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ON THE SIMPLIFICATION OF THE DIFFERENTIAL EQUATIONS
OF THE PRECESSION OF A TWO-DEGREE-OF-FREEDOM GYROSCOPE
IN THE STUDY OF THE ROTATIONAL MOTION OF A BODY
V. S. NOVOSELOV
UDC 517.917 : 531.383
- For controlling the rotational motion of a body, as well as for damping this motion, gyroscopes with two degrees of freedom are used. One degree of freedom corresponds to the rapid rotation of the rotor about its axis of symmetry. This rotation is produced by an electric motor and usually has a constant angular velocity. This rapid rotation leads to the fact that the intrinsic angular momentum of the gyroscope \(\bar H\) is a comparatively large quantity. The second degree of freedom corresponds to the rotation (precession) of the gyroscope frame relative to a certain axis rigidly connected with the body. By filling the space between the frame and the gyroscope casing with a viscous fluid, damping of the precessional motion of the gyroscope is obtained. Proceeding from the conditions of “dynamic balancing,” blocks of two or more gyroscopes are often used.
The equations of the rotational motion of a body in the form of the dynamic and kinematic Euler equations have the form
\[ J_i \dot{x}_i = X_i(x_j) + K_i + M_i(x_j, y_j, t), \]
\[ \dot{y}_i = \sum_{i,j=1}^{3} Y_{ij}(y_k)x_j + Y_{i0}(y_k) \quad (i, j, k = 1, 2, 3). \tag{1} \]
Here \(J_i\) are the axial moments of inertia of the body; \(x_1, x_2, x_3\) are the projections of the angular velocity of rotation of the body onto axes rigidly connected with it; \(y_i\) are the angles of orientation of the body; \(X_i\) are homogeneous quadratic forms in \(x_j\), whose coefficients are linear homogeneous functions of the axial moments of inertia \(J_i\) and the centrifugal moments \(J_{ij}\) of the body; \(Y_{ij}\) and \(Y_{i0}\) are trigonometric functions of the quantities \(y_k\); \(K_i\) are the moments of the reactions of the gyroscopic device; \(M_i\) are the moments of the external forces, including the moments of the action of the gravitational field.
Let \(x, y\), and \(z\) denote the principal axes of inertia of the frame with the rotor. Let the \(x\)-axis be the axis of precession, and the \(z\)-axis the axis of intrinsic rotation. By symmetry, the moments of inertia of the frame with the rotor relative to the axes \(x\) and \(y\) will be equal. Denote them by \(A\). Denote the moment of inertia relative to the \(z\)-axis by \(C\). For simplicity of exposition we shall consider only one gyroscope. However, the results obtained below also hold for a multi-gyroscope system.
The angular momentum \(\bar l\) of the frame with the gyroscope is written as
\[ \bar{l} = A(\dot{\gamma}+\omega_x)\bar{i} + A\omega_y\bar{j} + (C\omega_z+H)\bar{k}, \quad H=\mathrm{const}>0, \tag{2} \]
where \(\gamma\) is the angle of precession of the gyroscope; \(\omega_x,\omega_y\) and \(\omega_z\) are the projections of the angular velocity of rotation of the body; \(\bar i,\bar j\), and \(\bar k\) are the unit vectors of the axes \(x,y,z\); \(H\) is the magnitude of the intrinsic angular momentum.
The law of variation of the angular momentum \(\bar l\) in projections onto the axes \(x,y,z\) gives three differential equations:
\[ A(\ddot\gamma+\dot\omega_x)+(C\omega_z+H)\omega_y-A\omega_y\omega_z=-K_x, \tag{3} \]
\[ A\dot\omega_y+A(\dot\gamma+\omega_x)\omega_z-(C\omega_z+H)\omega_x=-K_y, \tag{4} \]
\[ C\dot\omega_z-A\dot\gamma\omega_y=-K_z. \tag{5} \]
The damping moment of the rotation of the frame with respect to the precession axis \(x\) will be equal to: \(-k_1^2\dot\gamma\), \(k_1=\mathrm{const}\). Usually an elastic suspension of the frame is used, and therefore an additional moment arises: \(-k_2^2\gamma-k_3\), where \(k_2^2\) is the stiffness coefficient and \(k_3\) is the preliminary tension of the spring; \(k_2=\mathrm{const}\) and \(k_3=\mathrm{const}\). Hence we obtain
\[ K_x=k_1^2\dot\gamma+k_2^2\gamma+k_3. \tag{6} \]
Expressing \(K_x\) in equation (3) by formula (6), and also \(K_x,K_y\), and \(K_z\) in equations (1) by formulas (4)—(6), we obtain the complete system of differential equations. We shall consider motion only under the action of such force moments \(M_i\) that the indicated system has a unique solution for any bounded initial conditions. The magnitude \(A\) is usually small in comparison with \(k_1^2\) and \(H\), which often have one and the same order and are small in comparison with the moments of inertia of the body \(J_i\) and \(J_{ij}\).
Let, in the adopted scale, the magnitude \(A\) be of order unity. Introduce the notation \(\omega_0=Ak_1^{-2}\). The quantity \(\omega_0\) will be small. We take \(\omega_0\) to be an infinitely small quantity of the first order. We assume \(O(H)=\omega_0^{-1}\), \(O(J_i)=\omega_0^{-2}\), \(O(\dot\gamma)=\omega_0\). Hence \(O(J_i\omega_0^2)=1\).
Since the moments of inertia of the body, by assumption, exceed by two orders the moments of inertia of the gyroscope with the frame, then, when substituting into equations (1) the expressions \(K_y\) and \(K_z\), equal up to sign to the left-hand sides of relations (4) and (5), it is inappropriate to write out in detail the terms containing the moments of inertia \(A\) and \(C\). We shall denote by \(Q\) the order of the largest of the quantities: \(\omega_x\omega_y,\omega_x\omega_z,\omega_y\omega_z,\dot\omega_x,\dot\omega_y,\dot\omega_z\). On the basis of formulas (4) and (5) we write
\[ K_y=H\omega_x-A\dot\gamma\omega_z+Q, \tag{4'} \]
\[ K_z=A\dot\gamma\omega_y+Q. \tag{5'} \]
Let \(a_{ij}\) denote the elements of the transformation matrix from the axes rigidly connected with the body to the axes \(x,y,z\). The quantities \(a_{ij}\) are functions of the angle \(\gamma\).
Equation (3), when formula (6) is used, makes it possible to obtain the equation of precession of the gyroscope
\[ A\ddot\gamma+k_1^2\dot\gamma+k_2^2\gamma+H(x_1a_{12}+x_2a_{22}+x_3a_{32})=k_3+Q. \tag{7} \]
- To determine the order of the individual terms in equations (1) and (7), it is useful to pass to a new independent variable \(u=\omega_0t\). We shall denote derivatives with respect to \(u\) by primes. We also introduce the notation: \(\dot x_i=x_i'\omega_0\). Equation (7) will be written as
\[ \omega_0^2\gamma''+\gamma'=f(x_j,y_j,\gamma)+Q, \tag{8} \]
where it is set that
\[ f=-A^{-1}\left[k_2^2\gamma+\omega_0 H\left(\tilde{x}_1 a_{12}+\tilde{x}_2 a_{22}+\tilde{x}_3 a_{32}\right)-k_3\right]. \tag{9} \]
Equation (8) contains a small parameter at the highest derivative. We shall now transform equations (1), in which the moments \(K_x\), \(K_y\), and \(K_z\) should be expressed by formulas \((4^1)\), \((5^1)\), and (6), to the new independent variable
\[ \tilde{x}_i' = X_i(\tilde{x}_j)+(J_i\omega_0^2)^{-1}K_i(\tilde{x}_j,y_j,\gamma',\gamma)+ \]
\[ +(J_i\omega_0^2)^{-1}M_i(x_j,y_j,t)+Q, \]
\[ y_i'=\sum_{j=1}^{3}Y_{ij}(y_k)\tilde{x}_j+\tilde{Y}_{i0}(y_k),\qquad \tilde{Y}_{i0}=Y_{i0}\omega_0^{-1}. \tag{10} \]
Let us note that \(Q\) will be the product of \(\omega_0^2\) by the order of the greatest of the quantities
\(\tilde{\omega}_x\tilde{\omega}_y\), \(\tilde{\omega}_x\tilde{\omega}_z\), \(\tilde{\omega}_y\tilde{\omega}_z\), \(\tilde{\omega}_x'\), \(\tilde{\omega}_y'\), \(\tilde{\omega}_z'\). Here, as before, the tilde indicates that the corresponding quantity has been referred to \(\omega_0\).
The presence of a small parameter at the highest derivative in equation (8) substantially complicates the investigation of the system of equations. The indicated system cannot be effectively integrated on electronic computers, since, in order to preserve the necessary accuracy, one has to choose a very small integration step, which leads to a large expenditure of machine time.
We shall state the basic assumption which is satisfied in real problems.
Let the order of the coefficients entering the right-hand sides of expressions (9) and (10), as well as the order of the initial values of the unknowns \(x_i^0\), \(y_i^0\), \(\gamma^0\), \(\gamma'{}^0\), be not less than unity.
Write the equation
\[ \sigma'=f(p_j,q_i,\sigma), \tag{11} \]
where the function \(f\) is determined by formula (9), in which the variables \(\tilde{x}_j\) and \(y_j\) are replaced by \(p_j\) and \(q_j\), and the variable \(\gamma\) is replaced by the variable \(\sigma\). We shall consider the solution of this equation together with the equations
\[ p_i'=X_i(p_j)+(J_i\omega_0^2)^{-1}K_i(p_j,q_j,\sigma',\sigma)+ \]
\[ +(J_i\omega_0^2)^{-1}M_i(p_j,q_j,t)+Q, \tag{12} \]
\[ q_i'=\sum_{j=1}^{3}Y_{ij}(q_k)p_j+\tilde{Y}_{i0}(q_k). \]
We take the following initial conditions: \(p_i^0=\tilde{x}_i^0\), \(q_i^0=y_i^0\), \(\sigma^0=\gamma^0\). Proceeding from the basic assumption, one may assert that the magnitudes of all variables and their derivatives with respect to \(u\) for the indicated solution will be of order unity.
Set \(\nu=\gamma-\sigma\), \(\varphi_i=\tilde{x}_i-p_i\), \(\psi_i=y_i-q_i\). On the basis of equations (7), (10)—(12) we obtain
\[ \omega_0^2\nu''+\nu'=-\omega_0^2\sigma''+ \]
\[ +f(\varphi_j+p_j,\psi_j+q_j,\nu+\sigma)-f(p_j,q_j,\sigma)+Q, \tag{13} \]
\[ \varphi_i'=\Phi_i+Q,\qquad \psi_i'=\Psi_i. \tag{14} \]
In equation (14) the notation adopted is
\[
\Phi_i=X_i(\varphi_j+p_j)-X_i(p_j)+(J_i\omega_0^2)^{-1}
\bigl[K_i(\varphi_j+p_j,\ \psi_j+q_j,\ \nu'+\sigma',\ \nu+\sigma)-
\]
\[
-K_i(p_j,\ q_j,\ \sigma',\ \sigma)+M_i(\varphi_j+p_j,\ \psi_j+q_j,\ t)-M_i(p_j,\ q_j,\ t)\bigr]+Q,
\tag{15}
\]
\[
\Psi_i=\sum_{j=1}^{3}Y_{ij}(\psi_k+q_k)(\varphi_j+p_j)+\tilde Y_{i0}(\psi_k+q_k)-
\]
\[
-\sum_{j=1}^{3}Y_{ij}(q_k)p_j-\tilde Y_{i0}(q_k).
\tag{16}
\]
The quantities \(\varphi_i,\ \psi_i,\ \nu\) are the differences of the variables of the basic equations (8) and (10) and of the simplified equations (11), (12). Equation (13) contains a small parameter at the highest derivative.
In studying the motion of gyroscopic systems [1–3], it proves useful, along with the “slow time” \(u=\omega_0 t\), to consider the “fast time” \(\tau=\omega_0^{-1}t\). Hence we obtain \(\tau=\omega_0^{-2}u\). We shall denote derivatives with respect to \(\tau\) by an asterisk. The variables \(\varphi_i,\ \psi_i\), and \(\nu\) have zero initial conditions; the initial condition for the derivative of \(\nu\) is equal to
\[ \nu_0^{*}=\omega_0^2\bigl[\gamma_0'-f(p_j^0,\ q_j^0,\ \sigma_0)\bigr]. \]
In order to have zero conditions for all variables, introduce the notation
\[ \theta=\nu-\tau\nu_0^{*}e^{-\tau}. \]
For \(\tau=0\) we obtain: \(\theta_0=0,\ \theta_0^{*}=0\).
Taking \(\tau\) as the independent variable, we write equation (13) as
\[
\theta^{**}+\theta^{*}-\omega_0^2\{f(\varphi_j+p_j,\ \psi_j+q_j,\ \theta+\tau\nu_0^{*}e^{-\tau}+\sigma)-
\]
\[
-f(p_j,\ q_j,\ \sigma)\}=-\omega_0^4\sigma''+\omega_0^2Q+\omega^2\nu_0'e^{-\tau}.
\tag{17}
\]
We transform equation (14) to the form
\[ \varphi_i^{*}=\omega_0^2\Phi_i(\varphi_j,\ \psi_j,\ p_j,\ q_j,\ \omega_0^{-2}\theta^{*}+\nu_0'(1-\tau)e^{-\tau},\ \sigma',\ \theta+\omega_0^2e^{-\tau},\ \sigma,\ t)+\omega_0^2Q, \tag{18} \]
\[ \psi_i^{*}=\omega_0^2\Psi_i(\varphi_j,\ \psi_j,\ p_j,\ q_j). \]
As formulas \((4^1),\ (5^1)\), and (6) show, the moments \(K_x,\ K_y\), and \(K_z\) depend linearly on \(\dot\gamma\). Therefore the functions \(\Phi_i\) will depend linearly on \(\omega_0^{-2}\theta^{*}\), and in equations (18) there will be a cancellation of the large parameter \(\omega_0^{-2}\) with the small parameter \(\omega_0^2\).
The system of equations (17), (18) contains no small parameters at the highest derivatives. When the quantities
\[ \lambda_1=\omega_0^4\sigma'',\qquad \lambda_2=\omega_0^2e^{-\tau},\qquad \lambda_3=\omega_0^2Q \]
are set equal to zero, the indicated system has the unique zero solution. As follows from Poincaré’s small-parameter method [4], the variables \(\varphi_i,\ \psi_i,\ \theta\), and \(\theta^{*}\) are equal to zero up to terms of the order of the largest of the quantities \(\lambda_1,\ \lambda_2\), and \(\lambda_3\).
Proceeding from the definition of the quantity \(Q\), from the changes of variables made, and also from the fact that the solution of the simplified equations has zero order in all variables, we obtain, up to terms of order \(\lambda_1\) and \(\lambda_2\), that the quantity \(Q\) has order \(\omega_0^2\). Differentiating equality (11) and taking equations (12) into account, we shall show that \(\sigma''\)
has order zero if the frequencies of variation of the moments under the fixation of \(x_j\) and \(y_j\) are of order \(\omega_0\) or lower.
For the indicated case the following estimates have been proved:
\[ O(\tilde{x}_j-p_j,\ y_j-q_j,\ \gamma-\sigma) =\omega_0^2\max\{\omega_0^2,\ e^{-\omega_0^{-1}t},\ \omega_0^{-1}te^{-\omega_0^{-1}t}\}, \]
\[ O(\gamma'-\sigma') =\max\{\omega_0^2,\ e^{-\omega_0^{-1}t},\ \omega_0^{-1}te^{-\omega_0^{-1}t}\}. \tag{19} \]
Formulas (19) show that, in investigating the rotational motion of a body, with the exception of a small interval of time at the beginning of the motion, one may pass to the simplified equations.
Remark 1. The estimates (19) also hold in the case where the terms \(-A\gamma\omega_z\) and \(A\gamma\omega_y\) in formulas \((4^1)\) and \((5^1)\) are omitted. Omitting the indicated terms when using equations (11) and (12), in which \(Q=0\), is equivalent to using, in deriving the equations, the precessional theory of gyroscopes.
Remark 2. If the frequencies \(\omega_M\) of variation of the moments of forces under the fixation of \(x_j\) and \(y_j\) are higher than \(\omega_0\), then \(\sigma''\) will be of order \(\omega_M\omega_0^{-1}\). The estimating formulas take the form
\[ O(\tilde{x}_j-p_j,\ y_j-q_j,\ \gamma-\sigma) =\omega_0^2\max\{\omega_0^2,\ \omega_M\omega_0,\ e^{-\omega_0^{-1}t},\ \omega_0^{-1}te^{-\omega_0^{-1}t}\}, \]
\[ O(\gamma'-\sigma') =\max\{\omega_0^2,\ \omega_M\omega_0,\ e^{-\omega_0^{-1}t},\ \omega_0^{-1}te^{-\omega_0^{-1}t}\}. \tag{20} \]
Formulas (20) should be applied, for example, when high-frequency vibrations of the body are present.
- We shall obtain a simplified system of increased accuracy for the case when the frequencies of variation of the moments \(M_i\) under the fixation of \(x_j\) and \(y_j\) are of order \(\omega_0\) or lower. We write equation (8) in the form
\[ \omega_0^2\gamma''+\gamma' =F(\tilde{x}_j,\ y_j,\ \gamma,\ \omega_0\tilde{x}'_j). \tag{21} \]
The right-hand side of equation (21) is obtained after substituting into equation (3) the expression for \(\omega_x\), computed with the aid of equations (1).
Let us pass in equation (21) to the integral representation
\[ \gamma'=e^{-\omega_0^{-2}u} \left(\gamma'_0+\omega_0^{-2}\int_0^u e^{\omega_0^{-2}u}F\,du\right). \tag{22} \]
Here \(\gamma'_0\) denotes the initial value of the variable \(\gamma'\). Applying integration by parts in formula (22), we obtain
\[ \gamma'=(\gamma'_0-F_0)e^{-\omega_0^{-2}u} +F-e^{-\omega_0^{-2}u}\int_0^u e^{\omega_0^{-2}u}F'\,du. \tag{23} \]
After \(n\)-fold integration by parts we write
\[ \begin{aligned} \gamma'={}&e^{-\omega_0^{-2}u} \left(\gamma'_0-F_0+\omega_0^2F'_0-\omega_0^4F''_0+\cdots +(-1)^n\omega_0^{2(n-1)}F_0^{\,n-1}\right) \\ &+F-\omega_0^2F'+\omega_0^4F''-\cdots +(-1)^{n-1}\omega_0^{2(n-1)}F^{n-1}+R_n . \end{aligned} \tag{24} \]
In formula (24), \(F_0^k\) denotes the initial value of the \(k\)-th derivative with respect to \(u\) of the function \(F\). The remainder term has the form
\[ R_n=(-1)^n\omega_0^{2(n-1)}e^{-\omega_0^{-2}u} \int_0^u e^{\omega_0^{-2}u}F^n\,du . \tag{25} \]
Since, under the assumptions made, \(F^n\) is a bounded quantity, let \(|F^n| < A_n\), \(A_n=\mathrm{const}>0\). From formula (25) we obtain
\[ |R_n| < A_n \omega_0^{2n}. \tag{26} \]
The equations of motion contain only the first derivatives of the variables \(x_j\); therefore it is expedient to restrict oneself in representation (24) only to terms with such derivatives. We shall have
\[ \gamma'=(\gamma_0' - F_0+\omega_0^2 F_0')e^{-\omega_0^{-2}u}+F-\omega_0^2F' + O(\omega_0^4). \tag{27} \]
Equation (27) replaces equation (8) with sufficiently high accuracy. To preserve this accuracy in equations (1), it will be necessary to abandon the simplified formulas \((4^1)\), \((5^1)\), and to compute the damping moments by the exact formulas (4) and (5).
If terms of order \(\omega_0^2\) are neglected, then formula (27) takes the form
\[ \gamma'=(\gamma_0' - f_0)e^{-\omega_0^{-2}u}+f+O(\omega_0^2). \tag{28} \]
When replacing the simplified equation (11) by equation (28), the terms with \(e^{-\omega_0^{-1}t}\) may be omitted in the right-hand sides of the estimate formulas (19). Therefore the solution of the joint system of simplified equations (12) and (28), if in the latter the variable \(\gamma\) is replaced by \(\sigma\), will coincide with the exact solution up to terms of order \(\omega_0^2\).
References
- Novoselov V. S. PMM, vol. XXIII, no. 1, 1959.
- Novoselov V. S. PMM, vol. XXIII, no. 5, 1959.
- Novoselov V. S. PMM, vol. XXIV, no. 6, 1960.
- Golubev V. V. Lectures on the Analytic Theory of Differential Equations. Moscow–Leningrad, GITTL, 1950.
Received by the editors
June 21, 1966
Leningrad State University
named after A. A. Zhdanov