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Mechanics of a Golf Swing
Introduction
How well you can strike a golf ball is not only determined by your muscle strength, but more importantly it is influenced by several other factors involved in the mechanics of the golf swing. The outcome of the golf stroke is basically determined by the movement of the club head just prior to the impact with the ball.
In this example, a multibody analysis of a golf swing is performed using the Multibody Dynamics interface in COMSOL Multiphysics. The aim of the analysis is to maximize the club head speed just prior to the impact with the ball. The wrist torque, which has an important role to play in improving the outcome of the stroke, is varied parametrically to see its effect on the club head speed.
Model Definition
Arm-Club (Two-Link) vs Shoulder–Arm–Club (Three-Link) Swing Model
A simple way to simulate the golf swing is by using a two-link model, where the arm and the club are the two links connected together by a hinge joint. In this model, the arm rotates about a fixed point, located at the base of the neck, and the club rotates about the wrist joint relative to the arm.
Figure 1: Diagram of two-link (left) and three-link (right) swing models.
The two-link model does not allow a sufficiently long backswing and is not actually a true representation of a real-life golf swing.
A better representation is the three-link model, which also includes the shoulder as a separate link. Adding one more link eliminates the problem related to the backswing. Hence, this three-link model is used in this analysis. To also consider the flexibility of the club shaft in a simple way, it is divided into two parts: the grip and the club. These parts are connected through a hinge joint with finite stiffness and damping.
Swing Mechanics
The torque profile applied by different body parts (shoulder, arms, and wrist) is assumed. It is limited by the maximum torque capacity of the respective parts. Among all applied torques, the wrist torque has quite an important role to play in getting the strike right.
While simulating the downswing of the club, the entire swing can be divided into two phases. In the first phase, arm and club rotate about the fixed point as a rigid assembly. In this phase, the arm and club are folded to minimize the inertia about the center of rotation, which allows the development of maximum angular velocity for the given arm-torque capacity. Here, the wrist is cocked to the maximum possible angle and the applied wrist torque tries to hold back the club in this position against the other two torques.
In the second phase, the wrist torque starts helping the shoulder and the arm torque by pushing the club forward to increase the club head speed to its maximum. The instance when the wrist torque changes its role is a crucial parameter in determining the stroke quality. To see its effect on the club head speed, the wrist torque switch time is varied parametrically.
Figure 2: Model geometry: center of mass of each link is also shown.
Geometry and Connections
The model geometry is shown in Figure 2, which consists of four links: the shoulder, arm, grip, and club. These links are connected to each other by four hinge joints. The initial values, also representing the limiting values in this case, of shoulder angle, arm swing angle, and the wrist-cock angle are also shown in Figure 2.
The geometric data, inertial properties, and joint properties are taken from Ref. 1. All the links are modeled as rigid bodies and their length, mass, and moment of inertia about the center of mass are given in Table 1.
Table 1: length, Mass, and Moment of inertia of linkages
Linkage
Length (m)
Mass (kg)
Moment of
inertia (kg·m2)
Shoulder
0.150
-
0.800
Arm
0.854
8.644
0.354
Grip
0.330
1.899
0.0136
Club
0.780
0.292
0.0257
The mass of the shoulder link is not required because this link is pivoted at its center of mass and has no translational motion.
Results and Discussion
Figure 3 shows the club head speed during the downswing for different switching times of the wrist torque. It can be observed that for tw = 0.15 s, the speed reaches its maximum before impact — this leads to early hitting. On the other hand, for tw = 0.23 s, the club head speed could not even reach its maximum at the time of impact.
For tw = 0.19 s, the club head speed is higher than the other two cases and close to the optimum value for the chosen geometrical parameters and muscle strength.
Figure 3: Club head speed during the downswing for different switching times of the wrist torque.
The driving torques, applied by the shoulder, arm, and the wrist are shown in Figure 4. The applied shoulder torque is assumed to start at its maximum positive value, after a short build-up time. The applied arm torque, which acts on the arm and reacts on the shoulder, increases linearly with time to its maximum positive value with a specified rate. The applied wrist torque, which acts on the club and reacts on the arm, is fully negative to start and switches to its maximum positive value at the specified time. The instance, when wrist torque switches to its maximum positive value, is the crucial parameter and hence its effect is studied in this analysis.
Figure 4: Time history of torque applied by the shoulder, arm, and wrist.
On the arm and wrist joint, the rotation is not fully free. It is limited in the forward and backward directions by the ligaments, muscles, joint shape, or a combination of all these. In our golf-swing analysis, the rotation limit in the backward direction is more important and this limiting value may vary from person to person.
In the beginning of the downswing, various parts try to rotate beyond their limiting values due to inertial forces. This is resisted by the stop which is modeled using the spring and damper on joints. The resistive moment applied by the stop changes the effective torque in the arm and the wrist as shown in Figure 4.
Figure 5 shows the relative rotation on the arm, wrist, and the shaft joints for tw = 0.19 s. It can be seen that during the downswing, the arm rotates with respect to the shoulder by approximately 20°. The wrist joint has the maximum rotation and the club (or the grip) rotates with respect to the arm by approximately 85°. The shaft joint has very little rotation compared to the wrist and the arm joint. Hence it can be said that, for the chosen values of stiffness and damping, the effect of shaft flexibility to the swing is negligible compared to other parameters.
Figure 5: Rotations on the arm, wrist, and the shaft joints.
Figure 6: Golf club trajectory for different values of wrist torque switch time. From left to right, results are displayed for the increasing values of the parameter.
Outcome of the Analysis
Maximum arm torque throughout the swing and very high arm speed in the beginning of the downswing can cause an early release, with the club head reaching its maximum speed before actually hitting the ball.
It can also be deduced that for the given torque capacity, it is potentially advantageous to have a long arm swing as well as a large wrist-cock limit angle. Furthermore, the extent to which the wrist can hold back the release is limited by its torque capacity. Therefore, your golfing skills are also strongly associated with the delayed release and the late hit.
Notes About the COMSOL Implementation
•
In this model, linkages are modeled as rigid elements using Rigid Domain node as we are only interested in the kinematics of the mechanism. Linkages can be modeled as flexible elements using the Linear Elastic Material node if the stresses and deformations in the linkages are also of interest.
•
The Mass and Moment of Inertia subnode of the Rigid Domain node is used to enter the inertia properties given at a certain point.
•
The connections set up in the model can be reviewed in the Joints Summary section at the physics node.
Reference
1. R.S. Sharp, “On the mechanics of the golf swing”, Proceedings of the Royal Society A, vol. 465, pp. 551–570, 2009.
Application Library path: Multibody_Dynamics_Module/Biomechanics/golf_swing_mechanics
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
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.
Global Definitions
Import the parameter list defining the mass and inertia properties of the linkages, joint stiffness, and maximum torque capacities.
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
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file golf_swing_mechanics_parameters.txt.
Geometry 1
Import 1 (imp1)
1
In the Home toolbar, click  Import.
2
In the Settings window for Import, locate the Import section.
3
Click Browse.
4
Browse to the model’s Application Libraries folder and double-click the file golf_swing_mechanics.mphbin.
5
Click Import.
Form Union (fin)
1
In the Model Builder window, under Component 1 (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 check box.
5
In the Home toolbar, click  Build All.
Definitions
Integration 1 (intop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, locate the Source Selection section.
3
From the Geometric entity level list, choose Point.
4
Select Point 14 only.
Variables 1
1
In the Model Builder window, right-click Definitions and choose Variables.
Import the expressions for the torque applied by different body parts.
2
In the Settings window for Variables, locate the Variables section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file golf_swing_mechanics_variables.txt.
Multibody Dynamics (mbd)
Rigid Domain: Shoulder
1
In the Model Builder window, under Component 1 (comp1) right-click Multibody Dynamics (mbd) and choose Rigid Domain.
2
In the Settings window for Rigid Domain, type Rigid Domain: Shoulder in the Label text field.
3
Select Domain 4 only.
Set the density to zero and use the Mass and Moment of Inertia subnode to add lumped mass and moment of inertia values.
4
Locate the Density section. From the ρ list, choose User defined.
Mass and Moment of Inertia 1
1
In the Physics toolbar, click  Attributes and choose Mass and Moment of Inertia.
2
In the Settings window for Mass and Moment of Inertia, locate the Mass and Moment of Inertia section.
3
In the Iz text field, type Ish.
Rigid Domain: Arm
1
In the Physics toolbar, click  Domains and choose Rigid Domain.
2
In the Settings window for Rigid Domain, type Rigid Domain: Arm in the Label text field.
3
Select Domain 2 only.
4
Locate the Density section. From the ρ list, choose User defined.
Mass and Moment of Inertia 1
1
In the Physics toolbar, click  Attributes and choose Mass and Moment of Inertia.
2
In the Settings window for Mass and Moment of Inertia, locate the Center of Mass section.
3
From the list, choose Centroid of selected entities.
4
From the Entity level list, choose Point.
5
Locate the Mass and Moment of Inertia section. In the m text field, type Ma.
6
In the Iz text field, type Ia.
Center of Mass: Point 1
1
In the Model Builder window, expand the Mass and Moment of Inertia 1 node, then click Center of Mass: Point 1.
2
Select Point 8 only.
Rigid Domain: Grip
1
In the Physics toolbar, click  Domains and choose Rigid Domain.
2
In the Settings window for Rigid Domain, type Rigid Domain: Grip in the Label text field.
3
Select Domain 1 only.
4
Locate the Density section. From the ρ list, choose User defined.
Mass and Moment of Inertia 1
1
In the Physics toolbar, click  Attributes and choose Mass and Moment of Inertia.
2
In the Settings window for Mass and Moment of Inertia, locate the Center of Mass section.
3
From the list, choose Centroid of selected entities.
4
From the Entity level list, choose Point.
5
Locate the Mass and Moment of Inertia section. In the m text field, type Mg.
6
In the Iz text field, type Ig.
Center of Mass: Point 1
1
In the Model Builder window, expand the Mass and Moment of Inertia 1 node, then click Center of Mass: Point 1.
2
Select Point 3 only.
Rigid Domain: Club
1
In the Physics toolbar, click  Domains and choose Rigid Domain.
2
In the Settings window for Rigid Domain, type Rigid Domain: Club in the Label text field.
3
Select Domain 3 only.
4
Locate the Density section. From the ρ list, choose User defined.
Mass and Moment of Inertia 1
1
In the Physics toolbar, click  Attributes and choose Mass and Moment of Inertia.
2
In the Settings window for Mass and Moment of Inertia, locate the Center of Mass section.
3
From the list, choose Centroid of selected entities.
4
From the Entity level list, choose Point.
5
Locate the Mass and Moment of Inertia section. In the m text field, type Mc.
6
In the Iz text field, type Ic.
Center of Mass: Point 1
1
In the Model Builder window, expand the Mass and Moment of Inertia 1 node, then click Center of Mass: Point 1.
2
Select Point 13 only.
Hinge Joint: Shoulder
1
In the Physics toolbar, click  Global and choose Hinge Joint.
2
In the Settings window for Hinge Joint, type Hinge Joint: Shoulder in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Fixed.
4
From the Destination list, choose Rigid Domain: Shoulder.
Center of Joint: Boundary 1
1
In the Model Builder window, expand the Hinge Joint: Shoulder node, then click Center of Joint: Boundary 1.
2
Select Boundary 13 only.
Hinge Joint: Shoulder
In the Model Builder window, click Hinge Joint: Shoulder.
Applied Force and Moment 1
1
In the Physics toolbar, click  Attributes and choose Applied Force and Moment.
2
In the Settings window for Applied Force and Moment, locate the Applied On section.
3
From the list, choose Joint.
4
Locate the Applied Force and Moment section. In the M text field, type Tsh.
Hinge Joint: Arm
1
In the Physics toolbar, click  Global and choose Hinge Joint.
2
In the Settings window for Hinge Joint, type Hinge Joint: Arm in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Rigid Domain: Shoulder.
4
From the Destination list, choose Rigid Domain: Arm.
Center of Joint: Boundary 1
1
In the Model Builder window, expand the Hinge Joint: Arm node, then click Center of Joint: Boundary 1.
2
Select Boundary 8 only.
Hinge Joint: Arm
In the Model Builder window, click Hinge Joint: Arm.
Spring and Damper 1
1
In the Physics toolbar, click  Attributes and choose Spring and Damper.
2
In the Settings window for Spring and Damper, locate the Spring and Damper: Rotational section.
3
In the kθ text field, type ka*(mbd.hgj2.th<0).
4
In the cθ text field, type ca*(mbd.hgj2.th<0).
Hinge Joint: Arm
In the Model Builder window, click Hinge Joint: Arm.
Applied Force and Moment 1
1
In the Physics toolbar, click  Attributes and choose Applied Force and Moment.
2
In the Settings window for Applied Force and Moment, locate the Applied On section.
3
From the list, choose Joint.
4
Locate the Applied Force and Moment section. In the M text field, type Ta.
Hinge Joint: Wrist
1
In the Physics toolbar, click  Global and choose Hinge Joint.
2
In the Settings window for Hinge Joint, type Hinge Joint: Wrist in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Rigid Domain: Arm.
4
From the Destination list, choose Rigid Domain: Grip.
Center of Joint: Boundary 1
1
In the Model Builder window, expand the Hinge Joint: Wrist node, then click Center of Joint: Boundary 1.
2
Select Boundary 1 only.
Hinge Joint: Wrist
In the Model Builder window, click Hinge Joint: Wrist.
Spring and Damper 1
1
In the Physics toolbar, click  Attributes and choose Spring and Damper.
2
In the Settings window for Spring and Damper, locate the Spring and Damper: Rotational section.
3
In the kθ text field, type kw*(mbd.hgj3.th<0).
4
In the cθ text field, type cw*(mbd.hgj3.th<0).
Hinge Joint: Wrist
In the Model Builder window, click Hinge Joint: Wrist.
Applied Force and Moment 1
1
In the Physics toolbar, click  Attributes and choose Applied Force and Moment.
2
In the Settings window for Applied Force and Moment, locate the Applied On section.
3
From the list, choose Joint.
4
Locate the Applied Force and Moment section. In the M text field, type Tw.
Hinge Joint: Shaft
1
In the Physics toolbar, click  Global and choose Hinge Joint.
2
In the Settings window for Hinge Joint, type Hinge Joint: Shaft in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Rigid Domain: Grip.
4
From the Destination list, choose Rigid Domain: Club.
Center of Joint: Boundary 1
1
In the Model Builder window, expand the Hinge Joint: Shaft node, then click Center of Joint: Boundary 1.
2
Select Boundary 9 only.
Hinge Joint: Shaft
In the Model Builder window, click Hinge Joint: Shaft.
Spring and Damper 1
1
In the Physics toolbar, click  Attributes and choose Spring and Damper.
2
In the Settings window for Spring and Damper, locate the Spring and Damper: Rotational section.
3
In the kθ text field, type ks.
4
In the cθ text field, type cs.
Definitions
Variables 1
Add a few more variables to use them in the postprocessing.
1
In the Model Builder window, under Component 1 (comp1)>Definitions click Variables 1.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Ta_eff
Ta-mbd.hgj2.sd1.Ms-mbd.hgj2.sd1.Md
 
Effective arm torque
Tw_eff
Tw-mbd.hgj3.sd1.Ms-mbd.hgj3.sd1.Md
 
Effective wrist torque
Study 1
Add a Parametric Sweep to solve the model for different values of wrist torque switch time.
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
tw (Wrist torque switch time)
0.15 0.19 0.23
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(0,2e-3,0.26).
Add a Stop Condition in the solver in order to stop the simulation just prior to the impact with the ball.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
Right-click Time-Dependent Solver 1 and choose Stop Condition.
4
In the Settings window for Stop Condition, locate the Stop Expressions section.
5
Click  Add.
6
In the table, enter the following settings:
Stop expression
Stop if
Active
Description
comp1.stop
True (>=1)
√
Stop expression 1
7
Locate the Output at Stop section. From the Add solution list, choose Step after stop.
8
In the Study toolbar, click  Compute.
Results
Follow the instructions below to add the point trajectory plot.
Displacement (mbd)
In the Model Builder window, under Results click Displacement (mbd).
Point Trajectories 1
1
In the Displacement (mbd) toolbar, click  More Plots and choose Point Trajectories.
2
Select Points 1, 9, and 14 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 check box.
5
In the Displacement (mbd) toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Follow the instructions below to generate the plot of club head speed shown in Figure 3.
Club Head Speed
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Club Head Speed in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
Click to expand the Title section. From the Title type list, choose None.
5
Locate the Plot Settings section. Select the y-axis label check box.
6
In the associated text field, type Club head speed (m/s).
7
Locate the Legend section. From the Position list, choose Upper left.
Point Graph 1
1
Right-click Club Head Speed and choose Point Graph.
2
Select Point 14 only.
3
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 1 (comp1)>Multibody Dynamics>Acceleration and velocity>mbd.vel - Velocity magnitude - m/s.
4
Click to expand the Coloring and Style section. In the Width text field, type 2.
5
Find the Line markers subsection. From the Marker list, choose Cycle.
6
Click to expand the Legends section. Select the Show legends check box.
7
From the Legends list, choose Manual.
8
In the table, enter the following settings:
Legends
tw=0.15000
tw=0.19000
tw=0.23000
9
In the Club Head Speed toolbar, click  Plot.
10
Click the  Zoom Extents button in the Graphics toolbar.
Follow the instructions below to generate the plot of driving torques as shown in Figure 4.
Driving Torques
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Driving Torques in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
From the Parameter selection (tw) list, choose From list.
5
In the Parameter values (tw (s)) list, select 0.19.
6
Locate the Title section. From the Title type list, choose None.
7
Locate the Plot Settings section. Select the y-axis label check box.
8
In the associated text field, type Driving torque (N-m).
Global 1
1
Right-click Driving Torques 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 (comp1)>Definitions>Variables>Tsh - Applied shoulder torque - N·m.
3
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Definitions>Variables>Ta - Applied arm torque - N·m.
4
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Definitions>Variables>Tw - Applied wrist torque - N·m.
5
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Definitions>Variables>Ta_eff - Effective arm torque - N·m.
6
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Definitions>Variables>Tw_eff - Effective wrist torque - N·m.
7
Click to expand the Coloring and Style section. In the Width text field, type 2.
8
Find the Line markers subsection. From the Marker list, choose Cycle.
Driving Torques
1
In the Model Builder window, click Driving Torques.
2
In the Settings window for 1D Plot Group, locate the Axis section.
3
Select the Manual axis limits check box.
4
In the y minimum text field, type -70.
5
In the y maximum text field, type 500.
6
In the Driving Torques toolbar, click  Plot.
Follow the instructions below to plot the relative rotation at the joints as shown in Figure 5.
Relative Rotation
1
Right-click Driving Torques and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Relative Rotation in the Label text field.
3
Locate the Plot Settings section. In the y-axis label text field, type Relative rotation (deg).
4
Locate the Legend section. From the Position list, choose Upper left.
Global 1
1
In the Model Builder window, expand the Relative Rotation node, then click Global 1.
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 (comp1)>Multibody Dynamics>Hinge joints>Hinge Joint: Arm>mbd.hgj2.th - Relative rotation - rad.
3
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
mbd.hgj2.th
deg
Arm joint
4
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Multibody Dynamics>Hinge joints>Hinge Joint: Wrist>mbd.hgj3.th - Relative rotation - rad.
5
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
mbd.hgj3.th
deg
Wrist joint
6
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Multibody Dynamics>Hinge joints>Hinge Joint: Shaft>mbd.hgj4.th - Relative rotation - rad.
7
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
mbd.hgj4.th
deg
Shaft joint
8
In the Relative Rotation toolbar, click  Plot.
9
Click the  Zoom Extents button in the Graphics toolbar.
Follow the steps below to plot the trajectory of shoulder-arm-club motion as shown in Figure 6.
Shoulder-arm-club Motion
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Shoulder-arm-club Motion in the Label text field.
Line Graph 1
1
Right-click Shoulder-arm-club Motion and choose Line Graph.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
From the Parameter selection (tw) list, choose From list.
5
In the Parameter values (tw (s)) list, select 0.15.
6
From the Time selection list, choose Interpolated.
7
In the Times (s) text field, type range(0,2e-2,0.26).
8
Locate the Selection section. Click  Paste Selection.
9
In the Paste Selection dialog box, type 2 6 10 14 in the Selection text field.
10
Click OK.
11
In the Settings window for Line Graph, locate the y-Axis Data section.
12
In the Expression text field, type y.
13
Locate the x-Axis Data section. From the Parameter list, choose Expression.
14
In the Expression text field, type x.
15
Click to expand the Coloring and Style section. From the Color list, choose Black.
16
In the Width text field, type 2.
Point Graph 1
1
In the Model Builder window, right-click Shoulder-arm-club Motion and choose Point Graph.
2
In the Settings window for Point Graph, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
From the Parameter selection (tw) list, choose From list.
5
In the Parameter values (tw (s)) list, select 0.15.
6
From the Time selection list, choose Interpolated.
7
In the Times (s) text field, type range(0,2e-2,0.26).
8
Locate the Selection section. Click  Paste Selection.
9
In the Paste Selection dialog box, type 6 14 18 in the Selection text field.
10
Click OK.
11
In the Settings window for Point Graph, locate the y-Axis Data section.
12
In the Expression text field, type y.
13
Locate the x-Axis Data section. From the Parameter list, choose Expression.
14
In the Expression text field, type x.
15
Locate the Coloring and Style section. From the Color list, choose Red.
16
Find the Line style subsection. From the Line list, choose None.
17
Find the Line markers subsection. From the Marker list, choose Point.
18
From the Positioning list, choose In data points.
Line Graph 2
1
In the Model Builder window, under Results>Shoulder-arm-club Motion right-click Line Graph 1 and choose Duplicate.
2
In the Settings window for Line Graph, locate the Data section.
3
In the Parameter values (tw (s)) list, select 0.19.
4
Locate the x-Axis Data section. In the Expression text field, type x+2.5.
Point Graph 2
1
In the Model Builder window, under Results>Shoulder-arm-club Motion right-click Point Graph 1 and choose Duplicate.
2
In the Settings window for Point Graph, locate the Data section.
3
In the Parameter values (tw (s)) list, select 0.19.
4
Locate the x-Axis Data section. In the Expression text field, type x+2.5.
Line Graph 3
1
In the Model Builder window, under Results>Shoulder-arm-club Motion right-click Line Graph 2 and choose Duplicate.
2
In the Settings window for Line Graph, locate the Data section.
3
In the Parameter values (tw (s)) list, select 0.23.
4
Locate the x-Axis Data section. In the Expression text field, type x+5.
Point Graph 3
1
In the Model Builder window, under Results>Shoulder-arm-club Motion right-click Point Graph 2 and choose Duplicate.
2
In the Settings window for Point Graph, locate the Data section.
3
In the Parameter values (tw (s)) list, select 0.23.
4
Locate the x-Axis Data section. In the Expression text field, type x+5.
5
In the Shoulder-arm-club Motion toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Shoulder-arm-club Motion
1
In the Model Builder window, click Shoulder-arm-club Motion.
2
In the Settings window for 1D Plot Group, locate the Title section.
3
From the Title type list, choose None.
4
Locate the Plot Settings section. Select the x-axis label check box.
5
In the associated text field, type x-coordinate (m).
6
Locate the Axis section. Select the Preserve aspect ratio check box.
7
In the Shoulder-arm-club Motion toolbar, click  Plot.
8
Click the  Zoom Extents button in the Graphics toolbar.
Finally, you can generate an animation of the shoulder-arm-club motion together with the trajectory of joints and club head for different values of wrist torque switch time.