Abstract
Full Text
MECHANICS
D. I. MANZHERON
ON REDUCED ACCELERATIONS OF ARBITRARY ORDER AND SOME OF THEIR EXTREMAL PROPERTIES
(Presented by Academician I. I. Artobolevskii, 6 VIII 1956)
Let (\mathbf w_M^{(n)}) be the acceleration of order (n) of a point (M) of a rigid body undergoing plane-parallel motion, expressed by Somov’s generalized formulas ((^1)), written in matrix notation:
[
\begin{Bmatrix}
x_1^{(n+1)}\
x_2^{(n+1)}
\end{Bmatrix}
=
\begin{Bmatrix}
x_{10}^{(n+1)}\
x_{20}^{(n+1)}
\end{Bmatrix}
+
\begin{Bmatrix}
-A_n & -B_n\
B_n & -A_n
\end{Bmatrix}
\begin{Bmatrix}
x_1-x_{10}\
x_2-x_{20}
\end{Bmatrix},
\tag{1}
]
where (A=A_n(t)) and (B_n=B_n(t)) are determined by the recurrence formulas
[
A_{n+1}=\dot A_n+\dot\theta B_n,\qquad
B_{n+1}=\dot B_n-\dot\theta A_n,\qquad
A_1=\dot\theta^2,\qquad
B_1=\ddot\theta,
\tag{2}
]
[
\left(\dot{\ } \equiv \frac{d}{dt}\right),\qquad
x_i^{(n+1)}=\frac{d}{dt}x_i^{(n)}
\quad (i=1,2;\ n=1,2,\ldots)
]
and (\theta=\theta(t)) is the angle of rotation of the body.
Theorem 1. The locus of points (M^), determined by the vector equation*
[
\mathbf r_{M^*}=\mathbf r_M+\lambda_n \mathbf w_M^{(n)},
\tag{3}
]
where (M) is a point lying on some line ((D_M)) taking part in plane-parallel motion, is a line ((D_{M^})) forming with the line ((D_M)) an angle (\varphi_n) determined by the relation*
[
\tg \varphi_n=\frac{\lambda_n B_n}{1-\lambda_n A_n}.
\tag{4}
]
It is obvious that the angle (\varphi_n) depends only on the state of motion of the link under consideration at the given instant of time and on the chosen value of the parameter (\lambda_n).
Theorem 2. The reduced accelerations of order (n), introduced in ((^2)) and determined by the relations
[
\mathbf w_r^{(n)}=\frac{\mathbf w^{(n)}}{A_n},\qquad
\mathbf w_r^{(1)}=\frac{\mathbf w^{(1)}}{A_1}\equiv \frac{\mathbf w}{A_1}
\quad (n=1,2,\ldots),
\tag{5}
]
where (A_n) (and (B_n)) are expressed by the recurrence formulas (2), are characterized by the extremal property of the function (4) of the angle (\varphi_n).
Theorem 3 (on the distribution of accelerations of arbitrary order). The locus of the endpoints of the reduced accelerations of order (n) of points of a line ((D)) undergoing plane-parallel motion is a line ((D_r)) perpendicular to the given one.
Theorem 4 (generalized Kotelnikov theorem ((^3))). Circles whose diameters are the reduced accelerations of order (n) of points of some rigid body undergoing plane-parallel motion, pro-
pass through the instantaneous center of accelerations of the same order (P_n(x_{1P_n}, x_{2P_n})), given in matrix notation by the following equality:
[
\begin{Bmatrix}
x_{1P_n}\
x_{2P_n}
\end{Bmatrix}
=
\begin{Bmatrix}
x_{10}\
x_{20}
\end{Bmatrix}
+
\frac{1}{A_n^2+B_n^2}
\begin{Bmatrix}
A_n & -B_n\
B_n & A_n
\end{Bmatrix}
\begin{Bmatrix}
x_{10}^{(n+1)}\
x_{20}^{(n+1)}
\end{Bmatrix}
\quad (n=1,2,\ldots).
\tag{6}
]
Theorem 5 (generalized theorem of similarity). The endpoints of the accelerations of the (n)-th order of the points of a rigid body undergoing plane-parallel motion, in the case when the origins of these accelerations are placed at one point, form a figure similar to this rigid body, rotated through the angle
[
\theta_n=\operatorname{arc\,tg}\frac{B_n}{A_n},
\tag{7}
]
where (A_n) and (B_n) are determined by relations (2).
The theorems established serve as the basis for a number of theorems ((^7,^8)) and lead to a new method, called the method of reduced accelerations of any order, developed by a collective under the author’s direction. The method of reduced accelerations of any order makes it possible, proceeding from the known classifications of I. I. Artobolevsky ((^4)), G. G. Baranov ((^5)), and others, and by studying the most characteristic Assur groups, to carry out a very simple systematic investigation, by a graphico-analytical method, of the classical problem of the distribution of first-order accelerations for all Assur groups belonging to plane mechanisms.
Iași Polytechnic Institute
Iași, Romanian People’s Republic
Received
11 VII 1956
REFERENCES
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- D. I. Manzheron, DAN, 102, No. 5, 897 (1955).
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