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Chapter 8 – Four-bar, Forces, Torques

HyperWorks 13.0 MotionView Introduction 163 Proprietary Information of Altair Engineering, Inc.

Chapter 8

Interactive Model Building and Simulation

Part 7– Four-bar, Forces, Torques

Introduction

In this chapter, you will continue to build on the four-bar model from the previous chapter:

• Deactivate motion and initial velocity

• Add torque to drive mechanism

• Add torque based on BISTOP function to limit joint motion

• Create Output to measure torque with VTORQ function

• Compare dynamics to kinematics simulation


MotionView Introduction 164 HyperWorks 13.0 Proprietary Information of Altair Engineering, Inc.

Model Building Process In the previous chapters, you have built up a working model of a four-bar mechanism. In this chapter, you will add torques to the model to drive the system, and another to limit the joint motion. You will also compare the difference between a kinematic simulation and a dynamic simulation.

Already created in previous chapter

To be created in this chapter

Modified in this chapter

1. Create points

• Needed for joints, center-of-mass locations, etc.

2. Create bodies/parts

• Define markers for mass, inertia, body coordinate system

• Define initial conditions

3. Create geometry

• Import CAD into MotionView. (this can also be used to automatically create parts)

• Use built-in primitive geometry.

4. Create joints

• Create revolute joint

5. Create actuators -- forces/motions

• Create torque to drive mechanism and another to limit joint motion

• Create time based motion

6. Create outputs

• Request measurement of torque and other mechanism states to define torque

7. Simulate via Run Panel



Force Overview The term force generically refers to both applied forces and applied moments. Force elements are commonly used for modeling:

• applied/driving forces

• compliant connections

• the effects of the environment on the system

• contact between bodies

Force Entity Toolbar

Compliant Connections Examples of compliant connections include:

• springs

• dampers

• bushings

• beams

Environmental Effects Examples of environmental effects include:

• gravity (which may or may not be constant)

• aerodynamic forces

• hydrodynamic pressure

Force Definition Force elements may be defined as arbitrary functions of any state variables in the system and time. You will define a force using system states of angular displacement and angular velocity later in the exercise.

The functional form of the force description may be:

• an interpolated curve (e.g., test data)

• an analytical expression

• a user-defined subroutine in C++, C, FORTRAN, python, or MATLAB scripts



Vector forces can be specified by their components in arbitrary reference frames (i.e., using a marker). Scalar forces act along the line connecting the origins of markers.

Forces Should Ideally Be “Smooth” This is an extremely open and powerful capability in MotionSolve that enables you to easily model most of the forces occurring mechanically. However, when modeling a force, you must make sure that the force is, at the very minimum, continuous. It is better if the forcing functions are smooth (without kinks) and do not change dramatically with respect to the independent variables (frequency content is not very high). If the force has undamped, high frequency content, the system motion may have higher frequencies as well, causing the integrator to take smaller step sizes, which ultimately results in slow run times.

F For more information, see any of the Force_* entities (e.g., Force_SpringDamper, Force_Bushing) in the MotionSolve Reference Manual.

Force Creation Methods Forces can be added using two methods you used previously:

1. Using the MotionView Toolbar

a. – forces (general applied forces/torques, of different types)

b. - bushings

c. – fields

d. –spring/dampers

e. –beams

f. – polybeams

g. –contacts

2. Using the MotionView Model Browser (right-click à Add Force Entity)



Example - Force Panel We will next investigate some of the highlights of the force panel.

The Forces panel allows you to edit the orientation and properties of forces.

Forces panel

Connectivity Tab The Connectivity tab on the Forces panel allows you to select the bodies and points to which a force is applied. The type of action and direction of the force are specified with option menus.

Forces can be action-only or action-reaction.

• An action-only force only acts on body 1.

• An action-reaction force exerts equal and opposite forces on body 1 and body 2.

Forces can have explicit translational and rotational components, or they can act along a line-of-action between two points.

Select: To:

Action only create an action only force.

Action reaction create an action-reaction force.

Use: To:

Body 1 select the body to which the action force is applied.

Body 2 select the body to which the reaction force is applied (action-reaction only).

Point(s) select the point or points at which the force is applied.

Ref Marker specify the local reference frame of the force.



Select:

To:

Translational apply a force that has only translational components.

Rotational apply a force that has only rotational components.

TransRotational apply a force that has both translational and rotational components.

Line Of Action Translational

apply a line-of-action force between two points. The force will only have components in the direction along the line connecting the two points. This option is only available for action-reaction forces.

Single Comp Rotational

apply a single-component-rotational force between two bodies about a specified axis. This option is only available for action-reaction forces.

User-defined Forces The User-defined properties option allows you to specify a force subroutine for writing the solver model (see SFOSUB, VFOSUB, VTOSUB, GFOSUB)

Trans Properties and Rot Properties Tabs The Trans Properties and Rot Properties tabs on the Forces panel allows you to define the components of a force with respect to a reference coordinate system.

Forces panel - Rot Properties tab

The magnitude of a force component can be defined using a constant value, math expression, or template expression.

For: Enter:

Linear constant torque



Curve torque expressed in the form of a torque vs. independent variable (2-D)

Spline 3D Expression

torque expressed in the form of a torque vs. independent variable1, independent variable 2 (3-D)

torque expressed in the form of a solver function

Functions Used in This Chapter The following functions will be necessary to complete the exercise at the end of this chapter:

• BISTOP

• AZ

• WZ

• VTORQ

The BISTOP function will be used to limit the motion of a joint in the four-bar mechanism. It will require the AZ and WZ functions to measure the angular displacement and velocity of the joint for which you will limit the motion.

As a second task, the VTORQ function will be used by an Output to measure the torque created using the BISTOP function.

BISTOP Function

Description The BISTOP function models a gap element with a spring/damper as the boundary to the gap. This can be used to limit the motion of a joint. The properties of the two boundary surfaces (spring/dampers) can be tuned as desired.

Example <Force_Vector_TwoBody

id = "30101"

type = "ForceOnly"

i_marker_id = "30102031"

j_floating_marker_id = "30101031"

ref_marker_id = "30101010"



fx_expression = BISTOP(DX(30102030,30101010,30101010), VX(30102030,30101010,30101010),0.5,9.5,10000000,2.1,1,0.001)"

fy_expression = "0"

fz_expression = "0"

/>

Arguments

The expression used for the independent variable. For example, to use the z-displacement of I marker with respect to J marker as resolved in the reference frame of RM marker as the independent variable, specify as DZ({marker_i.idstring}, {marker_j.idstring}, {marker_rm.idstring}).

The time derivative of the independent variable. For example, if is specified as above, then will be:

VZ({marker_i.idstring}, {marker_j. idstring}, {marker_rm.idstring}).

The lower bound of . If is less than , the bistop function returns a positive value. The value of must be less than the value of .

The upper bound of . If is greater than , the bistop function returns a negative value. The value of must be greater than the value of .

The stiffness of the boundary surface interaction. It must be non-negative.

The exponent of the force deformation characteristic. For a stiffening spring characteristic, must be greater than 1.0 and for a softening spring characteristic, must be less than 1.0.

It must always be positive.

The maximum damping coefficient. It must be non-negative.



The penetration at which the full damping coefficient is applied. It must be positive.

Definition

AZ, WZ Functions The AZ and WZ measure the rotational displacement and velocity, respectively of a marker “I” with respect to the marker, “J”. AZ has the following syntax:

AZ(I, J)

Arguments I

The marker whose rotational displacement is to be computed.

J

The marker with respect to which the rotational displacement is to be computed. This argument is optional. If omitted, it defaults to the ground coordinate system.

WZ(I, J, K)

Arguments I

The marker whose velocity is to be computed.

J

The marker with respect to which the velocity is to be computed. This argument is optional. If omitted, it defaults to the ground coordinate system.

K

The resultant velocity vector is resolved in the coordinate system of the K marker. This argument is optional. If omitted, it defaults to the ground coordinate system.

VTORQ Function This function returns the specified component of the torque applied by the Force_Vector_TwoBody or Force_Vector_OneBody element -- this is equivalent to the Action Reaction or Action Only forces in MotionView.

VTORQ(ID, JFLAG, COMP, RM)

Arguments id



The ID of the Force_Vector_TwoBody or Force_Vector_OneBody element.

jflag

jflag equal to 0 or 1 means that moments are reported at the I or J marker, respectively.

comp

5 - returns the torque magnitude.

6 - returns the torque x component.

7 - returns the torque y component.

8 - returns the torque z component.

RM

The reference frame in which the components are reported. RM=0 implies the global frame.

Kinematics vs. Dynamics Analysis Both kinematic and dynamic analysis use a method of solving non-linear algebraic equations called Newton-Raphson (N-R), or Newton’s Method.

Newton’s Method/Newton-Raphson This method repeatedly linearizes the non-linear algebraic equations at a given time step to try to find the solution of the system equations (the root is essentially the state of the system that solves the equations of motion at that time step). The slope of these linearized equations is basically a sensitivity (a partial derivative of the equation with respect to each state of the system), and all of the sensitivities make up what is called the Jacobian matrix.

Netwon’s method may not find a solution if the system starts too far away from the final solution. If this happens, the integrator may report that the solution has not converged, and will go back a take a smaller time step, hoping that this will be closer to the final solution so that N-R will converge.



Source: Wikipedia, http://en.wikipedia.org/wiki/Newton%27s_method, Author :Ralf Pfeifer

Kinematic Analysis Definition

• A model with zero degrees of freedom is defined as kinematic

• Determines the range of values for the displacement, velocity, and acceleration of any point of interest in the model, independent of forces applied to it.

• Position, velocity, and acceleration analyses through a purely joint and motion-driven system (i.e., “constraints define the motion” )

• Some of the constraints must be of type motion to drive the system

• Motion specified needs to be time based

• If you specify the mass and inertial properties of bodies in your model, MotionSolve calculates the corresponding applied and reaction forces required to generate the prescribed motions. Accurate mass and inertia is required for accurate kinematic reaction forces.

Solution Overview

• Non-linear algebraic equations solved by Newton-Raphson

• Forces calculated as a consequence of motion



• No need to solve differential equations à relatively robust solution!

Applications

• Mechanism design, Robot design and Task planning, Cam profile design

Dynamic Analysis Definition

• Study of motion of a system as a consequence of applied forces and inertia forces

• It is applicable to models with one or more degrees of freedom

• The dynamic simulation accounts for all the accelerations (linear, angular, centrifugal, and coriolis), forces, and constraints

• It is the most complex and computationally demanding simulation type

• This simulation enables you to develop accurate system level simulations of complex mechanical systems

Solution Overview

• Applied forces affect accelerations à differential equations

• Integrator converts differential equations into non-linear algebraic equations à solve for accel, vel, disp

• Netwon-Raphson is used to solver non-linear algebraic equations with prediction/correction, so non-smooth forces (like with intermittent contacts) are more difficult to solve since they are not “smooth”

Applications

• Anything that moves, usually “large” motion; studying motion and loads over a wide range of industries



Exercise 8.1: Adding Torques to the Four-bar mechanism In this exercise, you will add a driving torque and a torque to limit the motion of a joint to the four-bar mechanism you have built in previous chapters. You will also compare the results of a kinematic simulation to dynamic simulation to help illustrate their differences.

Step 1: Do one of the following: • Reuse your model from the previous exercise, or

• Open the file chapter8_exercise_start.mdl model file, which should start you at the same point.

Step 2: Disable the Motions and Initial Conditions 1. Right click on Motion 1 in the Project Browser. 2. Select Deactivate from the menu.

3. Click on Joint 0 in the Project Browser to bring up the Joints panel.

4. In the Initial Conditions tab, deselect the Rotation checkbox to remove the initial condition from the joint.

Step 3: Create a Driving Force and Run the Model

1. Right-click on the Forces panel button to create a new force named Driving Torque using the Add Force dialog box

2. Click OK to complete the creation of the Force.

3. Set the Force option to Action reaction

4. Set the Properties option to Rotational. 5. Click Body 1 next to Action Force on to activate the selector.

6. Select Body 1 for Action force on.

7. Select Ground Body for Reaction Force on.

8. Select Point F for Apply force at. 9. Select Global Frame for Local ref frame.

10. Click the Rot Properties tab to open it.



11. Enter a value of 150 for Ty as shown below.

Step 4: Save this model as four_bar_torque.mdl.

Step 5: Run the Analysis.

1. Click the Run panel icon

2. Save the model as four_bar_torque.xml and Run the model.

Step 6: Run the Animation 1. Load the results file into HyperView (animation window)

2. Click Start/Pause Animation to animate the model, ensuring that the small bar completes a full rotation.

Step 7: Create a BISTOP function to limit the motion

1. Right-click on the Forces panel button to create a new force named BISTOP Torque using the Add Force dialog box

2. Click OK to complete the creation of the Force.

3. Set Action reaction for the Force option.

4. Set the Properties option to Rotational. 5. Click the Body 1 selector for Action force on to activate it.

6. Select Free Body for Action force on.

7. Ensure Ground Body is selected for Reaction force on.

8. Select Point A for Apply force at.



9. Click on Rot Properties to enter the subpanel.

10. Change the selection under Ty from Linear to Expression. 11. Click in the Expression field for Ty to activate it.

12. Click the Expression Builder button to enter the Expression Builder. 13. In the Expression Builder window, click the General tab.

14. Click BISTOP to add the BISTOP function to the Ty expression.

The function will be added to the expression and the cursor will advance to the first variable spot within the function. For more information about the BISTOP function, please see the HyperWorks Help entries for BISTOP under RADIOSS, MotionSolve, and OptiStruct.

The first argument in the BISTOP function is the Z-angle of Joint 0 with respect to the Ground Body.

15. Click the Motion tab to find the AZ function.

16. Click AZ to add it to the BISTOP function with a set of curly brackets inside of it as shown below.

The AZ function needs two arguments here: the marker of the body whose angular displacement should be measured, and the marker to measure against.



17. With the cursor between the curly brackets { } for the AZ function, click on the Properties tab and expand and select the following in the model tree:

Joints > Joint 0 >Marker I > idstring

18. Click Add to place the information for that marker between the curly brackets in the AZ

function.

The AZ function will measure the Z-angular displacement of Joint 0, which is the first argument needed for the BISTOP function.

19. Following the first close bracket for the AZ function, add a comma and a second set of brackets and place the cursor between them as shown.

20. With the cursor between the new set of brackets, browse to the following element in the

Properties tab and add it to the expression.

Joints > Joint 0 > Marker J > idstring 21. Click Add to add to the function.

22. Move the cursor between the first and second commas in the BISTOP function to

prepare the editor to enter the next term.

The second term in the BISTOP function is the derivative of the first term.

23. Click the Motion tab.

24. Select the WZ function to add it as the basis for the second term.



The WZ function needs two arguments here: the marker of the body whose angular velocity should be measured, and the marker to measure against.

25. With the cursor between the curly brackets { } for the WZ function, click the Properties tab.

26. Expand and select the following tree and element:

Joints > Joint 0 >Marker I > idstring 27. Click Add to add the id string for the I marker for Joint 0 to the WZ function as shown

below.

The WZ function will measure the Z-angular velocity of Joint 0 with respect to ground, which is the second argument needed for the BISTOP function.

28. Following the first close bracket for the WZ function, add a comma and a second set of brackets and place the cursor between them as shown.

29. With the cursor between the new set of brackets, browse to the following element in the

Properties tab and add it to the expression.

Joints > Joint 0 > Marker J > idstring

The third argument in the BISTOP function is the lower bound of the allowable deflection before the resisting torque starts to be applied, entered in radians.

30. For an angle of -10 degrees, enter -10*PI/180 between the next set of commas.

The fourth argument in the BISTOP function is the upper bound of the allowable deflection before the resisting torque starts to be applied, also entered in radians.

31. For an angle of 10 degrees, enter 10*PI/180.

The fifth argument for the BISTOP is the Stiffness Value.

32. Enter a value of 1e4 in the next set of commas.

The sixth argument is the exponent of the deformation characteristic.

33. Enter a value of 2 in the next set of commas.

The seventh argument is the max damping coefficient.

34. Enter a value of 100 between the next set of commas.



The last argument is the penetration at which full damping is applied. In this example the measurement is millimeters.

35. Enter 0.1 after the last comma.

36. Click Evaluated to ensure the BISTOP function is populated with correct values. 37. Click OK to complete the function.

F If you have trouble getting the correct expression, you can copy/paste this

expression:

`BISTOP(AZ({j_0.joint_i.idstring},{j_0.joint_j.idstring}),WZ({j_0.joint_i.idstring},{j_0.joint_j.idstring}),-10*PI/180,10*PI/180,1e4,2,100,0.1)`

Step 8: Save the Model as four_bar_bistop.mdl

Step 9: Run the Analysis


2. Save the model as four_bar_bistop.xml and Run the model.

Step 10: Animate the Results 1. Load the results into HyperView for post-processing.

2. Click Start/Pause Animation to animate the model and see the bumper torque in action.

You should notice that the driving torque moves the mechanism until it hits the limit of the BISTOP torque, which then cause the mechanism to bounce back.



Step 11: Create an Output Measure for the BISTOP Function Create a new output to measure the torque applied by the BISTOP function called Bistop torque output.

1. Click to open the Add Ouput dialog.

2. Enter BISTOP_Torque_Output for the name of the output.

3. Select Expressions from the menu.

4. Click the F2 field to activate it.

5. Click the fx button to open the Expression Builder. 6. Click Force to open the tab.

7. Click the VTORQ button to add it to the expression.



First the ID of the body must be added.

8. Click the Properties tab to open it.

9. With the cursor between the new set of brackets, browse to the following element in the Properties tab and add it to the expression.

Forces > BISTOP Torque > idstring 10. Click Add to add the string.

The next argument is the jflag, which means that forces or moments are reported for I or J.

11. Enter 0 to report for I.

Next is the comp argument for the torque.

12. Enter 7 to measure the torque on the y-axis.

The final argument is for the reference frame.

13. Enter 0 to set the global frame.

14. Click Evaluated to check the function.

15. Click OK to complete the function.



2. Click Run to rerun the model.



Step 13: Plot the Torque Output 1. Click Plot to open HyperGraph.

2. Click in the HyperGraph window to activate it.

3. Select the following options:

Y Type: Expressions

Y Request: BISTOP_Torque_Output

Y Component: F2

4. Click Apply to create the curve.

The curve should look like the following:

5. Activate the HyperView window.

6. Click Start/Pause Animation to animate the model.

Step 14: Compare a Dynamic Simulation to a Kinematic Simulation Lastly, compare this dynamic simulation with a kinematic variant of the same model.

What you have been performing thus far in this tutorial is a dynamic simulation, which means the forces affect the motion of the mechanism. This is why the torque based on the BISTOP function will limit the motion of this joint. If however, a kinematic model is run by a driving motion, this BISTOP torque will no longer stop this motion, since the motion is now prescribed exactly, and the solver must follow the motion (motions are constraints).



1. Activate the MotionView window.

2. Right click on Driving torque and click Deactivate. 3. Right click Motion 1 and click Activate.

This recreates a kinematic system from our model.



2. Save the model as four_bar_bistop_motion.xml and Run the model.

Step 16: Animate the Results 1. Click Animate to load the model into HyperView for post-processing.

2. Click Start/Pause Animation to animate the model and see the bumper torque in action.

Notice that the joint with the motion keeps moving the mechanism, regardless of the BISTOP torque. This is how kinematic models behave -- the motion driving the mechanism must be followed. The forces and torques in the constraints will increase due to the BISTOP torque, but this does not change the overall motion of the mechanism.

F If your model does not behave how you expect, check that the initial conditions of the bodies and the joints have been deactivated and/or set back to zero.



Step 17: Save the Model 1. Click File > Save As > Model. 2. Save the model as four_bar_bistop.mdl.

interactive model building and simulation - forum.altair.com

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