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Torsionally-compliant driveshaft

The Flexible Shaft block models the torsional dynamics of a
flexible shaft. The shaft is represented by an approximate lumped
parameter circuit consisting of *N* segments. *N* is
a finite integer greater than zero.

You can parameterize the block in either the overall stiffness and inertia or the dimensions and material properties of the shaft. You can also apply optional viscous friction at the base and follower ports to represent bearing losses at the shaft ends.

B and F are rotational conserving ports associated with the shaft input and output sections, respectively.

**Parameterization**Select how to characterize the flexible shaft. The default is

`By stiffness and inertia`.`By stiffness and inertia`— Specify shaft characteristics by its inertia and elastic stiffness.`By material properties`— Specify shaft characteristics by its size and continuum properties. If you select this option, the panel changes from its default.automate

**Shaft Geometry**Select the geometry of the shaft. The default is

`Solid`.**Shaft length**Length

*L*of the shaft. Must be greater than 0. The default is`1`.From the drop-down list, choose units. The default is meters (

`m`).**Material density**Mass density

*ρ*of the shaft material. Must be greater than 0. The default is`7.8e+3`.From the drop-down list, choose units. The default is kilograms/meter

^{3}(`kg/m^3`).**Shear modulus**Shear modulus

*G*of the shaft material. Must be greater than 0. The default is`7.93e+10`.From the drop-down list, choose units. The default is pascals (

`Pa`).

**Damping ratio from internal losses**Damping ratio

*c*for the first flexible torsional mode. The default value is`0.01`.**Number of segments**Number

*N*of rigid segments into which the shaft is divided. The default is`1`.

**Viscous friction coefficients at base and follower**Viscous friction coefficients applied at the base and follower, respectively. The default is

`[0 0]`.From the drop-down list, choose units. The default is newton-meters/(radians/second) (

`N*m/(rad/s)`).

**Initial shaft angular deflection**Initial torsional angular deflection of the shaft. The default is

`0`.From the drop-down list, choose units. The default is radians (

`rad`).A positive initial deflection results in a positive torque action from the base (B) to the follower (F) port.

**Initial shaft angular velocity**Initial torsional angular velocity of the shaft. The default is

`0`.From the drop-down list, choose units. The default is revolutions/minute (

`rpm`).At the start of simulation, the entire shaft rotates collectively at this angular velocity, with no relative motion between the segments.

The Flexible Shaft block approximates the distributed, continuous
properties of a shaft by a lumped parameter model. The model contains
a finite number, *N*, of lumped inertia-damped spring
elements in series, plus a final inertia. The result is a series of *N*+1 inertias
connected by *N* rotational springs and *N* rotational
dampers. The block can also include viscous friction at the shaft
ends (base and follower ports) to represent bearing losses at these
points. Do not confuse this viscous friction at the shaft ends with
the internal material damping which corresponds to losses arising
in the shaft material itself.

The Flexible Shaft block model is parameterized in either the
shaft stiffness *k* and inertia *J* or
its dimensions and material properties.

The shaft stiffness and inertia are computed from the shaft dimensions and material properties by the following relationships:

*J*_{P} = (*π*/32)(*D*^{4} – *d*^{4})
,

*m* = (*π*/4)(*D*^{2} – *d*^{2})*ρ**L* ,

*J* = (*m*/8)(*D*^{2}+ *d*^{2})
= *ρ**L*·*J*_{P} ,

*k* = *J*_{P}·*G*/*L* ,

where:

J_{P} | Polar moment of inertia | ||

D | Shaft outside diameter | ||

d | Shaft inside diameter | ||

For solid shafts: | d = 0 | ||

For annular (hollow) shafts: | d> 0 | ||

L | Shaft length | ||

m | Shaft mass | ||

J | Moment of inertia | ||

ρ | Shaft material density | ||

G | Shear modulus of elasticity | ||

k | Shaft rotational stiffness |

For either shaft parameterization, the internal material damping
is defined by the damping ratio, *c*, for a single-segment
model. In this case, the damping torque is 2*c*/*ω*_{N}. *ω*_{N} is
the undamped natural frequency = √(2*k*/*J*).
For an *N*-segment model, the damping applied across
each of the *N* springs is 2*c**N*/*ω*_{N}.

The following figure shows an equivalent physical network constructed
from Simscape™ blocks only. There are *N* segments,
each consisting for a spring, damper, and inertia. A segment represents
a short section of the driveshaft, the spring representing torsional
compliance and the damper representing material damping. The total
shaft inertia is split into `N+1` parts, and partitioned
as shown in the figure.

The distributed parameter model of a continuous torsional shaft
is approximated by a finite number, *N*, of lumped
parameters.

The flexible shaft is assumed to have a constant cross-section along its length.

A larger number *N* of segments increases the
accuracy of the model, but reduces its speed. The single-segmented
model (*N*=1)
exhibits an eigenfrequency which is close to the first eigenfrequency
of the continuous, distributed parameter model.

For greater accuracy, you can select 2, 4, 8, or more segments. For example, the four lowest eigenfrequencies are represented with an accuracy of 0.1, 1.9, 1.6, and 5.3 percent, respectively, by a 16-segmented model.

[1] Bathe, K.-J., *Finite Element Procedures*,
Prentice Hall, Inc, 1996.

[2] Chudnovsky, V., D. Kennedy, A. Mukherjee, and J. Wendlandt, "Modeling
Flexible Bodies in SimMechanics and Simulink," *MATLAB
Digest* 14(3), May 2006.

Inertia | Rotational Damper | Rotational Spring | Torsional Spring-Damper

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