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Modeling Gyroscopic Effect
Introduction
Gyroscopes are used for measuring the orientation or maintaining the stability of airplane, spacecraft, and submarine vehicles in general. They are also used as sensors in inertial guidance systems.
This model demonstrates the modeling of a mechanical gyroscope. It analyzes the response of a spinning disc to an external torque coming on the disc due to the rotation of the frame. It is shown that the disc is able to maintain its orientation when spinning with a high speed. This fact can be explained with the principle of conservation of angular momentum.
In the second part of the model, the motion of a spinning top is analyzed. The external torque induced precession and nutation motion of the spinning top is computed.
Application Library path: Multibody_Dynamics_Module/Tutorials/gyroscope
Model Definition: Gyroscope
The geometry of a gyroscope, shown in Figure 1, consists of four parts: the frame, the outer gimbal, the inner gimbal, and the spinning disc. All these parts are assumed to be rigid and they are interconnected by hinge joints.
The frame is mounted on a platform which has a rotating motion with harmonically varying magnitude. The magnitude of rotation is 2 rad and the angular frequency is 4π Hz. The disc is spinning at 350 rad/s about its own axis. The orientation of the spinning disc is analyzed under the torques acting on it because of the frame motion.
Figure 1: Model geometry of a gyroscope.
Results and Discussion
The disc orientation and the position of the gimbals for the specified frame motion, can be seen in Figure 2 and Figure 3. It can be seen that the disc approximately maintains its orientation when it is spinning with high speed. In the case when the disc is not spinning, it does not offer much resistance to the torques acting on it and it fails to maintain its orientation. A comparison of the inclination angle of the disc for both the cases, spinning and not spinning, can be seen in Figure 4.
Figure 2: Displacement of the gyroscope components at t = 0.25 sec when the disc is not spinning.
Figure 3: Displacement of the gyroscope components at t = 0.25 sec when the disc is spinning.
Figure 4: Comparison of the inclination angle of the spinning disc.
Notes About the COMSOL Implementation
•
In this model, parts are modeled as rigid elements using Rigid Material nodes as we are only interested in the kinematics of the mechanism. Parts can be modeled as flexible elements using the Linear Elastic Material node if the stresses and deformations in the parts are also of interest.
•
Two different parts of the gyroscope are connected together with a pair of hinges. Both the hinges have the same axis and they are placed diagonally opposite to each other. As the parts are rigid and force computation is not of interest, it is sufficient to use only one hinge joint.
•
The center of hinge joint can be chosen anywhere on the joint axis. In this model, for the convenience of modeling, the center of joint for all the joints is chosen as the geometric center of the gyroscope.
Model Definition: Spinning Top (Building instructions given at the end)
The geometry of a spinning top is shown in Figure 5. This geometry is the spinning disc of the gyroscope assembly having an initial orientation of 20o from vertical.
The translational motion of the bottom point of the spinning top is constrained. The spinning top is rotating about its axis with an angular speed of 350 rad/s. The motion of the spinning top is analyzed under gravity load.
Figure 5: Model Geometry of a spinning top.
Instead of falling under gravity, the spinning top does have two types of motion:
Precession
If the axis of a spinning top is inclined to the vertical, the trajectory of the axis generates a vertical circular cone, so that the angle between the spinning top axis and the vertical remains constant during rotation. This kind of motion for a spinning top under an external torque is called forced or torque-induced regular precession.
Nutation
Precession is often accompanied by nutation which can be described as a fast shivering of the precessing axis.
Results and Discussion
Figure 6 and Figure 7 show the total displacement of the spinning top at the beginning and at t = 0.875 sec respectively. The trajectory of the topmost point of the spinning top is also shown in the latter plot.
Figure 6: Total displacement of the spinning top at the beginning.
Figure 8 shows the locus of the tip of the spinning top in the xy-plane. The precession and nutation motion exhibited by the spinning top can be seen in this plot.
The precession and nutation velocities of the tip of the spinning top are shown in Figure 9. Both the velocity components are varying harmonically with the same frequency, but they have a phase shift of 90o. It can also be seen that the mean value of the nutation velocity is zero whereas the precession velocity is having a nonzero mean value with the minimum value being zero. That means the spinning top is precessing continuously in one direction, however, it is nearly stopping periodically for a small duration.
Figure 7: Total displacement of the spinning top at t = 0.875 sec. The trajectory of the topmost point can also be seen.
Figure 8: Locus of the tip of the spinning top in the xy-plane.
Figure 9: Precession and nutation velocity of the tip of the spinning top.
Notes About the COMSOL Implementation
The angular velocity of spinning is an order of magnitude higher than the angular velocity of precession. Hence, care should be taken to take sufficiently small time steps to properly resolve the spinning motion in the time scale.
Modeling Instructions (Gyroscope)
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Structural Mechanics > Multibody Dynamics (mbd).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
Component 1: Gyroscope
1
In the Model Builder window, click Component 1 (comp1).
2
In the Settings window for Component, type Component 1: Gyroscope in the Label text field.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
omega
350[rad/s]
350 rad/s
Angular velocity
theta0
20[deg]
0.34907 rad
Inclination angle of spinning top
Geometry 1
Import 1 (imp1)
1
In the Geometry toolbar, click  Import.
2
In the Settings window for Import, locate the Source section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file gyroscope.mphbin.
5
Click  Import.
Form Union (fin)
1
In the Model Builder window, under Component 1: Gyroscope (comp1) > Geometry 1 click Form Union (fin).
2
In the Settings window for Form Union/Assembly, locate the Form Union/Assembly section.
3
From the Action list, choose Form an assembly.
4
Clear the Create pairs checkbox.
5
In the Geometry toolbar, click  Build All.
Definitions
Waveform 1 (wv1)
1
In the Definitions toolbar, click  More Functions and choose Waveform.
2
In the Settings window for Waveform, locate the Parameters section.
3
In the T text field, type 0.5[s].
4
In the A text field, type 2.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Aluminum.
4
Click the Add to Component button in the window toolbar.
5
In the tree, select Built-in > Structural steel.
6
Click the Add to Component button in the window toolbar.
7
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Structural steel (mat2)
Use Structural steel for the Spinning Disc to increase its inertia for a higher angular momentum.
1
Select Domain 3 only.
Multibody Dynamics (mbd)
Rigid Material: Frame
1
In the Physics toolbar, click  Domains and choose Rigid Material.
2
In the Settings window for Rigid Material, type Rigid Material: Frame in the Label text field.
3
Select Domain 2 only.
Prescribed Displacement/Rotation 1
1
In the Physics toolbar, click  Attributes and choose Prescribed Displacement/Rotation.
2
In the Settings window for Prescribed Displacement/Rotation, locate the Prescribed Displacement at Center of Rotation section.
3
Select the Prescribed in x direction checkbox.
4
Select the Prescribed in y direction checkbox.
5
Select the Prescribed in z direction checkbox.
6
Locate the Prescribed Rotation section. From the By list, choose Prescribed rotation.
7
Specify the Ω vector as
1
x
1
y
0
z
8
In the ϕ0 text field, type wv1(t).
Rigid Material: Outer Gimbal
1
In the Physics toolbar, click  Domains and choose Rigid Material.
2
In the Settings window for Rigid Material, type Rigid Material: Outer Gimbal in the Label text field.
3
Select Domain 1 only.
Rigid Material: Inner Gimbal
1
In the Physics toolbar, click  Domains and choose Rigid Material.
2
In the Settings window for Rigid Material, type Rigid Material: Inner Gimbal in the Label text field.
3
Select Domain 4 only.
Rigid Material: Spinning Disc
1
In the Physics toolbar, click  Domains and choose Rigid Material.
2
In the Settings window for Rigid Material, type Rigid Material: Spinning Disc in the Label text field.
3
Select Domain 3 only.
4
Locate the Initial Values section. From the list, choose Locally defined.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values: Rotational section.
3
Specify the ω vector as
0
x
0
y
omega
z
Rigid Material: Frame, Rigid Material: Inner Gimbal, Rigid Material: Outer Gimbal, Rigid Material: Spinning Disc
1
In the Model Builder window, under Component 1: Gyroscope (comp1) > Multibody Dynamics (mbd), Ctrl-click to select Rigid Material: Frame, Rigid Material: Outer Gimbal, Rigid Material: Inner Gimbal, and Rigid Material: Spinning Disc.
2
Right-click and choose Group.
Rigid Materials
In the Settings window for Group, type Rigid Materials in the Label text field.
Frame-Outer Gimbal
1
In the Physics toolbar, click  Global and choose Hinge Joint.
2
In the Settings window for Hinge Joint, type Frame-Outer Gimbal in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Rigid Material: Frame.
4
From the Destination list, choose Rigid Material: Outer Gimbal.
5
Locate the Center of Joint section. From the list, choose User defined.
Outer Gimbal-Inner Gimbal
1
Right-click Frame-Outer Gimbal and choose Duplicate.
2
In the Settings window for Hinge Joint, type Outer Gimbal-Inner Gimbal in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Rigid Material: Outer Gimbal.
4
From the Destination list, choose Rigid Material: Inner Gimbal.
5
Locate the Axis of Joint section. Specify the e0 vector as
0
x
1
y
0
z
Inner Gimbal-Spinning Disc
1
Right-click Outer Gimbal-Inner Gimbal and choose Duplicate.
2
In the Settings window for Hinge Joint, type Inner Gimbal-Spinning Disc in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Rigid Material: Inner Gimbal.
4
From the Destination list, choose Rigid Material: Spinning Disc.
5
Locate the Axis of Joint section. Specify the e0 vector as
0
x
0
y
1
z
Frame-Outer Gimbal, Inner Gimbal-Spinning Disc, Outer Gimbal-Inner Gimbal
1
In the Model Builder window, under Component 1: Gyroscope (comp1) > Multibody Dynamics (mbd), Ctrl-click to select Frame-Outer Gimbal, Outer Gimbal-Inner Gimbal, and Inner Gimbal-Spinning Disc.
2
Right-click and choose Group.
Hinge Joints
In the Settings window for Group, type Hinge Joints in the Label text field.
Mesh 1
1
In the Model Builder window, under Component 1: Gyroscope (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Fine.
4
In the table, clear the Use checkbox for Geometric Analysis, Detail Size.
5
Click  Build All.
Definitions
Define a variable for measuring the inclination angle of the spinning disc with the vertical axis. This variable can be written as a function of mbd.hgj3.e1z, the joint axis, z component variable for the Inner Gimbal-Spinning Disc node (listed in the Variables table of the node’s Equation View subnode).
Variables 1
1
In the Model Builder window, under Component 1: Gyroscope (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
theta
acos(mbd.hgj3.e1z)
rad
Inclination angle of spinning disc
View 1
1
In the Model Builder window, click View 1.
2
In the Settings window for View, locate the View section.
3
Clear the Show grid checkbox.
Study 1: Gyroscope
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1: Gyroscope in the Label text field.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
omega (Angular velocity)
0 350
rad/s
Step 1: Time Dependent
1
In the Model Builder window, click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,0.0025,0.25).
In order to obtain accurate results, limit the maximum time step by following the instructions below.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, click to expand the Time Stepping section.
4
From the Maximum step constraint list, choose Constant.
5
In the Maximum step text field, type 1e-4+9e-4*(omega==0).
6
In the Study toolbar, click  Compute.
Results
Displacement (mbd)
1
Click the  Zoom Extents button in the Graphics toolbar.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Parameter value (omega (rad/s)) list, choose 0.
4
In the Displacement (mbd) toolbar, click  Plot.
5
From the Parameter value (omega (rad/s)) list, choose 350.
6
In the Displacement (mbd) toolbar, click  Plot.
Animation: Not spinning gyro
1
In the Results toolbar, click  Animation and choose File.
2
In the Settings window for Animation, type Animation: Not spinning gyro in the Label text field.
3
Locate the Target section. From the Target list, choose Player.
4
Locate the Frames section. In the Number of frames text field, type 50.
Animation: Spinning gyro
1
In the Results toolbar, click  Animation and choose File.
2
In the Settings window for Animation, type Animation: Spinning gyro in the Label text field.
3
Locate the Target section. From the Target list, choose Player.
4
Locate the Frames section. In the Number of frames text field, type 50.
5
Locate the Animation Editing section. From the Parameter value (omega (rad/s)) list, choose 350.
To compare the above two cases, plot the inclination angle of the disc with time as shown in Figure 4.
Spinning disc orientation
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Spinning disc orientation in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1: Gyroscope/Parametric Solutions 1 (sol2).
4
Click to expand the Title section. From the Title type list, choose Label.
5
Locate the Legend section. From the Position list, choose Upper left.
Global 1
1
Right-click Spinning disc orientation and choose Global.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1: Gyroscope (comp1) > Definitions > Variables > theta - Inclination angle of spinning disc - rad.
3
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
theta
deg
Inclination angle of spinning disc
4
Click to expand the Coloring and Style section. From the Width list, choose 2.
5
Find the Line markers subsection. From the Marker list, choose Cycle.
6
From the Positioning list, choose Interpolated.
7
In the Number text field, type 25.
8
Click to expand the Legends section. From the Legends list, choose Manual.
9
In the table, enter the following settings:
Legends
Without spinning
With spinning
10
In the Spinning disc orientation toolbar, click  Plot.
11
Click the  Zoom Extents button in the Graphics toolbar.
Modeling Instructions (Spinning Top)
Add Component
In the Model Builder window, right-click the root node and choose Add Component > 3D.
Component 2: Spinning top
In the Settings window for Component, type Component 2: Spinning top in the Label text field.
Geometry 2
Import 1 (imp1)
1
In the Geometry toolbar, click  Import.
2
In the Settings window for Import, locate the Source section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file gyroscope.mphbin.
5
Click  Import.
Delete the frame, inner gimbal, and outer gimbal from the gyroscope assembly.
Delete Entities 1 (del1)
1
In the Model Builder window, right-click Geometry 2 and choose Delete Entities.
2
In the Settings window for Delete Entities, locate the Entities or Objects to Delete section.
3
From the Geometric entity level list, choose Domain.
4
On the object imp1(1), select Domain 1 only.
5
On the object imp1(2), select Domain 1 only.
6
On the object imp1(4), select Domain 1 only.
7
Click  Build Selected.
Rotate 1 (rot1)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
Select the object imp1(3) only.
3
In the Settings window for Rotate, locate the Rotation section.
4
In the Angle text field, type -theta0.
5
Locate the Point on Axis of Rotation section. In the z text field, type -0.0835.
6
Locate the Rotation section. From the Axis type list, choose x-axis.
7
Click  Build All Objects.
Define a rotated coordinate system with the same orientation as the spinning top.
Definitions (comp2)
Rotated System 3 (sys3)
1
In the Definitions toolbar, click  Coordinate Systems and choose Rotated System.
2
In the Settings window for Rotated System, locate the Rotation section.
3
Find the Euler angles subsection. In the β text field, type -theta0.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Aluminum.
4
Click the Add to Component button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Structural Mechanics > Multibody Dynamics (mbd).
4
Find the Physics interfaces in study subsection. In the table, clear the Solve checkbox for Study 1: Gyroscope.
5
Click the Add to Component 2: Spinning Top button in the window toolbar.
6
In the Home toolbar, click  Add Physics to close the Add Physics window.
Multibody Dynamics 2 (mbd2)
Rigid Material 1
1
In the Physics toolbar, click  Domains and choose Rigid Material.
2
Select Domain 1 only.
3
In the Settings window for Rigid Material, locate the Initial Values section.
4
From the list, choose Locally defined.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Coordinate System Selection section.
3
From the Coordinate system list, choose Rotated System 3 (sys3).
4
Locate the Initial Values: Rotational section. Specify the ω vector as
0
x1
0
x2
omega
x3
5
Locate the Center of Rotation section. From the list, choose Centroid of selected entities.
6
From the Entity level list, choose Point.
Center of Rotation: Point 1
1
In the Model Builder window, click Center of Rotation: Point 1.
2
Select Point 23 only.
It might be easier to select the correct point by using the Selection List window. To open this window, in the Home toolbar click Windows and choose Selection List. (If you are running the cross-platform desktop, you find Windows in the main menu.)
Rigid Material 1
In the Model Builder window, under Component 2: Spinning top (comp2) > Multibody Dynamics 2 (mbd2) click Rigid Material 1.
Prescribed Displacement/Rotation 1
1
In the Physics toolbar, click  Attributes and choose Prescribed Displacement/Rotation.
2
In the Settings window for Prescribed Displacement/Rotation, locate the Prescribed Displacement at Center of Rotation section.
3
Select the Prescribed in x direction checkbox.
4
Select the Prescribed in y direction checkbox.
5
Select the Prescribed in z direction checkbox.
6
Locate the Center of Rotation section. From the list, choose Centroid of selected entities.
7
From the Entity level list, choose Point.
Center of Rotation: Point 1
1
In the Model Builder window, click Center of Rotation: Point 1.
2
Select Point 23 only.
Mesh 2
1
In the Model Builder window, under Component 2: Spinning top (comp2) click Mesh 2.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
In the table, clear the Use checkbox for Geometric Analysis, Detail Size.
4
Click  Build All.
Multibody Dynamics 2 (mbd2)
Gravity 1
In the Physics toolbar, click  Global and choose Gravity.
Definitions (comp2)
Variables 2
1
In the Model Builder window, under Component 2: Spinning top (comp2) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
th
atan2(y,x)
rad
Precession angle
vp
cos(th+pi/2)*mbd2.u_tX+sin(th+pi/2)*mbd2.u_tY
m/s
Precession velocity
vn
cos(th)*mbd2.u_tX+sin(th)*mbd2.u_tY
m/s
Nutation velocity
View 2
1
In the Model Builder window, click View 2.
2
In the Settings window for View, locate the View section.
3
Clear the Show grid checkbox.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Time Dependent.
4
Find the Physics interfaces in study subsection. In the table, clear the Solve checkbox for Multibody Dynamics (mbd).
5
Click the Add Study button in the window toolbar.
6
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2: Spinning top
In the Settings window for Study, type Study 2: Spinning top in the Label text field.
Step 1: Time Dependent
1
In the Model Builder window, under Study 2: Spinning top click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,0.007,1.4).
In order to obtain accurate results, limit the maximum time step by following the instructions below.
Solution 5 (sol5)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 5 (sol5) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, locate the Time Stepping section.
4
From the Maximum step constraint list, choose Constant.
5
In the Maximum step text field, type 1e-4.
6
In the Study toolbar, click  Compute.
Results
Displacement (mbd2)
Add the trajectory of the topmost point of the spinning top to the default Displacement plot.
Point Trajectories 1
1
In the Displacement (mbd2) toolbar, click  More Plots and choose Point Trajectories.
2
Select Point 36 only.
3
In the Settings window for Point Trajectories, locate the Coloring and Style section.
4
Find the Line style subsection. From the Type list, choose Tube.
Color Expression 1
1
Right-click Point Trajectories 1 and choose Color Expression.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type t.
4
Locate the Coloring and Style section. Clear the Color legend checkbox.
5
In the Displacement (mbd2) toolbar, click  Plot.
Displacement (mbd2)
Follow the instructions below to plot the spinning top in various positions as shown in Figure 6 and Figure 7.
1
In the Model Builder window, under Results click Displacement (mbd2).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Time (s) list, choose 0.
4
In the Displacement (mbd2) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
6
From the Time (s) list, choose 0.875.
7
In the Displacement (mbd2) toolbar, click  Plot.
8
Click the  Zoom Extents button in the Graphics toolbar.
Animation: Spinning top
1
In the Results toolbar, click  Animation and choose File.
2
In the Settings window for Animation, type Animation: Spinning top in the Label text field.
3
Locate the Target section. From the Target list, choose Player.
4
Locate the Scene section. From the Subject list, choose Displacement (mbd2).
5
Locate the Frames section. In the Number of frames text field, type 50.
Plot the locus of a point on the upper surface of the spinning top as shown in Figure 8 by following the instructions below.
Tip locus (xy-plane)
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Tip locus (xy-plane) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2: Spinning top/Solution 5 (4) (sol5).
4
Locate the Title section. From the Title type list, choose Label.
5
Locate the Axis section. Select the Preserve aspect ratio checkbox.
6
Locate the Grid section. Select the Manual spacing checkbox.
7
In the x spacing text field, type 0.01.
8
In the y spacing text field, type 0.01.
Point Graph 1
1
Right-click Tip locus (xy-plane) and choose Point Graph.
2
Select Point 36 only.
3
In the Settings window for Point Graph, locate the y-Axis Data section.
4
In the Expression text field, type y.
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type x.
7
Click to expand the Coloring and Style section. From the Width list, choose 2.
8
In the Tip locus (xy-plane) toolbar, click  Plot.
9
Click the  Zoom Extents button in the Graphics toolbar.
Tip velocity (xy-plane)
1
In the Model Builder window, right-click Tip locus (xy-plane) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Tip velocity (xy-plane) in the Label text field.
3
Locate the Plot Settings section.
4
Select the y-axis label checkbox. In the associated text field, type Velocity (m/s).
5
Locate the Axis section. Clear the Preserve aspect ratio checkbox.
6
Locate the Grid section. Clear the Manual spacing checkbox.
Point Graph 1
1
In the Model Builder window, expand the Tip velocity (xy-plane) node, then click Point Graph 1.
2
In the Settings window for Point Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 2: Spinning top (comp2) > Definitions > Variables > vp - Precession velocity - m/s.
3
Locate the x-Axis Data section. From the Parameter list, choose Time.
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Cycle.
5
Click to expand the Legends section. Select the Show legends checkbox.
6
From the Legends list, choose Manual.
7
In the table, enter the following settings:
Legends
Precession
Point Graph 2
1
Right-click Results > Tip velocity (xy-plane) > Point Graph 1 and choose Duplicate.
2
In the Settings window for Point Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 2: Spinning top (comp2) > Definitions > Variables > vn - Nutation velocity - m/s.
3
Locate the Legends section. In the table, enter the following settings:
Legends
Nutation
4
In the Tip velocity (xy-plane) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.