Flexible Shaft

Driveline shaft with torsional compliance

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Couplings & Drives

Description

This block represents a driveline shaft with torsional compliance. The shaft consists of a flexible material that twists in response to an applied torque. The twisting action delays power transmission between the shaft ends, altering the dynamic response of the driveline system. The shaft twists but does not bend.

To represent the flexible shaft, the block uses a lumped-parameter model. This model divides the shaft into different elements that interconnect through parallel spring-damper systems. The elements provide the shaft inertia while the spring-damper systems provide the shaft compliance.

You specify the shaft inertia, compliance, and number of shaft elements directly in the block dialog box. Choosing from two parameterizations, you can specify the shaft compliance using stiffness and damping values or, alternatively, the shaft shear modulus. An additional parameter enables you to model the power losses due to viscous friction at the shaft ends.

Dialog Box and Parameters

Shaft

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.

     Stiffness and Inertia

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

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    Shaft Geometry

    Select the geometry of the shaft. The default is Solid.

    • Solid — Specify a solid shaft geometry.

       Solid

      Annular— Specify an annular , or hollow, shaft geometry. If you select this option, the panel changes from its default.

       Annular

    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/meter3 (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 Nof rigid segments into which the shaft is divided. The default is 1.

Viscous Bearing Losses

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 Conditions

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.

Flexible Shaft Model

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.

Shaft Characterized by Dimensions and Material Properties

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

JP = (π/32)(D4d4) ,

m = (π/4)(D2d2)ρL ,

J = (m/8)(D2+ d2) = ρL·JP ,

k = JP·G/L ,

where:

JPPolar moment of inertia
DShaft outside diameter
dShaft inside diameter
 For solid shafts:d = 0
 For annular (hollow) shafts:d> 0
LShaft length
mShaft mass
JMoment of inertia
ρShaft material density
GShear modulus of elasticity
kShaft rotational stiffness

Internal Material Damping

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 2c/ωN. ωN is the undamped natural frequency = √(2k/J). For an N-segment model, the damping applied across each of the N springs is 2cN/ωN.

Equivalent Physical Network

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.

Limitations

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.

Tradeoff Between Accuracy and Speed

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.

Ports and Conventions

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

References

[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.

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