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CAE and Multi Body Dynamics MBD Simulation with HyperWorks

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MBD Simulation with HyperWorks

We now have a fair understanding of both the design issues and the theoretical basis for MBD simulation. We also know that MBD problems

originate from 3D CAD in several, but not all, cases. We have seen the 3

distinct phases involved in the process. We start by creating the model, go on to solve it in the second step, and then move on to the third stage,

results-interpretation. Finally, we have seen that it would be nice to have the ability to optimize a design and to check how robust it is in terms of

tolerance to deviations in operating conditions or data.

With this background, let’s look at the problem from a HyperWorks point of view.

The Simulation Process The various modules of HyperWorks can be used in a variety of ways for

MBD Simulation, depending on the problem specification and complexity.

We’ll review the different approaches, and correlate these with our understanding of how MBD is used in the Product Design cycle.

Model Preparation – MotionView and HyperMesh The first stage, model preparation, can be done using either HyperMesh or MotionView. HyperMesh is perhaps better used if the components are

mechanical in nature and are flexible. That is, if stresses in the components are to be calculated in addition to the forces and velocities. MotionView is more of a “traditional” MBD preprocessor. It is less 3D-graphics-intensive

than HyperMesh, since it allows the definition of models using MDL – the Model Definition Language – in which graphical representations of the

surfaces and volumes of the bodies are an option, not a necessity. This capability, which is similar to using kinematic diagrams, is not available in

HyperMesh. Since MotionView also allows us to simulate the behavior of

flexible bodies and add graphics, our focus in this book and in the accompanying exercises is on MotionView, not on HyperMesh28.

28 See A Designer’s Guide to Finite Element Analysis – Student Project Summaries for an example of how HyperMesh can be used for the analysis of flexible MBD systems.

MBD Simulation with HyperWorks CAE and Multi Body Dynamics

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Solving the Equations – MotionSolve and OptiStruct Once the model has been built, the center of attention shifts to the solver.

This is where the equations of motion are drawn up, parameters chosen for

the numerical algorithms, and the numbers are crunched.

This is done by MotionSolve. It is usually invoked from within MotionView but can also be used as a standalone application that reads an input file and

generates output data.

OptiStruct, as detailed in other books in this series, is intended for

optimization of linear problems and for linear finite element analysis. However if a problem statement requires the kinds of analyses that are a

part of MBD, OptiStruct can call MotionSolve. The process by which OptiStruct invokes MotionSolve and interacts with it is transparent to the user: except for records in the log file, there is nothing you need to do to

manage this process. Since our goal is to see how MBD theory and MBD practice come together, we will restrict our attention to MotionSolve. If you’re comfortable with Finite

Element Analysis, the companion volumes in this series discuss the use of

OptiStruct for MBD problems.

Results – HyperView and HyperGraph The data generated by the solver depends not just on the statement of the

problem, but also on what you have asked to review. For a dynamics problem for example, you may choose to generate printed results at specific

time steps.

Both HyperView and HyperGraph are useful in this respect. HyperView

provides a variety of facilities for 3D viewing: animation, vector plots, and so on. HyperGraph comes in handy to generate plots. For cam design for

instance, you need to plot the velocity-vs. time.

Optimization – OptiStruct and HyperStudy As we have seen above, OptiStruct can invoke MotionSolve transparently, as

needed. This is useful not only to when the MBD model contains flexible

links, but also to run optimization. Look up the online documentation for a discussion on techniques like the Equivalent Static Load Method for the optimization of problems involving dynamic stresses.


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Optimization of MBD models, of course, is not always related to stress or

mass. In the case of synthesis of mechanisms, position control may be a more important objective. HyperStudy, which supports DOE and other techniques for non-linear optimization and robust design, can be used with

MotionSolve to address these requirements. A discussion of this approach is contained in CAE and Design Optimization – Advanced.

The Anatomy of a Model MotionView provides all the building blocks we listed as essential and desirable for MBD modeling. Remember that one of the recommended approaches in MBD is to build validated libraries of systems, and to use

these as sub-systems in the construction of more complex assemblies. To

promote this approach, MotionView stores model-definitions in files.

What the Files Contain The principle storage structure for MBD models follows the MDL (Model Definition Language) format. An MDL file, which usually carries the suffix mdl is an ASCII file that can be opened using any text

editor29. These files can be created without using

MotionView at all. This

approach requires that you be familiar with the syntax

of the MDL statements.

For instance, a revolute joint is defined using the

statement *RevJoint(…) where the items in brackets should be replaced with the

relevant values, as shown in the annotated file displayed alongside.

29 MDL files can be encrypted, to protect their contents.

Statements 1 to8 are definition-statements. Every entity is defined by at least a name and a label. Entities such as joints require additional data.

1. *BeginMDL (pendulum, "Pendulum Model") 2. *point (p_pendu_pivot, "Pivot Point") 3. *point ( p_pendu_cm, "Pendulum CM") 4. *Body (b_link, "Ball", p_pendu_cm)

The joint connects the ground (default name is B_Ground) and the body defined on line 4. The axis of rotation is the X axis, centered at the point defined on line 2

5. *RevJoint (j_joint, "New Joint", B_Ground, b_link, p_pendu_pivot, V_Global_X)

Graphics are purely for visual appeal. The sphere’s radius is set to 1, and the cylinder’s radius to 0.5.

6. *Graphic (gr_sphere, "pendulum sphere graphic", SPHERE, b_link, p_pendu_cm, 1 )

7. *Graphic (gr_link, "pendulum link graphic", CYLINDER, b_link, p_pendu_pivot, p_pendu_cm, 0.5, CAPBOTH )

This is where we specify the output: we want the displacement history of the link defined on line 4

8. *Output (o_pendu, "Disp Output", DISP, b_link)

Here we assign coordinates to the points, and mass and moments of inertia to the link

9. *setsystem (MODEL) 10. *setpoint (p_pendu_pivot, 0, 5, 5)

11. *setpoint (p_pendu_cm, 0, 10, 10) 12. *setbody ( b_link, 1, 1000, 1000, 1000, 0, 0, 0)

With this, we have finished defining our model

13. *EndMDL ()


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In the MDL syntax, the name is used by other MDL statements, while the label is used in the interactive-editor. For example, in the annotated MDL file, the name of the point defined on line 3 (p_pendu_cm) is used in the link definition on line 4. In MotionView, you would see it referred to by its label (that is, as Pendulum CM).

You will see that there are two types of statements for each entity. The first names it, the second assigns data to it. The definition statement must

always precede the assignation statement, of course. It is customary, but not essential, to group all definition statements followed by all assignment

statements. It is also customary, but not essential, that names follow a

pattern. This makes it easier to read an MDL file, as you will have to from time to time. In the annotated example, the first letter of the variable name

indicates its type – p for points, b for bodies, and so on.

Note that the “ball” of the pendulum is not modeled as a link at all from a kinematic point of view. To make the graphic display realistic, however,

graphic primitives are assigned to the link. In general, graphics can be assigned either from predefined primitives (such as the cylinder and sphere used in the example) or by importing graphics from files. The latter is

common for complex geometry, and ways to do this are covered in the accompanying projects.

Since MotionView is an interactive graphics editor, and since model construction may well take more than one session, it is often useful to save

the definition of the “desktop” – the windows, their contents, the last view of the model, and so on. These items are relevant only to the interactive

graphics environment. They are of no use to the construction of hierarchical systems (systems that are built using other systems).

So MotionView uses a different structure, the Session File format, to save this data. Session files usually have the suffix mvw, and contain the complete MDL definition of the model in addition to the state of the desktop. The MDL statements can be saved either in the MVW file or as a separate

MDL file that is referenced by the MVW file.

Summary of Modeling Entities MotionView and MotionSolve together provide several building blocks, some of which are more than the “basic” blocks we discussed in the earlier

chapter. Some of these are essential for the modeling of higher pairs. For


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example, you need point-to-surface or point-to-curve constraints30 to model

a cam and its follower. A complete list of the entities and their properties is contained in the online

documentation. The table below summarizes some of the more commonly used entities. Entity Data Required# Notes Point Coordinates Usually used to define nodes

Body Mass, Moments of Inertia Local Coordinate Systems can be defined if the moment of inertia’s origin is not the same as the center-of-mass

Spring Stiffness, Damping, Preload, Free length / angle

Springs can be either helical or torsional

Revolute Joint Names of the two bodies

connected by the joint, and the axis of revolution

Leaves only 1 free dof – rotation

about the axis

Translational Joint Names of the two bodies connected by the joint, and the axis of sliding

Leaves only 1 dof free – translation along the axis

Ball Joint Names of the two bodies

connected by the joint, and the center of rotation

Leaves 3 dofs free – rotations about

the 3 axes

Marker Body it is connected to, origin and orientation

A marker is a Local Coordinate System, but is treated as a distinct entity. You can attach a marker to a point on a link, and request output

for that marker!

System Points of attachment, orientation and initial conditions.

These are roughly similar to subroutines in a programming language. A system can be saved in a separate MDL file

# - All entities require a name and label

Several more entities – joints, bodies, forces, etc. – are supported. See the

online documentation for details.

Solution and Results MotionSolve can solve several different classes of problems. The most

general class consists of problems in dynamic analysis, where the system

can have more than one uncontrolled dof.

30 These are called PTSF and PTCV constraints, respectively.


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Static analysis is most often used to compute the equilibrium configuration

of a mechanism. Kinematic analysis, used for systems that have no uncontrolled dofs, is typically used early in the design cycle, at the concept stage. Quasi-static analysis is applicable when the forces change with time,

but do so slowly. This means inertial forces can be ignored, and the static-equilibrium equations can be solved at each instant of time. Stability analysis is a good example of its usage. MotionSolve does not read MDL or MVW files. Instead, MotionView creates

an XML file that is used as input by MotionSolve. Several different output files can be generated. The important ones are:

• Log files (<filename>.log) contain the history of the solution. It’s a good practice to review the log files after every analysis, checking for errors or warnings.

• Altair binary files (<filename>.abf) are used to generated

animations in MotionView

• HyperView 3D Player files (<filename>.h3d) can be viewed without

HyperWorks, using the free player

• Plot files (<filename>.h3d) are used to generate graphs with HyperGraph

Remember that MotionSolve has to solve non-linear equations, and has to numerically integrate the differential equations of motion. The numerical

solution of the non-linear equations is iterative. That is, the solver first guesses at a set of values and checks whether these form the solution at that instant of time or not. If there’s an error, the software corrects the

guess and repeats the cycle until the error is within an acceptable tolerance. Once this happens, the software concludes that the iteration has converged, and moves onto the next time step. If the error does not fall within the specified tolerance within a specified set of iterations, the software

concludes that the solution has diverged (i.e. not converged) and gives up the hunt for the answer. MotionSolve’s default settings for the numerical algorithms are usually adequate, but in advanced situations, you will need to choose between the

Maximum Kinetic Energy Attrition method and the Force Imbalance method,

between the Adams-Bashforth-Adams-Moulton and VSTIFF / MSTIFF integrators, the integration-time-step size, the iteration tolerance etc.


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Familiarity with the mathematics is, of course, essential for proper choice of

these settings. The online documentation and the references listed at the end of this book are a good place to cover these topics.

One warning, however, is that the default settings work well for a wide range of physically realistic problems. That is, for problems where the properties of various entities in the system are realistic. Entering

meaningless values, or neglecting to check for consistency in units are the first things to check for if MotionSolve fails to converge.

Integration with HyperWorks MotionView and MotionSolve are quite closely integrated with the other

HyperWorks applications. For instance, it is possible to use MotionView, together with HyperForm, to construct extremely realistic and useful

simulations of stamping transfer presses.

Such examples of integration are presented at technical conferences the world over. Several samples can be found on the website listed at the end of

this book.

Freedom is not worth having if it does not include the freedom to make mistakes.

Mahatma Gandhi

Advanced Topics CAE and Multi Body Dynamics

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Advanced Topics

Modeling physical systems can be extremely satisfying, but it is quite a challenging task. As Tom Clancy put it31, “The difference between fiction and reality? Fiction has to make sense.” All simulation models are, of course, fiction. Their proximity to reality is

limited by several things: the assumptions inherent in the model that generates the equations to be solved, the algorithms used to solve the

equations, the precision of the computer if numerical methods are used, and

so on. Since MBD models involve a high level of abstraction, several complications can be swept under the carpet. That is, the errors introduced by the

abstraction can be compensated for by tuning the model, as outlined earlier. However there are some situations in which more detail has to be included, for the simulation to be realistic and useful. Some of these are discussed

briefly in the pages that follow.

Flexibility When asked “Is light a particle or a wave?”, Einstein is supposed to have

answered, wholly seriously, “Yes”. Wondering whether a body is rigid or

flexible is a similar question, and deserves the same answer.

The sheer complexity of including the effects of the elasticity of links has led to the widely used assumption of rigid links but that is not always accurate

enough.

Compliant Joints The pin-joint in a link can be a major source of error, as any designer who

has analyzed tolerance stackup can attest. As we have seen, designers

usually try to reduce the number of links in a chain. In some applications such as the “scissor linkage” or in several open loop mechanisms, however,

the number of links is deliberately large. In cases like these, or in case where precision is extremely important, including the compliance of a joint in the model can make a significant difference.

31 In a similar vein, Mark Twain observed that “It's no wonder that truth is stranger than fiction. Fiction has to make sense.”

CAE and Multi Body Dynamics Advanced Topics

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High Speed Mechanisms At high speeds, the effects of inertial forces are large enough that the

deformation of the links may be significant. Neglecting these may well lead

to failure of the mechanism: it may jam, vibrate too much, generate too much noise, and so on.

Compliant Links An interest in high precision makes it preferable that the deflection of the links because of elastic deformation be included in the model. In

mechanisms that involve bodies of different materials, some materials may

be much less stiff than others – which means the stiffer ones can be considered rigid, while the more flexible ones should, preferably, be

compliant.

MBD and FEA Finite Element Analysis is widely used to calculate stresses and deformations

due to elastic effects, and it is only natural that an interface between FEA and MBD is the preferred way to include the effect of link-compliance in MD simulation.

One challenge, of course, is that the very approaches of FEM and MBD are different. One uses a distributed model while the other uses a lumped model. The first results in partial differential equations while the latter yields

ordinary differential equations.

But there is one approach, used even in “pure” FEA to reduce the size of the

problem, that allows us to elegantly mix the two methods. Called Component Mode Synthesis (CMS), it involves representing a set of elements as a black-box. That is, the set of elements is reduced to a matrix,

the size of which is defined by the number of modes that are employed in the abstraction. A complete discussion of the theory of the method is

beyond the scope of this book. An excellent description that is both complete and very comprehensible can be found in Structural Dynamics, by R.R.Craig32.

32 The Craig-Bampton and Craig-Chang methods, the most widely used CMS methods, both bear his name.


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Contact The very nature of MBD means that in many cases bodies move through large distances during the periods of interest. The movement may cause

contact to occur between different bodies, or between different surfaces of the same body. In turn, the contacts give rise to forces.

Examples of contact abound, of course. Electric switches, for instance, are designed to make and break contact. The duration of contact is a critical

parameter, particularly for high-voltage equipment.

The problem, then, is for the simulation tool to figure out whether contact

has been made or contact has been broken. This necessarily complicates the MBD modeling approach, since such a calculation is based on a knowledge

of where one body ends and another begins. In other words, the definitions of the surfaces that make up the outer volumes of the bodies are essential.

This is quite a departure from the approach we have seen so far, where the surface definitions of the body are dispensable for the calculations. In the

absence of contact, the inclusion of the surfaces is mainly to aid visualization.

If contact has to be included in the analysis, however, the boundary surfaces are no longer optional. They are an essential part of the problem definition.

One common mistake is to use contact where a constraint would suffice. If

you are sure that two links are always going to be in contact, then it is more efficient to use a constraint such as a point-to-curve or a point-to-surface

constraint. It is when you are unsure of whether the bodies will be in contact with each other or not that a full modeling of contact becomes

essential.

The coefficient of restitution (COR) is an important property in any collision.

And since the establishment of contact is a collision, the COR must be specified whenever contact is used.

There’s another bit of data that makes a larger difference to the solution than the COR. And unfortunately, is even harder to characterize. This is the

coefficient of friction.


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Slip and No-Slip When one object rolls against another, it is important to establish whether

slip is involved or the motion is pure rolling. For involute gears, for instance, slip is involved at all points of contact except at the pitch point. If slip is involved, the coefficient of friction is different than if the motion is pure

rolling.

Friction: Static, Dynamic and Stiction The “Laws” of solid friction are probably better referred to as “Theories” of

solid friction. The study of the mechanics of friction dates back at least to

Leonardo da Vinci’s times, but the accepted “Law” of friction is not as useful as we would like it to be. David Kessler put it quite clearly when he wrote33

“It is one of the dirty little secrets of physics that while we physicists can tell you a lot about quarks, quasars and other exotica, there is still no universally accepted explanation of the basic laws of friction."

Coulomb’s Law of friction is simple, and widely used. In this, friction is of two types: static and dynamic. Static Friction occurs when there is no relative motion between the surfaces in contact with each other, while

Dynamic Friction applies if there is relative motion. What of the transition

zone between static and dynamic friction? This is sometimes referred to as Stiction, probably derived from “sticky friction”, which is seen when a body

is just beginning to move: it is also sometimes called the Limiting Friction.

In any case, we calculate the frictional force using the formula

NRF •= µ

where µ is the coefficient of friction, and RN is the normal reaction at the

point or surface of contact.

33 In the journal Nature (413, 285-288, 20 September 2001)


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The approach used by MotionSolve to model friction is shown in the figure, where µs is the static coefficient of friction, µd is the dynamic coefficient of

friction, vs is the stiction transition velocity, and vd is the friction transition velocity.

Control Systems In 196os, several spacecraft, the Ranger series, were dispatched to explore the surface of the moon. The craft were supposed to rough-land on the moon, so needed some

way to stabilize and control their descent from second-stage

ejection till the lunar landing. Signals from Earth were used to control the system, but one of the problems34 in particular is

relevant to our discussion.

The craft was designed with a gyro as the control system.

Given the gyro’s time constant and inertia, the designer’s problem is to estimate the gain so that the response of the

Ranger to a step input (sent from earth) would overshoot by less than 5%.

For more down to earth (literally!) applications, consider the “automated

manual transmission” systems used in high-performance cars. This is of a

manual transmission, but without a clutch pedal. When the driver shifts gears, a control system manages the clutch – the actuation force is usually provided using either electronic or hydraulic actuators. This approach reduces the time it takes to change gears.

Problems such as these make it essential for us to include control systems in the MBD model. Including the control system and supplying input to the

system is more realistic than omitting it from the model and applying the motion or forces directly to the mechanical component.

For an effective use of control systems, a quick revision of two essential

concepts in Control Theory is in order.

34 Described here in simplified terms


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State Space Models If the behavior of a system is represented by an nth order differential equation, the State Space approach involves reducing this to a set of n coupled first order differential equations. The forms are entirely equivalent, but the latter is better suited for computer simulation.

For instance, the equilibrium equation for a damped spring-mass system is

)(tfkxxcxm =++ &&&

which is a second order differential equation.

The state-space model for this is the equations

vx =&

( ) ( )vmbxm

ktfv −−= )(&

Here, x and v are the state variables, and the set of equations involves only first derivatives of the state variables. (The second equation is obtained from

the equilibrium equation by simple substitution for x& and x&& , followed by rearrangement of terms to leave only v& on the left hand side).

Typical calculations performed using the state variables include evaluating

the response to specified inputs, and calculation of the transfer-function. It is particularly convenient when the system has multiple inputs and multiple outputs – a MIMO control system.

The state-space model, when written in a standard (or canonical) matrix form, uses 4 matrices named A, B, C and D. This nomenclature is used by MotionSolve for the definition of MIMO systems.

Laplace Transforms The Laplace transform of a function is defined by the equation

∫∞

−==0

)())(()( ττ τ deytyLsY s


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Transfer functions are often represented using Laplace transforms, which

have several advantages including the fact that they are distributive, which means

)}(()]([)]()([ tsLtrLtstrL +=+

Laplace transforms are particularly useful for control systems since differentiation of a signal is equivalent to multiplication of its Laplace transform by s, while its integration is equivalent to multiplying its transform

by s.

Block Diagrams The Laplace transform of the transfer functions of the various elements of a

control-loop are usually represented by a block diagram, such as that shown below:

Block diagrams are commonly used for modeling process-plants and

electrical-systems. They are less common in modeling of mechanical systems.

With MotionView and MotionSolve, you can include control systems in your model, though not as a block diagram. Look up the online documentation for

MotionSolve for details on how to build MBD models that include Single-Input-Single-Output (SISO) and Multiple Input Multiple Output (MIMO)

systems using the Laplace transform and state-space representations.

Cams, Gears and other Higher Pairs There are only 6 lower pairs, but any number of higher pairs can be constructed. Several higher pairs are fairly esoteric, which means their

applications are restricted to specific domains. Modeling elements for tires, for instance, are called for almost exclusively by vehicle-dynamics designers.

Some higher pairs can be constructed using simpler modeling elements, if the modeling tool supports programmatic control. For instance, a one-way


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clutch can be modeled using a bush together with an “if” statement to

change properties based on the direction of rotation35. Two higher-pairs that are extremely common are cams and gears.

Cams A cam rotates about an axis and pushes a follower. The cam usually rotates at a uniform speed, and the profile of the cam is chosen so as to deliver the required motion to the follower. There are various classifications of both

cams and followers, most of which reflect the topology or shape of the respective elements36. The follower is usually spring loaded to ensure that it

stays in contact with the cam all through the rotation cycle.

Design interest centers principally around two things:

1. the profile the cam should have to achieve a required motion – the rise,

dwell and return

2. the velocities and accelerations of the follower, and the resulting forces on the various components in the assembly

The first is usually the more interesting problem, but the second is no less challenging. Sometimes the cam profile is determined to match a specified

follower-motion, but such cams can be expensive to manufacture. Often a predetermined cam profile is chosen and the follower of the motion is to be determined so that the design of the rest of the assembly can be tailored

accordingly. In 4-stroke IC engines, for instance, designers need to determine the forces on the tappet.

The joint between the cam and its follower is maintained by contact. General

contact can be used, but this approach is subject to the difficulties discussed above, in the section on Contact. It is usually more computationally efficient

to use point-to-curve (PTCV) or point-to-surface (PTSF) constraints. This

approach does sacrifice some of the generality offered by a full-fledged contact model. For instance, the PTCV constraint does not allow for contact

to be broken. But at the concept design stage, the analysis is usually a kinematic analysis, since the goal is to derive the required profile of the cam.

Once this is done constraints like the PTCV can be used to verify that there

35 MotionView provides support both for bushes and for programmatic control. See the companion volume Managing the CAE Process – Basics. 36 Details can be found in any undergraduate-text on Machine Design.

Positive Return Cam, from the KMODDL


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has been no loss of contact. If there is indeed loss of contact, full fledged

contact modeling is essential. Contact between the cam and follower can break if the spring-load is not

enough to compensate for the inertial forces (that is, forces due to the accelerations the bodies experience). In engine-design this commonly called valve float, because cams are mainly used in the engine to control the valve-timing of four-stroke engines. The term lift-off is also used in several applications.

Gears There are two distinct problems posed by gears, which serve to transmit torque between different axes of rotation.

The transmission of torque is by positive engagement of the teeth. Accordingly, the tooth itself needs to be designed for strength. The design of

gear teeth is a subject that is normally not covered by MBD simulation. MBD analysis can help calculate the tooth-loads, and these loads can then be

used as input for a stress analysis program – usually using Finite Element Analysis.

The other main class of problems deals with the design of the gear train itself. Gear trains range from the aptly named simple gear trains to the amazingly complex epicyclic gear trains. In these cases, analyzing the motion of the output shaft and calculating the ratio of input and output torques are the main areas of interest. An excellent range of models and

animations at the KMODDL shows how complex the motion of gear trains can be. The images of a 4-bar mechanism with two gears, taken from an

animation at the KMODDL, illustrate how complex the motion can be.

Designers of planetary gear trains need to calculate the loads on each gear. Several gearboxes allow for multiple inversions of the gear train – that is,

different gears are held “fixed” to generate different motion. MBD models go

a long way towards eliminating the tedium and error in this demanding task.

MBD models also make it easier to estimate the efficiency of the gear train. A detailed discussion of this aspect is beyond the scope of this book37.

37 See, for instance, Gear Handbook: The Design, Manufacture, and Application of Gears by Dudley, D


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Epicyclic gears are over 300 years old, and are widely used today in a

variety of applications, ranging from almost all propeller and turbine driven aircraft to lawn-mowers. While they are more challenging to design, the present a host of advantages, principally a lower weight and volume.

Calculating the efficiency of the gear train is an important but tedious task even for gears whose axes of rotation are fixed, like the worm-driven helical-

rack-and-pinion shown alongside38.

Gear models in MBD are relatively easy to build. Revolute joints define the axes of rotation of the

shafts, while the gear joint represents the constraint between the two revolute joints.

38 This image too, is from a model at the KMODDL.

If everything seems under control, you're just not going fast enough

Mario Andretti

Glossary and References CAE and Multi Body Dynamics

58

Glossary And References

References Applied Kinematics, Kurt Hain

Modern Control Engineering, Katsuhiko Ogata

Mechanism Design: Analysis and Synthesis, Volume 1, A.G.Sandor, G.N.Erdman

Advanced Mechanism Design: Analysis and Synthesis, Volume 2, A.G.Sandor, G.N.Erdman

Design of Machinery, An Introduction to the Synthesis and Analysis of Machines and Mechanisms , Robert L. Norton

Handbook of Numerical Applications, Jaroslav Pachner

Other Resources www.altair-india.com/edu, which is periodically updated, contains case

studies of actual usage. It also carries tips on software usage.

The Kinematic Models for Design - Digital Library (http://kmoddl.library.cornell.edu) is an excellent resource both for a

historical coverage of mechanisms, animations of models and for several e-books, including da Vinci’s Codex Madrid I and Hartenburg’s Kinematic Synthesis of Linkages.

Types of Analyses The table below39, is a convenient way to summarize the types of analyses, the data required for each, the principles involved in finding the solutions,

and the types of results that can be calculated.

Method

Statics Kinetostatics Dynamics

39 From Advanced Mechanism Design, Erdman and Sandor

CAE and Multi Body Dynamics Glossary and References

59

Masses / Inertias

Weight of links

may be required but the inertia is

not

Required Required

Loading Specified Specified at each position

Specified in terms of position,

velocity, and / or time

Input Information and assumptions

Motion Positions specified

Position,

velocity and acceleration specified

Unknown

Output Information

Force required to

balance load, mechanical advantage at

each position, reactions in joints

Force required

to sustain assumed motion,

reactions in joints

Position, velocity and acceleration of each member as a function of time – that is, the actual motion

Required analytical tools Statics, Linear Algebra

D’Alembert’s principle,

statics, linear algebra

Differential equations of motion

Formulae for the Moments of Inertia In these days of 3D CAD, we often pay little attention to the geometric and mass properties of the bodies we’re working with. Most CAD packages can

quickly and accurately give you these properties even for complicated

shapes.

However this reliance on CAD calculations often leads to mistakes which can critically affect the analysis. The most common mistake is to forget that the

Moments of Inertia are strongly orientation dependent. A moment’s reflection will remind you that this is only to be expected, since Mass Moments of Inertia are related to angular acceleration by40

T = Iα

40 Similar to F = ma for linear acceleration. Look up Euler’s Equations of Motion for a more complete treatment of the variables involved.

Glossary and References CAE and Multi Body Dynamics

60

where T is the torque, I is the moment of inertia and α is the angular acceleration. Which Moment of Inertia should be chosen depends on the axis of rotation. The equations for the mass Moments of Inertia are

∫∫∫ += dmzyI xx )( 22

∫∫∫ += dmzxI yy )( 22

∫∫∫ += dmxyI zz )( 22

∫∫∫= dmzI xy2

∫∫∫= dmyI xz2

∫∫∫= dmxI yz2

Ixx = Izx + Ixy, Iyy = Iyz + Ixy, Izz = Iyz + Izx The radius of gyration is given by

mass

Ir x

x =

When you build a model, it’s useful to run a first analysis with approximate bodies – cylinders, boxes, etc. – both to reduce computation time and to

verify that the range that the properties lie in is acceptable to the Solver’s default settings.

The Moments of Inertia of some “primitives” are listed below. All the values

are about the center of gravity. Refer to any text on Statics for details – see,

for example, Theoretical Mechanics by P.F.Smith and W.R.Longley. Note that the units are mass*length2. In SI units, therefore, the mass moment of inertia would be in kg-m2.

Mass moments of inertia should not be confused with the area moments of

inertia, used for example in the formulae for beam bending. The area moment of inertia uses a different formula, and has the units m4.

CAE and Multi Body Dynamics Glossary and References

61

Cylinder with open ends

The z axis is along the axis of the cylinder. The x and y axes are any diameters.

)(2

1 22

21 rrmI z +=

)33(12

1 222

21 hrrmII yx ++==

where m is the mass, r1 is the inner diameter, r2 is the outer diameter, and h is the height.

Solid Sphere

5

2 2mrI =

where m is the mass and r is the radius.

Cuboid

)(12

1 22 dwmI h +=

)(12

1 22 hwmI d +=

)(12

1 22 dhmI w +=

where m is the mass, and h, d and w are the dimensions along the 3 principal directions. The origin of the 3 axes is at the center of mass of the

cuboid.

Common Coefficients of Friction Friction coefficients are extremely sensitive to the presence / absence of lubrication, as well as to other factors like the pressure between the surfaces, surface finish, etc. The values in this table should be treated with

corresponding care. Several websites provide similar information (see, for instance, http://www.roymech.co.uk/Useful_Tables/Tribology/co_of_frict.htm)

which are useful for preliminary design. For further analyses, nothing beats lab tests.

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