A double disc rotor system consisting of a rotating shaft tw

A double disc rotor system, consisting of a rotating shaft, two discs and bearings represented as clamped supports at each end of the shaft, is modeled as shown in FIGURE Q1. Assume G_1 = G_2 = G_3 = 80 times 10^9 N/m^2, J_1 = J_2 = J_3 = 1.5 times 10^-8 m^4, L_1 = L_3 = 1.5L and L = 0.4 m. The mass moments of inertia of disc 1 and disc 2 are 1.5 and 0.81 kg.m^2 respectively. Model the system as two degree-of-freedom system and determine the torsinal stiffness k_g k_12 k_3 Derive the differential equations of the system theta _1 and theta _2 as genararalised coordinates. Determine the natural frequencies and mode shapes.

Solution

Cannot read the individual parameters, so giving a general analysis, you have to put in the numbers.

1)Torsional stiffness: Kt =GJ/l

2)let phi i be the angle at disci

I₁ phi₁\" +Kt( ph1 1-phi2) =0, Kt is the torsion stiffness of the shaft

I ₂ phi ₂ \" +Kt( phi2 -phi1) =0

for free vibrations assume a soln phi\" = - w² phi

substitute in the eqns,

coupled differential eqns are reduced to eigenvalue problem:

K _t - I₁ w² -K_t

=0

-K_t Kt-I₂w²

3) solve for w²

from [D- w²I]U`= w² U , where U is the mode shape solve knowing w²

Best!