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 Incoming links: Frequency ratio

For two masses, connect each to a fixed anchor and to the other mass with equal stiffness

k

m_{1} = 1, k = 1

to choose units. The squared angular frequencies have characteristic equation

m_{2} z^{2} - 2 (m_{2} + 1) z + 3 = 0

. Requiring a frequency ratio of two means that these roots are

z

and

4 z

. Their sum and product give

16 m_{2}^{2} - 43 m_{2} + 16 = 0

, yielding the two reciprocal ratios in the header. Restoring units multiplies all squared frequencies by

k / m_{1}

, leaving the permitted mass ratios unchanged. The term octave refers here only to this two-to-one ratio.

Let

z = ω_{1}^{2}

, so

ω_{2}^{2} = 4 z

. The sum and product of roots of the characteristic polynomial in (i) give

5 z = \frac{2 ( m _{2} + 1 )}{m _{2}}, 4 z^{2} = \frac{3}{m _{2}} .

Eliminating

z

yields

16 (m_{2} + 1)^{2} = 75 m_{2}, 16 m_{2}^{2} - 43 m_{2} + 16 = 0.

m_{2} = \frac{43 \pm 5 33}{32} ≃ 0.446162086 or 2.241337914.

The two values are reciprocals. Both meet the requested frequency ratio; substituting either into the expressions in (i) gives

ω_{2} / ω_{1} = 2