Flexible Shaft
Shaft with torsional and bending compliance
Description
The Flexible Shaft block represents a driveline shaft with
torsional and bending compliance. The shaft consists of a flexible material that twists
in response to an applied torque and bends in response to static mass unbalance. The
twisting action delays power transmission between the shaft ends, altering the dynamic
response of the driveline system.
To represent a torsionflexible shaft, the block uses a lumped mass method. 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.
The block provides four parameterization methods that allow you to model compliance in
either a homogeneous or an axially inhomogeneous shaft. An axially inhomogeneous shaft
is one for which any of these attributes vary along the length of the shaft:
Torsional stiffness
Torsional inertia
Bending rigidity
Density
Shear modulus
Young's modulus
Outer diameter
Inner diameter
An additional parameter enables you to model the power losses in the bearings due to viscous
friction at the shaft ends. For more information, see Torsion Model.
Note
The viscous friction at the shaft ends is different from the internal material
damping, which corresponds to losses arising in the shaft material
itself.
To represent the bendingflexible shaft, the block uses either a lumped mass method or an
eigenmodes method. While the lumped mass method is simpler to configure, the eigenmodes
method tends to simulate faster.
Tip
If simulation speed, is a high priority, first simulate using the lumped mass
method, adjusting parameters as needed until the results match your mathematical
models or experimental data. Next, simulate using the eigenmodes method. Again,
adjust the parameters as needed until the results mathematical models or
experimental data. For an example that uses both methods, see Shaft with Torsional and Transverse Flexibility.
For the lumped mass method, the number of bending shaft elements is the same as the number
of torsion shaft elements. The model divides the shaft into a series of such elements.
The elements provide the shaft inertia, while the stiffness matrices provide the shaft
compliance. The eigenmodes method computes effective massspringdamper systems that
represent the bending modes of the shaft. You can specify the number of modes to include
and the precision of the mode shapes. Both the lumped mass and eigenmodes methods allow
you to model:
Excitational static unbalances
Concentrically attached rigid masses
Up to four support locations along the shaft
Linear damping proportional to shaft inertia
Linear damping proportional to shaft stiffness
Note
The eigenmodes method assumes that the support damping is light compared to
support stiffness.
Static unbalances, which excite bending, occur when the center of mass of the shaft or an
attached rigid mass is not aligned with the shaft principal axis. You can vary the
locations, magnitudes, and angle offsets of static unbalances on the shaft.
You can represent concentrically attached rigid masses as disks or idealized point masses. A
concentric disk adds diametric and polar moments of inertia to the shaft and mass to the
translation degree of freedom of the shaft nodes. The model assumes that the disk is
thin, so the shaft can still bend on either side of the axial location with the disk.
The polar moment of inertia couples the two planes of bending. A concentric point mass
is an idealized version of a concentric disk. A concentric point mass adds mass to the
translation degrees of freedom of the shaft nodes but does not have rotational moments
of inertia. You can vary the locations and inertias of concentric disks or point masses
that are attached to the shaft.
You can model the supports as ideal or by using stiffness and damping matrices. For
each support, you can vary the:
Location — Any point along the shaft length.
Type — Ideal clamp, ideal pin, free, constant bearing stiffness and
damping, or speeddependent stiffness and damping.
Number — Two, three, or four.
For both bending methods, you can specify the shaft bending compliance by using either
bending rigidity and linear mass density or Young’s modulus and shaft diameter.
You can parameterize the torsional model by using either stiffness k and
inertia J or the dimensions and material properties of the
shaft.
Torsion Model
For the torsion model, the Flexible Shaft block approximates
the distributed, continuous properties of a shaft by using a lumped mass method. The
model contains a finite number, N, of lumped inertiadamped
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 models the shaft as an equivalent physical network of N
flexible elements. Each flexible element,
FE_{i}, represents a short section of the
driveshaft and contains:
One spring, k_{FE_i}, for
torsional compliance. The network has a total of N
springs.
One damper, b_{FE_i}, for
material damping. The network has a total of N
dampers.
Two inertias, I_{FE_iC} and
I_{FE_iR}, for rotational
resistance. The inertias of neighboring flexible elements are
consolidated together so that the network has a total of $$N+1$$ inertias.
For an axially homogeneous shaft, the flexible element lengths, compliance, damping, and
distributed inertias in the physical network are equal, such that:
For an axially inhomogeneous shaft, the amount of compliance, damping, and
Rnode and Cnode inertia can differ
for individual flexible elements in the physical network model.
Node Placement Algorithm
The balance between model fidelity and simulation speed depends on
N, the number of flexible elements that the block uses to
represent the shaft. For information on balancing simulation speed and model
fidelity, see Improve Simulation Speed or Accuracy.
The block allows you to specify a minimum number of flexible elements,
N_{min}, as the value for the
Minimum number of flexible elements parameter. However,
the number of flexible elements that the block actually uses depends on the
complexity of the shaft that it is modeling. If the block requires more flexible
elements than you specify to solve a model that contains axial inhomogeneity,
intermediate supports, concentric disks or masses, or static unbalances, then $$N\ge {N}_{\mathrm{min}}$$.
For example, suppose that, for the complex shaft in the diagram, you specify axial
locations for the supports, static unbalance, larger diameter section, and
concentric disk. You set the parameter for
N_{min} to 7
.
If model bending is on, the torsion model flexible element locations account
for the locations of static unbalances and concentric rigid masses, so that the
torsion flexible elements align with the bending flexible elements. During
simulation, the torsion model is independent of any static unbalances or
concentric rigid masses.
The algorithm for the block determines the number of flexible elements and the length of
the individual elements that are required to solve the simulation:
The block places one node at the base and follower ends of the
shaft. These nodes are considered fixed in
axial location because they represent physical entities along the
shaft axis. In the diagram, fixed nodes are shown in red. The block
evenly distributes the other five
(N_{min}2) internal
nodes along the length of the shaft. It then places a flexible
element between each consecutive pair of nodes.
For an endsupported, axially homogenous shaft, with no static
unbalances or attached concentric disks, depending on the other
parameter options and values that you specify, the block might be
able to solve the simulation using only
N_{min} flexible
elements of equivalent lengths:
In most cases, however, the block can only solve the simulation if
it adds more flexible elements.
To add more flexible elements, the block places fixed internal nodes at these locations:
Each shaft support location. The block allows you to
specify the number and location of shaft supports. For
the shaft in the diagram, there are supports at
z_{1} and
z_{6}.
Each static unbalance. For the shaft in the diagram,
there is a static unbalance at
z_{2}.
Each rigid mass. Rigid masses are concentrically
attached disks or point masses. For the shaft in the
diagram, there is a rigid mass, represented as a disk,
at z_{5}.
Each parameterization segment boundary.
Parameterization boundaries are
locations along an axially inhomogeneous shaft where two
neighboring sections of the shaft vary in stiffness,
inertia, or geometry. The block allows you to define the
parameterization segment boundary locations. For the
shaft in the diagram, there are segment boundaries at
z_{3} and
z_{4}.
Note that the block did not add a node at
z_{4}
because a node was already added in the previous step of
the algorithm. However, the node is now fixed because it
represents a physical entity along the shaft
length.
The block adjusts the nonfixed node locations between the fixed
nodes so that they are evenly distributed.
Finally, the block places flexible elements between each node. The length of each
flexible element corresponds to the centertocenter distances
between the neighboring nodes. The block distributes the inertia
among the flexible elements based on the length of the individual
element and the corresponding shaft geometry. Ultimately, this
complex shaft is represented by 12 flexible elements, with $${l}_{1}={z}_{1}$$, $${l}_{2}={l}_{3}=\frac{\left({z}_{2}{z}_{1}\right)}{2}$$, $${l}_{4}={l}_{5}=\frac{\left({z}_{3}{z}_{2}\right)}{2}$$, $${l}_{6}={l}_{7}=\frac{\left({z}_{4}{z}_{3}\right)}{2}$$, $${l}_{8}={l}_{9}=\frac{\left({z}_{5}{z}_{4}\right)}{2}$$, $${l}_{10}={l}_{11}=\frac{\left({z}_{6}{z}_{5}\right)}{2}$$, and $${l}_{12}={z}_{7}{z}_{6}$$.
If N_{min} is large enough
to yield a number of unfixed nodes that is greater than the number
of fixed nodes, the block distributes more than one unfixed node
between each set of neighboring fixed nodes.
Dimensions and Material Properties
You can parameterize the torsion model by using either stiffness, k,
and the polar moment of inertia, J, or the dimensions and
material properties of the shaft.
The stiffness and inertia for each element are computed from the shaft dimensions and
material properties as:
where:
J_{P} is the polar moment
of inertia of the shaft at the flexible element location.
D is the outer diameter of the shaft at the
flexible element location.
d is the inner diameter of the shaft at the
flexible element location. For a solid shaft, $$d=0$$. For an annular shaft, $$d>0$$.
$$l$$ is the flexible element length.
m is the mass of the shaft at the flexible
element location.
J is the moment of inertia of the shaft at the
flexible element location.
ρ is the density of the shaft material.
G is the shear modulus of elasticity of the
shaft material.
k is the rotational stiffness of the flexible
element.
Internal Material Damping
For either torsional parameterization, the internal material damping is defined by the
damping ratio, c, for a singleflexible element model with
the equivalent torsional stiffness and inertia. The damping torque is then $$2\frac{ck}{{\omega}_{N}}$$, while the undamped natural frequency is $${\omega}_{N}=\sqrt{\frac{2k}{J}}$$. The damping applied across an individual flexible element of
a lumped mass model is equal to the product of the damping coefficient and the
relative rotational velocity of that flexible element.
Bending Models
The Shaft Geometry, Support Loading, and Motion figure shows how to measure:
The static unbalance offset angle, which is the angle of a static
unbalance about the shaft axis relative to the x
axis
The distances of a support, a rigid mass, and a static unbalance,
relative to the base end of the shaft, B
The parameterization of the segment lengths
In the figure, the shaft has three fixed supports:
B_{1} — Base end
support
I_{1} — Intermediate
support
F_{1} — Follower end
support
The shaft has translational velocity V, rotational
velocity W, and exerts forces F, and moments
M, on the supports. The curved arrows and sign conventions
follow the righthand rule. The signs of the physical signals that the block outputs
correspond to the arrows that represent the forces, moments, and velocities of the
shaft acting on the supports.
The vector signals are:
Force, $$Fr=\left[{F}_{xB1},{F}_{yB1},{F}_{xI1},{F}_{yI1},{F}_{xF1},{F}_{yF1}\right]$$
Moment, $$M=\left[{M}_{xB1},{M}_{yB1},{M}_{xI1},{M}_{yI1},{M}_{xF1},{M}_{yF1}\right]$$
Translational velocity, $$V=\left[{V}_{xB1},{V}_{yB1},{V}_{xI1},{V}_{yI1},{V}_{xF1},{V}_{yF1}\right]$$
Rotational velocity, $$M=\left[{M}_{xB1},{M}_{yB1},{M}_{xI1},{M}_{yI1},{M}_{xF1},{M}_{yF1}\right]$$
If the shaft has two supports, each vector signal has a length of four. Force, for example,
is then $$Fr=\left[{F}_{xB1},{F}_{yB1},{F}_{xF1},{F}_{yF1}\right]$$.
If the shaft has four supports, each vector signal has a length of eight. Force, for
example, is then $$Fr=\left[{F}_{xB1},{F}_{yB1},{F}_{xI1},{F}_{yI1},{F}_{xI2},{F}_{yI2},{F}_{xF1},{F}_{yF1}\right]$$.
Bending Model Lumped Mass Method
Like the torsion model, the lumped mass method for the bending model discretizes the
distributed, continuous properties of the shaft into a finite number,
N, of flexible elements. The N
flexible elements correspond to $$N+1$$ lumped inertias connected in series by damping and spring
elements. However, for the bending model, each mass has four degrees of freedom:
translation and rotation in both the x and
y directions perpendicular to the shaft axis.
The lumped mass equation of motion ^{[1]} is
where:
M is the $4\left(N+1\right)\times 4\left(N+1\right)$ matrix that represents the mass of the
shaft.
B is the $4\left(N+1\right)\times 4\left(N+1\right)$ matrix for the internal damping and support
damping.
G_{Disk} is the $4\left(N+1\right)\times 4\left(N+1\right)$ matrix that accounts for disk gyroscopics
Ω is the shaft torsional velocity during simulation.
K is the $4\left(N+1\right)\times 4\left(N+1\right)$ matrix for the spring stiffness.
$\overrightarrow{x}$ is the $4\left(N+1\right)\times 1$ vector that represents the degrees of freedom for
all nodes.
$\overrightarrow{f}$ is the $4\left(N+1\right)\times 1$ vector that represents external forces due to the
application of static mass unbalance.
The equation for the mass matrix [4] is
where:
$${M}_{i/\left(i+1\right)}$$ is the mass matrix for an individual flexible
element. For each flexible element, half of the mass and moment of
inertia is transferred to the nodes at both ends of the flexible
element. The $${M}_{i/\left(i+1\right)}$$ matrix has nonzero elements in the $\left(4i3\right):\left(4i+4\right)$ rows and the $\left(4i3\right):\left(4i+4\right)$ columns:
where:
$l$ is the flexible element length along
the shaft between internal nodes. To determine the
length of each flexible element, the block uses the
algorithm that is described in Node Placement Algorithm. Each flexible element contains two inertias. Each
inertia has two translational degrees of freedom, two
rotational degrees of freedom, and one stiffness
matrix.
Each flexible element in the equivalent physical model for bending in the
XZplane (the beam translation in
the Xdirection and rotation about
the Yaxis) and in the physical model
for bending in the YZplane (the beam
translation in the Ydirection and rotation about the
Xaxis) then contains two masses, two inertias, and a
stiffness matrix.
To determine the internal node locations, and therefore the number and lengths of
the flexible elements, the block uses the same
nodeplacement algorithm as it uses for the torsion
model. For more information, see Node Placement Algorithm.
m is the flexible element mass.
m depends on the outer,
D, and inner,
d, diameters, the density,
ρ, of the shaft and the length of
the flexible element, such that $m=\left(\frac{\pi}{4}\right)\left({D}^{2}{d}^{2}\right)\rho l$.
I_{d}, the
halfelement mass moment of inertia about an axis
perpendicular to the shaft axis, depends on the mass,
m, length, $l$, and torsion moment of inertia,
J, of the flexible element, such
that ${I}_{d}=\frac{J}{4}+\frac{m}{6}{\left(\frac{l}{2}\right)}^{2}$.
${{\displaystyle \sum}}^{\text{}}{M}_{disk,i}$ is the summed mass matrices of the rigid masses
concentrically attached to the shaft.
The mass properties of each rigid mass that is concentrically
attached to the shaft are added to the closest node, $i$, such that
where
I_{D,disk,i} is the mass
diametric moment of inertia about an axis perpendicular to the shaft
for a rigid disk attached to the
i^{th} node. The
model assumes that the disk is thin, so the shaft can still bend on
either side of the axial location with the disk. A concentric point
mass has ${I}_{D,disk,i}\text{=0}$.
The equation for the damping matrix is
where:
α is the damping constant proportional to
mass.
β is the damping constant proportional to
stiffness.
B_{support} is the
damping coefficient at each support. For a support at the
i^{th} node, the
damping matrix, in terms of global coordinates, is
where:
${G}_{disk,i}$ accounts for the gyroscopic effects of any
concentrically attached disks, and is defined as
where
I_{P,disk,i} is the mass
polar moment of inertia about the shaft axis for the disk attached
to the i^{th} node. The
mass polar moment of inertia for a concentric point mass is ${I}_{P,disk,i}\text{=0}$.
The equation for the bearing stiffness matrix is
where:
${K}_{i/i+1}$ is the stiffness matrix for an individual shaft
flexible element. The stiffness matrix for the ${i}^{th}$ shaft flexible element, between the
i^{th} and the ${\left(i+1\right)}^{th}$ nodes, has nonzero elements in the $\left(4i3\right):\left(4i+4\right)$ rows and the $\left(4i3\right):\left(4i+4\right)$ columns, such that
where:
K_{support} is the
stiffness at each support. For a support at the
i^{th} node, the
stiffness matrix, in terms of global coordinates, is
where:
The support stiffness matrix, K_{support}, is
nonzero only if you select Bearing matrix
or
Speeddependent bearing matrix
for the support.
If you select the Clamped
mounting type, the
kinematic conditions of zero rotation and translation are applied to the degrees
of freedom that correspond to the support node (B1,
I1, I2, or F1). If
you select the Pinned
mounting type, the kinematic
conditions of zero translation are applied to the translational degrees of
freedom that correspond to the support node (B1,
I1, I2, or
F1).
The table includes the boundary conditions applied to the lumped mass nodes with
supports.
Support Type  Boundary Condition for the Lumped Mass Equation 

Clamped  ${x}_{i}=0,{y}_{i}=0,{\theta}_{i}=0,{\phi}_{i}=0$ 
Pinned  ${x}_{i}=0,{y}_{i}=0$ 
Bearing Matrix  K_{support} is
nontrivial. 
Speeddependent bearing
matrix  K_{support} is nontrivial and depends on
shaft rotation speed. At each time step,
K_{Support}
is calculated as:
where: Ω_{Ref} is the bearing speed,
as specified, in the Supports
settings, for the Bearing speed
[s1,...,sS] parameter. For each support,
K_{Support,Ref}
is the bearing speeddependent translational
stiffness, which you specify in the
Supports settings. The lookup table uses linear interpolation and
nearest extrapolation for the shaft rotation
speed.

The matrix that represents the degrees of freedom for all nodes, $\overrightarrow{x}$, is calculated such that the degrees of freedom for the
i^{th} and the ${\left(i+1\right)}^{th}$ nodes are
External forces due to each static mass unbalance are applied to the closest node. The
forcing at the ${i}^{th}$ node is
where:
mε_{j} is the
j^{th}static
unbalance, located at the
i^{th}node.
Ω_{i} is the shaft rotational velocity during
simulation for the i^{th}
node.
φ_{shaft, i} is the torsion
lumped mass rotation angle for the
i^{th}
node.
Bending Model Eigenmodes Method
For the eigenmodes method, the block reduces the bending dynamics from the $$4(N+1)$$ degrees of freedom that the bending model lumped mass method
provides, to M degrees of freedom, where M
is the number of modes.
The block computes the bending mode properties of the shaft during model compilation, then
solves the modal massspringdamper systems during model simulation.
Reducing the degrees of freedom in the model dynamics and separating the calculations into
compiletime and runtime tasks improves simulation performance. The eigenmodes
method assumes the mode shapes are unaffected by damping. Therefore, the method
is best suited to models that include limited disk gyroscopic and support
damping.
During compilation, the block computes the approximate damped eigenmodes using these
steps:
The block computes the matrices using the same lumped mass equation of motion that it
uses for the bending model lumped mass method:
For more information, see Bending Model Lumped Mass Method.
When determining the node axial locations for $$\overrightarrow{x}$$, the block uses one of two variations of the Node Placement Algorithm that it uses for the torsion model
and the bending model lumped mass method. The variation that the block
uses depends on whether, in the Advanced Bending
settings, the Bending mode determination parameter
is set to Simscape determined
or to
User defined
.
If the Bending mode determination parameter is set to
Simscape determined
, instead of using the
Minimum number of flexible elements parameter
for N_{min}, as the lumped mass
methods do, the eigenmodes method calculates
N_{min} as
where:
L is the specified value, in the
Shaft settings, for the
Shaft length parameter.
dz is the specified value, in the
Advanced Bending settings, for the
Shaft length increments for mode shape
computations parameter.
To compute the m undamped eigenmodes and
eigenfrequencies, the block uses the eigs
function. The
equation takes the form
:
[H, λ] = eigs( sparse(K), sparse(M), mMax, 'smallestabs’ ),
where:
H is
the $4\left(N+1\right)\times M$ eigenvector matrix. Each column is an
eigenmode in the $$\overrightarrow{x}$$ coordinates.
λ are the eigenvalues, which are the
square of the eigenfrequencies.
m_{Max} is the
specified value, in the Advanced
Bending settings, for the Limit
number of modes parameter.
The number of eigenmodes computed, m, is less than
m_{Max} if:
There are modes with eigenfrequencies that exceed the
specified value, in the Advanced
Bending settings, for the
Eigenfrequency upper limit
parameter. The block discards these modes.
The eigenvalues fail to converge. For more information,
see eigs
.
If the Bending mode determination parameter is set to
User defined
, the block computes the
eigenvector matrix H
from the specified values, in the Advanced Bending
settings, for these parameters:
Xdirection mode shapes
Ydirection mode shapes
Shaft position
To determines the node axial locations for $$\overrightarrow{x}$$, the block uses the elements specified for the
Shaft position parameter as the primary
nodes.
To compute the modal rotation, θ and φ, for each node, the block uses
the gradient
function. The
equations take the form:
θ = gradient(Y direction mode shapes)
φ = gradient(X direction mode shapes)
The block assembles the Xdirection mode shapes,
Ydirection mode shapes, and modal rotations, θ
and φ, into $$\overrightarrow{x}$$ coordinates for each column of H.
The block computes the modal matrices, M_{Modal},
K_{Modal},
B_{Modal},
G_{Modal}, and
f_{Modal}, as:
Although the block computes undamped eigenmodes, H, in step 1, the modal damping matrix,
B_{Modal}, and modal
gyroscopics matrix, G_{Modal}, may model
light damping. The block normalizes the matrices so that
M_{Modal} is the
identity matrix.
During simulation, the block simulates the eigenmode equation of
motion:
where the modal degrees of freedom, $$\overrightarrow{\eta}$$, relate to the node degrees of freedom by:
SpeedDependent Eigenmodes Method
The support stiffness and support damping vary if, in the Supports
settings, the mounting type parameter for any of the supports is set to
Speeddependent bearing matrix
. The
speeddependent eigenmodes model accounts for these effects by varying the modal
properties, H,
B_{Modal}, K_{Modal}, and
f_{Modal} as the shaft speed
changes. M_{Modal} is normalized to the
identity matrix for all shaft speeds, so it does not depend on shaft
speed.
If the shaft has speeddependent bearing supports, then the block repeats the bending mode
eigenmodes method steps for each element in the shaft speed vector. The shaft
vector elements are the specified values, in the Supports
settings, for the Bearing speed [s1,...,sS] parameter.
During simulation, the modal stiffness, damping, and forcing magnitude are
adjusted based on lookup tables of the properties versus the shaft speed.
That is, the block simulates the eigenmode equation of motion as:
where K_{Modal},
B_{Modal}, and
f_{Modal} have the form:
where:
Ω_{Ref} is the specified value, in the
Supports settings, for the
Bearing speed [s1,...,sS] parameter.
K_{Modal,Ref} is
the table of modal stiffnesses at each
Ω_{Ref}.
B_{Modal,Ref} is the table
of support damping at each Ω_{Ref}.
f_{Modal,Ref} is
the table of modal forcing at each
Ω_{Ref}.
The block correlates the mode shape similarity at different values of
Ω_{Ref} and reorders modes, if necessary, so that each
modal degree of freedom, $$\overrightarrow{\eta}$$, has properties that gradually change with the shaft
speed.
Improve Simulation Speed or Accuracy
The balance between simulation accuracy and performance depends on N,
the number of flexible elements that the block uses to represent the shaft.
Simulation accuracy is a measure of how much the simulation results agree with
mathematical and empirical models. Generally, as N increases, so
does model fidelity and simulation accuracy. However, the computational cost of the
simulation is also correlated to N, and as computational cost
increases, performance decreases. Conversely, as N decreases,
simulation speed increases but simulation accuracy decreases.
To increase simulation accuracy for the lumped mass method for either a torsion or bending
model, increase the minimum number of flexible elements,
N_{min}. The singleflexibleelement
torsion model exhibits a torsional eigenfrequency that is close to the first
eigenfrequency of the continuous, distributed parameter model. For greater accuracy
you can select 2, 4, 8, or more flexible elements. For example, the four lowest
torsional eigenfrequencies are represented with an accuracy of 0.1, 1.9, 1.6, and
5.3 percent, respectively, by a 16flexibleelement model.
To increase simulation accuracy for the eigenmodes method to a bending model:
If simulating with static eigenmode dependency on rotation speed,
verify that the Nominal shaft speed for bending
modes parameter is close to the simulation shaft speed.
This parameter may affect model results if you parameterize a rigid disk
attached to the shaft with a large mass moment of inertia about the
shaft axis or specify any speeddependent bearing matrix
supports.
If simulating with dynamic eigenmode dependency on rotation speed,
verify that, in the Supports settings, the
specified values for the Bearing speed [s1,...,sS]
span the shaft speed range of the simulation or that saturation of the
support stiffness and damping at shaft speeds outside the range is an
acceptable approximation.
In the Advanced Bending settings, decrease the
value of the Shaft length increments for mode shape
computations parameter. Reducing the value can increase
the accuracy of modal frequencies and shapes.
Decrease the support damping and disk polar moment of inertia about
the shaft axis. Simscape™ computations of the mode shapes and frequencies before
simulation do not account for this damping.
Check the sensitivity to the Advanced Bending
settings by using your parameters in the flexible shaft model in the
Shaft with Torsional and Transverse Flexibility example. Adjust the parameters and use the links provided in the
example to examine how the values affect the eigenmode frequencies and
shapes. Adjust the parameter values in your model accordingly.
Increase the values of the Eigenfrequency upper
limit and Limit number of modes
parameters. The highest modal frequency in the simulation must be
significantly larger than the shaft rotation frequency.
Limitations and Assumptions
The distributed parameter model of a continuous torsional shaft is
approximated by a finite number, N, of lumped masses.
Shaft rotation and torsion flexibility excite shaft bending, but bending does
not affect shaft rotation and torsion flexibility.
Rigid point masses or disks attached to the shaft have thin lengths parallel
to the shaft axis.
For the eigenmodes bending model, damping does not affect the
eigenfrequencies.
Shaft bending is not transmitted between Flexible
Shaft blocks.
Relative to the shaft length, the shaft outer diameter is small.
Relative to the shaft length, the bending deflection is small.
Static mass unbalances are the only shaftbending external exciting loads.
Shaft supports are stationary.
Gyroscopic effects of the rigid disks are considered; gyroscopic effects of
the shaft itself are neglected.
Static mass unbalance forcing in the eigenmodes method uses the rotation speed
at the shaft midpoint.
If the shaft models torsion only and uses the parameterization options
By stiffness and inertia or By segment
stiffness and inertia, the block uses only two supports, one each
at the B and F ends.
Ports
Output
expand all
Fr
— Force on bearing supports
physical signal
Physical signal outport associated with the force that the shaft
exerts on the bearing supports.
Dependencies
This port is visible if, in the Shaft settings, the
Model bending parameter is set to
On
. For more information, see Model bending.
M
— Moment on bearing supports
physical signal
Physical signal outport associated with the moment that the shaft
exerts on the bearing supports.
Dependencies
This port is visible if, in the Shaft settings, the
Model bending parameter is set to
On
. For more information, see Model bending.
V
— Shaft translational velocity
physical signal
Physical signal outport associated with the translational velocity of
the shaft at the bearing supports.
Dependencies
This port is visible if, in the Shaft settings, the
Model bending parameter is set to
On
. For more information, see Model bending.
W
— Shaft angular velocity
physical signal
Physical signal outport associated with the angular velocity of the
shaft at the bearing supports.
Dependencies
This port is visible if, in the Shaft settings, the
Model bending parameter is set to
On
. For more information, see Model bending.
Conserving
expand all
B
— Base
rotational mechanical
Rotational conserving port associated with the shaft base.
F
— Follower
rotational mechanical
Rotational conserving port associated with the shaft follower.
Parameters
expand all
Shaft
For certain parameters in the Shaft settings, the option that you
choose affects the visibility of:
Other parameters in the Shaft settings.
Parameters in the Torsion settings.
Supports settings and parameters.
Bending settings
Advanced Bending settings
These output ports:
The table shows how the options that you choose for the Shaft settings
affect the visibility of other parameters in the Shaft
settings. To learn how to read the table, see Parameter Dependencies.
Shaft Parameter Dependencies
Shaft Setting Parameters and
Values 

Model
bending 
Off  On 
Minimum number of flexible
elements  Minimum number of flexible
elements 
Parameterization  Parameterization 
By stiffness and
inertia  By material
and geometry  By segment stiffness
and inertia  By material
and segment geometry  By stiffness
and inertia  By material
and geometry  By segment
stiffness and inertia  By material
and segment geometry 
Shaft
Length  Segment lengths
[B,...,F]  Shaft
Length  Shaft
Length  Segment lengths
[B,...,F]  Segment lengths
[B,...,F] 
Torsional
stiffness  Segment torsional
stiffness [B,...,F]  Torsional
stiffness  Segment
torsional stiffness [B,...,F] 
Torsional inertia  Segment torsional inertia
[B,...,F]  Torsional
inertia  Segment
torsional inertia [B,...,F] 
Bending
rigidity  Segment bending
rigidity [B,...,F] 
Material density  Material density  Linear
density  Material
density  Segment linear
density [B,...,F]  Material
density 
Shear
modulus  Shear
modulus  Shear modulus  Shear modulus 
Young's modulus  Young's modulus 
Shaft geometry  Shaft geometry  Shaft geometry  Shaft geometry 
Solid  Annular  Solid  Annular  Solid  Annular  Solid  Annular 
Shaft outer diameter  Shaft outer diameter  Segment outer diameter [B,...,F]
 Segment outer diameter
[B,...,F]  Shaft outer diameter  Shaft outer diameter  Segment outer diameter
[B,...,F]  Segment outer diameter
[B,...,F] 
Shaft inner diameter  Segment inner diameter [B,...,F]  Shaft inner diameter  Segment inner diameter
[B,...,F] 
Model bending
— Bending model option
Off
(default)  On
Option to model shaft bending.
Dependencies
These parameters, settings, and ports are affected by the Model
bending parameter.
For the Shaft settings, the
visibility of dependent parameters is tabulated in Shaft Parameter Dependencies.
In the Torsion settings:
The Viscous friction coefficients
at base (B) and follower (F) parameter
is visible if, in the Shaft
settings, Model bending is
set to Off
and
Parameterization is set to
By stiffness and
inertia
or By segment
stiffness and inertia
.
The Viscous friction coefficients at each support [B1,...,F1]
parameter is visible if, in the
Shaft settings, one of these
conditions is met:
These settings are exposed only if Model
bending is set to
On
:
Supports
Bending
Advanced bending
These ports are exposed only if Model
bending is set to
On
:
Minimum number of flexible elements
— Minimum number of flexible elements
1
(default)  positive integer
Minimum number of flexible elements, N_{min}, for
the approximation.
It is possible that the flexible elements have different lengths or that the simulated
number of flexible elements, N, is larger than
N_{min}. For more
information, see Node Placement Algorithm.
A larger number of flexible elements, N, increases the fidelity of
the model, but reduces simulation performance. The single flexible
element model (N=1) exhibits a torsion eigenfrequency that is close to
the first eigenfrequency of the continuous, distributed parameter
model.
If model fidelity is more important than performance, select 2, 4, 8, or more flexible
elements. For example, the four lowest torsion eigenfrequencies are
represented with an accuracy of 0.1, 1.9, 1.6, and 5.3 percent,
respectively, by a model with 16 flexible elements. Generally, more
flexible elements are required for accurately modeling bending dynamics
than are required for accurately modeling torsion dynamics.
For more information, see Improve Simulation Speed or Accuracy.
Parameterization
— Parameterization method
By stiffness and
inertia
(default)  By material and geometry
 By segment stiffness and inertia
 By material and segment geometry
Parameterization method. You can model a homogeneous shaft or one that
is axially inhomogeneous for any of these attributes:
Torsional stiffness
Torsional inertia
Bending rigidity
Density
Shear modulus
Young's modulus
Outer diameter
Inner diameter
The parameterization options for a homogeneous shaft model are:
By stiffness and inertia
— Specify the torsional stiffness and inertia, and
the density per unit length of the shaft. For the bending
model, also specify the bending rigidity and length of the
shaft.
By material and geometry
—
Specify the length and axial crosssectional
geometry, in terms of inner and outer diameters, of the
shaft. For the shaft material, specify the density and shear
modulus. For the bending model, also specify Young's modulus
for the material of the shaft.
The parameterization options for an axially inhomogeneous shaft model are:
By segment stiffness and
inertia
— For each segment of the
shaft, specify the torsional stiffness, torsional inertia,
and density per unit length. For the bending model, also
specify the bending rigidity and length for each
segment.
By material and segment
geometry
— For each segment of the shaft,
specify the length and axial crosssectional geometry, in
terms of inner and outer diameters. For the material of the
shaft, specify the density and shear modulus. For the
bending model, also specify Young's modulus for the material
of the shaft.
Dependencies
Each Parameterization option affects the
visibility of:
Dependent parameters in the Shaft
settings. For more information, see Shaft Parameter Dependencies.
Dependent parameters in the Torsion settings:
The Viscous friction coefficients
at base (B) and follower (F) parameter
is visible if, in the Shaft
settings, Model bending is
set to Off
and
Parameterization is set to
By stiffness and
inertia
or By segment
stiffness and inertia
.
The Viscous friction coefficients
at each support [B1,...,F1] parameter
is visible if, in the Shaft
settings, one of these conditions is met:
The Supports settings. When the
Model bending parameter is set
to Off
, the
Supports settings are visible
when the Parameterization parameter
is set to By material and
geometry
or By material
and segment geometry
.
Shaft length
— Shaft length
1
m
(default)  positive scalar
Length of the shaft.
Dependencies
This parameter is visible when one of these conditions are met:
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Segment lengths [B,...,F]
— Length of each shaft segment
[1, .5, .25]
(default)  vector
Length of each shaft segment that the shaft is divided into lengthwise for modeling an
axially inhomogeneous shaft. The number of elements in the vector is
equal to the number of segments that you use to model the inhomogeneous
shaft. The order of the elements in the vector corresponds to the
segment order relative to B, the base end of the
shaft.
Dependencies
This parameter is visible when one of these conditions are met:
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Torsional stiffness
— Material stiffness
2e5
N*m/rad
(default)  positive scalar
Torque per radian twist of the shaft.
Dependencies
This parameter is visible when Parameterization is set to
By stiffness and inertia
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Segment torsional stiffness [B,...,F]
— Material stiffness for each shaft segment
[400000, 200000, 100000]
N*m/rad
(default)  positive vector
Torque per radian twist for each segment of the shaft. The number of elements in the
vector must be the same as the number of elements specified for the
Segment lengths
[B,...,F] parameter. The order of the elements
in the vector corresponds to the segment order relative to
B, the base end of the shaft.
Dependencies
This parameter is visible when Parameterization is set to
By segment stiffness and
inertia
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Torsional Inertia
— Polar moment of inertia
0.02
kg*m^2
(default)  positive scalar
Ability of the shaft to resist torsional acceleration.
Dependencies
This parameter is visible when Parameterization is set to
By stiffness and inertia
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Segment torsional inertia [B,...,F]
— Polar moment of inertia for each shaft segment
[.025, .02, .015]
kg*m^2
(default)  positive vector
Ability of the each shaft segment to resist torsional acceleration. The number of
elements in the vector must be the same as the number of elements
specified for the Segment lengths
[B,...,F] parameter. The order of the elements
in the vector corresponds to the segment order relative to,
B, the base end of the shaft.
Dependencies
This parameter is visible when Parameterization is set to
By segment stiffness and
inertia
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Bending rigidity
— Bending rigidity
5e5
m^4*Pa
(default)  positive scalar
Bending rigidity for the shaft material.
Dependencies
This parameter is visible when Model bending is set to
On
and
Parameterization is set to By
stiffness and inertia
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Segment bending rigidity [B,...,F]
— Bending rigidity of each segment
[600000, 500000, 400000]
m^4*Pa
(default)  positive vector
Bending rigidity for the material of each sequential segment of the shaft. The number of
elements in the vector must be the same as the number of elements
specified for the Segment lengths
[B,...,F] parameter. The order of the elements
in the vector corresponds to the segment order relative to
B, the base end of the shaft.
Dependencies
This parameter is visible when Model bending is set to
On
and
Parameterization is set to By
segment stiffness and inertia
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Linear density
— Density per unit length
19
kg/m
(default)  positive scalar
Density of the shaft material per unit length of the shaft.
Dependencies
This parameter is visible when Parameterization is set to
By stiffness and inertia
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Segment linear density [B,...,F]
— Density per unit length for each segment
[20, 19, 18]
kg/m
(default)  positive vector
Density of the shaft material per unit length of each segment of the shaft. The number
of elements in the vector must be the same as the number of elements
specified for the Segment lengths
[B,...,F] parameter. The order of the elements
in the vector corresponds to the segment order relative to
B, the base end of the shaft.
Dependencies
This parameter is visible when Parameterization is set to
By segment stiffness and
inertia
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Material density
— Material density
7.8e3
kg/m^3
(default)  positive scalar
Density of the shaft material.
Dependencies
This parameter is visible when Parameterization is set to
By material and geometry
or
By material and segment
geometry
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Shear modulus
— Shear modulus
7.93e9
Pa
(default)  positive scalar
Shear modulus for the shaft material.
Dependencies
This parameter is visible when Parameterization is set to
By material and geometry
or
By material and segment
geometry
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Young's modulus
— Young's modulus
200e9
Pa
(default)  positive scalar
Young's modulus for the material.
Dependencies
This parameter is visible when Model bending is set to
On
and
Parameterization is set to By
material and geometry
or By material
and segment geometry
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Shaft geometry
— Crosssectional geometry
Solid
(default)  Annular
Crosssectional geometry along the length of the shaft. If the shaft
or segments of the shaft are hollow, select
Annular
. Otherwise, select
Solid
.
Dependencies
This parameter is visible when Parameterization is set to
By material and geometry
or
By material and segment
geometry
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Shaft outer diameter
— Outer diameter of the shaft
0.075
m
(default)  positive scalar
Outer diameter of the shaft.
Dependencies
This parameter is visible when Parameterization is set to
By material and geometry
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Segment outer diameter [B,...,F]
— Outer diameter for each shaft segment
[.085, .075, .065]
m
(default)  positive vector
Outer diameter of each shaft segment. The number of elements in the vector must be the
same as the number of elements specified for the Segment lengths
[B,...,F] parameter. The order of the elements
in the vector corresponds to the segment order relative to
B, the base end of the shaft.
Dependencies
This parameter is visible when Parameterization is set to
By material and segment
geometry
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Shaft inner diameter
— Inner diameter of the shaft
0.05
m
(default)  positve scalar
Inner diameter of the annular shaft. The value must be smaller than
the value specified for the Shaft outer diameter
parameter.
Dependencies
This parameter is visible when Parameterization is set to
By material and geometry
and
Shaft geometry is set to
Annular
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Segment inner diameter [B,...,F]
— Inner diameter for each shaft segment
[.055, .05, .045]
m
(default)  vector
Inner diameters of the shaft segments. The number of elements in the vector must be the
same as the number of elements specified for the Segment lengths
[B,...,F] parameter. The order of the elements
in the vector corresponds to the segment order relative to
B, the base end of the shaft. Each value must
be smaller than the corresponding value specified for the
Segment outer diameter [B,...,F] parameter. If
a shaft segment is solid, specify 0
for the
corresponding vector element. At least one element in the vector must be
positive.
Dependencies
This parameter is visible when Parameterization is set to
By material and segment geometry
and
Shaft geometry is set to
Annular
.
For more information on how other parameters affect the visibility of this parameter,
see Shaft Parameter Dependencies.
Torsion
Damping ratio from internal losses
— Material damping ratio
0.01
(default)  positive scalar
Viscous friction coefficients at base (B) and follower (F)
— Viscous friction coefficients
[0, 0]
N*m/(rad/s)
(default)  nonnegative vector
Viscous friction coefficients at the base, B, and
follower, F, ends of the shaft. The vector must
contain two elements.
Dependencies
This parameter is visible if, in the Shaft settings,
Model bending is set to
Off
and
Parameterization is set to By
stiffness and inertia
or By segment
stiffness and inertia
.
Viscous friction coefficients at each support [B1,...,F1]
— Viscous friction coefficients
[0, 0]
N*m/(rad/s)
(default)  nonnegative vector
Viscous friction coefficients at each support. The number of elements in the vector must
be the same as the number specified in the Supports
settings for the Number of
supports parameter. The order of the elements must
correspond to the sequential position of each support
B, the base end of the shaft.
Dependencies
This parameter is visible if, in the Shaft settings, one of these
conditions is met:
Initial shaft torsional deflection
— Initial shaft torsional deflection
0
rad
(default)  nonnegative scalar
Angular deflection of the shaft at the start of simulation.
A positive initial deflection results in a positive rotation of B,
the base end of the shaft, relative to F, the
follower end of the shaft.
Initial shaft angular velocity
— Initial angular velocity
0
rpm
(default)  nonnegative scalar
Angular velocity of the shaft at the start of simulation.
Supports
The Supports settings are visible if, in the
Shaft settings, one of these conditions is met:
Model bending is set to
Off
and
Parameterization is set to By
material and geometry
or By material and
segment geometry
.
For this condition, only these parameters in the
Supports settings are visible:
Model bending is set to
On
.
The visibility of the Supports settings depends
on the values of several parameters in the Supports
settings.
In the Supports settings, if the Number of
supports parameter is set to 2
, only mounting
types and the dependent parameters for the B1 and
F1 supports are visible. The B1 support is
the support that is closest to B, the base end of the shaft.
The F1 support is the support that is closest to
F, the follower end of the shaft. If Number of
supports is set to a value greater than 2
, the
mounting type and dependent parameters are exposed for the intermediate supports,
I_{N}, where N
is the number of intermediate supports. For example, for 3
supports, the mounting type and dependent parameters for the B1,
I1, and F1 support are exposed.
These tables show the parameter dependencies for the Lumped Mass
and Eigenmodes
bending vibration analysis methods for
each mounting type. The parameters are visible for the B1 and
F1 supports and for any intermediate supports,
I1 and I2, that you specify. The names of
the dependent parameters are prefixed by the name of the corresponding support. For
example, for the B1 support, the dependent parameter
Rotational damping [xx,yy] is named Base (B1)
rotational damping [xx,yy]. The parameter prefixes are:
Base (B1)
Intermediate (I1)
Intermediate (I2)
Follower (F1)
Mounting Type Parameter Dependencies
Supports Mounting Types and
Dependent Parameters 

Clamped  Pinned  Free  Bearing
matrix  Speeddependent bearing
matrix 

  Rotational damping [xx,yy]  Rotational damping [xx,yy]  Rotational damping
[xx,yy]  Rotational damping
[xx,yy] 
Translational damping
[xx,xy,yx,yy]  Speeddependent translational damping
[xx1,xy1,
yx1,yy1;...xxS,xyS,yxS,yyS] 
Rotational stiffness
[xx,yy]  Rotational stiffness
[xx,yy] 
Translational stiffness [xx,xy,yx,yy]  Speeddependent translational stiffness
[xx1,xy1,yx1,
yy1;...xxS,xyS,yxS,yyS] 
Bearing speed
[s1,…,sS] 
Number of supports
— Number of supports
2
(default)  3
 4
Number of shaft supports.
Dependencies
This parameter is visible if, in the Shaft settings, one of these
conditions is met:
If this parameter is set to:
2
— Parameters for
the B1 and F1
supports are exposed. The B1 support
is the support that is closest to B
the base end of the shaft. The F1
support is the support that is closest to
F, the follower end of the
shaft.
3
— Parameters for
the B1, I1, and
F1 supports are exposed.
4
— Parameters for
the B1, I1,
I2, and F1
supports are exposed.
Support locations relative to base (B)
— Support locations
[0, 1]
m
(default)  nonnegative increasing vector
Support locations relative to B, the base end of the shaft. The
number of elements must be the same as the number specified for the
Number of supports parameter. The order of the
elements corresponds to the sequential position of each support relative
to the base end of the shaft. The largest value must be no larger than
the length of the shaft. For a segmented shaft model, the shaft length
is equal to the sum of the individual segment lengths.
Dependencies
This parameter is visible if, in the Shaft settings, one of these
conditions is met:
Base (B1) mounting type
— Support B1 mounting type
Clamped
(default)  Pinned
 Free
 Bearing matrix
 Speeddependent bearing matrix
Type of mounting at the base end of the shaft.
Dependencies
This parameter is visible when, in the Shaft settings the
Model bending parameter is set to
On
. For more information, see Shaft Parameter Dependencies.
Setting this parameter to Pinned
,
Free
, Bearing
matrix
, or Speeddependent bearing
matrix
exposes related parameters. For more
information, see Mounting Type Parameter Dependencies.
Base (B1) rotational damping [xx,yy]
— Support B1 rotational damping
[1e3, 1e3]
N*m/(rad/s)
(default)  vector
Rotational damping for the B1 support. B1 is the
support that is closest to B, the base end of the
shaft. The elements of the twoelement vector are:
Dependencies
This parameter is visible when all these conditions are met:
Base (B1) translational damping [xx,xy,yx,yy]
— Support B1 translational damping
[2e4, 4.4e4, 4.4e4, 3.5e5]
N/(m/s)
(default)  vector
Translational damping for the B1 support. The
elements of the fourelement vector are:
xx — Damping in the
xaxis direction
xy — Damping in the
xaxis direction coupled with motion
in the yaxis direction
yx — Damping in the
yaxis direction coupled with motion
in the xaxis direction
yy — Damping in the
yaxis direction
Dependencies
This parameter is visible when all these conditions are met:
Base (B1) speeddependent translational damping [xx1,xy1,yx1,yy1;...xxS,xyS,yxS,yyS]
— Support B1 speeddependent translational
damping
[2e4, 4.4e4, 4.4e4, 3.5e5; 1e4, 2.5e4,
2.5e4, 1.4e5; 5e3, 1.0e4, 1.0e4, 2.0e4]
N/(m/s)
(default)  matrix
Speeddependent translational damping for the B1
support. The number of rows in the matrix must equal the number of
elements in the vector specified for the Bearing speed [s1, …
sS] parameter. Each row contains four elements:
xx_{s} —
Damping in the xaxis direction at the
s^{th}
speed
xy_{s} —
Damping in the xaxis direction coupled
with motion in the yaxis direction at
the s^{th}
speed
yx_{s} —
Damping in the yaxis direction coupled
with motion in the xaxis direction at
the s^{th}
speed
yy_{s} —
Damping in the yaxis direction at the
s^{th}
speed
Dependencies
This parameter is visible when all these conditions are met:
Base (B1) rotational stiffness [xx,yy]
— Support B1 rotational stiffness
[1e4, 1e4]
N*m/rad
(default)  nonnegative vector
Rotational stiffness for the B1 support. The
elements of the twoelement vector are:
Dependencies
This parameter is visible when all these conditions are met:
Base (B1) translational stiffness [xx,xy,yx,yy]
— Support B1 translational stiffness
[1e6, 2e5, 3e5, 5e]
N/m
(default)  vector
Translational stiffness for the B1 support. The
elements of the fourelement vector are:
xx — Stiffness in the
xaxis direction
xy — Stiffness in the
xaxis direction coupled with motion
in the yaxis direction
yx — Stiffness in the
yaxis direction coupled with motion
in the xaxis direction
yy — Stiffness in the
yaxis direction
Dependencies
This parameter is visible when all these conditions are met:
Base (B1) speeddependent translational stiffness [xx1,xy1,yx1,yy1;...xxS,xyS,yxS,yyS]
— Support B1 speeddependent translational
stiffness
[1e6, 6e5, 3.5e6, 8e6; 1e6, 2e5, 3.0e6,
5e6; 1e6, 5e4, 2.5e6, 3e6]
N/m
(default)  matrix
Speeddependent translational stiffness for the B1
support. The number of rows in the matrix must equal the number of
elements in the vector specified for the Bearing speed [s1, …
sS] parameter. Each row contains four elements:
xx_{s} —
Stiffness in the xaxis direction at the
s^{th}
speed
xy_{s} —
Stiffness in the xaxis direction coupled
with motion in the yaxis direction at
the s^{th}
speed
yx_{s} —
Stiffness in the yaxis direction coupled
with motion in the xaxis direction at
the s^{th}
speed
yy_{s} —
Stiffness in the yaxis direction at the
s^{th}
speed
All xx and yy stiffness values
must be positive. All xy and yx
values must be zero or nonzero at all speeds.
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I1) mounting type
— Support I1 mounting type
Clamped
(default)  Pinned
 Free
 Bearing matrix
 Speeddependent bearing matrix
Type of mounting at the I1 support. The I1 support
is the closest intermediate support to the B1
support.
Dependencies
This parameter is visible when both of these conditions are met:
In the Shaft settings,
Model bending is set to
On
. For more information,
see Shaft Parameter Dependencies.
In the Supports settings,
Number of supports is set to
3
or
4
.
Setting this parameter to Pinned
,
Free
, Bearing
matrix
, or Speeddependent bearing
matrix
exposes related parameters. For more
information, see Mounting Type Parameter Dependencies.
Intermediate (I1) rotational damping [xx,yy]
— Support I1 rotational damping
[1e3, 1e3]
N*m/(rad/s)
(default)  vector
Rotational damping for the I1 support. The elements of the
twoelement vector are:
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I1) translational damping [xx,xy,yx,yy]
— Support I1 translational damping
[2e4, 4.4e4, 4.4e4, 3.5e5]
N/(m/s)
(default)  vector
Translational damping for the I1 support. The
elements of the fourelement vector are:
xx — Damping in the
xaxis direction
xy — Damping in the
xaxis direction coupled with motion
in the yaxis direction
yx — Damping in the
yaxis direction coupled with motion
in the xaxis direction
yy — Damping in the
yaxis direction
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I1) speeddependent translational damping [xx1,xy1,yx1,yy1;...xxsS,xyS,yxS,yyS]
— Support I1 speeddependent translational
damping
[2e4, 4.4e4, 4.4e4, 3.5e5; 1e4, 2.5e4,
2.5e4, 1.4e5; 5e3, 1.0e4, 1.0e4, 2.0e4]
N/(m/s)
(default)  matrix
Speeddependent translational damping for the I1
support. The number of rows in the matrix must equal the number of
elements in the vector specified for the Bearing speed [s1, …
sS] parameter. Each row contains four elements:
xx_{s} —
Damping in the xaxis direction at the
s^{th}
speed
xy_{s} —
Damping in the xaxis direction coupled
with motion in the yaxis direction at
the s^{th}
speed
yx_{s} —
Damping in the yaxis direction coupled
with motion in the xaxis direction at
the s^{th}
speed
yy_{s} —
Damping in the yaxis direction at the
s^{th}
speed
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I1) rotational stiffness [xx,yy]
— Support I1 rotational stiffness
[1e4, 1e4]
N*m/rad
(default)  nonnegative vector
Rotational stiffness for the I1 support. The
elements of the twoelement vector are:
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I1) translational stiffness [xx,xy,yx,yy]
— Support I1 translational stiffness
[1e6, 2e5, 3e5, 5e]
N/m
(default)  vector
Translational stiffness for the I1 support. The
elements of the fourelement vector are:
xx — Stiffness in the
xaxis direction
xy — Stiffness in the
xaxis direction coupled with motion
in the yaxis direction
yx — Stiffness in the
yaxis direction coupled with motion
in the xaxis direction
yy — Stiffness in the
yaxis direction
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I1) speeddependent translational stiffness [xx1,xy1,yx1,yy1;...xxS,xyS,yxS,yyS]
— Support I1 speeddependent translational
stiffness
[1e6, 6e5, 3.5e6, 8e6; 1e6, 2e5, 3.0e6,
5e6; 1e6, 5e4, 2.5e6, 3e6]
N/m
(default)  matrix
Speeddependent translational stiffness for the I1
support. The number of rows in the matrix must equal the number of
elements in the vector specified for the Bearing speed [s1, …
sS] parameter. Each row contains four elements:
xx_{s} —
Stiffness in the xaxis direction at the
s^{th}
speed
xy_{s} —
Stiffness in the xaxis direction coupled
with motion in the yaxis direction at
the s^{th}
speed
yx_{s} —
Stiffness in the yaxis direction coupled
with motion in the xaxis direction at
the s^{th}
speed
yy_{s} —
Stiffness in the yaxis direction at the
s^{th}
speed
All xx and yy stiffness values
must be positive. All xy and yx
values must be zero or nonzero at all speeds.
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I2) mounting type
— Support I2 mounting type
Clamped
(default)  Pinned
 Free
 Bearing matrix
 Speeddependent bearing matrix
Type of mounting at the I2 support. The I2 support
is located between the I1 and F1
supports.
Dependencies
This parameter is visible when both of these conditions are met:
In the Shaft settings,
Model bending is set to
On
. For more information,
see Shaft Parameter Dependencies.
In the Supports settings,
Number of supports is set to
4
.
Setting this parameter to Pinned
,
Free
, Bearing
matrix
, or Speeddependent bearing
matrix
exposes related parameters. For more
information, see Mounting Type Parameter Dependencies.
Intermediate (I2) rotational damping [xx,yy]
— Support I2 rotational damping
[1e3, 1e3]
N*m/(rad/s)
(default)  vector
Rotational damping for the I2 support. The elements
of the twoelement vector are:
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I2) translational damping [xx,xy,yx,yy]
— Support I2 translational damping
[2e4, 4.4e4, 4.4e4, 3.5e5]
N/(m/s)
(default)  vector
Translational damping for the I2 support. The
elements of the fourelement vector are:
xx — Damping in the
xaxis direction
xy — Damping in the
xaxis direction coupled with motion
in the yaxis direction
yx — Damping in the
yaxis direction coupled with motion
in the xaxis direction
yy — Damping in the
yaxis direction
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I2) speeddependent translational damping [xx1,xy1,yx1,yy1;...xxS,xyS,yxS,yyS]
— Support I2 speeddependent translational
damping
[2e4, 4.4e4, 4.4e4, 3.5e5; 1e4, 2.5e4,
2.5e4, 1.4e5; 5e3, 1.0e4, 1.0e4, 2.0e4]
N/(m/s)
(default)  matrix
Speeddependent translational damping for the I2
support. The number of rows in the matrix must equal the number of
elements in the vector specified for the Bearing speed [s1, …
sS] parameter. Each row contains four elements:
xx_{s} —
Damping in the xaxis direction at the
s^{th}
speed
xy_{s} —
Damping in the xaxis direction coupled
with motion in the yaxis direction at
the s^{th}
speed
yx_{s} —
Damping in the yaxis direction coupled
with motion in the xaxis direction at
the s^{th}
speed
yy_{s} —
Damping in the yaxis direction at the
s^{th}
speed
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I2) rotational stiffness [xx,yy]
— Support I2 rotational stiffness
[1e4, 1e4]
N*m/rad
(default)  nonnegative vector
Rotational stiffness for the I2 support. The
elements of the twoelement vector are:
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I2) translational stiffness [xx,xy,yx,yy]
— Support I2 translational stiffness
[1e6, 2e5, 3e5, 5e]
N/m
(default)  vector
Translational stiffness for the I2 support. The
elements of the fourelement vector are:
xx — Stiffness in the
xaxis direction
xy — Stiffness in the
xaxis direction coupled with motion
in the yaxis direction
yx — Stiffness in the
yaxis direction coupled with motion
in the xaxis direction
yy — Stiffness in the
yaxis direction
Dependencies
This parameter is visible when all these conditions are met:
Intermediate (I2) speeddependent translational stiffness [xx1,xy1,yx1,yy1;...xxS,xyS,yxS,yyS]
— Support I2 speeddependent translational
stiffness
[1e6, 6e5, 3.5e6, 8e6; 1e6, 2e5, 3.0e6,
5e6; 1e6, 5e4, 2.5e6, 3e6]
N/m
(default)  vector
Speeddependent translational stiffness for the I2
support. The number of rows in the matrix must equal the number of
elements in the vector specified for the Bearing speed [s1, …
sS] parameter. Each row contains four elements:
xx_{s} —
Stiffness in the xaxis direction at the
s^{th}
speed
xy_{s} —
Stiffness in the xaxis direction coupled
with motion in the yaxis direction at
the s^{th}
speed
yx_{s} —
Stiffness in the yaxis direction coupled
with motion in the xaxis direction at
the s^{th}
speed
yy_{s} —
Stiffness in the yaxis direction at the
s^{th}
speed
All xx and yy stiffness values
must be positive. All xy and yx
values must be zero or nonzero at all speeds.
Dependencies
This parameter is visible when all these conditions are met:
Follower (F1) mounting type
— Support F1 mounting type
Clamped
(default)  Pinned
 Free
 Bearing matrix
 Speeddependent bearing matrix
Type of mounting at the follower end of the shaft.
Dependencies
This parameter is visible when, in the Shaft settings, the
Model bending parameter is set to
On
. For more information, see Shaft Parameter Dependencies.
Setting this parameter to Pinned
,
Free
, Bearing
matrix
, or Speeddependent bearing
matrix
exposes related parameters. For more
information, see Mounting Type Parameter Dependencies and Mounting Type Parameter Dependencies.
Follower (F1) rotational damping [xx,yy]
— Support F1 rotational damping
[1e3, 1e3]
N*m/(rad/s)
(default)  vector
Rotational damping for the F1 support, which is the support that is
located closest to F, the follower end of the
shaft. The elements of the twoelement vector are:
Dependencies
This parameter is visible when all these conditions are met:
Follower (F1) translational damping [xx,xy,yx,yy]
— Support F1 translational damping
[2e4, 4.4e4, 4.4e4, 3.5e5]
N/(m/s)
(default)  vector
Translational damping for the F1 support, which is the support that
is closest to F, the follower end of the shaft. The
elements of the fourelement vector are:
xx — Damping in the
xaxis direction
xy — Damping in the
xaxis direction coupled with motion
in the yaxis direction
yx — Damping in the
yaxis direction coupled with motion
in the xaxis direction
yy — Damping in the
yaxis direction
Dependencies
This parameter is visible when all these conditions are met:
Follower (F1) speeddependent translational damping [xx1,xy1,yx1,yy1;...xxS,xyS,yxS,yyS]
— Support F1 speeddependent translational
damping
[2e4, 4.4e4, 4.4e4, 3.5e5; 1e4, 2.5e4,
2.5e4, 1.4e5; 5e3, 1.0e4, 1.0e4, 2.0e4]
N/(m/s)
(default)  matrix
Speeddependent translational damping for the F1 support, which is
the support that is closest to F, the follower end
of the shaft.
The number of rows in the matrix must equal the number of elements in
the vector specified for the Bearing speed [s1, …
sS] parameter. Each row contains four elements:
xx_{s} —
Damping in the xaxis direction at the
s^{th}
speed
xy_{s} —
Damping in the xaxis direction coupled
with motion in the yaxis direction at
the s^{th}
speed
yx_{s} —
Damping in the yaxis direction coupled
with motion in the xaxis direction at
the s^{th}
speed
yy_{s} —
Damping in the yaxis direction at the
s^{th}
speed
Dependencies
This parameter is visible when all these conditions are met:
Follower (F1) rotational stiffness [xx,yy]
— Support F1 rotational stiffness
[1e4, 1e4]
N*m/rad
(default)  nonnegative vector
Rotational stiffness for the F1 support, which is the support that is
closest to F, the follower end of the shaft. The
elements of the twoelement vector are:
Dependencies
This parameter is visible when all these conditions are met:
Follower (F1) translational stiffness [xx,xy,yx,yy]
— Support F1 translational stiffness
[1e6, 2e5, 3e5, 5e]
N/m
(default)  vector
Translational stiffness for the F1 support, which is the support that
is closest to F, the follower end of the shaft. The
elements of the fourelement vector are:
xx — Stiffness in the
xaxis direction
xy — Stiffness in the
xaxis direction coupled with motion
in the yaxis direction
yx — Stiffness in the
yaxis direction coupled with motion
in the xaxis direction
yy — Stiffness in the
yaxis direction
Dependencies
This parameter is visible when all these conditions are met:
Follower (F1) speeddependent translational stiffness [xx1,xy1,yx1,yy1;...xxS,xyS,yxS,yyS]
— Support F1 speeddependent translational
stiffness
[1e6, 6e5, 3.5e6, 8e6; 1e6, 2e5, 3.0e6,
5e6; 1e6, 5e4, 2.5e6, 3e6]
N/m
(default)  matrix
Speeddependent translational stiffness for the F1 support, which is
the support that is closest to F, the follower end
of the shaft.
The number of rows in the matrix must equal the number of elements in
the vector specified for the Bearing speed [s1, …
sS] parameter. Each row contains four elements:
xx_{s} —
Stiffness in the xaxis direction at the
s^{th}
speed
xy_{s} —
Stiffness in the xaxis direction coupled
with motion in the yaxis direction at
the s^{th}
speed
yx_{s} —
Stiffness in the yaxis direction coupled
with motion in the xaxis direction at
the s^{th}
speed
yy_{s} —
Stiffness in the yaxis direction at the
s^{th}
speed
All xx and yy stiffness values
must be positive. All xy and yx
values must be zero or nonzero at all speeds.
Dependencies
This parameter is visible when all these conditions are met:
Bearing speed [s1,...,sS]
— Support bearing rotational speed
[200, 500, 1500]
rpm
(default)  nonnegative vector
Support bearing rotational speed.
Dependencies
This parameter is visible when all these conditions are met:
In the Shaft settings,
Model bending is set to
On
. For more information,
see Shaft Parameter Dependencies.
In the Support settings, at least
one of these parameters is set to
Speeddependent bearing
matrix
:
Base (B1) mounting
type
Intermediate (I1) mounting
type
Intermediate (I2) mounting
type
Follower (F1) mounting
type
For more information, see Mounting Type Parameter Dependencies and Mounting Type Parameter Dependencies.
Bending
The Bending settings are visible when, in the
Shaft settings, the Model bending
parameter is set to On
. For more information, see Model bending.
The table shows how, in the Bending settings, the specified
value for the Rigid masses concentrically attached to shaft
parameter affects the visibility of related parameters. To learn how to read the
table, see Parameter Dependencies.
Rigid Masses Concentrically Attached to Shaft Parameter
Dependencies
Rigid Masses Concentrically Attached to
Shaft 

None  Point mass  Disk 
  Rigid mass distances from base
(B)  Rigid mass distances from
base (B) 
Rigid masses  Rigid masses

Rigid mass diametric
moments of inertia about axis perpendicular to
shaft 
Rigid mass polar moments
of inertia about shaft axis 
Damping constant proportional to mass
— Damping constant α
10
1/s
(default)  positive scalar
Damping constant, α, proportional to mass.
When the eigenmodes bending model is enabled, a translation damper in
each modal massspringdamper system has the damping coefficient
aM_{Mode}, where
M_{Mode} is the modal
mass.
When the lumped mass bending model is enabled, a damping matrix,
αM is added
to the system. M is
the equation of motion mass matrix.
Dependencies
This parameter is visible when, in the Shaft
settings, the Model bending parameter is set to
On
. For more information, see Model bending.
Damping constant proportional to stiffness
— Damping constant β
1e8
s
(default)  positive scalar
Damping constant, β, proportional to stiffness.
When the lumped mass bending model is enabled, a damping matrix, βK is added to the system.
K is the
equation of motion stiffness matrix. When the eigenmodes bending model
is enabled, a damping βK_{Modal} is added
to the system.
Dependencies
This parameter is visible when, in the Shaft
settings, the Model bending parameter is set to
On
. For more information, see Model bending.
Rigid masses concentrically attached to shaft
— Type of rigid masses attached to shaft
None
(default)  Point mass
 Disk
Type, if any, of rigid masses attached to shaft.
Dependencies
This parameter is visible when, in the Shaft settings, the
Model bending parameter is set to
On
. For more information, see Model bending.
The specified value for this parameter affects the visibility of
related parameters. For more information, see Rigid Masses Concentrically Attached to Shaft Parameter Dependencies.
Rigid mass distances from base (B)
— Rigid mass locations along shaft in distance from base node
0.5
m
(default)  nonnegative scalar or vector
Rigid mass locations along the shaft in distance from B, the base
end of the shaft. For multiple masses, specify an increasing row vector.
The number of elements in the vector must be equal to the number of
masses that are attached to the shaft. The value of the scalar or, for
multiple masses, the largest value in the vector must not exceed the
length of the shaft.
Dependencies
This parameter is visible when both of these conditions are met:
Rigid masses
— Mass of rigid masses concentrically attached to shaft
2
kg
(default)  positive scalar or vector
Mass of rigid masses concentrically attached to shaft. For multiple masses, specify a
row vector. The number and order of the elements in the vector must
correspond to the elements in the vector specified for the
Rigid mass distances from base (B)
parameter.
Dependencies
This parameter is visible when both of these conditions are met:
Rigid mass diametric moments of inertia about axis perpendicular to shaft
— Diametric moments of inertia about axis perpendicular to shaft
0.025
kg*m^2
(default)  positive scalar or vector
Rigid mass moments of inertia about the axis perpendicular to the shaft. For multiple
masses, specify a row vector. The number and order of elements in the
vector must correspond to the elements in the vector specified for the
Rigid mass distances from base (B)
parameter.
Dependencies
This parameter is visible when both of these conditions are met:
Rigid mass polar moments of inertia about shaft axis
— Polar moments of inertia about shaft axis
0.050
kg*m^2
(default)  positive scalar or vector
Rigid polar mass moments of inertia about the principle shaft axis. For multiple masses,
specify a row vector. The number and order of the elements in the vector
must correspond to the elements in the vector specified for the
Rigid mass distances from base (B)
parameter.
Dependencies
This parameter is visible when both of these conditions are met:
Static unbalances that excite bending
— Excitational static unbalances
[.01, .01]
m*kg
(default)  nonnegative vector
Static unbalances that excite bending.
Dependencies
This parameter is visible when, in the Shaft settings, the
Model bending parameter is set to
On
. For more information, see Model bending.
Static unbalance distances from base (B)
— Static unbalance distances from base
[.33, .67]
m
(default)  nonnegative vector
Distance of excitational static unbalances from B, the base end of
the shaft.
Dependencies
This parameter is visible when, in the Shaft settings, the
Model bending parameter is set to
On
. For more information, see Model bending.
Static unbalance offset angles
— Static unbalance offset angles
[0, pi]
rad
(default)  nonnegative vector
Initial angle, about the center line of the shaft relative to the
xaxis, of the excitational static unbalances.
Dependencies
This parameter is visible when, in the Shaft settings, the
Model bending parameter is set to
On
. For more information, see Model bending.
Advanced Bending
The Advanced Bending settings are exposed when, in the
Shaft settings, the Model bending
parameter is set to On
. For more information, see Model bending.
In the Advanced Bending settings, the value of the Bending
vibration analysis method parameter affects the visibility of
parameters in the Supports settings. For more information, see
Bending vibration analysis method.
The Advanced Bending Parameter Dependencies table shows how the
visibility of certain parameters in the Advanced Bending
settings depends on the values for a combination of parameters in one or more of
these settings:
Shaft
Supports
Advanced Bending
For example, the Simulated eigenmode dependency on rotation speed
parameter is visible if all these conditions are met:
In the Shaft settings, Model
bending is set to On
.
In the Supports settings, at least one
Mounting type parameter is set to
Speeddependent bearing matrix
.
In the Advanced Bending settings,
Bending vibration analysis method is set to
Eigenmodes
.
Advanced Bending Parameter Dependencies
Setting > Parameter  Specified Parameter Values and Dependent
Advanced Bending Parameters 

>  Off  On 

>    Any combination of
Clamped , Pinned ,
Free , or Bearing
matrix  At least one
Speeddependent bearing matrix and any
combination of Clamped ,
Pinned , Free , or
Bearing matrix 

The columns to the right show the Advanced
Bending parameters that are visible for the parameter
configuration set shown in the corresponding column in the heading rows of this
table.    Bending vibration analysis
method  Bending vibration analysis
method 
Lumped mass  Eigenmodes  Lumped mass  Eigenmodes 
Bending mode
determination  Bending mode
determination 
Simscape determined  User Defined  Simscape
determined  User
Defined 
Limit number of modes  Limit number of
modes 
Simulated eigenmode dependency
on rotation speed  Simulated eigenmode dependency
on rotation speed 
Static  Dynamic  Static  Dynamic 
Nominal shaft speed for bending modes  Nominal shaft speed for bending
modes 
Eigenfrequency upper limit  Eigenfrequency upper limit  Eigenfrequency upper limit 
Shaft length increments for mode shape
computations  Shaft length increments for mode shape
computations  Shaft length increments for mode shape
computations 
Modal frequencies [m]  Modal frequencies [s, m]  Modal frequencies [s, m] 
Shaft position [z]  Shaft position [z]  Shaft position [z] 
Xdirection mode shapes [z,m]  Xdirection mode shapes [z,m,s]  Xdirection mode shapes [z,m,s] 
Ydirection mode shapes [z,m]  Ydirection mode shapes [z,m,s]  Ydirection mode shapes [z,m,s] 
Bending mode determination
— Bending mode determination method
Simscape determined
(default)  User defined
Method for determining the eigenmode frequencies and shapes:
Dependencies
This parameter is visible when both of these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Advanced Bending settings,
the Bending vibration analysis
method parameter is set to
Eigenmodes
. For more
information, see Advanced Bending Parameter Dependencies.
The value of this parameter affects the visibility of other parameters in the
Advanced Bending settings. For more
information, see Mounting Type Parameter Dependencies.
Limit number of modes
— Maximum number of modes
6
(default)  positive integer scalar
Maximum number of modes that Simscape determines.
Dependencies
This parameter is visible when both of these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Advanced Bending settings,
the Bending vibration analysis
method parameter is set to
Eigenmodes
and the
Bending mode determination
parameter is set to Simscape
determined
. For more information, see
Advanced Bending Parameter Dependencies.
Simulated eigenmode dependency on rotation speed
— Eigenmode dependency
Static
(default)  Dynamic
Simulated dependency of the eigenmode properties on shaft rotation speed:
Static — The bending analysis holds the eigenmode
properties constant during changes in the rotational speed
of the shaft.
Dynamic — The bending analysis adjusts the
eigenmode properties as the rotational speed of the shaft
changes. The block uses the elements in the vector specified
for the Bearing speed [s1,…, sS]
parameter as lookup table reference points. For this model,
the relative magnitudes of speeddependent translational
stiffness elements may not change at each bearing
speed.
Dependencies
This parameter is visible when all these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Supports settings, at
least one mounting type parameter is set to
Speeddependent bearing
matrix
.
In the Advanced Bending settings,
the Bending vibration analysis
method parameter is set to
Eigenmodes
. For more
information, see Advanced Bending Parameter Dependencies.
Nominal shaft speed for bending modes
— Nominal shaft speed
5000
rpm
(default)  scalar
Rated shaft speed for the bending mode analysis.
Dependencies
This parameter is visible if all these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
.
In the Supports settings,
at least one mounting type
parameter are set to
Speeddependent bearing
matrix
.
In the Advanced Bending settings:
For more information, see Advanced Bending Parameter Dependencies.
Eigenfrequency upper limit
— Eigenfrequency upper limit
1e8
rpm
(default)  positive integer scalar
Eigenfrequency upper limit.
Dependencies
This parameter is visible when both of these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Advanced Bending settings,
the Bending vibration analysis
method parameter is set to
Eigenmodes
and the
Bending mode determination
parameter is set to Simscape
determined
. For more information, see
Advanced Bending Parameter Dependencies.
Shaft length increments for mode shape computations
— Shaft length increments
0.01
m
(default)  positive scalar
Shaft length increments used for mode mass and shape computations. For more information,
see Bending Model Eigenmodes Method.
Dependencies
This parameter is visible when both of these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Advanced Bending settings,
the Bending vibration analysis
method parameter is set to
Eigenmodes
and the
Bending mode determination
parameter is set to Simscape
determined
. For more information, see
Advanced Bending Parameter Dependencies.
Modal frequencies [m]
— Bearingspeedindependent modal frequencies
[280, 644, 615, 1285]
rad/s
(default)  nonnegative vector
Modal frequencies for the bearingspeedindependent model. For more
information, see Bending Model Eigenmodes Method.
Dependencies
This parameter is visible when both of these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Supports settings, no
mounting types is set to Speeddependent
bearing matrix
.
In the Advanced Bending settings:
For more information, see Advanced Bending Parameter Dependencies.
Modal frequencies [s,m]
— Bearingspeeddependent modal frequencies
[257, 780, 580, 1560; 280, 644, 615, 1285;
294, 513, 636, 1020]
rad/s
(default)  nonnegative matrix
Modal frequencies for the bearingspeeddependent model. For more
information, see Bending Model Eigenmodes Method.
Dependencies
This parameter is visible when both of these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Supports settings, at
least one mounting type is set to
Speeddependent bearing
matrix
.
In the Advanced Bending settings:
For more information, see Advanced Bending Parameter Dependencies.
Shaft position [z]
— Shaft position
[0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8,
0.9, 1]
(default)  increasing nonnegative vector
Shaft position for mode shapes. The number of elements in the vector correspond to the
number of rows in Xdirection and
Ydirection mode shapes. For more information,
see Bending Model Eigenmodes Method.
Dependencies
This parameter is visible when both of these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Advanced Bending settings:
For more information, see Advanced Bending Parameter Dependencies.
Xdirection mode shapes [z,m]
— Bearingspeedindependent Xdirection mode shapes
[.1648, .0086, .2759, .0162; .1671, .0093,
.2231, .0133; .1691, .01, .1689, .0102; .1707, .0105, .1135,
.007; .1717, .0108, .057, .0035; .1721, .0109, 0, 0; .1717, .0108,
.057, .0035; .1707, .0105, .1135, .007; .1691, .01, .1689, .0102;
.1671, .0093, .2231, .0133; .1648, .0086, .2759,
.0162]
(default)  matrix
The Xdirection mode shapes matrix for the
bearingspeedindependent model. For more information, see Bending Model Eigenmodes Method.
The matrix must have dimensions
zbym, where:
The mode shape matrix has the form
[U_{1x},
U_{2x}, …,
U_{mx}], where each
column is the mode shape deflection in the X
direction for the m^{th}
mode. The algorithm computes the modal properties based on the
parameters in the Shaft and
Bending settings.
Dependencies
This parameter is visible when all these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Supports settings, no
mounting type is set to Speeddependent
bearing matrix
.
In the Advanced Bending settings:
For more information, see Advanced Bending Parameter Dependencies.
Xdirection mode shapes [z,m,s]
— Bearingspeeddependent Xdirection mode shapes
cat(3, [.1842, .0134, .3072, .0272; .1863,
.0152, .2482, .0226; .1882, .0169, .1878, .0176; .1897, .0182,
.1261, .0121; .1906, .019, .0634, .0062; .191, .0193, 0, 0; .1906,
.019, .0634, .0062; .1897, .0182, .1261, .0121; .1882, .0169,
.1878, .0176; .1863, .0152, .2482, .0226; .1842, .0134, .3072,
.0272], [.1648, .0086, .2759, .0162; .1671, .0093, .2231, .0133;
.1691, .01, .1689, .0102; .1707, .0105, .1135, .007; .1717, .0108,
.057, .0035; .1721, .0109, 0, 0; .1717, .0108, .057, .0035; .1707,
.0105, .1135, .007; .1691, .01, .1689, .0102; .1671, .0093, .2231,
.0133; .1648, .0086, .2759, .0162], [.1291, .0046, .2166, .0082;
.131, .0048, .1751, .0067; .1327, .005, .1327, .0051; .1341,
.0052, .0891, .0035; .135, .0053, .0448, .0017; .1353, .0053, 0,
0; .135, .0053, .0448, .0017; .1341, .0052, .0891, .0035; .1327,
.005, .1327, .0051; .131, .0048, .1751, .0067; .1291, .0046,
.2166, .0082])
(default)  matrix
The Xdirection mode shapes matrix for the
bearingspeeddependent model. For more information, see Bending Model Eigenmodes Method.
The matrix must have dimensions
zbymbys,
where:
z is the number of elements in the
specified vector for the Shaft position
[z] parameter.
m is the number of coplumns in the
specified vector for the Modal frequencies
[z,m] parameter.
s is the number of elements in the
specified vector for the Bearing speed [s1,…,
sS] parameter.
The mode shape matrix has the form
cat(3,[U_{1x1},
U_{2x1}, …,
U_{mx1}], …, [
U_{1xs},
U_{2xs}, …,
U_{mxs}]), where each
column is the mode shape deflection in the x
direction, for the m^{th}
mode. Each page corresponds to an element in the vector specified for
the Bearing speed [s1,…, sS] parameter. The
algorithm computes the modal properties based on the parameters in the
Shaft and Bending
settings.
Dependencies
This parameter is visible when all these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Supports settings, at
least one mounting type is set to
Speeddependent bearing
matrix
.
In the Advanced Bending settings:
For more information, see Advanced Bending Parameter Dependencies.
Ydirection mode shapes [z,m]
— Bearingspeedindependent Ydirection mode shapes
[.1193, .1784, .1997, .3348; .1209,
.1933, .1615, .2751; .1224, .2068, .1223, .2119; .1235,
.2176, .0821, .1445; .1243, .2245, .0413, .0734; .1246,
.2269, 0, 0; .1243, .2245, .0413, .0734; .1235, .2176, .0821,
.1445; .1224, .2068, .1223, .2119; .1209, .1933, .1615, .2751;
.1193, .1784, .1997, .3348]
(default)  matrix
The Ydirection mode shapes matrix for the
bearingspeedindependent model. For more information, see Bending Model Eigenmodes Method.
The matrix must have dimensions
zbym, where:
The mode shape matrix has the form
[U_{1y},
U_{2y}, …,
U_{my}], where each
column is the mode shape deflection in the Y
direction, for the m^{th}
mode. The algorithm computes the modal properties based on the
parameters in the Shaft and
Bending settings.
Dependencies
This parameter is visible when both of these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Supports settings, no
mounting type is set to Speeddependent
bearing matrix
.
In the Advanced Bending settings:
For more information, see Advanced Bending Parameter Dependencies.
Ydirection mode shapes [z,m,s]
— Bearingspeeddependent ydirection mode shapes
cat(3, [.0885, .1625, .1475, .3301; .0895,
.1847, .1192, .2745; .0904, .2048, .0902, .2141; .0911,
.2209, .0606, .1475; .0916, .2313, .0304, .0754; .0917,
.2349, 0, 0; .0916, .2313, .0304, .0754; .0911, .2209, .0606,
.1475; .0904, .2048, .0902, .2141; .0895, .1847, .1192, .2745;
.0885, .1625, .1475, .3301], [.1193, .1784, .1997, .3348; .1209,
.1933, .1615, .2751; .1224, .2068, .1223, .2119; .1235,
.2176, .0821, .1445; .1243, .2245, .0413, .0734; .1246,
.2269, 0, 0; .1243, .2245, .0413, .0734; .1235, .2176, .0821,
.1445; .1224, .2068, .1223, .2119; .1209, .1933, .1615, .2751;
.1193, .1784, .1997, .3348], [.1566, .1901, .2628, .3377; .1589,
.1994, .2125, .2753; .161, .2079, .1609, .2103; .1627, .2146,
.1081, .1424; .1638, .219, .0544, .0719; .1642, .2205, 0, 0;
.1638, .219, .0544, .0719; .1627, .2146, .1081, .1424; .161,
.2079, .1609, .2103; .1589, .1994, .2125, .2753; .1566, .1901,
.2628, .3377])
(default)  matrix
The Ydirection mode shapes matrix for the bearingspeeddependent
model. For more information, see Bending Model Eigenmodes Method.
The matrix must have dimensions
zbymbys,
where:
z is the number of elements in the
specified vector for the Shaft position
[z] parameter.
m is the number of columns in the
specified vector for the Modal frequencies
[z,m] parameter.
s is the number of elements in the
specified vector for the Bearing speed [s1,…,
sS] parameter.
The mode shape matrix has the form
cat(3,[U_{1y1},
U_{2y1}, …,
U_{my1}], …, [
U_{1ys},
U_{2ys}, …,
U_{mys}]), where each
column is the mode shape deflection in the y
direction, for the m^{th}
mode. Each page corresponds to an element in the vector specified for
the Bearing speed [s1,…, sS] parameter. The
algorithm computes the modal properties based on the parameters in the
Shaft and Bending
settings.
Dependencies
This parameter is visible when all these conditions are met:
In the Shaft settings, the
Model bending parameter is set
to On
. For more information,
see Model bending.
In the Supports settings, at
least one mounting type is set to
Speeddependent bearing
matrix
.
In the Advanced Bending settings:
For more information, see Advanced Bending Parameter Dependencies.
References
[1] Adams, M.L. Rotating
Machinery Vibration. CRC Press, NY: 2010.
[2] Bathe, K. J. Finite
Element Procedures. Prentice Hall, 1996.
[5] Muszynska, A.
Rotordynamics. Taylor & Francis, 2005
[6] Rao, S.S. Vibration of
Continuous Systems. Hoboken, NJ: John Wiley & Sons,
2007.
Introduced in R2018b