Documentation

# Flexible Shaft

Shaft with torsional and bending compliance

• Library:
• Simscape / Driveline / Couplings & Drives

## 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 the torsion-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 by using stiffness and damping values or by using the shaft shear modulus. An additional parameter enables you to model the power losses due to viscous friction at the shaft ends. For more information, see Torsional Model.

To represent the bending-flexible shaft, the block uses your choice of a lumped parameter model or an eigenmodes model. While the lumped parameter approach is simpler to configure, the eigenmodes approach tends to simulate faster. For more information on both models, see Bending Model.

For the lumped parameter model, the number of bending shaft elements is the same as the number of torsion shaft elements, and additional parameters enable you to model linear damping proportional to shaft inertia and at the shaft ends. The model divides the shaft into different elements. The elements provide the shaft inertia, while the stiffness matrices provide the shaft compliance.

The eigenmodes model computes effective mass spring damper systems that represent the bending modes of the shaft. You can specify the number of modes to include, and additional parameters enable you to model linear damping proportional to modal mass and stiffness.

For both models, you specify the shaft bending compliance by using either bending rigidity and linear mass density or Young’s modulus. Additional parameters enable you to model:

• Shaft mounting types. Options are ideal clamp, pin, free, or bearing stiffness.

• Locations and inertia of either rigid point masses or disks that are attached to the shaft.

• Locations, magnitudes, and angle offsets of static unbalances on the shaft.

Since the two modeling approaches are different, convergence between them lends confidence to their results. A recommended practice is to verify agreement for the lumped parameter and eigenmodes responses by using the parameter values from your own flexible shaft model in the Shaft with Torsional and Transverse Flexibility example. Then, for faster simulation of your own model, use the eigenmodes approach.

### Torsional Model

For the torsional 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 equivalent physical network contains $N$ segments, each consisting of a spring, damper, and inertia. A segment represents a short section of the driveshaft, the spring represents torsional compliance, and the damper represents material damping. The total shaft inertia is split into $N+1$ parts, and distributed evenly along the length of the shaft.

The block can also include viscous friction at the shaft ends (base and follower ports) to represent bearing losses at these points.

### 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.

You can parameterize the torsional model using either stiffness k and inertia J or the dimensions and material properties of the shaft.

#### Dimensions and Material Properties

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

`${J}_{p}=\frac{\pi }{32}\left({D}^{4}-{d}^{4}\right)$`
`$m=\frac{\pi }{4}\left({D}^{2}-{d}^{2}\right)\rho L$`
`$J=\frac{m}{8}\left({D}^{2}+{d}^{2}\right)=\rho L\cdot Jp$`
`$k=Jp\cdot \frac{G}{L}$`

Where:

• JP is the polar moment of inertia of the shaft.

• D is the outer diameter of the shaft.

• d is the inner diameter of the shaft. For a solid shaft, $d=0$. For an annular shaft, $d>0$.

• L is the shaft length.

• m is the mass of the shaft.

• J is the moment of inertia of the shaft.

• ρ is the density of the shaft material.

• G is the shear modulus of elasticity of the shaft material.

• k is the shaft rotational stiffness.

#### Internal Material Damping

For either torsional parameterization, the internal material damping is defined by the damping ratio, c, for a single-segment model. In this case, the damping torque is $2\frac{ck}{{\omega }_{N}}$. Undamped natural frequency ${\omega }_{N}=\sqrt{\frac{2k}{J}}$. The damping applied across an individual segment of a segmented model is equal to the product of the damping coefficient and the relative rotational velocity of that segment.

### Bending Model

The next figure shows how to measure:

• The static unbalance offset angle, that is the angle of a static unbalance about the shaft axis relative to the x axis.

• The distances of a rigid mass and a static unbalance, relative to the base, B.

In the figure, the shaft has fixed supports at the base and follower. The shaft exerts forces, F, and moments, M, on the supports. The translational velocity, V, and rotational velocity, W, of the base and follower ends are relative to their respective supports. All curved arrows and sign conventions follow the right-hand 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 = [Fxb, Fyb, Fxf, Fyf]

• Moment, M = [Mxb, Myb, Mxf, Myf]

• Translational velocity, V = [Vxb, Vyb, Vxf, Vyf]

• Rotational velocity, W = [Wxb, Wyb, Wxf, Wyf]

Fixed-Support Shaft Forces, Moments, and Phases

The partial differential equations governing the bending motion of a rotating shaft with points of static mass unbalance can be solved using a lumped parameter approach or an eigenmodes approach. For both models, there are two defining equations:

Where:

• $EI$ is the shaft bending rigidity.

• $\rho A$ is the shaft linear mass density.

• $m{\epsilon }_{j}$ is the ${j}^{th}$ static unbalance.

• ${\phi }_{offset,i}$ is the initial angle offset of the ${j}^{th}$ static unbalance.

• $n$ is the number of static unbalances.

• $\Omega$ is the shaft rotation speed.

• ${\phi }_{shaft}$ is the instantaneous shaft rotation angle.

• $z$ is the coordinate along the shaft axis.

• ${u}_{x}\left(t,z\right)$ is the shaft bending deflections in the x direction perpendicular to the shaft axis.

• ${u}_{y}\left(t,z\right)$ is the shaft bending deflections in the y direction perpendicular to the shaft axis.

#### Lumped Parameter Approach

Like the torsion model, the lumped parameter approach for the bending model discretizes the distributed, continuous properties of the shaft into a finite number, $N$, of shaft segments. The $N$ shaft segments 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 parameter equation [1] of motion is

`$M\stackrel{¨}{\stackrel{\to }{x}}+B\stackrel{˙}{\stackrel{\to }{x}}+K\stackrel{\to }{x}=\stackrel{\to }{f}$`

Where:

• $M$ is the $4\left(N+1\right)×4\left(N+1\right)$ matrix that represents the mass of the shaft.

• $\stackrel{\to }{x}$ is the $4\left(N+1\right)×1$ vector that represents the degrees of freedom for all nodes, such that the degrees of freedom for the ${i}^{th}$ and the ${\left(i+1\right)}^{th}$ nodes are

`$\stackrel{\to }{x}=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}⋮\\ \begin{array}{c}{x}_{i}\\ {y}_{i}\\ {\theta }_{i}\end{array}\\ {\phi }_{i}\\ \begin{array}{c}\begin{array}{c}{x}_{i+1}\\ {y}_{i+1}\\ {\theta }_{i+1}\end{array}\\ {\phi }_{i+1}\\ ⋮\end{array}\end{array}\end{array}\end{array}\end{array}\right]$`

• $K$ is the $4\left(N+1\right)×4\left(N+1\right)$ matrix for the spring stiffness.

• $B$ is the $4\left(N+1\right)×4\left(N+1\right)$ matrix for the damping constant.

• $\stackrel{\to }{f}$ is the $4\left(N+1\right)×1$ 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 shaft segment. For each shaft segment, half of the mass and moment of inertia is transferred to the nodes at both ends of the segment. The ${M}_{i/\left(i+1\right)}$ matrix has nonzero elements in the $\left(4i-3\right):\left(4i+4\right)$ rows and the $\left(4i-3\right):\left(4i+4\right)$ columns:

Where:

• $l$ is the segment length along the shaft axis. $l$ depends on both the shaft length, $L$, and the number of segments, $N$, such that $l=\frac{L}{N+1}$.

• $m$ is the segment mass. $m$ depends on the outer, $D$, and inner, $d$, diameters, the density, $\rho$, of the shaft and the length of the segment, such that .

• , the half-segment mass moment of inertia about an axis perpendicular to the shaft axis, depends on the mass and length of the segment, such that ${I}_{d}=m\frac{{l}^{2}}{24}+\frac{m}{2}{\left(\frac{l}{2}\right)}^{2}$.

• is the summed mass matrices of the rigid masses concentrically attached to the shaft.

• The mass properties of each rigid mass 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 the disk.

The equation for the damping matrix is

Where:

• $\alpha$ is the damping constant proportional to mass.

• $\left[{b}_{B},{b}_{F}\right]$ are the translation damping coefficients at the base, B, and follower, F.

• $\left[{\delta }_{B},{\delta }_{F}\right]$ are the rotation damping coefficients at the base, B, and follower, F.

• 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.

The equation for the bearing stiffness matrix is

Where:

• ${K}_{i/i+1}$ is the shaft segment stiffness matrix for an individual shaft segment. The stiffness matrix for the ${i}^{th}$ shaft segment, between the ${i}^{th}$ and the ${\left(i+1\right)}^{th}$ nodes, has nonzero elements in the $\left(4i-3\right):\left(4i+4\right)$ rows and the $\left(4i-3\right):\left(4i+4\right)$ columns, such that

Where:

• $l$ is the segment length.

• $EI$ is the shaft rigidity.

• ${K}_{B}$ is the bearing stiffness matrices at the base, B, applied to the first node, such that

Where:

• $\left[\begin{array}{cc}{K}_{Bxx}& {K}_{Byy}\end{array}\right]$ is the base translational stiffness in the x and y directions.

• $\left[\begin{array}{cc}{K}_{B\theta \theta }& {K}_{B\phi \phi }\end{array}\right]$ is the base rotational stiffness about the x and y directions.

• ${K}_{F}$ is the bearing stiffness matrices at the follower, F, applied to the first and node, such that

Where:

• $\left[\begin{array}{cc}{K}_{Fxx}& {K}_{Fyy}\end{array}\right]$ is the follower translational stiffness in the x and y directions.

• $\left[\begin{array}{cc}{K}_{F\theta \theta }& {K}_{F\phi \phi }\end{array}\right]$ is the follower rotational stiffness about the x and y directions

The base and follower bearing stiffness matrices, ${K}_{B}$ and ${K}_{F}$, are nonzero only if you select ```Bearing matrix``` for the Base (B) mounting type parameter or Follower (F) mounting type parameter, respectively. If you select the `Clamped` mounting type, the kinematic conditions of zero rotation and translation are applied to the first or last four rows of $\stackrel{\to }{x}$ in the lumped parameter equation of motion. If you select the `Pinned` mounting type, the kinematic conditions of zero translation are applied to base of follower nodes in the lumped parameter equation of motion.

NodeMounting TypeBoundary Condition for the Lumped Parameter Equation
Base, B`Clamped`$\stackrel{\to }{x}\left(1:4\right)=0$
`Pinned`$\stackrel{\to }{x}\left(1:2\right)=0$
`Bearing Matrix`${K}_{B}\left(1:4,1:4\right)$ are nontrivial.
Follower, F`Clamped`$\stackrel{\to }{x}\left(end-3:end\right)=0$
`Pinned`$\stackrel{\to }{x}\left(end-3:end-2\right)=0$
`Bearing Matrix`${K}_{F}\left(end-3:end,end-3:end\right)$ are nontrivial

External forces due to each static mass unbalance are applied to the closest node. The forcing at the ${i}^{th}$ node is

Where:

• ${L}_{m\epsilon j}$ is the distance of the ${j}^{th}$ static unbalance from the closest ${i}^{th}$ node along the shaft axis.

• $m{\epsilon }_{j}$ is the static unbalance.

• ${\phi }_{shaft,i}$ is the torsion lumped parameter angle for the ${i}^{th}$ node.

#### Eigenmodes Approach

For the eigenmodes approach, the block computes the bending mode properties of the shaft during compilation time and then simulates the modal effective mass spring damper systems using these steps:

1. Determine the eigenmode shapes by treating the deflection of each flexible shaft segment between rigid masses as a spatial function with unknown coefficients:

1. Generate equations for the flexible shaft segment boundary conditions.

2. Numerically solve the system of equations for the modal frequencies.

3. Determine the mode shape at each modal frequency using nonlinear least squares.

### Note

If you input the mode shapes directly, the block skips this step.

2. Determine these dynamic properties for each eigenmode:

• Effective modal mass

• Stiffness

• Forcing coefficients

3. Simulate the independent responses of the effective mass spring damper system for each eigenmode.

4. Obtain the total shaft response by summing the deflections of each eigenmode. Loads at the supports are proportional to the spatial derivatives of the mode shapes.

The equations for each step in the eigenmodes approach are shown. For information on the derivation of the equations, see Muszynska[5], Rao[6], and Wu[7].

The next figure shows the deflection angles and variables that the block uses to solve the eigenmodes approach boundary condition equations.

Neighboring Shaft Segments and Rigid Disk Motion

In step 1, to solve for the eigenmodes, the block considers the unforced bending equations for a beam in the two directions perpendicular to the shaft axis, using these equations:

The block assumes that solutions to the two equations have this form:

Where:

• ${U}_{x}$ is the mode shape deflection in the x direction.

• ${U}_{y}$ is the mode shape deflection in the y direction.

• $j=\sqrt{-1}$.

• $\omega$ is a modal frequency.

• are coefficients.

• $EI$ is the shaft bending rigidity.

• $\rho A$ is the shaft linear mass density.

In the eigenmodes approach equations, $\text{'}$ indicates a derivative with respect to $z$, the axis along the shaft, such that:

• $\text{'}$ indicates a first derivative with respect to $z$.

• $\text{'}\text{'}$ indicates a second derivative with respect to $z$.

• $\text{'}\text{'}\text{'}$ indicates a third derivative with respect to $z$.

In step 1a, to generate equations for the flexible shaft segment boundary conditions, the block divides the shaft into $N$ segments between the shaft ends and $N-1$ rigid masses attached to the shaft. For each segment, the block defines the ${k}^{th}$ mode shape in a piecewise manner along the entire shaft as ${U}_{kx}\left(z\right)$ and ${U}_{ky}\left(z\right)$.

For each individual shaft segment, , the ${k}^{th}$ mode shapes, ${U}_{kxi}\left({z}_{i}\right)$ and ${U}_{kyi}\left({z}_{i}\right)$, are defined by the solutions to the unforced bending equations. For each shaft segment, $i$, refers to an origin at the segment end closest to the shaft base, B, end.

Each shaft segment mode shape has eight unknown coefficients, . Mode shapes ${U}_{x}$ and ${U}_{y}$ satisfy boundary conditions at the disk interfaces and at both shaft ends.

The boundary conditions depend on the mounting types and are applied to both modes shapes. The boundary conditions at the base end of the shaft apply to the first segment. The boundary conditions at the follower end of the shaft apply to segment $N$.

Equations at Shaft B and F Ends

NodeMounting TypeBoundary Condition Equation
Base, B`Clamped` — Zero deflection and slope
`Pinned` — Zero deflection and moment
`Free` — Zero moment and force
`Bearing Matrix` — With bearing stiffness matrix ${K}_{B}$

Where:

• ${K}_{B}{\stackrel{\to }{X}}_{b}$ is the spring load at the base end as a function of the mode shape.

• correspond to the deflections of variables in the Fixed-Support Shaft Forces, Moments, and Phases figure.

Follower, F`Clamped` — Zero deflection and slope
`Pinned` — Zero deflection and moment
`Free` — Zero moment and force
`Bearing Matrix` — With bearing stiffness matrix ${K}_{F}$

Where:

• ${K}_{F}{\stackrel{\to }{X}}_{f}$ is the spring load at the follower end as a function of the mode shape

• correspond to the deflections of variables in the Fixed-Support Shaft Forces, Moments, and Phases figure.

At each rigid mass, the shaft segment on either side of the rigid mass must satisfy continuity in displacement and slope, and balance of forces and moments. To satisfy the boundary conditions, the block uses the equations in the table.

Equations at Rigid Masses Concentrically Attached to the Shaft

ConditionRigid masses concentrically attached to shaft
Point massDisk
Displacement continuity
• ${U}_{xi}\left({l}_{i}\right)={U}_{x\left(i+1\right)}\left(0\right)$

• ${U}_{yi}\left({l}_{i}\right)={U}_{y\left(i+1\right)}\left(0\right)$

Slope continuity
• ${{U}^{\prime }}_{xi}\left({l}_{i}\right)=-{{U}^{\prime }}_{x\left(i+1\right)}\left(0\right)$

• ${{U}^{\prime }}_{yi}\left({l}_{i}\right)=-{{U}^{\prime }}_{y\left(i+1\right)}\left(0\right)$

Force balance
• $-{\omega }^{2}{M}_{disk,i}{U}_{xi}\left({l}_{i}\right)=EI{{U}^{‴}}_{xi}\left({l}_{i}\right)-EI{{U}^{‴}}_{x\left(i+1\right)}\left(0\right)$

• $-{\omega }^{2}{M}_{disk,i}{U}_{yi}\left({l}_{i}\right)=EI{{U}^{‴}}_{yi}\left({l}_{i}\right)-EI{{U}^{‴}}_{y\left(i+1\right)}\left(0\right)$

Moment balance
• $0=-EI{{U}^{″}}_{xi}\left({l}_{i}\right)+EI{{U}^{″}}_{x\left(i+1\right)}\left(0\right)$

• $0=-EI{{U}^{″}}_{yi}\left({l}_{i}\right)+EI{{U}^{″}}_{y\left(i+1\right)}\left(0\right)$

• $-{\omega }^{2}{I}_{D,disk,i}\Theta -j\omega {\Omega }_{shaft}{I}_{P,disk,i}{\Phi }_{disk\left(i+1\right)}=-EI{{U}^{″}}_{yi}\left({l}_{i}\right)+EI{{U}^{″}}_{y\left(i+1\right)}\left(0\right)$

• $-{\omega }^{2}{I}_{D,disk,i}\Phi +j\omega {\Omega }_{shaft}{I}_{P,disk,i}{\Theta }_{disk\left(i+1\right)}=-EI{U}^{\prime \text{​}\prime }{}_{xi}\left({l}_{i}\right)+EI{{U}^{″}}_{x\left(i+1\right)}\left(0\right)$

The variables for the equations in this table are:

• ${\Omega }_{shaft}$ is the nominal shaft rotation speed at the shaft midpoint.

• ${I}_{P,disk,i}$ is the rigid mass polar moment of inertia about shaft axis.

• $\omega$ is the modal frequency.

• is the rigid mass diametric moment of inertia about shaft axis.

• $EI$ is the shaft bending rigidity.

• $j=\sqrt{-1}$

In step 1b, the block numerically solves for modal frequencies by combining shaft segment boundary conditions at shaft B and F ends with the shaft segment boundary conditions at the rigid masses concentrically attached to the shaft. The resulting equation, in matrix form, is

`$H\left(\omega \right)\stackrel{\to }{S}=0$`

Where:

• $\stackrel{\to }{S}$ contains the modal coefficients for the $N$ shaft segments, such that

For nontrivial modal solutions, the system of equations must satisfy $|H\left(\omega \right)|=0$.

To determine the modal frequency solutions numerically, the block:

1. Computes the initial eigenfrequency solutions, ${\omega }_{kINITIAL}$, by taking the local minima of $abs\left(|H\left(\omega \right)|\right)$ over $\omega =\left[0:d\omega :{\omega }_{Max}\right]$, where:

• $d\omega$ is the eigenfrequency search resolution.

• ${\omega }_{\mathrm{max}}$ is the maximum eigenfrequency considered.

2. The block refines each ${\omega }_{k}{}_{INITIAL}$ by replacing it with the minimum of $abs\left(|H\left(\omega \right)|\right)$ over $\omega ={\omega }_{k}{}_{INITIAL}+d\omega \left(\left[-1:d{\omega }_{RefineFactor}:1\right]\right)$

The block then uses nonlinear least squares to determine the mode shape coefficients $\stackrel{\to }{S}$ at each modal frequency.

In step 2, the block determines these eigenmode dynamics properties:

• Modal mass

• Modal stiffness

• Modal damping

• Modal forcing coefficients

The block reduces the dynamics of each mode, $k$, to an independent system of equations:

`${M}_{kx}{\stackrel{¨}{\eta }}_{kx}+{B}_{kdx}{\stackrel{˙}{\eta }}_{kx}+{B}_{kpy}{\stackrel{˙}{\eta }}_{ky}+{K}_{kx}{\eta }_{kx}={\text{f}}_{kx}$`

Where:

• ${\eta }_{kx}$ is the time-dependent coefficient of the ${k}^{th}$ mode in the $x$ direction.

• ${\eta }_{ky}$ is the time-dependent coefficient of the ${k}^{th}$ mode in the $y$ direction.

The equations for the ${k}^{th}$ modal mass are:

The equations for the ${k}^{th}$ modal stiffness are:

Where:

• L is the shaft length.

• ${K}_{B}$ and ${K}_{F}$ are only used when the shaft boundary is defined by a bearing matrix.

• Only rows that contribute to loading in each modal direction play a role in that direction’s modal stiffness expression.

The equations for modal damping due to the disk polar moment of inertias are:

The equations for the dissipative modal damping are:

Where:

• $\alpha$ is the damping constant proportional to mass.

• $\beta$ is the damping constant proportional to stiffness.

The equations for modal forcing due to the mass static unbalances are:

`${f}_{kx}={\Omega }_{shaft}^{2}{\sum }^{\text{​}}\left(m{\epsilon }_{offset,j}\right){U}_{kx}\left({z}_{offset,j}\right)\mathrm{cos}\left({\phi }_{shaft}+{\phi }_{offset,j}\right)$`

`${f}_{ky}={\Omega }_{shaft}^{2}{\sum }^{\text{​}}\left(m{\epsilon }_{offset,j}\right){U}_{ky}\left({z}_{offset,j}\right)\mathrm{sin}\left({\phi }_{shaft}+{\phi }_{offset,j}\right)$`

Where:

• The forces of each static unbalance, $j$, are summed for the total force on the ${k}^{th}$ mode.

• ${\Omega }_{shaft}$ is the instantaneous rotation speed at the shaft midpoint.

• ${\phi }_{shaft}$ is the instantaneous angle at the shaft midpoint.

• ${\phi }_{offset,j}$ is the initial angle of the excitational ${j}^{th}$ static unbalance.

In step 4, the block sums the deflections of each eigenmode to get the total shaft response. The equations for the total instantaneous deflection along the shaft are:

`${u}_{x}\left(t,z\right)={\sum }^{\text{​}}{U}_{kx}\left(z\right){\eta }_{kx}\left(t\right)$`

`${u}_{y}\left(t,z\right)={\sum }^{\text{​}}{U}_{ky}\left(z\right){\eta }_{ky}\left(t\right)$`

The equations for the instantaneous force acting on the supports at the shaft B and F ends in the x and y directions are:

The equations for the instantaneous moment acting on the supports at the shaft ends are:

### Improve Simulation Speed or Accuracy

To increase simulation accuracy for the lumped parameter approach for a torsional or bending model, increase the number of segments . However, as the number of segments increases, simulation performance, that is simulation speed, may decrease. The single-segmented torsional model exhibits an 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 segments. 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 16-segmented model.

To increase simulation accuracy for the eigenmodes approach to a bending model:

• 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 mass attached to the shaft with a large mass moment of inertia about shaft axis.

• Decrease the values of the Eigenfrequency search resolution and Eigenfrequency refine factor parameters. Reducing these values may increase the accuracy of modal frequencies and shapes.

• Decrease the damping constants. Simscape™ computations of the modal properties before simulation do not account for damping.

• Swap the values for the base, B and follower, F, boundary condition parameters and flip the block in the Simscape model. Eigenmodes shapes depend on sinh and cosh functions. Therefore, errors may occur near the F end.

• Check the sensitivity to the Advanced Bending parameters 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 sufficiently 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 parameters.

• The flexible shaft has a constant axisymmetric cross-section along its length.

• Shaft rotation and torsion flexibility excites 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.

• Bearing supports are located at the shaft endpoints.

• Shaft bending is not transmitted between Flexible Shaft blocks.

• Relative to the shaft length, the shaft outer diameter is small. The ratio of the shaft length to outer diameter is greater than 20.

• Relative to the shaft length, the bending deflection is small.

• Static mass unbalances are the only shaft-bending 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 approach uses the rotation speed at the shaft midpoint.

• To satisfy the shaft mode shape boundary conditions problem, the block uses the Levenberg-Marquardt algorithm.

## Ports

### Output

expand all

Physical signal outport associated with the force that the shaft exerts on the bearing supports.

Physical signal outport associated with the moment that the shaft exerts on the bearing supports.

Physical signal outport associated with the translational velocity of the shaft at the bearing supports.

Physical signal outport associated with the angular velocity of the shaft at the bearing supports.

### Conserving

expand all

Rotational conserving port associated with the shaft base.

Rotational conserving port associated with the shaft follower.

## Parameters

expand all

### Shaft

The table shows how the visibility of some parameters depends on the option that you choose for other parameters. To learn how to read the table, see Parameter Dependencies.

Shaft Parameter Dependencies

Shaft
Model bending — Choose `Off` or `On`
OffOn

Exposes block outports:

• Fr

• M

• V

• W

Exposes parameters for:

• Bending

Parameterization — Choose `By stiffness and inertia` or `By material properties`
By stiffness and inertiaBy material propertiesBy stiffness and inertiaBy material properties
Number of segments
Torsional stiffnessShaft lengthTorsional stiffnessShaft length
Torsional inertiaShaft geometry — Choose `Solid` or `Annular`Torsional inertiaShaft geometry — Choose `Solid` or `Annular`
SolidAnnularShaft lengthSolidAnnular
Shaft outer diameterBending rigidityShaft outer diameter
-Shaft inner diameterLinear density Shaft inner diameter
Material densityMaterial density
Shear modulusShear modulus
Young's modulus

Option to model bending.

#### Dependencies

Setting the Model bending parameter to `On` exposes these ports and parameters, which are hidden by default:

• Fr outport

• M outport

• V outport

• W outport

• Bending parameters

Parameterization method.

#### Dependencies

Each Parameterization option exposes related parameters and hides unrelated parameters. For more information, see Shaft Parameter Dependencies.

Number of segments, $N$, for the approximation.

A larger number of segments, $N$, increases the accuracy of the model, but reduces simulation performance, that is, simulation speed. The single-segmented model ($N$=1) exhibits a torsion eigenfrequency that is close to the first eigenfrequency of the continuous, distributed parameter model.

If accuracy is more important than performance, select 2, 4, 8, or more segments. 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 16-segmented model. Accurately modeling bending dynamics generally requires more segments than torsion dynamics.

Material stiffness.

#### Dependencies

Setting the Model bending parameter to `Off` and the Parameterization parameter to `By stiffness and inertia` or setting the Model bending parameter to `On` exposes this parameter. For more information, see Shaft Parameter Dependencies.

Shaft inertia.

#### Dependencies

Setting the Parameterization parameter to `By stiffness and inertia` exposes this parameter. For more information, see Shaft Parameter Dependencies.

Shaft length.

#### Dependencies

This parameter is visible when you set the Model bending parameter to `Off` and the Parameterization parameter to `By stiffness and inertia` or when you set the Model bending parameter to `On`. For more information, see Shaft Parameter Dependencies.

Bending rigidity for the shaft material.

#### Dependencies

Setting the Model bending parameter to `On` and the Parameterization parameter to `By stiffness and inertia` exposes this parameter. For more information, see Shaft Parameter Dependencies.

Linear density for the shaft material.

#### Dependencies

Setting the Model bending parameter to `On` and the Parameterization parameter to `By stiffness and inertia` exposes this parameter. For more information, see Shaft Parameter Dependencies.

Shaft cross-sectional geometry.

#### Dependencies

Setting the Parameterization parameter to `By material properties` exposes this parameter. Each Shaft geometry option exposes other parameters. For more information, see Shaft Parameter Dependencies.

Shaft outer diameter.

#### Dependencies

Setting the Parameterization parameter to `By material properties` exposes this parameter. For more information, see Shaft Parameter Dependencies.

Shaft inner diameter. If the shaft is solid, specify `0`.

#### Dependencies

Setting the Parameterization parameter to `By material properties` and the Shaft geometry parameter to `Annular` exposes this parameter. For more information, see Shaft Parameter Dependencies.

Material density.

#### Dependencies

Setting the Parameterization parameter to `By material properties` exposes this parameter. For more information, see Shaft Parameter Dependencies.

Shear modulus for the shaft material.

#### Dependencies

Setting the Parameterization parameter to `By material properties` exposes this parameter. For more information, see Shaft Parameter Dependencies.

Young's modulus for the material.

#### Dependencies

Setting the Model bending parameter to `On` and the Parameterization parameter to `By material properties` exposes this parameter. For more information, see Shaft Parameter Dependencies.

### Torsion

Material damping ratio.

Viscous friction coefficients at base port, B, and the follower port, F.

### Bending

Bending parameters are exposed only when the Shaft parameter Model bending is set to `On`. For more information, see Shaft Parameter Dependencies. The visibility of some Bending parameters also depends on the option that you choose for the Advanced Bending parameter Bending vibration analysis method.

The table shows how the visibility of some parameters depends on the option that you choose for other parameters. To learn how to read the table, see Parameter Dependencies.

Bending Parameter Dependencies

Bending
Base (B) mounting type — Choose `Clamped`, `Pinned`, `Free`, or ```Bearing matrix```
Clamped, Pinned, or FreeBearing matrix

Base (B) translational stiffness [x y]

Base (B) rotational stiffness [x y]

Follower (F) mounting type — Choose `Clamped`, `Pinned`, `Free`, or ```Bearing matrix```
Clamped, Pinned, or FreeBearing matrix

Follower (F) translational stiffness [x y]

Follower (F) rotational stiffness [x y]

Damping constant proportional to mass

Damping constant proportional to stiffness

Translation damping coefficient at base (B) and follower (F)*

Rotation damping coefficient at base (B) and follower (F)*

Damping constant proportional to stiffness**

Rigid masses concentrically attached to shaft — Choose `None`, ```Point mass```, or `Disk`
NonePoint massDisk
Rigid mass locations along shaft (distance from B node)
Rigid masses
-Rigid mass diametric moments of inertia about axis perpendicular to shaft
Rigid mass polar moments of inertia about shaft axis
Static unbalances that excite bending
Static unbalance distances from base (B)
Static unbalance offset angle

* This parameter is visible only when the Advanced Bending parameter Bending vibration analysis method is set to `Lumped`

** This parameter is visible only when the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`

Type of mounting at the base end of the shaft.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`. For more information, see Shaft Parameter Dependencies.

Setting this parameter to ```Bearing matrix``` exposes related parameters. For more information, see Bending Parameter Dependencies.

Translational stiffness for the mounting at the base end of the shaft, $\left[\begin{array}{cc}{K}_{Bxx}& {K}_{Byy}\end{array}\right]$. The first element is the support stiffness in the x direction. The second element is the support stiffness in the y direction.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Bending parameter Base (B) mounting type is set to ```Bearing matrix```. For more information, see Shaft Parameter Dependencies and Bending Parameter Dependencies.

Rotational stiffness for the mounting at the base end of the shaft, $\left[\begin{array}{cc}{K}_{B\theta \theta }& {K}_{B\phi \phi }\end{array}\right]$. The first element is the support stiffness about the x direction. The second element is the support stiffness about the y direction.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Bending parameter Base (B) mounting type is set to ```Bearing matrix```. For more information, see Shaft Parameter Dependencies and Bending Parameter Dependencies.

Type of mounting at the follower end of the shaft.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`. For more information, see Shaft Parameter Dependencies.

Setting this parameter to ```Bearing matrix``` exposes related parameters. For more information, see Bending Parameter Dependencies.

Translational stiffness for the mounting at the follower end of the shaft, $\left[\begin{array}{cc}{K}_{Fxx}& {K}_{Fyy}\end{array}\right]$. The first element is the support stiffness in the x direction. The second element is the support stiffness in the y direction.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Bending parameter Follower (F) mounting type is set to ```Bearing matrix```. For more information, see Shaft Parameter Dependencies and Bending Parameter Dependencies.

Rotational stiffness for the mounting at the follower end of the shaft, $\left[\begin{array}{cc}{K}_{F\theta \theta }& {K}_{F\phi \phi }\end{array}\right]$. The first element is the support stiffness about the x direction. The second element is the support stiffness about the y direction.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Bending parameter Follower (F) mounting type is set to ```Bearing matrix```. For more information, see Shaft Parameter Dependencies and Bending Parameter Dependencies.

Damping constant, a, proportional to mass.

When the eigenmodes bending model is enabled, a translation damper in each modal mass spring damper system has the damping coefficient aMMode, where MMode is the modal mass.

When the lumped parameter bending model is enabled, each lumped mass element has a translation damping coefficient aMelement and rotation damping coefficient aIelement. Melement is the combined mass of the shaft lumped mass element and a rigid point mass or disk, if a rigid point mass or disk is attached to that element. Ielement is the combined mass moment of inertia of the shaft lumped mass element and a rigid disk, if a rigid disk is attached to that element.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`. For more information, see Shaft Parameter Dependencies.

Damping coefficients for a damper that acts on shaft translation motion at the base and follower.

The first element is for damping at the base and the second element is for damping at the follower. X and Y directions have the same damping.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Shaft parameter Bending vibration analysis method is set to `Lumped mass`. For more information, see Shaft Parameter Dependencies and Bending Parameter Dependencies.

Damping coefficient for a damper that acts on shaft rotation motion at the base and follower ends in both the X and Y directions. The first element is for damping at the base and the second element is for damping at the follower. X and Y directions have the same damping.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Shaft parameter Bending vibration analysis method is set to `Lumped mass`. For more information, see Shaft Parameter Dependencies and Bending Parameter Dependencies.

When the eigenmodes bending model is enabled, a translation damper in each modal mass spring damper system has the damping coefficient bKMode, where KMode is the modal stiffness.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Shaft parameter Bending vibration analysis method is set to `Eigenmodes`. For more information, see Shaft Parameter Dependencies and Bending Parameter Dependencies.

Type, if any, of rigid masses attached to shaft.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Shaft parameter Bending vibration analysis method is set to `Eigenmodes`. For more information, see Shaft Parameter Dependencies.

Setting this parameter to `Point mass` or `Disk` exposes related parameters. For more information, see Bending Parameter Dependencies.

Rigid mass locations along shaft in distance from base node. For multiple masses, specify a row vector.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Bending parameter Rigid masses attached to shaft is set to ```Point mass```. For more information, see Shaft Parameter Dependencies and Bending Parameter Dependencies.

Mass of rigid masses concentrically attached to shaft. For multiple masses, specify a row vector.

Rigid mass moments of inertia about axis perpendicular to the shaft. For multiple masses, specify a row vector.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Bending parameter Rigid masses attached to shaft is set to `Disk`. For more information, see Shaft Parameter Dependencies and Bending Parameter Dependencies.

Rigid polar mass moments of inertia about axis perpendicular to shaft.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Bending parameter Rigid masses attached to shaft is set to `Disk`. For more information, see Shaft Parameter Dependencies and Bending Parameter Dependencies.

Static unbalances that excite bending.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`. For more information, see Shaft Parameter Dependencies.

Distance of excitational static unbalances from the base, B end of the shaft.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`. For more information, see Shaft Parameter Dependencies.

Initial angle, about centerline of shaft relative to the x-axis, of the excitational static unbalances.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`. For more information, see Shaft Parameter Dependencies.

Advanced Bending parameters are exposed only when the Shaft parameter Model bending is set to `On`. For more information, see Shaft Parameter Dependencies.

The table shows how the visibility of some parameters depends on the option that you choose for other parameters. To learn how to read the table, see Parameter Dependencies.

Bending vibration analysis method — Choose `Lumped mass` or `Eigenmodes`

The option that you choose for this parameter affects the visibility of some Bending parameters. For more information, see Shaft Parameter Dependencies.

Lumped massEigenmodes
Bending mode determination — Choose `Simscape determined` or `User defined`
Simscape determinedUser defined
Limit number of modes
Nominal shaft speed for bending modes

Eigenfrequency initial search resolution

Modal frequencies
Eigenfrequency refine factorShaft position
Eigenfrequency upper limitMode shape matrix

Shaft length increments for mode shape computations

Method for analyzing the bending vibration.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`. For more information, see Shaft Parameter Dependencies.

Setting this parameter to `Eigenmodes` exposes related Advanced Bending parameters. For more information, see Advanced Bending Parameter Dependencies.

The option that you choose for this parameter also affects the visibility of some Bending parameters. For more information, see Bending Parameter Dependencies.

Option to have Simscape determine the eigenmode frequencies and shapes or to define the bending mode frequencies and shapes using parameters that you specify.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`. Each option for this parameter exposes related parameters and hides unrelated parameters. For more information, see Shaft Parameter Dependencies and Advanced Bending Parameter Dependencies.

Maximum number of modes that Simscape determines.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`, the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`, and the Advanced Bending parameter Bending mode determination is set to `Simscape determined`. For more information, see Shaft Parameter Dependencies and Advanced Bending Parameter Dependencies.

Nominal shaft speed for bending modes.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On` and the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`. For more information, see Shaft Parameter Dependencies and Advanced Bending Parameter Dependencies.

Initial resolution for the eigenfrequency search.

The algorithm for determining the eigenfrequencies considers the values [0::ɷMax] as possible eigenfrequencies, where is the eigenfrequency search resolution and ɷMax is the eigengrequency upper limit.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`, the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`, and the Advanced Bending parameter Bending mode determination is set to `Simscape determined`. For more information, see Shaft Parameter Dependencies and Advanced Bending Parameter Dependencies.

Refinement factor for the resolution for the eigenfrequency search.

For each eigenfrequency, ɷkINITIAL, that is found using the eigenfrequency search resolution, the algorithm refines with eigenfrequency value by repeating the solution procedure when considering ɷkINITIAL+*[-1:RefineFactor:1] as possible eigenfrequencies. This refinement procedure may be important for obtaining the correct eigenmode shape.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`, the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`, and the Advanced Bending parameter Bending mode determination is set to `Simscape determined`. For more information, see Shaft Parameter Dependencies and Advanced Bending Parameter Dependencies.

Eigenfrequency upper limit.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`, the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`, and the Advanced Bending parameter Bending mode determination is set to `Simscape determined`. For more information, see Shaft Parameter Dependencies and Advanced Bending Parameter Dependencies.

Shaft length increments used for mode mass and shape computations.

The mode shapes are defined at the distances from the base, [0:dz:L], where L is the shaft length. These mode shape vectors are numerically integrated to determine the modal mass, stiffness, and damping.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`, the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`, and the Advanced Bending parameter Bending mode determination is set to `Simscape determined`. For more information, see Shaft Parameter Dependencies and Advanced Bending Parameter Dependencies.

Modal frequencies.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`, the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`, and the Advanced Bending parameter Bending mode determination is set to `User defined`. For more information, see Shaft Parameter Dependencies and Advanced Bending Parameter Dependencies.

Shaft position.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`, the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`, and the Advanced Bending parameter Bending mode determination is set to `User defined`. For more information, see Shaft Parameter Dependencies and Advanced Bending Parameter Dependencies.

The mode shape matrix has dimensions n-by-2m, where n is the length of elements in the Shaft position vector and m is the length of Modal frequencies vector. The mode shape matrix has the form , where the column vectors Uix and Uiy are the mode shape deflections in the x and y directions, respectively, for the ${i}^{th}$ node. The algorithm accounts for the properties specified for the Shaft and Bending parameters that are used to compute the modal properties.

#### Dependencies

This parameter is visible only when the Shaft parameter Model bending is set to `On`, the Advanced Bending parameter Bending vibration analysis method is set to `Eigenmodes`, and the Advanced Bending parameter Bending mode determination is set to `User defined`. For more information, see Shaft Parameter Dependencies and [6].

### Initial Conditions

Angular deflection of the shaft at the start of simulation.

A positive initial deflection results in a positive rotation of the base, B, end of the shaft relative to the follower, F, end of the shaft.

Angular velocity of the base, B, end of the shaft relative to the follower, F, end of the shaft at the start of simulation.

## References

[1] Adams, M.L., Rotating Machinery Vibration. CRC Press, New York, 2010.

[2] Bathe, K. J., Finite Element Procedures. Prentice Hall, Inc, United States, 1996.

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

[4] Miller, S., Soares, T., Van Weddingen, Y., Wendlandt, J., Modeling Flexible Bodies with Simscape Multibody. The MathWorks, 2017.

[5] Muszynska, A., Rotordynamics, Taylor & Francis, 2005

[6] Rao, S.S., Vibration of Continuous Systems. John Wiley & Sons, Hoboken, NJ, 2007.

[7] Wu, J.S., Lin, F.T., Shaw, H.J., Analytical Solution for Whirling Speeds and Mode Shapes of a Distributed-Mass Shaft With Arbitrary Rigid Disks. ASME Journal of Applied Mechanics, Volume 81, 2014.