Abstract Generated abstract
This paper treats the determination of small free longitudinal vibration frequencies in finite rectilinear chains composed of diatomic and triatomic molecules under exact boundary conditions. Using previously introduced polynomial systems satisfying recurrence relations, the equations of motion are reduced to characteristic equations expressed through a parameter gamma for chains with both ends fixed, one end fixed and one free, and both ends free. For diatomic chains, the work derives frequency equations for fixed and mixed boundaries and gives explicit frequency values for the free end case, including the zero frequency mode. The same method is then extended to triatomic chains, yielding final equations from which the corresponding vibration frequencies can be obtained.
Full Text
MATHEMATICAL PHYSICS
A. F. POZUBENKOV
FREQUENCIES OF FREE VIBRATIONS OF RECTILINEAR CHAINS
(Presented by Academician A. A. Lebedev on 31 XII 1964)
At the present time, the problem of vibrations of rectilinear chains is of great interest. Usually the frequencies of small free vibrations of chains of identical atoms or of diatomic molecules are calculated using periodicity conditions \((^{1,2})\). Sometimes solutions are found for exact boundary conditions \((^3)\).
To determine the frequencies of free vibrations of diatomic, triatomic, or, in the general case, \(k\)-atomic molecules, we introduced systems of polynomials \((^2)\)
\[ {}^{k}n_{km+l}(a_1,a_2,\ldots,a_k,\alpha_1,\alpha_2,\ldots,\alpha_k), \tag{1} \]
\[ l=1,2,\ldots,k;\quad m=0,1,2,\ldots, \]
satisfying the recurrence relation
\[ {}^{k}n_{km+l} - f_{1k}(a_1,a_2,\ldots,a_k,\alpha_1,\alpha_2,\ldots,\alpha_k)\, {}^{k}n_{(k-1)m+l} + \]
\[ + f_{2k}(a_1,a_2,\ldots,a_k)\, {}^{k}n_{(k-2)m+l} =0. \tag{2} \]
For \(k=1,\ a_1=2x,\ \alpha_1=-1\), the polynomials (1) coincide with the Gegenbauer polynomials \(C_n^1(x)\) \((^1)\).
Let us determine the frequencies of free vibrations of a rectilinear chain of diatomic molecules \(M_1M_2M_1\ldots,\ \alpha_1\alpha_2\alpha_1\ldots\), in which both ends are fixed (Fig. 1, I).
The equations of motion for free longitudinal vibrations with small amplitudes about positions of stable equilibrium have the form
\[ M_1\ddot U_1=\alpha_1(U_2-U_1)-\alpha_2U_1, \]
\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]
\[ M_1\ddot U_{2n+1} = \alpha_1(U_{2n+2}-U_{2n+1}) + \alpha_2(U_{2n}-U_{2n+1}), \]
\[ M_2\ddot U_{2n+2} = \alpha_2(U_{2n+3}-U_{2n+2}) + \alpha_1(U_{2n+1}-U_{2n+2}), \tag{3} \]
\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]
\[ M_2\ddot U_N=\alpha_1(U_{N-1}-U_N)-\alpha_2U_N. \]
Substituting into (3) the particular solution
\[ U_n=|U_n|e^{i\omega t}, \]
we obtain the equation for determining the frequencies
\[ {}^{2}n_N(a_1,a_2,\alpha_1,\alpha_2)=0, \tag{4} \]
where \(a_1=-M_1\omega^2+\alpha_1+\alpha_2,\ a_2=-M_2\omega^2+\alpha_1+\alpha_2\).
For the polynomials \({}^{2}n_N(a_1,a_2,\alpha_1,\alpha_2)\), the recurrence relations (2) have the form
\[ {}^{2}n_N-(a_1a_2-\alpha_1^2-\alpha_2^2)\,{}^{2}n_{N-2} +\alpha_1^2\alpha_2^2\,{}^{2}n_{N-4}=0. \]
Let us introduce a quantity \(\gamma\) such that
\[ a_1a_2-a_1^2-a_2^2=2a_1a_2\operatorname{ch}2\gamma; \tag{5} \]
then equation (4) can be reduced to the form
\[ \frac{1}{\operatorname{sh}2\gamma} \left[ \operatorname{sh}(N+2)\gamma+\frac{a_2}{a_1}\operatorname{sh}N\gamma \right]=0. \tag{6} \]
If one end is fixed and the other is free (Fig. 1, II), then
\[ \frac{1}{\operatorname{sh}2\gamma} \left[ \operatorname{sh}(N+2)\gamma+\frac{a_2-a_1}{a_1}\operatorname{sh}N\gamma \right]=0; \tag{7} \]
if both ends are free (Fig. 1, III), then
\[ \frac{\operatorname{sh}N\gamma}{\operatorname{sh}2\gamma} \left[ a_1a_2-a_2(a_1+a_2)-a_1^2+a_2^2 \right]=0. \tag{8} \]
Fig. 1. Rectilinear chain of diatomic molecules
In equations (7) and (8), \(\gamma\), as in equation (6), is determined by relation (5).
For the case in which both ends are fixed, or one is fixed and the other is free, we obtain the following equation for determining the frequencies:
\[ \omega^4-\omega^2 \frac{(M_1+M_2)(a_1+a_2)}{M_1M_2} + \frac{2a_1a_2}{M_1M_2}(1-\operatorname{ch}2\gamma)=0, \tag{9} \]
where \(\gamma\) are the roots of equations (6) and (7).
For a chain with free ends, equation (8) is solved exactly, and for the frequencies one obtains the values
\[ \omega^2= \frac{(M_1+M_2)(a_1+a_2)}{2M_1M_2} \pm \left\{ \frac{\left[(M_1+M_2)(a_1+a_2)\right]^2}{4M_1^2M_2^2} - \frac{2a_1a_2}{M_1M_2} \left(1-\cos^2\frac{k_2\pi}{N}\right) \right\}^{1/2}, \]
\[ \omega^2=\alpha_1(M_1+M_2)/M_1M_2,\qquad k_2=1,2,\ldots,N/2-1, \tag{10} \]
\[ \omega^2=0. \]
The calculation of the vibration frequencies of a chain of triatomic molecules is carried out in an analogous manner; therefore we give at once the final form of the equations for \(\gamma\):
\[ \frac{1}{\operatorname{sh}3\gamma} \left[ \operatorname{sh}(N+3)\gamma+ \frac{a_2a_3}{a_1a_2}\operatorname{sh}N\gamma \right]=0, \tag{11} \]
if both ends are fixed;
\[ \frac{1}{\operatorname{sh}3\gamma} \left[ \operatorname{sh}(N+3)\gamma+ \frac{a_2a_3-a_1a_2+a_1^2}{a_1a_2}\operatorname{sh}N\gamma \right]=0, \tag{12} \]
if the first end is fixed and the second is free;
\[ \frac{\operatorname{sh}N\gamma}{\operatorname{sh}3\gamma} \left[ a_1a_2a_3-a_3(a_1a_2+a_2a_3)-a_1a_2^2-a_2a_3^2-a_3a_1^2+a_1^2a_3+a_2^2a_3 \right]=0, \tag{13} \]
if both ends are free.
Having determined the quantity \(\gamma\) from (11), (12), and (13) and substituting its value into the equality
\[ a_1a_2a_3-a_1a_2^2-a_2a_3^2-a_3a_1^2 =2a_1a_2a_3\operatorname{ch}3\gamma, \tag{14} \]
we obtain the corresponding frequency values.
Received
24 XII 1964
CITED LITERATURE
- L. Brillouin, M. Parodi, Wave Propagation in Periodic Structures, IL, 1959.
- A. F. Pozubenko, Proceedings of the State Optical Institute, issue 159, 127 (1963).
- M. Parodi, Mém. Sci. Phys. Paris, F. 47 (1944).