Abstract Generated abstract
This paper generalizes Huygens’ reciprocity theorem for the physical pendulum to a heavy rigid body undergoing Hessian motion about a fixed point. It introduces the kinetic axis and kinetic center of the body, relates them to the central axis of the momentum screw, and analyzes the Hess structural condition through circular sections of the gyration ellipsoid. The argument shows that, for bodies satisfying Hess’s conditions, the kinetic center lies on the same perpendicular to the relevant circular section as the center of gravity, and the angular equations can be written as if the mass were concentrated at this point. It follows that if the suspension point is moved to the kinetic center, the new kinetic center becomes the former suspension point and, under identical initial conditions, the motion remains unchanged.
Full Text
MECHANICS
V. A. CHIKIN
A GENERALIZATION OF A THEOREM OF HUYGENS
(Presented by Academician A. Yu. Ishlinskii on 20 X 1961)
§ 1. Let a rigid body move about a fixed point \(O\). We shall consider the quantity of motion \(\mathbf Q\) and the kinetic moment \(\mathbf K_0\) of the body, according to their definition, as the principal vector and the principal moment of the system of vectors of the momenta \(m_i\mathbf v_i\) of its individual particles. The central axis of the screw of this system of vectors will be called the kinetic axis of the body. The point of intersection of this axis with the plane \(\alpha_\omega\), passing through the center of gravity of the body \(G\) and the vector of the instantaneous angular velocity \(\vec\omega\), will be called the kinetic center of the body \(A\). If the radius vector of the point \(A\) is denoted by \(\mathbf r_A\), then, according to the theory of reduction of a system of vectors \((^1)\),
\[ \mathbf r_A=\frac{\mathbf Q\times \mathbf K_0}{Q^2} =\frac{(\vec\omega\times \mathbf r_G)\times \mathbf K_0}{M v_G^2}, \tag{1} \]
where \(M\) is the mass of the body; \(\mathbf r_G\) and \(\mathbf v_G\) are the radius vector and velocity of the center of mass \(G\).
§ 2. Let us take a rectangular coordinate system \(Gx_1y_1z_1\) with origin at the center of gravity of the body \(G\) and with the axis \(Gy_1\) directed along the middle semiaxis of the central ellipsoid of inertia \(E_G\). Then the equation of this ellipsoid can be written in the form
\[ I_x^{(G)}x_1^2+I_y^{(G)}y_1^2+I_z^{(G)}z_1^2-2I_{zx}^{(G)}z_1x_1=1, \tag{2} \]
where \(I_x^{(G)}\), \(I_y^{(G)}\), \(I_z^{(G)}\) are the axial moments, and \(I_{zx}^{(G)}\) is the centrifugal moment of inertia of the body with respect to the axes \(Gx_1y_1z_1\).
The equation of the gyration ellipsoid \(E_G^{-1}\), corresponding to the ellipsoid \(E_G\), in the same axes will be
\[ \frac{I_z^{(G)}x_1^2}{I_x^{(G)}I_z^{(G)}-I_{zx}^{(G)2}} +\frac{y_1^2}{I_y^{(G)}} +\frac{I_x^{(G)}z_1^2}{I_x^{(G)}I_z^{(G)}-I_{zx}^{(G)2}} +\frac{2I_{zx}^{(G)}z_1x_1}{I_x^{(G)}I_z^{(G)}-I_{zx}^{(G)2}} =1. \tag{3} \]
If the axis \(Gz_1\) is made to coincide with the perpendicular to one of the circular sections of the gyration ellipsoid, then the plane \(Gx_1y_1\) coincides with the plane of this section, and in equation (3) the coefficients of \(x_1^2\) and \(y_1^2\) become equal to each other.
Thus we obtain the relation
\[ I_z^{(G)}\bigl(I_x^{(G)}-I_y^{(G)}\bigr)-I_{zx}^{(G)2}=0, \tag{4} \]
which we shall call the Hess structural condition. When condition (4) is fulfilled, equations (2) and (3) are transformed into the form
\[ I_y^{(G)}(x_1^2+y_1^2)+\frac{1}{I_z^{(G)}}\bigl(I_z^{(G)}z_1-I_{zx}^{(G)}x_1\bigr)^2=1, \tag{5} \]
\[ \frac{1}{I_y^{(G)}}(x_1^2+y_1^2) +\frac{I_x^{(G)}}{I_y^{(G)}I_z^{(G)}}z_1^2 +\frac{2I_{zx}^{(G)}}{I_y^{(G)}I_z^{(G)}}z_1x_1=1. \tag{6} \]
Now take at any point \(O\) of the perpendicular to the circular section \(E_G^{-1}\) chosen by us a coordinate system \(Oxyz\) with axes parallel to
axes \(Gx_1y_1z_1\), setting the distance \(GO=z_G\). In this case, on the basis of the theorems on moments of inertia with respect to parallel axes, the axial and centrifugal moments of inertia with respect to the system \(Oxyz\) will be
\[ I_x=I_x^{(G)}+Mz_G^2,\qquad I_y=I_y^{(G)}+Mz_G^2,\qquad I_z=I_z^{(G)}, \tag{7} \]
\[ I_{yz}=0,\qquad I_{zx}=I_{zx}^{(G)},\qquad I_{xy}=0. \]
It follows from (7) that the axis \(Oy\) will be a principal axis of inertia, the structural condition (4) at the point \(O\) will remain exactly the same, and, consequently, the equation of the gyration ellipsoid \(E_0^{-1}\) at the point \(O\), according to (6), will have the form
\[ \frac{1}{I_y}(x^2+y^2)+\frac{I_x}{I_y I_z'}z^2+ \frac{2I_{zx}}{I_y I_z'}zx=1. \tag{8} \]
Comparing (8) with (6), we arrive at the conclusions:
1) The planes of the circular sections of the gyration ellipsoids constructed at the points of the perpendicular to the circular section of the central gyration ellipsoid are parallel to one another \(\left({}^{2}\right)\).
2) Hess’s structural conditions for any heavy rigid body will be satisfied if it is fixed at any point lying on the perpendicular to one of the circular sections of the central gyration ellipsoid \(E_G^{-1}\) \(\left({}^{3}\right)\).
§ 3. Let a heavy rigid body, fixed at a stationary point \(O\), perform Hessian motion. Then the structural condition (4) must be satisfied, and the kinetic moment of the body \(\mathbf K_0\) will lie in the plane of the circular section of the gyration ellipsoid (8) \(\left({}^{3}\right)\). These conditions, in the coordinate system \(Oxyz\) chosen by us, will be written in the form
\[ x_G=y_G=0,\qquad z_G\ne0;\qquad K_z=-I_{zx}p+I_z r=0. \tag{9} \]
Here \(x_G,y_G,z_G\) are the coordinates of the center of gravity of the body \(G\); \(p,q,r\) are the projections of the angular velocity \(\vec\omega\).
Taking into account conditions (4) and (9), we find the position of the kinetic center according to (1):
\[ \mathbf r_A=\frac{I_y}{Mz_G}\,\mathbf k =\frac{I_y}{Mz_G^2}\,\mathbf r_G =\frac{1}{\lambda}\,\mathbf r_G, \tag{10} \]
where \(\mathbf k\) is the unit vector of the axis \(Oz\); \(\lambda\) is a constant coefficient. Thus, in the motion of the body in the Hess case, the kinetic center will lie on the same perpendicular to the circular section \(E_0^{-1}\) as the center of gravity \(G\).
It follows from (10) that the velocity of the kinetic center of the body is found as
\[ \mathbf v_A=\vec\omega\times\mathbf r_A=\frac{1}{\lambda}\mathbf v_G. \tag{11} \]
Now multiplying (1) vectorially on the right by \(M\mathbf v_G\), we obtain
\[ \mathbf K_0=\mathbf r_A\times M\mathbf v_G \]
or, taking (10) into account,
\[ \mathbf K_0=\mathbf r_A\times \lambda M\mathbf v_A=\lambda \mathbf K_0^A, \tag{12} \]
where \(\mathbf K_0^A\) denotes the moment of momentum of the kinetic center of the body relative to \(O\), under the assumption that the mass of the body is concentrated in it.
Similarly, we express the moment of the weight of the body \(\mathbf M_0\) relative to the point \(O\):
\[ \mathbf M_0=-Mg\,(\mathbf r_G\times \vec\xi^{\,0}) =-\lambda Mg\,(\mathbf r_A\times \vec\xi^{\,0}) =\lambda \mathbf M_0^A. \tag{13} \]
In this formula \(g\) denotes the acceleration of gravity; \(\vec\xi^{\,0}(\gamma_1,\gamma_2,\gamma_3)\) is the unit vector of the upward-directed vertical and \(\mathbf M_0^A\) is the moment of the weight of the ki-
of the kinetic center \(A\) relative to \(O\), on the assumption that the mass of the body is concentrated in it.
§ 4. Using the preceding conclusions, we shall prove a theorem which is a generalization of Huygens’ well-known theorem on the reciprocity of the point of suspension and the center of oscillation of a physical pendulum.
Theorem. If the point of suspension \(O\) of a heavy rigid body satisfying Hesse’s conditions is moved to the kinetic center \(A\), then the kinetic center moves to the point \(O\), and the motion of the body remains unchanged under the same initial conditions.
Substituting into the vector equation of motion of the body
\[ \dot{\mathbf K}_0+\vec{\omega}\times \mathbf K_0=\mathbf M_0 \]
the values of \(\mathbf K_0\) and \(\mathbf M_0\) according to (12) and (13), we obtain, after cancellation by \(\lambda\),
\[ \dot{\mathbf K}_0^A+\vec{\omega}\times \mathbf K_0^A=\mathbf M_0^A . \tag{14} \]
We now project (14) onto the moving coordinate axes \(Oxyz\)
\[ \dot p-qr=\frac{g}{r_A}\gamma_2,\qquad \dot q+pr=-\frac{g}{r_A}\gamma_1 . \tag{15} \]
As the third equation we take Hesse’s fourth integral (9), which we write in the form
\[ r=\frac{I_{zx}}{I_z}\,p . \tag{16} \]
In order that the system of equations (15) and (16) be closed, we add to them Poisson’s equations
\[ \dot\gamma_1=r\gamma_2-q\gamma_3,\qquad \dot\gamma_2=p\gamma_3-r\gamma_1,\qquad \dot\gamma_3=q\gamma_1-p\gamma_2 . \tag{17} \]
We now take the kinetic center \(A\) as the fixed point. In this case the distance from \(A\) to the center of gravity of the body will be, taking into account (10) and (7),
\[ r_A-r_G=\frac{I_y}{Mz_G}-z_G =\frac{I_y-Mz_G^2}{Mz_G} =\frac{I_y^{(G)}}{Mz_G}. \tag{18} \]
If we denote by \(r'_A\) the distance from \(A\) to the new kinetic center and by \(I'_y\) the moment of inertia of the body about an axis passing through \(A\) parallel to the axis \(Oy\), then, according to (10), (7), and (18), we obtain
\[ r'_A=\frac{I'_y}{M(r_A-r_G)} =\frac{I_y^{(G)}+M(r_A-r_G)^2}{M(r_A-r_G)} =r_A, \tag{19} \]
i.e. the kinetic center coincides with the point \(O\).
It is also evident from (7) that, when the point of suspension is displaced along the perpendicular to the circular section of the ellipsoid of gyration, \(I_{zx}\) and \(I_z\) will not change.
Thus, from the system of equations (15), (16), and (17) we conclude that, when the body is fixed at the point \(A\), the general solution remains the same as when it is fixed at the point \(O\). If, moreover, the same initial conditions remain, then the particular solution of this system will not change either.
Ryazan Radio Engineering Institute
Received
23 V 1960
REFERENCES
- G. K. Suslov, Theoretical Mechanics, 1946.
- S. A. Chaplygin, “On Hesse’s loxodromic pendulum,” Collected Works, 1, 1948.
- N. E. Zhukovsky, “Hesse’s loxodromic pendulum,” Collected Works, 1, 1948.