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Implement mechanical shaft
The model outputs the transmitted torque through the shaft regarding the speed difference between the driving side and the loaded side of the shaft.
The transmitted torque T_{l} is given by the following equation:
$${T}_{l}=K{\displaystyle \int \left({\omega}_{m}-{\omega}_{l}\right)\text{\hspace{0.17em}}dt+B}\left({\omega}_{m}-{\omega}_{l}\right),$$
where K (N.m) is the shaft stiffness, B (N.m.s) is the internal damping, and ω_{m} and ω_{l} are the speeds (rad/s) of the driving side and the loaded side, respectively. The following figure shows the internal schematic of the model. In this model the speeds are converted from rpm to rad/s.
Mechanical Shaft Model Schematic
The stiffness is defined as
K = T / θ,
where T is the torsional torque applied to the shaft and θ the resulting angular deflection (rad).
The stiffness can also be determined by
K = GJ / l,
where G is the shear modulus, J the polar moment of inertia, and l the length of the shaft.
For steel, the shear modulus G is usually equal to about 80 GPa, and the polar moment of inertia J of a shaft with a circular section of diameter D is given by
J = πD^{4} / 32.
Mechanical shafts have very small angular deflections to avoid bearing problems. As an example, the following table gives the corresponding stiffness for angular deflections of 0.1 degrees at maximum torque with respect to the power and speed of electrical motors connected to the driving end of the shaft. The maximum torque is here assumed to be 1.5 times bigger than the nominal torque.
Shaft Stiffness K
P (HP) | N (rpm) | T (N.m) | Tmax (N.m) (=1.5 T) | K (N.m) |
---|---|---|---|---|
5 | 1750 | 20 | 30 | 17190 |
200 | 1750 | 815 | 1223 | 700730 |
200 | 1200 | 1190 | 1785 | 1022730 |
The damping factor B represents internal friction. This factor increases with the shaft stiffness. As an example, the following table gives some values of B for the stiffness of the preceding table.
Shaft Internal Damping B
K (N.m) | B (N.m.s) |
---|---|
17190 | 600 |
700730 | 24460 |
1022730 | 35700 |
The stiffness must be high enough to avoid large angular deflections that could cause misalignment inside the bearings and damage.
For proper simulation results, the internal damping must be high enough to avoid undesired transient speed and torque oscillations.
The model is discrete. Good simulation results have been obtained with a 10 µs time step.
The block has two inputs: Nm and Nl.
The first input, Nm, is the speed (rpm) of the driving end of the shaft.
The second input, Nl, is the speed (rpm) of the load connected to the second end of the shaft.
The block has one output: Tl.
The Tl output is the torque transmitted from the driving end of the shaft to the load.
The library contains three preset models. The nominal torques of these mechanical shaft models are shown in the following table:
Preset Mechanical Shaft Models
1st | 2nd | 3rd | |
Nominal torque (N.m) | 20 | 815 | 1190 |
The preset models have been designed in order to present 0.1 degrees of angular deflection at maximum torque (supposed to be 1.5 times the nominal torque).
The shaft_example example illustrates the mechanical shaft model.
The shaft is driven by a variable speed source and is connected to a load. The load has an inertia of 0.35 kg.m2 and a viscous friction term of 0.006 N.m.s.
The shaft has a stiffness of 17190 N.m and an internal damping factor of 600 N.m.s. This shaft is designed to have 0.1 degree of angular deflection for a 30 N.m load torque.
At t = 0 s, the driving speed starts climbing to 1750 rpm with a 500 rpm/s acceleration ramp. The angular deflection jumps to about 0.06 degree and the shaft transmits about 18.5 N.m to the load in order to accelerate it. At t = 0.2 s, the driving and load speeds tend to equalize. During the acceleration phase, the angular deflection increases slowly in order to transmit a higher torque to compensate the viscous friction increase.
At t = 3.5 s, the driving speed settles at 1750 rpm. This reduces the angular deflection and also the transmitted torque, which settles around 1.1 N.m to compensate the viscous friction of the load.
At t = 5 s, the driving speed lowers towards 0 rpm with a −500 rpm/s deceleration ramp. The angular deflection becomes negative and thus the transmitted torque in order to decelerate the load. During the deceleration phase, the angular deflection increases in order to transmit a higher deceleration torque to compensate the reduction of viscous friction.
At t = 8.5 s, the driving speed stabilizes at 0 rpm. This causes the angular deflection to decrease to 0 degree, the transmitted torque becomes null, and the load stops.
The following figure shows the speeds of the driving and loaded sides, the speed difference between both sides, the angular deflection, and the transmitted torque.