UDC 621.8.034.4

A. F. POZUBENKOV

FREQUENCIES OF FREE VIBRATIONS OF BRANCHED CHAINS

(Presented by Academician A. A. Lebedev on 6 VII 1965)

Branched chains of diatomic molecules may be of two types (see Fig. 1).

Fig. 1. Branched chains of diatomic molecules

We choose the coordinate system in such a way that the (x)-axis is parallel to the chain. The (y)-axis is perpendicular to the (x)-axis and lies in the plane of the chain, and the (z)-axis is perpendicular to the (x)- and (y)-axes. The components of the displacements of atoms from their positions of stable equilibrium along the (x)-axis are (U_n), along the (y)-axis (V_n), and along the (z)-axis (W_n).

Small free vibrations can be divided into two groups: 1) vibrations in which the atoms do not leave the plane of the chain; 2) vibrations in which the atoms leave the plane of the chain. The equations of motion for small free vibrations of the (U_n) components of a branched chain of the first type (Fig. 1a) have the form

[
\begin{aligned}
M_1 \ddot U_1 &= \beta_1(U_2 - U_1),\
M_2 \ddot U_2 &= \alpha_2(U_4 - U_2) + \beta_1(U_1 - U_2) - \alpha_2 U_2,\
& \tag{1}\
M_1 \ddot U_3 &= \beta_1(U_4 - U_3),\
&\cdots\
M_1 \ddot U_{2n+1} &= \beta_1(U_{2n+2} - U_{2n+1}),\
M_2 \ddot U_{2n+2} &= \alpha_2(U_{2n+4} + U_{2n} - 2U_{2n+2}) + \beta_1(U_{2n+1} - U_{2n+2}),\
&\cdots\
M_1 \ddot U_{N-1} &= \beta_1(U_N - U_{N-1}),\
M_2 \ddot U_N &= \alpha_2(U_{N-2} - U_N) + \beta_1(U_{N-1} - U_N) - \alpha_2 U_N .
\end{aligned}
]

We seek the solution of system (1) in the form

[
U_n = |U_n| e^{i\omega t}.
]

For the vibration frequencies we obtain the equation

[
\left|
\begin{array}{cccccccccc}
a_1 & -\beta_1 & 0 & 0 & 0 & 0 & \cdots & \cdots & \cdots & 0\
-\beta_1 & a_2 & 0 & -\alpha_2 & 0 & 0 & \cdots & \cdots & \cdots & 0\
0 & 0 & a_1 & -\beta_1 & 0 & 0 & \cdots & \cdots & \cdots & 0\
0 & -\alpha_2 & -\beta_1 & a_2 & 0 & -\alpha_2 & \cdots & \cdots & \cdots & 0\
0 & 0 & 0 & 0 & a_1 & -\beta_1 & \cdots & \cdots & \cdots & 0\
0 & 0 & 0 & -\alpha_2 & -\beta_1 & a_2 & \cdots & \cdots & \cdots & 0\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\
0 & \cdots & \cdots & \cdots & \cdots & \cdots & a_1 & -\beta_1 & 0 & 0\
0 & \cdots & \cdots & \cdots & \cdots & -\beta_1 & a & 0 & -\alpha_2\
0 & \cdots & \cdots & \cdots & \cdots & 0 & 0 & a_1 & -\beta_1\
0 & \cdots & \cdots & \cdots & \cdots & 0 & -\alpha_2 & -\beta_1 & a_2
\end{array}
\right| = 0,
\tag{2}
]

where

[
a_1 = -M_1 \omega^2 + \beta_1, \qquad
a_2 = -M_2 \omega^2 + 2\alpha_2 + \beta_1.
]

The polynomials ({}^{2}n_N(a_1,a_2,\beta_1,\alpha_2)) satisfy the recurrence relations

[
{}^{2}n_N-(a_1a_2-\beta_1^2){}^{2}n_{N-2}+a_1^2a_2^2{}^{2}n_{N-4}=0,
\tag{3}
]

therefore they can be represented in the form of functions of the chain parameters ((^1))

[
{}^{2}n_N=(a_1a_2)^{N/2}\frac{\operatorname{sh}(N+2)\gamma}{\operatorname{sh}2\gamma}
\tag{4}
]

for (a_1a_2-\beta_1^2=2a_1a_2\operatorname{ch}2\gamma).

From (2) we obtain the following values of the vibration frequencies of the chain:

[
\omega^2=
\frac{(M_1+M_2)\beta_1+4M_1\alpha_2\sin^2 k\pi/(N+2)}
{2M_1M_2}
\pm
\left{
\left[
\frac{(M_1+M_2)\beta_1+4M_1\alpha_2\sin^2 k\pi/(N+2)}
{2M_1M_2}
\right]^2
-4\alpha_2\beta_1\sin^2\frac{k\pi}{N+2}
\right}^{1/2},
]

[
k=1,2,\ldots,\frac{N}{2}.
\tag{5}
]

The frequencies of the free vibrations making up (V_n) and (W_n) are obtained from (5) by replacing (\alpha_2,\beta_1) by (\beta_2,\alpha_1) and by (\beta_2,\beta_1), respectively.

The frequencies of the free vibrations of a chain of the second type (Fig. 1b) are determined from the equation

[
{}^{4}n_N(a_1,a_2,a_3,a_4,\beta_1,\alpha_1,\beta_1,\alpha_1)=
]

[

\left|
\begin{array}{cccccccccc}
a_1 & -\beta_1 & 0 & 0 & 0 & 0 & \ldots & \ldots & \ldots & 0\
-\beta_1 & a_2 & 0 & -\alpha_1 & 0 & 0 & \ldots & \ldots & \ldots & 0\
0 & 0 & a_3 & -\beta_1 & 0 & 0 & \ldots & \ldots & \ldots & 0\
0 & -\alpha_1 & -\beta_1 & a_4 & 0 & -\alpha_1 & \ldots & \ldots & \ldots & 0\
0 & 0 & 0 & 0 & a_1 & -\beta_1 & \ldots & \ldots & \ldots & 0\
0 & 0 & 0 & -\alpha_1 & -\beta_1 & a_2 & \ldots & \ldots & \ldots & 0\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\
0 & \ldots & \ldots & \ldots & \ldots & \ldots & a_1 & -\beta_1 & 0 & 0\
0 & \ldots & \ldots & \ldots & \ldots & -\beta_1 & a_2 & 0 & -\alpha_1\
0 & \ldots & \ldots & \ldots & \ldots & 0 & 0 & a_3 & -\beta_1\
0 & \ldots & \ldots & \ldots & \ldots & 0 & -\alpha_1 & -\beta_1 & a_4
\end{array}
\right|
=0.
\tag{6}
]

In equation (6) it is assumed that

[
a_1=-M_1\omega^2+\beta_1,\qquad
a_2=-M_2\omega^2+2\alpha_1+\beta_1,
]

[
a_3=-M_2\omega^2+\beta_1,\qquad
a_4=-M_1\omega^2+2\alpha_1+\beta_1.
\tag{7}
]

The polynomials ({}^{4}n_N) satisfy a recurrence relation of the form

[
{}^{4}n_N-\left[(a_1a_2-\beta_1^2)(a_3a_4-\beta_1^2)-2a_1a_3\alpha_1^2\right]{}^{4}n_{N-4}
+a_1^2a_3^2\alpha_1^4{}^{4}n_{N-8}=0.
\tag{8}
]

If

[
(a_1a_2-\beta_1^2)(a_3a_4-\beta_1^2)-2a_1a_3\alpha_1^2
=
2a_1a_3\alpha_1^2\operatorname{ch}4\gamma,
\tag{9}
]

then

[
{}^{4}n_N=(a_1a_3\alpha_1^2)^{N/4}
\frac{2\operatorname{sh}(N+2)\gamma\,\operatorname{ch}2\gamma}{\operatorname{sh}4\gamma}.
\tag{10}
]

From (6) the following values for (\gamma) are obtained:

[
\gamma=k\pi i/(N+2),\qquad k=1,2,\ldots,N/4.
\tag{11}
]

Knowing (\gamma), from equation (8) one can determine the values of the frequencies.

Received
25 VI 1965

CITED LITERATURE

1. A. F. Pozubenko, Tr. Gos. optich. inst. im. S. I. Vavilova, 30, no. 159, 127 (1963).