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ω_n+ = [ ( (1/2) ( ( k_t1/ I₁ ) + ( k_t2 / I₂ ) ( 1 + ( I₂/I₁ ) )

+ ( ( k_t1 / I₁ + ( k_t2 / I₂ ) ( 1 + I₁ / I₁) )² - ( (4k_t1 k_t2 ) / (I₁ I₂ ) ^(1/2) ] ^(1/2)

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ω_n- = [ ( (1/2) ( ( k_t1/ I₁ ) + ( k_t2 / I₂ ) ( 1 + ( I₂/I₁ ) )

- ( ( k_t1 / I₁ + ( k_t2 / I₂ ) ( 1 + I₁ / I₁) )² - ( (4k_t1 k_t2 ) / (I₁ I₂ ) ^(1/2) ] ^(1/2)

Where:

k_{t1, t2} = Torsional Stiffness of Shaft ( lb-in/rad )
I_{1, 2} = Rotary Mass Moment of Inertia of Mass ( lb-in-sec² )
ω_{n+, n-} = Angular Natural Frequency ( rad/sec )

Reference Harris, Shock and Vibration Handbook