperbolic relation \(I_{\mathrm{cp}}\delta=\mathrm{idem}\) is approximately valid also in the case when \(M=M(\varphi,\omega)\). Then we obtain
\[ I_M^{\mathrm{ref}} \approx \frac{\hat{\delta}}{[\delta]}\left(I_M+I_{\mathrm{red.\,s}}\right)-I_{\mathrm{red.\,av}}. \]
After determining the value \(I_M^{\mathrm{ref}}\), it is necessary once again to determine the course of the curve \(\Delta\omega(\varphi)\) by means of formulas (10)–(14), which will require only the simplest and least laborious additional calculations.
In conclusion, we note that the need to correct the value \(I_M\) and to carry out the above-mentioned refinement of the course of the curve \(\Delta\omega(\varphi)\) practically does not arise if, as an approximate value, one takes
\[ I_M \approx \frac{L_{\max}-L_{\min}}{[\delta]\,\omega_{\mathrm{cp}}^2}-I_{\mathrm{red.\,av}}, \]
which corresponds to the known method of N. I. Mertsalov and K. E. Rerikh (for \(M^{II}=-M_{\mathrm{cp}}^{I}\)) in the author’s interpretation [3].
Thus, using the function \(L(\varphi)\), first introduced into practice by N. I. Mertsalov and K. E. Rerikh for the case \(M=M(\varphi)\) and called above the dynamic work, one can very effectively solve the entire range of problems in the dynamics of steady-state motion also for \(M=M(\varphi,\omega)\).
The method presented is easily extended to the general case of specifying the dependence \(M(\varphi,\omega)\), when it cannot be represented as the sum \(M^{I}(\varphi)+M^{II}(\omega)\).
LITERATURE
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I. I. Artobolevskii. Theory of Mechanisms. Moscow, “Nauka,” 1967.
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G. G. Baranov. Course in the Theory of Mechanisms and Machines. Moscow, “Mashinostroenie,” 1967.
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Yu. Ya. Kovylin. On calculating a flywheel by the method of Prof. N. I. Mertsalov and Prof. K. E. Rerikh. Izvestiya VUZov SSSR, “Mashinostroenie,” Moscow, No. 10, 1960.