Bushing

Applies translational and rotational stiffness and damping in three directions between two frames

 Description The equations for the Bushing component are given below. Reaction Forces: where $\mathbf{F}$is the reaction force vector, $\mathbf{r}={\mathbf{r}}_{b}\mathbf{-}{\mathbf{r}}_{a}=⟨\left({x}_{b}-{x}_{a}\right),\left({y}_{b}-{y}_{a}\right),\left({z}_{b}-{z}_{a}\right)⟩$, is the relative displacement vector, ${\mathbit{r}}_{0}$, is the undeformation distance frame_b with respect to frame_a expressed along Inboard frame (frame_a), and $\mathbf{v}=\frac{d}{\mathrm{dt}}\mathbf{r},$ ${\mathbf{K}}_{s}$$=\left[\begin{array}{ccc}{k}_{x}& 0& 0\\ 0& {k}_{y}& 0\\ 0& 0& {k}_{z}\end{array}\right],$ is the diagonal stiffness coefficient matrix, and, ${\mathbf{K}}_{d}$$=\left[\begin{array}{ccc}{d}_{x}& 0& 0\\ 0& {d}_{y}& 0\\ 0& 0& {d}_{z}\end{array}\right],$ is the diagonal damping coefficient matrix.   Reaction Torques: T where $\mathbf{T}=⟨{\mathrm{\tau }}_{x},{\mathrm{\tau }}_{y},{\mathrm{\tau }}_{z}⟩,$is the reaction torque vector, $\mathbf{\theta }$are the Euler angles - defined by Rotation Sequence parameter - calculated from the relative rotation matrix of frame_b with respect to frame_a (see Euler Angle Sensor), ${\mathbf{\theta }}_{0}$ designates the undeformed rotation of frame_b with respect to frame_a, $\mathbf{\omega }$$=⟨{\mathrm{ω}}_{x},{\mathrm{ω}}_{y},{\mathrm{ω}}_{z}⟩,$ is the relative angular velocity vector, ${\mathbf{K}}_{\mathrm{θ}}$$=\left[\begin{array}{ccc}{k}_{\mathrm{a1}}& 0& 0\\ 0& {k}_{\mathrm{a2}}& 0\\ 0& 0& {k}_{\mathrm{a3}}\end{array}\right],$ is the diagonal angular stiffness coefficient matrix, and, ${\mathbf{K}}_{\mathrm{ω}}$$=\left[\begin{array}{ccc}{d}_{\mathrm{ax}}& 0& 0\\ 0& {d}_{\mathrm{ay}}& 0\\ 0& 0& {d}_{\mathrm{az}}\end{array}\right],$ is the diagonal angular damping coefficient matrix.

Connections

 Name Description Modelica ID ${\mathrm{frame}}_{a}$ Inboard frame frame_a ${\mathrm{frame}}_{b}$ Outboard frame frame_b $K$ Real signal of dimension 3x1. Three elements of diagonal translational stiffness matrix $\left\{{k}_{x},{k}_{y},{k}_{z}\right\}.$ K_in $D$ Real signal of dimension 3x1. Three elements of diagonal translational damping matrix $\left\{{d}_{x},{d}_{y},{d}_{z}\right\}.$ D_in ${K}_{a}$ Real signal of dimension 3x1. Three elements of diagonal rotational stiffness matrix $\left\{{k}_{\mathrm{a1}},{k}_{\mathrm{a1}},{k}_{\mathrm{a1}}\right\}.$ Ka_in ${D}_{a}$ Real signal of dimension 3x1. Three elements of diagonal rotational damping matrix $\left\{{d}_{\mathrm{ax}},{d}_{\mathrm{ay}},{d}_{\mathrm{az}}\right\}.$ Da_in $r$ Real signal of dimension 3x1. Relative displacement vector of frame_b with respect to frame_a, expressed in the frame selected by the  parameter. r_out $v$ Real signal of dimension 3x1. Relative velocity vector of frame_b with respect to frame_a, expressed in the frame selected by the $\mathrm{Resolved}$ parameter. v_out θ Real signal of dimension 3x1. Relative Euler angles describing the rotation of frame_b with respect to frame_a. Rotation sequence is defined by  parameter. th_out ω Real signal of dimension 3x1. Relative angular velocity vector of frame_b with respect to frame_a, expressed in the frame selected by the  parameter. w_out

Nonlinear Options

The translational and rotational spring and damping coefficients can be time varying. By selecting the nonlinear Boolean parameter (checked = true), the user is given four options to define variable coefficients:

 Data Source option Description 1 $\mathrm{inline}$ The tri-axial coefficients are interpolated from an n by 4 table entered by user. For ${\mathbf{K}}_{s}$, the first column is displacement [m]. Columns 2, 3, and 4 correspond to ${k}_{x}$, ${k}_{y}$, and ${k}_{z}$ ([N/m]), respectively. For ${\mathbf{K}}_{d}$, the first column is velocity [m/s]. Columns 2, 3, and 4 correspond to ${d}_{x}$, ${d}_{y}$, and ${d}_{z}$ ([N.s/m]), respectively. For ${\mathbf{K}}_{\mathrm{\theta }}$, the first column is angular displacement [rad]. Columns 2, 3, and 4 correspond to ${k}_{\mathrm{a1}}$, ${k}_{\mathrm{a2}}$, and ${k}_{\mathrm{a3}}$ ([N.m/rad]), respectively. For ${\mathbf{K}}_{\mathrm{\omega }}$, the first column is angular velocity [rad/s]. Columns 2, 3, and 4 correspond to ${d}_{\mathrm{ax}}$, ${d}_{\mathrm{ay}}$, and ${d}_{\mathrm{az}}$ ([N.m.s/rad]), respectively. To change the dimensions of a table, right-click (Control-click for Mac) and select Edit Matrix Dimensions. You can then specify the number of rows and columns to include in the table. 2 $\mathrm{attachment}$ The tri-axial coefficients are interpolated from a n by 4 attachment table. The data can be retrieved from an attached .csv, .xls, or .xlsx file. For more information, see Attaching a File to a Model. Data columns in the attached file are assumed to have the same order and attributes as those described in the inline option. 3 $\mathrm{file}$ The tri-axial coefficients are interpolated from a n by 4 table stored on disk. The data can be retrieved from an attached .csv, .xls, or .xlsx file. Data columns in the attached file are assumed to have the same order and attributes as those described in the inline option. 4 $\mathrm{input}$ The coefficients are defined via Real input signal ports.

Parameters

Modeling Parameters

 Symbol Default Units Description Modelica ID [1,2,3] $-$ Euler angles rotation sequence RotSeq ($\mathbf{false}$) $-$ Enables relative displacement sensor output sensor_T ($\mathbf{false}$) $-$ Enables relative velocity sensor output sensor_Tv ($\mathbf{false}$) $-$ Enables relative angular displacement sensor output (expressed using the Euler angles selected by ) sensor_R ($\mathbf{false}$) $-$ Enables relative angular velocity sensor output sensor_Rw ${r}_{0}$ [0,0,0] [m] Undeformed distance of frame_b with respect to frame_a (expressed in Inboard frame: frame_a) r0 ${\mathrm{θ}}_{0}$ [0,0,0] [rad] Undeformed rotation of frame_b with respect to frame_a (expressed using the Euler angles selected by ) theta0

Translational Stiffness

 Symbol Condition Default Units Description Modelica ID $\mathrm{Nonlinear}$ - ($\mathbf{false}$) $-$ When checked, activates options to define variable coefficients. nonlinear_TS $\mathrm{inline}$ $-$ Enumeration defining the data source of variable coefficients. See Nonlinear Options section above. DCM_TS ${K}_{\mathrm{s}}$ [4.5e6,4.5e6,8.0e5] [N/m] Spring constants: ${k}_{x}$, ${k}_{y}$, and ${k}_{z}$ TS [0,4.5e6,4.5e6,8.0e5] - Table for displacement-dependent coefficients. See Nonlinear Options section above. TS_table - - MapleSim attachment where data for displacement-dependent coefficients is stored. See Nonlinear Options section above. TS_data - - Path to a file where data for displacement-dependent coefficients is stored.  See Nonlinear Options section above. TS_fileName or $\mathrm{attachment}$ or 0 - Number of rows that are skipped from the top of the data table. sr_TS $\mathrm{Smoothness}$ or $\mathrm{attachment}$ or Linearly interpolate table points - Determines whether the data points will be interpolated linearly or with a cubic spline. sm_TS

Translational Damping

 Symbol Condition Default Units Description Modelica ID $\mathrm{Nonlinear}$ - ($\mathbf{false}$) $-$ When checked, activates options to define variable coefficients. nonlinear_TD $\mathrm{inline}$ $-$ Enumeration defining the data source of variable coefficients. See Nonlinear Options section above. DCM_TD ${K}_{\mathrm{d}}$ [1e4,1e4,1e4] [N.s/m] Damping constants: ${d}_{x}$, ${d}_{y}$, and ${d}_{z}$ TD [0,1e4,1e4,1e4] - Table for displacement-dependent coefficients. See Nonlinear Options section above. TD_table - - MapleSim attachment where data for displacement-dependent coefficients is stored. See Nonlinear Options section above. TD_data - - Path to a file where data for displacement-dependent coefficients is stored.  See Nonlinear Options section above. TD_fileName or $\mathrm{attachment}$ or 0 - Number of rows that are skipped from the top of the data table. sr_TD $\mathrm{Smoothness}$ or $\mathrm{attachment}$ or Linearly interpolate table points - Determines whether the data points will be interpolated linearly or with a cubic spline. sm_TD

Rotational Stiffness

 Symbol Condition Default Units Description Modelica ID $\mathrm{Nonlinear}$ - ($\mathbf{false}$) $-$ When checked, activates options to define variable coefficients. nonlinear_RS $\mathrm{inline}$ $-$ Enumeration defining the data source of variable coefficients. See Nonlinear Options section above. DCM_RS ${K}_{\mathrm{θ}}$ [2.6e3,2.6e3,1e2] [N.m/rad] Rotational spring constants: ${k}_{\mathrm{ax}}$, ${k}_{\mathrm{ay}}$, and ${k}_{\mathrm{az}}$ RS [0,2.6e3,2.6e3,1e2] - Table for displacement-dependent coefficients. See Nonlinear Options section above. RS_table - - MapleSim attachment where data for displacement-dependent coefficients is stored. See Nonlinear Options section above. RS_data - - Path to a file where data for displacement-dependent coefficients is stored.  See Nonlinear Options section above. RS_fileName or $\mathrm{attachment}$ or 0 - Number of rows that are skipped from the top of the data table. sr_RS $\mathrm{Smoothness}$ or $\mathrm{attachment}$ or Linearly interpolate table points - Determines whether the data points will be interpolated linearly or with a cubic spline. sm_RS

Rotational Damping

 Symbol Condition Default Units Description Modelica ID $\mathrm{Nonlinear}$ - ($\mathbf{false}$) $-$ When checked, activates options to define variable coefficients. nonlinear_RD $\mathrm{GUI}$ $-$ Enumeration defining the data source of variable coefficients. See Nonlinear Options section above. DCM_RD ${K}_{\mathrm{ω}}$ [26,26,5] [N.m.s/rad] Rotational damping constants: ${d}_{\mathrm{ax}}$, ${d}_{\mathrm{ay}}$, and ${d}_{\mathrm{az}}$ RD [0,26,26,5] - Table for displacement-dependent coefficients. See Nonlinear Options section above. RD_table - - MapleSim attachment where data for displacement-dependent coefficients is stored. See Nonlinear Options section above. RD_data - - Path to a file where data for displacement-dependent coefficients is stored.  See Nonlinear Options section above. RD_fileName or $\mathrm{attachment}$ or 0 - Number of rows that are skipped from the top of the data table. sr_RD $\mathrm{Smoothness}$ or $\mathrm{attachment}$ or Linearly interpolate table points - Determines whether the data points will be interpolated linearly or with a cubic spline. sm_RD

Initial Conditions

 Symbol Default Units Description Modelica ID ${\mathrm{IC}}_{r,v}$ Ignore - Indicates whether MapleSim ignores, tries to enforce, or strictly enforces the translational initial conditions MechTranTree ${\mathrm{s&conjugate0;}}_{0}$ [0,0,0] [m] Initial displacement of the center of mass frame at the start of the simulation. These values are expressed along the x-, y- and z-axis of the inboard frame respectively InitPos ${\stackrel{&conjugate0;}{v}}_{0}$ [0,0,0] [m/s] Initial velocity of the center of mass frame at the start of the simulation. These values are expressed along the x-, y- and z-axis of the inboard frame respectively InitVel ${\mathrm{IC}}_{\mathrm{θ},\mathrm{ω}}$ Ignore - Indicates whether MapleSim ignores, tries to enforce, or strictly enforces the rotational initial conditions MechRotTree ${\stackrel{&conjugate0;}{\theta }}_{0}$ [0,0,0] [rad] Initial rotation of the center of mass frame at the start of the simulation, based on the ${\mathrm{Type}}_{\mathrm{\theta }}$ parameter values InitAng ${\stackrel{&conjugate0;}{\omega }}_{0}$ [0,0,0] [rad/s] Initial velocity of the center of mass frame at the start of the simulation, based on the ${\mathrm{Type}}_{\mathrm{ω}}$ parameter values InitAngVel ${\mathrm{Type}}_{\mathrm{ω}}$ Euler - Indicates whether the initial angular velocity is expressed in the inboard or outboard frame.  If Euler is selected, the initial angular velocities are assumed to be the direct derivatives of the Euler angles defined by . AngVelType