Geared systems
In some machine the shaft may not be continuous from one end of the machine to the other, but may have a gearbox installed at one or more locations. So shafts will be having different angular velocities as shown in Figure 1(a).
For the purpose of analysis the geared system must be reduced to a system with a continuous shaft so that they may be treated as described in the preceding section.
In the real geared system as shown in Figure 1(a) the shaft torsional stiffness is k2 and the rotor moment of inertia is 
.
Let the equivalent general system as shown in Figure 1(b) has the shaft torsional stiffness ke and disc moment of inertia . The strain and kinetic energies must be the same in both the real and equivalent systems for the theoretical model to be valid.
Strain energy : By imagining the rotor to be held rigidly whilst shaft 1 is rotated through some angle θ1 at the gearbox.
The shaft 2 is rotated through an angle θ1/n at the gear box, where n is the gear ratio
The strain energy stored in shaft 2 is given as
While applying the same input at the gear location to the equivalent system results in the stain energy stored in the equivalent shaft and can be expressed as
Equating equations (1) and (2), it gives
If consideration is now given to kinetic energies of both the real and equivalent systems, which must also be equal
Where 
and 
are are angular frequencies of disc polar moment of inertias of real & equivalent systems (i.e. 
and 
), respectively.
Equation (4) can be written as
Where θ2 and θe are the angle of twist of the shaft 2 in the actual and equivalent system respectively.
It can be seen from Figure 1(b) that θe = θ1 and ω1and ω2 are the angular frequencies of shafts 1 and 2, respectively.
Noting that
equation (5) can be written as
Where T is the torque input to the gearbox pinion (shaft 1). On substituting equation (3) in equation (6),
we get
which implies to
In equations (3) and (8) ke and Ie are the equivalent shaft stiffness and rotor moment of inertia of the geared system referred to the ‘reference shaft' speed (i.e. shaft 1). The general rule, for forming the equivalent system for the purpose of analysis, is to divide all shaft stiffness and rotor inertias of the geared system by n2(where n is the gear ratio, i.e.
When the analysis is completed, it should be remembered that the elastic line of the real system is modified (as compared to with that of the equivalent system) by dividing the displacement amplitudes for the equivalent shaft by the gear ratio n, as shown in Figure 2.
Branched Rotor System:
For some type of applications such as marine vessel power transmission shaft or machine tool drives, there may be many rotor inertias in the system and the gear box may be a branch point where more than two shafts are attached as shown in Figure 3.
For Figure 3 having branched system, state vectors for different branch can be written as
For branch A , taking θ0A = 1 as reference value for the angular displacement and we have T0A= 0 for free vibrations since the left hand end of the branch A is free end and hence equation (9) can be written as
which can be expanded as
TnA = a21 and θnA = a11 (13, 14)
At the branch point, between shaft A and B, we have
Where nAB is the gear ratio between shaft A and B . For branch B, TnB = 0 since the end of branch is free.
For branch B from equation (10), and noting the condition of equation (15), we have
Equation (16) can be expanded making use of eq (13)
At branch C, we have the following condition (noting equation (14))
On substituting equation (13) in equation (19), we get
Substituting equations (19) and (20) into equation (11), we get
where, Tnc = 0, is the boundary condition describing the free end of branch C.
From equation (21) the second equation will give the frequency equation as
Where a’s, b’s and c’s are function of the natural frequency ωn. Roots of equation (22) are system natural frequencies. As before, these frequencies may then be substituted back into transfer matrices for each station considered, where upon the state vector at each station may be evaluated. The plot of the angular displacement against the shaft position then indicates the system mode shapes.
Using this method, there will not be any change in elastic line due to gear ratio since these have now already been allowed for in the analysis. Moreover, for the present case we have not gone for equivalent system at all. For the case when the system can be converted to a single shaft the equivalent system approach has advantage.
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