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ωn = { (1/2) [ B ± ( B2 - (( 4 kt1kt2 ) / ( I1 I2 I3 )) ( I1 + I2 + I3) ) (1/2) ] }(1/2)

or

ωn+ = { (1/2) [ B + ( B2 - (( 4 kt1kt2 ) / ( I1 I2 I3 )) ( I1 + I2 + I3) ) (1/2) ] }(1/2)

ωn- = { (1/2) [ B - ( B2 - (( 4 kt1kt2 ) / ( I1 I2 I3 )) ( I1 + I2 + I3) ) (1/2) ] }(1/2)

and

B = kt1/ I1 + kt2/ I3 + ( kt1 + kt2 ) / I2

Where:

kt1, t2 = Torsional Stiffness of Shaft ( lb-in/rad )
I1, 2 = Rotary Mass Moment of Inertia of Mass ( lb-in-sec2 )
ωn+, n- = Angular Natural Frequency ( rad/sec )

Reference Harris, Shock and Vibration Handbook

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