MECHANICS

I. I. Metelitsyn

THE INFLUENCE OF CHANGES IN THE PARAMETERS OF LINEAR GYROSCOPIC SYSTEMS ON THE FREQUENCIES OF OSCILLATIONS AND THE DAMPING COEFFICIENTS

(Presented by Academician A. Yu. Ishlinskii, 11 VI 1963)

The roots of the characteristic equation of a gyroscopic system satisfy equality (1)

\[ T^* \mu^2 + (D^* + i\Gamma^*)\mu + V^* + iE^* = 0, \tag{1} \]

where \(T^*, D^*, \Gamma^*, V^*, E^*\) are real numbers:

\[ T^*=\sum\sum a_{ik}A_iA'_k,\qquad V^*=\sum\sum c_{ik}A_iA'_k,\qquad D^*=\sum\sum b_{ik}A_iA'_k, \]

\[ i\Gamma^*=\sum\sum \gamma_{ik}A_iA'_k,\qquad iE^*=\sum\sum \varepsilon_{ik}A_sA'_k, \tag{2} \]

where

\[ a_{ik}=a_{ki},\qquad b_{ik}=b_{ki},\qquad c_{ik}=c_{ki},\qquad \gamma_{ik}=-\gamma_{ki},\qquad \varepsilon_{ik}=-\varepsilon_{ki}. \]

The conjugate constants \(A_i, A'_i\) satisfy the system of equations

\[ \sum\left(a_{ik}\mu^2+(b_{ik}+\gamma_{ik})\mu+c_{ik}+\varepsilon_{ik}\right)A_k=0 \]

or

\[ \sum\left[a_{ik}{\mu'}^2+(b_{ik}+\gamma_{ik})\mu' + c_{ik}+\varepsilon_{ik}\right]A'_k=0 \qquad (i=1,2,\ldots,n). \tag{3} \]

From (3) it is clear that \(A_i\) depend on the roots of the characteristic equation, and from (1) we find that the roots of the characteristic equation depend on \(A_k\) and on the coefficients \(a_{ik}, \ldots, \varepsilon_{ik}\).

It is easy, however, to prove that the roots of the characteristic equation do not depend on \(A_k, A'_k\); therefore, when determining the differential \(d\mu\) from (1), one should take into account the dependence of \(T^*, \ldots, E^*\) and \(\mu\) only on the coefficients \(a_{ik}, \ldots, \varepsilon_{ik}\); this is readily checked directly if one takes into account relations (3).

This observation is the basis of a method for investigating the influence of changes in the coefficients \(a_{ik}, \ldots, \varepsilon_{ik}\) on the frequencies of oscillations and on the damping coefficients.

Restricting ourselves to such gyroscopic systems in which the gyroscopic forces have a dominant significance \((^1)\), let us recall that the roots of equation (1)

\[ \mu=\frac{-(D+i\Gamma)\pm(x+iy)}{2T}, \]

\[ \mu_1=\frac{-(D-x)-i(\Gamma-y)}{2T},\qquad \mu_2=\frac{-(D+x)-i(\Gamma+y)}{2T}, \tag{4} \]

have negative coefficients of the imaginary unit if the system is statically unstable \((V^*<0)\). If the system is statically stable \((V^*>0)\),

then the root \(\mu_1\) may be positive, since

\[ \begin{aligned} \mu_1 &\simeq -\frac{D^*-x}{2T}+\frac{4V^*T^*-D^{*2}}{4TT^*}\,i,\\ \mu_2 &\simeq \frac{D^*+x}{2T^*}-\left(\frac{\Gamma^*}{T^*}+\frac{4V^*T^*-D^{*2}}{4T^*T^*}\right)i . \end{aligned} \tag{5} \]

As for \(\Gamma^*\) and \(E^*\), both functions may be regarded as positive or negative, but necessarily of the same sign \((^1)\). With simultaneous changes in the signs of \(\Gamma^*\) and \(E^*\), the roots of equation (1) do not change.

Fig. 1

Fig. 2

A change in the potential energy of the system entails a change in the roots of the characteristic equation, which can be determined from equality (1).

If the gyroscopic system is conservative \((D^*\equiv 0\) and \(\Gamma^*\equiv 0)\), then

\[ \frac{\partial \mu_1}{\partial c} = \frac{\partial V^*/\partial c}{+\sqrt{\Gamma^{*2}+4V^*T^*}}\,i, \qquad \frac{\partial \mu_2}{\partial c} = \frac{\partial V^*/\partial c}{-\sqrt{\Gamma^{*2}+4V^*T^*}}\,i, \]

where \(c=c_{ik}\).

From the last equalities it follows: with an increase in the stiffness of an ideal gyroscopic system, its oscillation frequencies increase.

The proof is readily seen from Fig. 1a.

If the masses (the coefficients \(a_{ik}\)) in an ideal gyroscopic system are increased, then

\[ \frac{\partial \mu_1}{\partial a} = +\frac{\dfrac{\partial T^*}{\partial a}\,\mu^2}{\sqrt{\Gamma^{*2}+4V^*T^*}}\,i, \qquad \frac{\partial \mu_2}{\partial a} = -\frac{\dfrac{\partial T^*}{\partial a}\,\mu^2}{\sqrt{\Gamma^{*2}+4V^*T^*}}\,i . \]

Assuming that the system is statically stable, and noting that \(\mu^2=-\beta^2<0\), we arrive at the conclusion: with an increase in the masses of the system, the oscillation frequencies decrease, if the system is statically stable.

For a statically unstable system \((V^*<0)\), the frequencies of the slow oscillations increase, while the frequencies of the fast oscillations decrease (Fig. 1b) with increasing masses of the system.

The influence of a change in stiffness in a nonconservative gyroscopic system \((D^*\ne 0,\ E^*\ne 0)\) can be investigated if \(\partial\mu/\partial c\) \((c=c_{ik})\) is determined from equality (1):

\[ \begin{aligned} \frac{\partial \mu_1}{\partial c} &= -\frac{\partial V^*/\partial c}{2T^*\mu+D^*+i\Gamma^*} = -\frac{\partial V^*/\partial c}{x^2+y^2}(x-iy),\\ \frac{\partial \mu_2}{\partial c} &= -\frac{\partial V^*/\partial c}{-(x+iy)} = \frac{\partial V^*/\partial c}{x^2+y^2}(x-iy). \end{aligned} \tag{6} \]

Figure 2 presents the vectors \(\partial\mu_1/\partial c\) and \(\partial\mu_2/\partial c\) for a statically stable system \((V^* > 0)\). Hence one may draw the following conclusion: with an increase in stiffness, the damping coefficients and the frequencies of the slow oscillations increase; the frequencies of the fast oscillations increase, while the damping coefficients decrease.

Replacing in equalities (6) the derivative \(\dfrac{\partial V^*}{\partial c}\) by \(\dfrac{\partial E^*}{\partial \varepsilon} i\) \((\varepsilon = \varepsilon_{ik})\), we obtain

Fig. 3

\[ \begin{aligned} \frac{\partial \mu_1}{\partial \varepsilon} &= -\frac{\dfrac{\partial E^*}{\partial \varepsilon} i}{x^2+y^2}(x-iy),\\[6pt] \frac{\partial \mu_2}{\partial \varepsilon} &= -\frac{\dfrac{\partial E^*}{\partial \varepsilon} i}{x^2+y^2}(x-iy). \end{aligned} \tag{7} \]

Using Fig. 3, which shows the vectors \(\partial\mu_1/\partial\varepsilon\) and \(\partial\mu_2/\partial\varepsilon\), we find: with an increase in the intrinsic nonconservative forces, the oscillation frequencies decrease; the damping coefficients of the slow oscillations increase, while those of the fast oscillations decrease.

Obviously, by the indicated method one can also prove a number of other, more complicated theorems, if various assumptions are made concerning changes in the parameters of the system.

Received
21 V 1963

REFERENCES CITED

1. I. I. Metelitsyn, DAN, 86, No. 1 (1952).