Altair Students Guide - CAE and Multi Body Dynamics

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CAE and Multi Body Dynamics Introduction

Introduction

About This Series

To make the most of this series you should be an engineering student, in your third or final year of Mechanical Engineering. You should have access to licenses of HyperWorks, to the Altair website, and to an instructor who can guide you through your chosen projects or assignments.

Each book in this series is completely self-contained. References to other volumes are only for your interest and further reading. You need not be familiar with the Finite Element Method, with 3D Modeling or with Finite Element Modeling. Depending on the volumes you choose to read, however, you do need to be familiar with one or more of the relevant engineering subjects: Design of Machine Elements, Strength of Materials, Kinematics of Machinery, Dynamics of Machinery, Probability and Statistics, Manufacturing Technology and Introduction to Programming. A course on Operations Research or Linear Programming is useful but not essential.

About This Book

This volume introduces techniques to model and analyze mechanisms, which lie at the heart of machines.

If product design is your area of interest, you will find the companion volumes, CAE And Design Optimization – Basics and CAE And Design Optimization – Advanced useful. The techniques outlined in this book are usually applied at the very early stage in product design, to be followed up at a later stage in the design cycle with detailed analyses and optimization, both to improve peak performance and to introduce robustness.

While it’s not essential, a good grasp of the basic principles of vector mathematics will help you tremendously. Several essential aspects are covered in this book, although in a qualitative fashion. You may want to treat the chapter titled Theory: Basic, Essential and Advanced as a

reference. If you choose to adopt this approach, at least a cursory reading of this chapter is strongly recommended.

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CAE and Multi Body Dynamics Introduction The various references cited in the book will probably be most useful after

you have worked through your project and are interpreting the results.

Supporting Material

Your instructor will have the Instructor’s Manual that accompanies these volumes – it should certainly be made use of. Further reading and

references are indicated both in this book and in the Instructor’s Manual. If you find the material interesting, you should also look up the HyperWorks On-line Help System. The Altair website, www.altair.com, is also likely to be of interest to you, both for an insight into the evolving technology and to help you present your project better.

My robots were machines designed by engineers, not pseudo-men created by blasphemers.

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Mechanics, Mechanisms and Machines CAE and Multi Body Dynamics

The South Pointing Chariot is widely regarded as the most complex geared mechanism of the ancient Chinese civilization. Invented sometime around 2600BC in China by the Yellow Emperor Huang Di, the first historical version was created by Ma Jun (c. 200-265 AD). The chariot is a two-wheeled vehicle, upon which is a pointing figure connected to the wheels by means of differential gearing. Through careful selection of wheel size, track and gear ratios, the figure atop the chariot will always point in the same direction, hence acting as a non-magnetic compass vehicle.

After being mocked that he could not reproduce a non-historical and nonsensical pursuit, Ma Jun retorted "Empty arguments with words cannot (in any way) compare with a test which will show practical results". After inventing the device and proving those who were doubtful wrong, he was praised by many.

The differential in the gear system integrates the difference in wheel rotation between the two wheels and thus detects the rotation of the base of the chariot. The mechanism

compensates this rotation by rotating the pointer in the opposite direction.

Adapted from The Wikipedia See Wikipedia, South Pointing Chariot

Mechanics, Mechanisms and Machines

The study of mechanisms can be a joy, if done properly.

The devil, unfortunately, lies in that “if”. Much of the mathematics of the subject is tedious when done by hand, and a beginner can be excused for feeling lost in the headlong rush of vector notations and vector manipulations. Most engineers end up treating mechanisms like poisonous snakes: worthy of a great deal of respect, and safe only when viewed from a distance.

Even more unfortunately for a core discipline in mechanical engineering, the study of mechanisms at the undergraduate level has probably benefited the least from the widespread developments in CAE1 software and related technologies. In fact, a mechanical engineer who uses an Internet-search engine to look for material on “mechanisms” is likely to give up the exercise as counterproductive. Most search engines return references from economics and game theory, making a challenging subject even more confusing!

This is a shame.

The use of the word “mechanisms” in Game Theory is a very good illustration of the power of the various theories and approaches used in the design of machines. The methods used to construct

building blocks that allow the modeling of complicated machines are appealing in their

simplicity, and often stunning in their power. Hence their use in fields as far flung as Economics, where a mechanism is simply “the agency or means by

1_{Short for}

Computer Aided Engineering – a term that usually covers design, analysis, 3D modeling, and testing in the course of product-design.

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CAE and Multi Body Dynamics Mechanics, Mechanisms and Machines which an effect is produced or a purpose is achieved”.

What is Multi-Body Dynamics?

At first glance, there are few design issues common between a fighter plane in supersonic flight, a car rolling over as it crashes, a ship pitching in the stormy seas and the micro-precision movement of the read / write head of a hard-disk drive. If we apply the wider definition of “mechanisms” that we have just seen, the effects are different, the purposes are different and the agencies used are different.

A little consideration, however, shows that all of these involve the

investigation of the movement of, and impact of, multiple bodies. The jet plane shoots missiles at other targets, and may even be a target itself. The car bounces off the road or crashes into other vehicles or obstacles. The ship contains several bodies – machinery, passengers, cargo, etc. And, as anyone who has mistimed a dive into a swimming pool knows, at high velocity water can be “hard” enough that it can be treated as a single body, rather than a collection of droplets. Every computer owner knows that sooner or later the disk-drive will “crash” – one form of this is a literal crash, when the head makes contact with the platters themselves, destroying the disk and any data the unfortunate user has placed there.

While the scale of movements, the sizes of the bodies and the forces

involved vary widely between these applications, in all these cases designers need to understand how the forces affect the movement of the body, and

vice versa. And, of course, there are multiple bodies involved.

This aspect, together with advances in software technology over the recent past has, in fact, led to the widespread adoption of the title Multi-body Dynamics2 in the place of phrases like “Rigid Body Mechanics” and “Mechanism Design”.

MBD finds applications in almost any field where there are moving mechanical components: machine tools, packaging equipment, conveyor belts, engines, road vehicles, elevators, railways, stereos, washing

machines, aircraft, spacecraft, pumps, robotics – the list can go on almost indefinitely. One application that’s sometimes dismissed as trivial but throws up several exquisite applications of this remarkable science is the design of

2_{Often abbreviated to “MBD”}

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Mechanics, Mechanisms and Machines CAE and Multi Body Dynamics toys, as can be attested to by anyone who has puzzled over the internal

workings of Rubik’s Cube.

The problems that designers grapple with are introduced in the next

chapter, but common to all of them is the need to deal with one or more of the forces, displacements, velocities and accelerations of different parts of the system. Some designers analyze mechanisms: that is, they find out the values of parameters of interest under different operating conditions. Still others synthesize mechanisms: they come up with designs that will provide required movement.

The subject is often multi-disciplinary. For instance, the source of motion – the actuator – could be hydraulic. Study of the mechanical links or

components requires a strong hold on mechanics. The control system could be electronic, while the sensors could be piezoelectric.

Learning MBD - Different Approaches

There are two ways, then, to gain a command over the capabilities of MBD tools. One approach is to focus on the theory, drawing comfort from the fact that a robust theory can be applied widely, provided the fine-print is

followed meticulously. Another approach is to pick a specific application and pay attention to the assumptions and data specific to this application.

The use of general-purpose MBD software for CAE mirrors these

approaches. At the “theoretical” level, all bodies can be modeled using a few basic building blocks. At the applied level, each of these building blocks is adapted to the requirements of the specific field. For instance pneumatics and hydraulics both use similar building blocks – valves, pistons, etc. – but the specific behaviors of the fluids varies.

In several industries this “specific behavior” is treated as intellectual

property. It is fiercely guarded, since it is arrived at over the course of much trial and error, and can make the critical difference between performance that’s “just good enough” and performance that makes the product a pleasure to use!

Putting It All Together

MBD, then, is about more than just machines and mechanisms, but

definitely involves mechanics. Unlike “simple” mechanism design, though, it often involves a lot more. If the study of the approaches that MBD tools take

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CAE and Multi Body Dynamics Mechanics, Mechanisms and Machines seems overwhelming, it is useful to remember the old saying take care of

the pennies and the pounds take care of themselves.

Analysis problems are solved using the divide-and-rule approach: break down complex objects into simpler blocks, and these into even simpler blocks, and so on. The synthesis problems are solved by starting with known blocks, and looking for ways to put them together to achieve complex

behaviors.

Our goal, then, is easy to state and vast in it’s coverage: we expect to be able to design any mechanical system with moving parts!

The secret of getting ahead is getting started. The secret of getting started is breaking your complex overwhelming tasks into small manageable tasks, and then starting on the first one.

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Typical Design Issues CAE and Multi Body Dynamics

Typical Design Issues

Practitioners in the engineering industry often complain that software tools are used just for the sake of using the tools. All too often, these complaints are justified. What is the point of spending time and effort, in addition to money, on usage of tools if the results do not help the designer? To make matters worse, the usage of the tools may even draw resources away from the actual design goals!

Of course, this potential criticism of CAE tools applies to all the tools covered in this series of books. What’s special about MBD? Why should we pay special attention to this aspect when studying MBD?

As we have already seen, the spread of applications that MBD addresses is extremely wide. And we will see later that the MBD modeling approach takes a relatively abstract view of the behavior. It’s this abstraction that makes it that much more important for you to keep track of what the design issues are. Your entire model-building approach and results-interpretation should be tailored to suit these.

Accordingly, before reviewing the underlying theory, it is useful to review some areas of application and the related design issues as relevant to MBD modeling and analysis.

Product Liability

The lot of a product designer is often stressful, and not just because of pressures on time, cost and quality. Laws in several countries are extremely demanding, and the trend is towards stronger legal safeguards against faulty products. In a review of the impact of legislative reforms on product-liability risks in the Asia-Pacific region3,

100% of insurers/brokers thought that there had been an increase in the number of product liability claims in the Asia-Pacific region since the Reforms. One hundred percent reported that there had been an increase in settlements.

3_{Kellam, J and Nottage, L: "Report on Clayton Utz Asia-Pacific Product Liability}

Survey" (2006) 17 (9) APLR 121, published in the Australian Product Liability Reporter.

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CAE and Multi Body Dynamics Typical Design Issues Failures that can cause loss of life or grievous harm are often identified and

publicized voluntarily by the manufacturers themselves. Such product recalls can be expensive both in terms of actual expenditure to fix the flaws and in terms of the damage to the reputation of the company involved.

Legal protection for consumers mean that designers need to be alert even to failures that are potentially less exacting, as illustrated in the extracts from US Consumer Product Safety Commission’s recall notice reproduced below4:

In cooperation with the U.S. Consumer Product Safety Commission (CPSC), JB Research Inc., of Bellflower, Calif., had voluntarily recalled about 15,000 back massagers sold under the Relaxor, Deep Knead™ Shiatsu brand name. The motor for the massager's Deep Knead™ mechanism can jam and overheat. This will cause scorching to the foam and fabric on back of the unit, presenting a potential fire hazard to consumers…JB Research Inc. has received 46 reports of units overheating. No fires or injuries have been reported.

Consumers should stop using the recalled massagers immediately. Since JB Research Inc. is no longer in business, recalled massagers should be discarded or destroyed to prevent fires and injuries.”

Some Application Areas

Machine Tools

Machine tools are often thought of as “old” technology, which in a sense they are: the growth in use of various types of machine tools dates back to the 100 years between 1860 and 1960. But that does not mean the

technology is trivial or that design is easy. High-precision jig-boring

machines, even in the 1940s, were designed so as to account for the effects of heat generated by human operators and even the slightest tremors of the earth.

Common to all machine tools is the principal goal: a specified degree of precision.

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Typical Design Issues CAE and Multi Body Dynamics To achieve this, the designer should produce variable movements, and

provide control over these movements. Conditions of operation can usually be maintained within specified ranges, particularly if the designer can demonstrate a link between the operating conditions and the

precision of the machine. Attendant design issues are the life of the machine, and its cost.

From a designer’s point of view, models that can calculate and predict forces, loci of various points, and times of motion are particularly critical.

Packaging Machinery

The term itself usually covers machines that can do one or more of wrapping, palletizing, taping, capping, filling, labeling and printing. If you consider that almost any goods – from toothpaste to automobiles – need to be packed, the size of the industry is extremely large. Environmental

concerns are prompting changes in the materials used, prompting designers to exercise their ingenuity.

From one point of view, packaging machinery is similar to machine tools. Conditions of operation can usually be controlled, and the movement needs to be controlled automatically. The differences stem mainly from the scale of usage. Packaging machinery is usually critical for mass-produced items, where the volume of production is extremely high.

This means the design approach can often afford to sacrifice versatility of motion for economy and precision – in a sense, this is similar to the design approach that underlies Special Purpose Machines. And since the scale of production of the goods being packaged is very large, a lot of design focus is on the time of motion. A design that can reduce the filling time by 1 second can be much more attractive if the filling time per package is of the order of seconds!

An example of this approach is highlighted by a product manufacturer’s brochure, which says “machine kinematics have been designed to allow a very high speed, while at the same time leaving more time for the most delicate phases of the filling process”. In the pharmaceutical industry, speeds of 200,000 capsules per hour are not uncommon.

Engines

Most mechanical engineers are familiar with, if not extremely comfortable with the working of, IC engines. What we sometimes fail to remember is

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CAE and Multi Body Dynamics Typical Design Issues that engines themselves can vary tremendously: from the enormous diesel

engines to the rotary-piston Wankel engine. The picture shows engineers installing the thin-shell bearings for the Wartsila-Sulzer RTA96-C

turbocharged two-stroke diesel engine – an engine with a stroke of over 8 feet. The technicians shown in the picture show its size!

Engine design is a multi-disciplinary area, covering heat transfer, vibration, combustion, etc. Conditions of operation are less predictable than for machine tools, so designers often have to investigate and allow for harsh operations. Recommended ranges of operation are usually provided, such as the “red-line” speed limit for IC engines.

From an MBD perspective, the red-line speed is an interesting parameter. If a 4-stroke engine is run at a higher-than-recommended speed, the force exerted on the return-springs may be high enough that the valves “float”. That is, the valve-lifters lose contact with the lobes of the cam. This, in turn, leads to lost horse-power.

Unlike machine tools, the interest is not so much in providing variable movement as in calculating component forces at various operating conditions. These forces are then used to

perform stress and fatigue verifications. Advanced engines also require quite sophisticated forms of motion-control. High performance engines, for example, alter the valve timings and lift as the engine speed changes.

Vehicles

Lumping cars, trucks, buses, motorcycles, bicycles, ships, aircraft and spacecraft, etc. into one group is obviously a simplification, but one that is quite effective from our current perspective. The degrees of complexity vary – from a few dozen parts in a bicycle to several thousand parts in larger vehicles – but the vehicles themselves have fairly similar requirements: stability, safety, comfort and (in most, but not all, cases) economy of operation.

The baseline science scenario [for the design of a Mars airplane mission architecture] requires completion of a controlled aerial survey, spanning a flight range of 500 km at an altitude below 2 km. These requirements drive selection of a powered airplane as well as the airplane propulsion and navigation systems and aerodynamic configuration. The entire sequence of events (including pullout) is approximately 5 minutes in duration. Airplane extraction is initiated 7 seconds after heatshield release.

Six-degree of freedom multi-body simulations have confirmed the 7-second delay is

sufficient to mitigate the potential for re-contact between the airplane and the heatshield.

From The Mars Airplane IEEE, 2004

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Typical Design Issues CAE and Multi Body Dynamics The first requirement, stability, is of particular interest. Since the operating

conditions tend to vary widely, a lot of design effort centers around issues of providing proper control over the behavior of the motion, rather than the motion itself. Remember that the motion is reasonably predictable. What is unpredictable is the acceleration that the driver applies, the conditions that the surface provides, and so on.

Car designers, for example, pay attention to the “toe curve”, since it is a critical measure of drive quality of the vehicle. Modern designs have seen a steady increase in the amount of “on board” electronics used to help steer the vehicle safely. Many road vehicles, for instance, come with anti-lock brakes, where a control system senses the motion and automatically adjusts the brake pressure to prevent a skid.

Vehicle stability, in fact, is currently receiving the attention of lawmakers in some countries. Electronics stability controls may soon be mandatory in many vehicles5.

Robotics

In his book “Inside the Robot Kingdom”, F.L.Schodt paints an impressive picture of the Fanuc factory in Japan, where, under Mount Fiji, robots work unattended at night – making other robots! The point, of course, is that robots are not just inhabitants of Science-Fiction worlds. They are very much here to stay.

Robotics, as a discipline, poses problems that are hard to categorize, since robots themselves are so hard to categorize. Some work in controlled environments, such as factories, while others are designed for harsh and unpredictable environments such as the depths of the oceans. Some require precision, such as those designed to help weak or ill humans, while others are designed to work with more forgiving payloads.

The mechanical side of the robot is the classical mechanisms problem of synthesis: how to assemble mechanical elements that can describe various motions. In several cases, inspiration is drawn from biology, mimicking human or animal joints.

5_{See http://www.consumeraffairs.com/news04/2006/09/nhtsa_stability.html, for}

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CAE and Multi Body Dynamics Typical Design Issues A multi-body designer, then, needs to generate paths of motion, predict

velocity of different parts of the assembly, and to predict forces that will be experienced and that can be generated by the robot. A large degree of integration with electronic control systems is also essential, given the current state of technology in robotics design. Issues of stability, which have to be handled by the control systems, have recently been addressed well enough for a two-legged robot to climb stairs or catch a ball thrown at it.

The MBD Modeling Philosophy

There are many other areas, of course, where designers seek to understand the forces experienced by and caused by multiple bodies. There are also many modeling and analysis techniques other than MBD, some of which are covered in other books in this series. As the figure below6 emphasizes, any model is a part of a larger system, and can in turn be broken into smaller sub-systems – all the way down to quantum mechanics

MBD methods are reduced order models that are best applied at the “Product” and “Assembly” stages7.

6_{Courtesy of Prof. Bert Bras, George W. Woodruff School of Mechanical Engineering,}

Georgia Institute of Technology, Atlanta, GA 30332-0405, USA.

7_{The Finite Element Method, covered in}

A Designr’s Guide to FEA, is most relevant at the “Component” and “Assembly” stages.

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Typical Design Issues CAE and Multi Body Dynamics From a product design perspective, just as product design follows a

sequence – starting with a concept, going on to a rough design, and finally the detailed design – it is useful to group the design issues into system level

issues and component level issues.

Simulation of component behavior is often done using the Finite Element method. Here, the analyst requires the forces on the component as data for the model.

Simulation of system level behavior is best done using the MBD approach. Obviously, one benefit is that the forces calculated from an MBD analysis can be used to provide data for a Finite Element analysis. However, there are other reasons that make this a natural way to address several complex design issues.

For one, MBD models take a ”lumped” approach. That is, the behavior of an arbitrarily complex component or assembly is abstracted as a single

element. The abstraction may represent a single rigid link, the suspension assembly of an automobile, or the undercarriage of an aircraft. In all these cases, some accuracy is traded for speed of analysis. Where a Finite Element analysis frequently requires minutes, if not hours or days of CPU time, an MBD analysis is often complete in seconds.

Next, simple MBD models are used to build more complex models. In an approach that follows the engineering practice of using simple tools to build more complex tools, this provides the capability to quickly build complex models that yield useful results without taking an inordinate amount of time. Take, for example, a bearing that supports a rotating shaft. Both theory and practice tell us that there are losses within the shaft, but it is well near impossible to get an accurate model that can predict the losses in a production-quality bearing within reasonable times and at reasonable

expense. The MBD approach is quite practical, yielding a usable model while taking into account the absence of detailed mechanics8:

The bearing module is used to determine the bearing moments due to friction for all of the bearings in the transmission. The input signals to the bearing module consist of the torque at each shaft location. The module then calculates the torque loss due to friction with the relation:

8_From

Design And Analysis Of A Modified Power Split Continuously Variable Transmission, A.J.Fox, West Virginia University

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CAE and Multi Body Dynamics Typical Design Issues

Tloss,n = Fndµ

where Tloss,n is the torque loss in bearing n, Fn is the force on bearing n, and µ is the bearing coefficient of friction. Fn is calculated from the force analysis on the shaft due to tangential gear forces and component weight. The bearing module solves this relation for each bearing location, and the output signals consist of the torque loss at each bearing location.

Note how simple the equation is. The simplicity is justified because there is no complete theory of the specific mechanics that also lends itself to quick calculations. Actual usage of this bearing module would only require that the coefficient of friction be fed in, since the forces are calculated using the equations of equilibrium. The model is not only simple, it is effective, since several of these bearing modules can be employed in the model of the overall transmission.

Another example of this approach is the construction of computer models for animated movies – such as the dinosaurs in Jurassic Park and its sequels. Designers concentrate on capturing an adequate behavior of selected joints, not on the body as a whole. Once they have the individual joints behaving the way they want them to, they can assemble these to get the complete body – and can be sure that the assembly will move in a “realistic” fashion. A publication from the Aalborg University9 puts it well:

Depending on the type of joint, kinematic pairs are either referred to as force-closed or form-closed, where the fastening of the pair in a form-closed pair is maintained by the shape of the bones themselves, and the fastening of a force-closed pair is maintained by a superficially applied forces, such as a tendon. […] To limit the amount of joints, [… some] joints have been combined. Bone structure of [the figure] refers to the bone controlling the belly of the orc, which makes it possible to animate a “jumping” belly whenever the orc moves, walks, runs or jumps.

This approach – complex models constructed from simpler models – lies at the heart of multi-body dynamics. Once we look at the building blocks that are commonly available, our study will be almost complete!

9

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Typical Design Issues CAE and Multi Body Dynamics

Summary

Engineers working on CAE have several sources of worry, ranging from potential legal complications to essential product performance. To make things worse, the expectations change almost continuously during any product-design project. Almost invariably, the quality of results expected from the designers tends to get raised as the project progresses.

One benefit, however, is that early in the design cycle the analyses need not be very precise. Later, when the detailed-design phases is undertaken, results need to be more accurate – but at the early stage, quick results are often of more value than accurate results.

The MBD approach is tailor-made for this. And if it can be coupled with detailed-design tools such as Finite Element Analysis and Design

Optimization, the design engineer really can’t ask for very much more.

The fact that the man who gave the world electric light, motion pictures, talking machines, and the Edison storage battery was responsible for this utterly useless device should encourage inventors whose first attempts have failed.

George Lee Dowd Jr in Popular Science Monthly,1930, on Edison’s unsuccessful Helicopter

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CAE and Multi Body Dynamics Theory: Basic, Essential and Advanced

Theory: Basic, Essential and Advanced

The availability of easy-to-use and reliable simulation software, coupled with high quality graphics, makes it easier to grasp the principles embodied in much of engineering and engineering mathematics. A picture can be worth a thousand lectures.

This does not mean, of course, that the software can be used without a good grasp of the underlying theory. The previous chapter outlined the importance of focusing on the requirements of design and on the

importance of proper abstraction of behavior. Using software without an understanding of the fundamentals is an invitation to disaster, not to mention being a waste of time, effort and money!

This chapter is intended to serve as a quick reference, not as a complete discussion. Our emphasis is on providing definitions with a minimum of equations or other mathematical notations. The references listed at the end of this book are an excellent (and strongly recommended) source of

complete theory, and should be referred to if any of the intentionally brief definitions presented below are ambiguous or incomplete.

A quick review of the adoption of 3D CAD tools by the industry is

illuminating. In the early years, users had to understand internal details like the equations of splines or the algorithms used to calculate surface

intersections. As usage and software matured, a lot of this could be taken for granted: just as the way you can today drive a car without

understanding how an IC engine works. Of course, if the engine breaks down, you either need to call for an expert, or develop the expertise yourself! In a similar fashion, if the software fails to achieve a particular task, a grasp of the theory used always helps.

While MBD tools have not been adopted as widely by the industry as CAD tools, for a variety of reasons, the fact is that MBD tools today are both capable and robust. The benefits are clear, the applications are clear, and the tools are available.

To use MBD tools effectively, of course, you should make sure that you pay attention to detail. When you work through the assignments that accompany this book, you may want to turn back either to this chapter or to the

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Theory: Basic, Essential and Advanced CAE and Multi Body Dynamics references listed at the end of this book to make sure that you are correctly

correlating the software’s features with the theoretical underpinnings. Finally, classroom studies are often limited to problems that can be solved using trigonometry and vector algebra. Usually, this means the coverage is restricted to Cams, Gears and 4-bar linkages. Non-linear equations, complex numbers and mechanisms with more links are often omitted simply because they are not tractable enough for hand-calculations. MBD software makes such problems tractable, thereby making it easy for meaningful problems to be modeled and analyzed even at the learning level.

Theory …

Basic Definitions

Statics, Kinetics, Kinematics and Dynamics

Mechanics (or, more correctly, Solid Mechanics10) has three branches: Statics, Kinetics and Kinematics. Statics covers the effects of forces on bodies in the absence of motion. Kinetics is the study of the action of forces on bodies in motion. Kinematics is the study of the relative motion between bodies. Kinetics and kinematics together are often referred to as dynamics. Often designers use kinematics to determine the initial design to achieve the required motion. Kinetics is then applied to investigate and improve weight, stability, cost, control, etc. Kinematics is sometimes called the geometry of pure motion – because there is no reference to mass or forces. Most CAD packages use this approach to animate assemblies. Kinematics, for example, can be used to calculate the motion required for a robot to pick up an

object, while dynamics can tell you the forces required for this.

A kinematic diagram is used to represent a mechanism – the figure on the right is the kinematic diagram of the folding steel chair on the left. The procedure to derive a kinematic diagram from a mechanism can be

confusing, as can the process of visualizing a mechanism represented by a given kinematic diagram. MBD software is very useful here, since it allows you to add graphics to the kinematic diagram.

10_{The kinematics of fluids is normally not called kinematics, and is considerably more}

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Mechanisms

Engineering Mechanics differentiates between a structure and a mechanism by calculating the mobility of the assembly. If the assembly is such that no movement is possible, it is a structure. A good example is a simple plane truss, where the dimensions of the links define the only relative position that can be achieved.

A mechanism is also sometimes defined as a device that transfers force / motion from a source to a destination. From our point of view, a mechanism consists of links and joints.

Machines

There is no clear-cut difference between a mechanism and a machine. Some define a machine as a mechanism that does useful work, but that distinction is not relevant to our study. One of the dictionary meanings for “machine” is, not surprisingly, “mechanism”.

Conservation of Linear Momentum

In the absence of external forces, the momentum of a body or set of bodies remains constant. When applied to linear motion, this results in the familiar equation

F

=

m

• a

. Another way of stating the same, of course, is to say that force involved in a collision is equal to the rate of change of

momentum.

Collisions can be either elastic or inelastic. Elastic collisions conserve kinetic energy but inelastic collisions don’t. Both, of course, conserve momentum. The coefficient of restitution is a measure of the elasticity of the collision. It has no units, since it is the ratio of the differences in velocities before and after collision. It is 1 for a perfectly elastic collision and 0 for a perfectly inelastic collision – that is, one in which the bodies stick together after the collision.

Conservation of Angular Momentum

If the law of conservation of momentum is applied to angular motion, it leads the equation

T

=

I

• α

, where T is the torque, I is the mass moment of inertia about the axis of rotation, and α is the angular acceleration. Calculation of the mass moments of inertia is often confusing, particularly to beginners who often forget the reliance on the axes (or coordinate systems) used, particularly if the values are calculated automatically by a CAD

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Theory: Basic, Essential and Advanced CAE and Multi Body Dynamics package. See the Glossary and References section for more details on

Moments of Inertia.

Links

A link is a body that is a part of a mechanism. Some definitions of links require that they be treated as rigid bodies (i.e. those that cannot deform under the action of forces) but MBD removes this necessity.

Links are often classified as binary, ternary and quaternary, depending on how many other links they’re attached to. A binary link is attached to two other links, a ternary link to 3 other links, and a quaternary link to 4.

Nodes

A node is the point at which one link is attached to another in a kinematic drawing. (Do not confuse this with a node in a Finite Element model!). A binary link has 2 nodes, a ternary link has 3, and a quaternary link has 4.

Degrees of Freedom

The DOFs (which is how the phrase degrees of freedom is usually abbreviated) of a link is the number of independent inputs required to determine its position with respect to the ground. The DOFs of a mechanism are the number of independent inputs required to determine the positions of all links (with respect to the ground) that make up the mechanism.

Note that a joint “eliminates” one or more degrees of freedom, as described below.

Calculating the DOFs of a mechanism is not a trivial task, as we will see when we discuss Gruebler’s Equation. A rigid body in 3D space has 6 DOFs – translations along the 3 axes, and rotations about the 3 axes. In 2D space (textbooks often refer to planar mechanisms, which are mechanisms restricted to 2D space) a rigid body has 3 degrees of freedom: translations along the 2 axis and rotation about the third axis (which is the cross product of the two axes of translation).

Constraints

A constraint is a condition that removes one or more DOFs. In MBD, a constraint is usually imposed by defining a joint.

For instance, if a system consists of 2 links that are not connected to each other, the system has 12 dofs (6 for each link). If they are connected by a

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CAE and Multi Body Dynamics Theory: Basic, Essential and Advanced joint, however, the dofs will be less than 12. Which dofs are eliminated by

the joint is often a source of confusion to a beginner.

If the number of constraints is more than the dofs of the system, the system is described as over-constrained. An over-constrained system cannot be analyzed using MBD. If presented with an over-constrained system, many programs arbitrarily discard as many constraints as necessary. A designer should beware of such situations! It is far better to correct the joint definitions yourself than to leave it to the software.

Joints

From a mathematical perspective, a joint is just a constraint – it relates the motion between one or more DOFs of one or more bodies. In the context of MBD modeling, a joint is usually defined using a physical equivalent.

Most joints eliminate one or more DOFs. However if the joint is redundant, it does not affect the dofs of the system. Redundant constraints are also called

passive constraints: their presence or absence does not make any difference to the behavior of the mechanism11.

Pairs, Higher and Lower

Two links connected by a joint are called a pair. Pairs are classified as lower pairs and higher pairs. A lower pair is one where interchanging the links does not alter relative motion. For a higher pair, exchanging the links alters

relative motion.

There are only 6 lower pairs, while there are infinite types of higher pairs. The 6 lower pairs are revolute (or pin-joint), prismatic (or slider joint), helical

(as in a nut-and-screw), cylindrical (as a shaft in a collar), spherical (or ball joint) and planar12_.

Cams, gears, belt-drives, etc. are higher pairs. Higher pairs normally need additional equations, such as the gear-ratio, to fully-define them. Higher pairs are also more susceptible to drawbacks such as backlash, slip, creep and friction losses. Of course, manufacturing tolerances can introduce error into lower pairs too.

11_{Passive constraints can cause trouble when manufacturing tolerances are taken}

into account.

12_{Planar, Revolute and Prismatic pairs can be treated as special cases of helical}

pairs. A zero lead makes it a revolute joint, an infinite lead makes it a prismatic joint. Moving the center of the helical pair to infinity gives a planar joint.

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Theory: Basic, Essential and Advanced CAE and Multi Body Dynamics

Closure

Some joints ensure contacts between the links by means of the elements themselves – a revolute joint is a good example. Such joints have form closure. Other joints, such as cams, require external forces to maintain contact, and have force closure. The external force can be supplied via a spring, or by gravity, etc.

Chains

A chain is a series of pairs connected together, without a grounded link. A chain is called a mechanism only if at least one link is grounded. This is because force-transmission makes sense only if the “ground” provides the support for the reactions that Newton’s Third Law guarantees. A chain can be either closed or open. Two binary links connected by a joint are called a

dyad.

Inversion

The behavior of some mechanisms can change dramatically depending on which links in the chain are fixed and which are left free to move. An excellent example of this is the epicyclic gear train.

Gruebler’s Equation and the Kutzbach Criterion

Calculating the DOFs of an assembly is not easy. If movement is restricted to a plane (that is, if we have a planar mechanism), we can use Gruebler’s Equation:

h

l

n

F

=

3 (

−

1 )

−

2 −

where F is the total degrees of freedom of the mechanism, n is the number of links (remember to include the ground or frame), l is the number of lower pairs and h is the number of higher pairs.

Be careful when using the formula – it is not foolproof in the sense that it cannot be applied blindly, but needs some judgment. The mechanism shown below has 1 DOF although Gruebler’s equation would say it has none!13

13

n = 5 since there are 5 links including the ground, l = 6 since there are 6 lower pairs. The formula fails due to redundancy: removal of the middle link has no affect on the mechanism. The correct values of n and l should be 4 and 4, respectively, which gives 1 dof.

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The DOFs of a mechanism are also called its mobility. This term is used when we want to count the number of input parameters that must be controlled independently to achieve a particular motion or position. The

Kutzbach Criterion, which is used to calculate the mobility allows for the elimination of partial DOFs by a joint.

Also remember that there’s a difference between mechanisms an 3D space and mechanisms in 2D space. The equation used in 3D has a slightly different form14.

If the 2D equation is applied to a 3D mechanism, the answer can be misleading. Take, for instance, the slider-crank mechanism. If restricted to 2D, there are 4 links in all, with 3 dof each, for a total of 12 dof for the system. If link is grounded, that leaves 9 dofs. The three revolute joints remove 2 dof each, since they only permit rotation about the axis. This leaves 9 – 6 = 3 dof. The slider joint too removes 2 dofs, since it only permits translation along one axis. The system, then, has 1 dof.

If the same calculation is conducted in 3D space, we start with 18 dofs (6 dofs for each of the free links). The 3 revolute joints and the 1 slider joint remove 5 dof each. As a result, the mechanism is over-constrained!

14_{For details on the 3D form, see page 551 of}

Advanced Mechanism Design, Volume

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Theory: Basic, Essential and Advanced CAE and Multi Body Dynamics It is easy to see that a slider-crank mechanism in 3D should, of course, use

spherical joints to avoid this situation15.

Essential Theory

Analysis vs. Synthesis

Analysis involves calculating items of interest for a given mechanism or system. Synthesis, on the other hand, involves finding a mechanism that provides a required behavior.

Synthesis can be extremely challenging, since it means choosing both the types and dimensions of links and joints – called type synthesis and

dimensional synthesis. Type synthesis is

sometimes referred to as number synthesis since it determines the number of links in the

mechanism.

There are infinite possible mechanisms that can satisfy any given set of behaviors, so the chosen mechanism usually depends on the experience of the designer or the available links and joints. Complexity of synthesis rises dramatically as the number of links increases. Even for 4-bar

mechanisms, Hrones and Nelson’s Atlas of Curves, which presented several thousand coupler loci, is often the starting point even today. The Atlas itself was compiled in the 1950s!

Depending on the specifications for synthesis, the goal of the designer is one or more of function,

path and motion. Function generation involves synchronization of the motion of input and output links, as in a Pantograph, for example. In path generation, a point is required to trace a path with respect to a reference frame. In some cases, timings can also be a part of the specification. A cricket-bowling machine is a good example of

15_{See, for example, page 612 of}

Advanced Mechanism Design, Volume 2, Erdman and Sandor.

Computational complexity theory is the study of the complexity of problems - that is, the difficulty of solving them. Some problems are difficult to solve, while others are easy. Take the traveling salesman problem, for example. If the network of cities grows by 1, the time needed to solve the problem - that is,

construct the shortest route that visits every city exactly once - is multiplied by a factor of c, hence the time needed to find the route grows exponentially.

Even though a problem may be computationally solvable in principle, in actual practice it may not be that simple. These problems might require large amounts of time or an

inordinate amount of space.

There exist a certain class of problems that require so much time or space that it is not practical to attempt to solve them although they are solvable in principle. These problems are called Intractable.

From The Wikipedia

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CAE and Multi Body Dynamics Theory: Basic, Essential and Advanced this. Motion generation involves guiding the entire body (or link) through a

prescribed sequence of motion. Consider the movement of the bucket of a tipper machine. Not only must the bucket follow a particular path, the rotation of the bucket must also be controlled.

Types of Analysis

Depending on the scenario being investigated, the analysis is classified as one of the following:

• Static – usually used to find the equilibrium position of a mechanism

• Quasi-Static – used when the inertial forces are not important

• Kinematic – used if there are zero dofs in the system. That is, all possible movements are specified either by joints or by input motion.

• Dynamic (or Transient Dynamic)

• Linear – most MBD models are non-linear. Linear analyses are used mainly to calculate eigenvalues16 or for design of control systems.

The data required to construct the model, the methods used to solve the problem, and the type of results that can be computed vary according to the type of the analysis. The last chapter of this book, Glossary and References, contains a table that summarizes these.

Forward and Inverse Kinematics

In forward kinematics, given the forces and positions of some links, we want to estimate the location / velocity / acceleration of the bodies or points of interest. In inverse kinematics, we want to know what forces to apply to achieve a required position / velocity / acceleration. The latter is often required in robotics.

16_{Eigenvalues are not discussed in this book. See the companion volume}

A Designer’s Guide To Finite Element Analysis for more details on how and why we calculate eigenvalues.

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Quaternions and Euler Angles

Any spatial movement can be expressed as a combination of rotations and translations along 3 axes. Rotations, unfortunately, are not commutative – that is, the final configuration depends on the order of the rotations about different axes.

Rotations and translations are usually represented by 3x3 transformation matrices, and Euler Angles are one convention used to specify angles of rotation. A quaternion17 is an alternative representation of the

transformation.

Instantaneous Center of Rotation

This is useful to determine the relative velocities between two bodies. The instantaneous center of rotation for two bodies in plane motion is a point, common to both bodies, which has the same instantaneous velocity in each body. The point can be a “virtual” point, located off the two bodies.

Damping Coefficient

Vibrating bodies experience damping, a force that retards movement. While this is sometimes an adverse affect, in other cases it can be useful – as in the case of shock-absorbers on a car. Damping coefficients are hard to characterize. Testing is frequently used to establish reliable values. A damped system is non-conservative. This means that energy is “lost” (from the mechanical system) – it is converted to other forms such as sound or heat. Numerical calculations sometimes introduce numerical damping. This is a loss of energy due to the finite precision of computer-arithmetic or because of other truncation errors.

Numerical Integration

The differential equation of motion is

p

ku

cv

ma

+

=

where m is the mass, c is the damping coefficient, and k is the stiffness. a, v

and u are the acceleration, velocity and displacement, respectively, while p

is the external force.

17_{Discovered, invented or defined by W.R.Hamilton in 1843. The utility of the}

quaternion has been a subject of lively debate since then, but most engineers encounter H, the set of all quaternions, named for the illustrious mathematician.

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CAE and Multi Body Dynamics Theory: Basic, Essential and Advanced A common form of the same equation is

)

(t

f

kx

x

c

x

m

&

+

&

+

=

where x(t) is the displacement vector.

Given the initial state of the body or bodies, we need to calculate the configuration of the bodies as we move forwards in time. The initial state is called the initial condition.

This is normally done by numerically integrating the equation of motion. That is, we use the finite difference equations to replace the derivatives with differences. For instance, we can write

i j i j

t

u

t

u

dt

du

v

−

=

∆

=

and a similar equation between a and u. To get started, i is at initial time. That is, at t = 0. At the initial time, if we know the velocity vt=o, we can solve the equation for uj. Then, substituting for in the equilibrium equation we solve for acceleration. Now that the values at time tj are known, we follow the same method to step forward.

There are many numerical integration schemes available to solve such problems. The numerical integration methods are classified as explicit and

implicit, single-step and multi-step, and corrector-predictor methods. For a complete discussion, see the references listed at the end of this book.

… and Practice

Remember that MBD models are models of physical systems. Their value lies in their ability to represent the behavior of systems in the real world. The models, however, are built using theory, and as we know theory is always built on some assumptions19. As the bumble-bee paradox illustrates, our knowledge of theory, while powerful and useful, is far from complete. It’s important, therefore, to keep in mind the reasons that a model of the physical world can differ from actual behavior.

Precision Points

Consider, for instance, function generation. That is, you have to correlate the motion of input and output links. A 4-bar mechanism is often chosen because it can be synthesized easily (relative to linkages with more links, that is!), and is often easy to construct.

Unfortunately, a 4-bar mechanism is not capable of error-free generation of arbitrary curves.

As an acceptable solution, we settle for correlation at a selected set of points. These points are called the precision points. The location and spacing of these points can be

19_See

Godel, Escher, Bach: an Eternal Golden Braid by D.R.Hofstadter for an entertaining, challenging and comprehensive discussion of the Incompleteness Theorem.

“Conventional aerodynamics seemed to suggest that the insect should not generate enough lift to fly. The bees stayed resolutely airborne and the sums caused

consternation.

The underlying problem turned out to be treating a wing as if it was fixed, like in an aeroplane and, thanks to studies over the past few years, including the construction of robotic bees, this "bumble-bee paradox" has been solved: extra lift comes when flexible insect wings slice through the air at a high angle of attack, creating a large swirling vortex at their leading edge.

In this way, insect wings produce the vortices – spinning masses of air – which generate lift and help them move. Today, Prof Ismet Gursul of the University of Bath will describe another step on the way for engineers to make air vehicles smaller than a human hand that can be used for detecting chemicals leaks and reconnaissance.”

Roger Highfield, Science Editor The Telegraph

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CAE and Multi Body Dynamics Theory: Basic, Essential and Advanced calculated by a variety of methods, none of which can be addressed in this

book20.

Remember to keep this in mind when working with either the synthesis of a mechanism or the verification of a proposed mechanism. Since a linkage only has finite significant dimensions, there can only be a finite number of precision points.

Engineering Data and Robust Design

Beginners have a tendency to take published data as the Gospel Truth. This is, quite simply, wrong. At the other end of the spectrum is a stubborn and unreasonable refusal to build a model unless all data is available at the required precision.

Designers, like the rest of the human race, have to live in a world that is less than perfect. Data is not always available at the right time. It may be

insufficient. It may be unreliable. And so on.

To deal with this, one approach is to look for robust designs. In this

approach, we look for a design that will produce the required output even if the specified inputs vary. Obviously, this is not always possible. This is particularly true if we are looking for an optimum design – one that provides the best possible performance at the least possible cost.

Various techniques are available to verify the robustness of designs, as well as to build robust and optimal designs. More details on these can be found in the companion volumes in this series.

One approach in particular is very useful for MBD modeling: the method called parameter identification21. This refers to the extraction of information about a system using measured input-and-output data. It is particularly useful when the transfer-function approach is used, or if a high degree of abstraction is involved.

The Virtuous Circle

MBD tools are easier to use if the fundamentals are clear, and fundamentals are easier to grasp if MBD tools can be used to demonstrate the

20_See

Advanced Mechanism Design: Analysis and Synthesis by G.N.Sandor and A.G.Erdman for an excellent discussion.

21_{Covered in the companion volume}

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Theory: Basic, Essential and Advanced CAE and Multi Body Dynamics applications. It’s this aspect that makes it so enjoyable and so challenging to

study and use MBD tools for CAE!

In theory there is no difference between theory and practice. In practice there is.

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CAE and Multi Body Dynamics Working with MBD Models

Working with MBD Models

Based on the previous chapters, we now have an idea as to the design issues involved in several different types of MBD scenarios, and are familiar with the terminology and underlying theory. In this chapter, we’ll look at how software helps us eliminate a lot of the tedium from the process of applying the theory to design problems.

Different Strokes for Different Folks

A Machine-design Approach

MCAE, short for Mechanical Computer Aided Engineering, covers the various tasks involved with design of mechanical components. Solid modeling, CNC tool-path generation, creation of manufacturing drawings, Finite Element Analysis, etc. For all of these, the 3D solid model usually serves as the basis. That is, the 3D model is the central source of data on which all other

applications draw.

MBD too falls under the umbrella of MCAE, particularly when the approach or goal is to design a mechanism.

It’s only natural, then, for a designer to expect to follow the same approach: to expect to use the 3D model as the starting point for MBD modeling and analysis.

As we have seen in the previous chapter, solid models are indeed useful when it comes to tasks like the calculation of the mass moments of inertia of geometrically complex objects. But as we have also seen, a lot of

mechanisms-theory uses kinematic representations to perform various calculations. In this approach, the detailed shape of the body is immaterial. You only need to specify the locations of the nodes and the moments of inertia of each of the links22.

This means the detailed 3D model is often a dispensable overhead. There are situations where the 3D model is essential, as will see

subsequently when we study problems that involve contact. But MBD models

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Working with MBD Models CAE and Multi Body Dynamics for many scenarios can be built without using any 3D graphics (of the

“shaded” variety) whatsoever.

The use of 3D graphics, though, often helps interpretation dramatically. Provided the tools support it, it is useful to include 3D graphics even if the modeling does not require it, simply because it makes it so much easier to visualize the performance and to catch modeling errors.

A Control-System Approach

There are some elements, however, particularly those representing electrical systems such as motors, where 3D graphics hurts more than it helps. It makes little sense to take the effort of building even a representative model of a motor when the only data to be visualized is the rotation of a shaft! Also, unlike mechanical power transmission which involves visible, tangible conduits like links or pipes, electrical power transmission cables need no 3D modeling. Electrical designers, in fact, often prefer to use symbolic tools for design simulation.

And Ever The Twain Shall Meet

The thing to remember, then, is that there are two distinct parts to any MBD model, and effective usage brings both together.

One is the equivalent of the kinematic model: locations of nodes, properties of links or transfer functions, and constraints between them. This is

sufficient for all electrical components, and is often used by experienced mechanical designers to quickly build MBD models of mechanical parts too. The other is the 3D graphics, consisting of visually appealing images of the bodies that are represented by the links. This is rarely, if ever, required for an electrical component. The graphics are usually derived from a CAD model, but the task of integrating these into the MBD model requires an understanding of the abstractions involved in and represented by the various elements of the model.

Basic Building Blocks

Just as CAD modelers give you the power to use simple construction primitives to build intricate and accurate models of almost any geometry, one of the best things about MBD tools for CAE is that we can start with

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CAE and Multi Body Dynamics Working with MBD Models simple systems and put these together to achieve remarkably useful

simulations of very complicated assemblies.

Once you understand the data required for the “primitives”, it is easy to work forwards. Pay attention, therefore, not only to the building blocks required, but also to the data that is required for these.

Bodies

A body is the same as a link. Graphics can be associated with a body if required, but it’s not essential. The mass properties of the body are

essential. These properties consist of the mass and the 6 mass moments of inertia and the coordinates of the center of gravity of the body.

Further, for a dynamic analysis, the initial velocity of the body must be specified. The initial position is defined by the joints, while the accelerations are computed as a part of the solution.

In some cases, the body may have no appreciable moment of inertia. This occurs when the mass is so closely concentrated at the center of gravity compared to the overall dimensions involved in the model.

In conventional analyses, bodies are considered to be rigid, but current technology includes the capability to define bodies as flexible23.

Constraints or Joints

A joint represents a constraint on the bodies that are connected to it. A revolute joint, for example, only leaves 1 dof free – the bodies can only rotate with respect to each other about the axis of rotation of the joint. The 6 lower pairs are essential for modeling. Higher pairs are not essential, since they do not form a finite set. In the absence of available elements, they can sometimes be constructed using combinations of other building blocks.

Forces

A concentrated force, at its most basic, can de defined as a vector: the magnitude, orientation and point of application are enough to completely specify the force. Other forces, require more general definitions, since not all forces can be modeled as point forces.