Downforce is a downwards thrust created by the aerodynamic characteristics of a car. The purpose of downforce is to allow a car to travel faster through a corner by increasing the vertical force on the tires, thus creating more grip; the same principle that allows an airplane to rise off the ground by creating lift from its wings is used in reverse to apply force that presses the race car against the surface of the track. This effect is referred to as "aerodynamic grip" and is distinguished from "mechanical grip", a function of the car's mass and suspension; the creation of downforce by passive devices can be achieved only at the cost of increased aerodynamic drag, the optimum setup is always a compromise between the two. The aerodynamic setup for a car can vary between race tracks, depending on the length of the straights and the types of corners; because it is a function of the flow of air over and under the car, downforce increases with the square of the car's speed and requires a certain minimum speed in order to produce a significant effect.
Some cars have had rather unstable aerodynamics, such that a minor change in angle of attack or height of the vehicle can cause large changes in downforce. In the worst cases this can cause the car to experience lift, not downforce. Two primary components of a racing car can be used to create downforce when the car is travelling at racing speed: the shape of the body, the use of airfoils. Most racing formulae have a ban on aerodynamic devices that can be adjusted during a race, except during pit stops; the downforce exerted by a wing is expressed as a function of its lift coefficient: D = 1 2 W H F ρ v 2 where: D is downforce W is wingspan H is the chord of the wing if F is wing area basis, or the thickness of the wing if using frontal area basis F is the lift coefficient ρ is air density v is velocity In certain ranges of operating conditions and when the wing is not stalled, the lift coefficient has a constant value: the downforce is proportional to the square of airspeed. In aerodynamics, it is usual to use the top-view projected area of the wing as a reference surface to define the lift coefficient.
The rounded and tapered shape of the top of the car is designed to slice through the air and minimize wind resistance. Detailed pieces of bodywork on top of the car can be added to allow a smooth flow of air to reach the downforce-creating elements; the overall shape of a street car resembles an airplane wing. All street cars have aerodynamic lift as a result of this shape. There are many techniques. Looking at the profile of most street cars, the front bumper has the lowest ground clearance followed by the section between the front and rear tires, followed yet by a rear bumper with the highest clearance. Using this method, the air flowing under the front bumper will be constricted to a lower cross sectional area, thus achieve a lower pressure. Additional downforce comes from the rake of the vehicles' body, which directs the underside air up and creates a downward force, increases the pressure on top of the car because the airflow direction comes closer to perpendicular to the surface. Volume does not affect the air pressure because it is not an enclosed volume, despite the common misconception.
Race cars will exemplify this effect by adding a rear diffuser to accelerate air under the car in front of the diffuser, raise the air pressure behind it to lessen the car's wake. Other aerodynamic components that can be found on the underside to improve downforce and/or reduce drag, include splitters and vortex generators; some cars, such as the DeltaWing, do not have wings, generate all of their downforce through their body. The amount of downforce created by the wings or spoilers on a car is dependent on two things: The shape, including surface area, aspect ratio and cross-section of the device, The device's orientation. A larger surface area creates greater drag; the aspect ratio is the width of the airfoil divided by its chord. If the wing is not rectangular, aspect ratio is written AR=b2/s, where AR=aspect ratio, b=span, s=wing area. A greater angle of attack of the wing or spoiler, creates more downforce, which puts more pressure on the rear wheels and creates more drag; the function of the airfoils at the front of the car is twofold.
They create downforce that enhances the grip of the front tires, while optimizing the flow of air to the rest of the car. The front wings on an open-wheeled car undergo constant modification as data is gathered from race to race, are customized for every characteristic of a particular circuit. In most series, the wings are designed for adjustment during the race itself when the car is serviced; the flow of air at the rear of the car is affected by the front wings, front wheels, driver's helmet, side pods and exhaust. This causes the rear wing to be less aerodynamically efficient than the front wing, because it must generate more than twice as much downforce as the front wings in order to maintain the handling to balance the car, the rear wing typicall
Air suspension is a type of vehicle suspension powered by an electric or engine-driven air pump or compressor. This compressor pumps the air into a flexible bellows made from textile-reinforced rubber; the air pressure inflates the bellows, raises the chassis from the axle. Air suspension is used in place of conventional steel springs in heavy vehicle applications such as buses and trucks, in some passenger cars, it is used on semi trailers and trains. The purpose of air suspension is to provide a smooth, constant ride quality, but in some cases is used for sports suspension. Modern electronically controlled systems in automobiles and light trucks always feature self-leveling along with raising and lowering functions. Although traditionally called air bags or air bellows, the correct term is air spring. In 1901 an American, William W. Humphreys, patented an idea - a'Pneumatic Spring for Vehicles'; the design consisted of a left and right air spring longitudinally channeled nearly the length of the vehicle.
The channels were concaved to receive two long pneumatic cushions. Each one was provided with an air valve at the other end. From 1920, Frenchman George Messier provided aftermarket pneumatic suspension systems, his own 1922-1930 Messier automobiles featured a suspension "to hold the car aloft on four gas bubbles."During World War II, the U. S. developed the air suspension for heavy aircraft in order to save weight with compact construction. Air systems were used in heavy trucks and aircraft to attain self-levelling suspension. With adjustable air pressure, the axle height was independent of vehicle load. In 1946, American William Bushnell Stout built a non-production prototype Stout Scarab that featured numerous innovations, including a four-wheel independent air suspension system. In 1950, Air Lift Company patented a rubber air spring, inserted into a car's factory coil spring; the air spring expanded into the spaces in the coil spring, keeping the factory spring from compressing, the vehicle from sagging.
In 1954, Frenchman Paul Magès developed a functioning air/oil hydropneumatic suspension, incorporating the advantages of earlier air suspension concepts. Citroën replaced the conventional steel springs on the rear axle of their top-of-range model, the Traction Avant 15 Hydraulique. In 1955, the Citroën DS incorporated four wheel hydropneumatic suspension; this combined a soft, comfortable suspension, with controlled movements, for sharp handling, together with a self-levelling suspension. In 1956 air suspension was used on EMD's experimental Aerotrain. In the U. S. General Motors built on its World War II experience with air suspension for airplanes, it introduced air suspension as standard equipment on the new 1957 Cadillac Eldorado Brougham. An "Air Dome" assembly at each wheel included sensors to compensate for uneven road surfaces and to automatically maintain the car's height. For 1958 and 1959, the system continued on the Eldorado Brougham, was offered as an extra cost option on other Cadillacs.
In 1958, Buick introduced an optional "Air-Poised Suspension" with four cylinders of air for automatic leveling, as well as a "Bootstrap" control on the dashboard to raise the car 5.5 inches for use on steep ramps or rutted country roads, as well as for facilitating tire changes or to clean the whitewall tires. For 1959, Buick offered an optional "Air Ride" system on all models that combined "soft-rate" steel coil springs in the front with air springs in the rear. An optional air suspension system was available on the 1958 and 1959 Rambler Ambassadors, as well as on all American Motors "Cross Country" station wagon models; the "Air-Coil Ride" utilized an engine-driven compressor, air bags within the coil springs, a ride-height control, but the $99 optional system was not popular among buyers and American Motors discontinued it for 1960. Only Cadillac continued to offer air suspension through the 1960 model year, where it was standard equipment on the Eldorado Seville and Brougham. In 1960, the Borgward P 100 was the first German car with self-levelling air suspension.
In 1962, the Mercedes-Benz W112 platform featured an air suspension on the 300SE models. The system used a Bosch main valve with two axle valves on one on the rear; these controlled a cone-shaped air spring on each wheel axle. The system maintained a constant ride height utilizing an air reservoir, filled by a single-cylinder air compressor powered by the engine. In 1964, the Mercedes-Benz 600 used larger air springs and the compressed air system powered the brake servo. Rolls-Royce incorporated self-levelling suspension on the 1965 Rolls-Royce Silver Shadow, a system built under license from Citroën. In 1975, the Mercedes-Benz 450SEL 6.9 incorporated a hydropneumatic suspension when the patents on the technology had expired. This design replaced the expensive and problematic compressed air system, still used on the 600 models until 1984. Air suspension was not included in standard production American-built cars between 1960 and 1983. In 1984, Ford Motor Company incorporated a new design as a feature on the Lincoln Continental Mark VII.
In 1986, Toyota Soarer introduced the first electronically controlled, a semi-active full air suspension. Dunlop Systems Coventry UK were pioneers of Elecronically Controlled Air Suspension for off-road vehicles - the term ECAS was trade marked; the system was first fitted to the 93MY Land Rover Range Rover. In 1989, Arnott Air Suspension Products is founded, eventuall
W124 is the Mercedes-Benz internal chassis-designation for the 1984/85 to 1995/96 version of the Mercedes-Benz E-Class, as well as the first generation to be referred to as E-Class. The W124 models replaced the W123 models after 1985 and were succeeded by the W210 E-Class after 1995. In North America, the W124 was launched in early November 1985 as a 1986 model and sold through the 1995 model year, through November 7, 1995. Series production began at the beginning of November 1984, with press reveal taking place on Monday, November 26, 1984 in Sevilla, with customer deliveries and European market launch starting in January 1985; the W124 was a mid-sized vehicle platform, which entered planning in the autumn of 1976 under development Hans Scherenberg. In July 1977, the W124 program began, with R&D commencing work under newly appointed Werner Breitschwerdt. In April 1978, decisions were made to base it on the Mercedes-Benz W201 model program. By April 1979, a package plan was completed for the program, laying out the guidelines of the project.
During the winter of 1980–1981, the final exterior for the W124 program was completed, chosen as the leading proposal by design director Bruno Sacco, approved by the board of management in early 1981. By mid-1982, the first prototypes reflective of the production design, were assembled and sent to testing. In March 1984, pilot production commenced and development of the sedan concluded with engineering sign-off. Front suspension used a separate damper with a rubber top mount; the rear suspension of the W124 featured the Mercedes multi-link axle introduced in 1982 with the Mercedes W201 and, now standard on many modern cars. Estate cars had Citroen-like rear self-leveling suspension with suspension struts rather than shock absorbers, gas-filled suspension spheres to provide damping and an under bonnet pressurizing pump. Unlike the traditional Citroën application Mercedes opted for a fixed ride height and employed rear coil springs to maintain the static ride height when parked; the R129 SL roadster was based on the W124 platform, in return, W124 was equipped with one of the roadster's engines, creating the 500 E.
Much of the 124's engineering and many of its features were advanced automotive technology at its introduction, incorporating innovations that have been adopted throughout the industry. It had one of the lowest coefficient of drag of any vehicle of the time due to its aerodynamic body, that included plastic molding for the undercarriage to streamline airflow beneath the car, reducing fuel consumption and wind noise, it had a single windscreen wiper that had an eccentric mechanism at its base that extended the wiper's reach to the top corners of the windscreen. The saloon/sedan, coupés and convertibles had optional rear headrests that would fold down remotely to improve rearward visibility when required; this feature was not available for the T-model because of its specific layout, but the estate serially came with a "neighbour-friendly" rear door, pulled in the shut-position silently and automatically by a sensor-controlled servomotor. This allowed the use of a tighter fitting rear gate, minimizing the cabin noise in the T-model - sometimes an area of concern for station wagons.
With the exception of the 200, equipped with a Stromberg or Pierburg carburetor but was not available to the United States, fuel injection was standard, the engines incorporated features that maximised performance. The most notable such feature was the addition of an oxygen sensor in the exhaust system which, in conjunction with a semi-electronic fuel injection system, could make the engine run more efficiently; this improved fuel consumption while meeting stricter emission regulations. Mercedes-Benz's four-wheel drive system, the 4Matic was first introduced on the W124 in 1987; the estate cars came in 5 or 7-seat models, the 7-seater having a rear-facing bench seat that folded flush luggage compartment cover and an optional retractable cargo net. To provide a flat loading floor with the seat folded down, the T-model's rear seat squab was mounted about 10 cm higher than in saloons, robbing rear seat passengers of some head room; the S124 estate continued in production alongside the new W210 until the S210 estate launched more than a year later.
A two-door coupé version was built, with the chassis designation C124. Mercedes launched a cabriolet version in Europe in 1991, the 300CE-24 cabriolet, in the UK and Japan; the 320CE, North America, the 300CE, in 1992. These versions were re-designated as the E 320 in 1993, complemented by the less powerful, but less expensive E 220 in 1993, the mainland-Europe-only E 200 in 1994. Mercedes brought the E 320 cabriolet to the USA and Japan from 1993 to 1995. There were 68 E 36 AMG cabriolets built from 1993 until 1996 to complement the rare E 36 AMG coupé, estate. Aproximately 171 estate cars were produced for the Japanese market; the pre-merger AMG coupés are based on the 124 series 2 update. The AMG 3.4 CE were all LHD, 25 were produced from 1988 until 1993. There were 7 cabriolets built, eleven saloons. AMG Japan carried out such conversions locally; the E 320, E 220, E 200 cabriolets ceased production in 1997. Indian assembly began in March 1995. Offered with five-cylinder diesel engines built by Mercedes' Indian partner Bajaj Tempo, the W124
The Mercedes-Benz W201 was the first compact executive car manufactured by German automotive manufacturer Mercedes-Benz. Introduced in 1982, it was positioned in the size category below the E-Class and marketed under variants of the Mercedes-Benz 190 nameplate; the W201 featured innovative rear 5-link suspension, subsequently used in E and C class models and rear anti-roll bars, anti-dive and anti-squat geometry—as well as airbags, ABS brakes and seatbelt pretensioners. The W201 fared poorly in the United States. Series production ended on 13 April 1993 after the manufacture of 1.8 million examples. The 190 and its variants were succeeded in the compact executive car segment by the C-Class, a newly-created nameplate. From January 1974 to January 1982, Mercedes spent over £600 million researching and developing the 190 and subsequently said it was'massively over-engineered'; the first test mules were put into testing in 1978, with final styling being approved on March 6, 1979. The first prototypes based on that design were tested that year, with pilot production beginning in February 1982, following engineering sign-off.
It marked a new venture for Mercedes-Benz giving it a new smaller model to compete with the likes of the Audi 80, BMW 3 Series and Saab 900, as well as the more expensive versions of the many medium-sized saloons and hatchbacks from mainstream brands. The W201-based 190 was unveiled on December 8, 1982, being launched in Germany the next day on December 9, 1982 and was sold in right-hand drive for the UK market from September 1983. Local red tape in Bremen prevented Daimler-Benz from building the 190 there, so production was started in Sindelfingen at a capacity of just 140,000 units per year. After just the first year, Bremen was cleared for production of the 190, replacing its commercial vehicle lines, there the 190 was built with the first running modifications since release; the 190 E model uses the Bosch KE-Jetronic Multi-Point Fuel Injection to meter fuel instead of the carburetor of 190 models. Thanks to their fuel injection system, 190 E models made more power and were more fuel efficient when compared to non-fuel injected 190 models.
In 1982, the first available models were the 190 and 190 E. Each was fitted with an M102 1,997 cc inline-4 engine; the 190 was fitted with an M102.921 engine producing 90 hp and the 190 E fitted with an M102.962 engine producing 122 hp. In September 1983, the 190 E 2.3 was launched for the North American market only, fitted with a 113 hp M102.961 engine. This reduction in power was due to the emissions standards in the North American market at the time; the intake manifold and fuel injection system were refined in 1984, the engine produced 122 hp. The carbureted 190 was revised in 1984 as well, receiving a power increase to 105 hp. 1984 saw the arrival of the 2.3-16 "Cosworth" variant. In 1985, the 190 E 2.3 now came fitted with the M102.985 engine, producing 130 hp until it was revised in 1987, now utilising the Bosch KE3-Jetronic Injection system, a different ignition system, a higher compression ratio, producing 136 hp. 1987 marked the arrival of the first 190 equipped with an Inline-six engine, the 190 E 2.6.
Fitted with the M103.940 engine, the 190 E 2.6 had a maximum power output of 160 hp with a catalytic converter and 164 hp without it. In the North American market, the 190 E 2.6 was sold until 1993, the end of the W201's production run. From 1992–1993 the 2.6 was available as a special "Sportline" model, with an upgraded suspension and interior. The 190 E 2.3 was sold until 1988 went on a brief hiatus until it was sold again from 1991 until 1993. The 190 D was available with three different engines; the 2.0 L inline-4 engine was the base engine, was never marketed in North America. The 2.2 L, with the same power as the 2.0 L, was introduced in September 1983. It was only available in model years 1984 and 1985, only in the USA and Canada; the 2.5 L inline-5 engine was available in the late early 90's. The 2.5 L Turbo engine, sold in mainland Europe, but not the UK for many years, was available to American buyers only in 1987 and is now somewhat of a collectors item. The exterior of the 2.5 Turbo model is different from other models in that it has fender vents in the front passenger side fender to feed air to the turbocharger.
For the UK and Irish market a special edition 190 was produced for the 1993 model year. The car was given the badge name 190LE, though on the rear boot lid it read 190 E and LE on the right hand side. 1,000 cars were produced and each one came with a large A3 sized certificate giving each car a unique number. The 190 LE was available in three colours only; the Azzuro blue coloured cars came with a grey checked cloth interior, the silver coloured cars with a black checked cloth interior and the Rosso Red coloured cars with a biscuit/cream checked cloth interior. The LE was equipped with extra features, options on the other models and were only available with the 1.8 or 2.0-litre engine. Both the 1.8 and 2.0-litre models were equipped with a standard electric tilt and slide steel sunroof, four electric windows, electric aerial, 8-hole alloy wheels, Blaupunkt Verona CR43 Radio/cassette player and walnut wood trim. The 2.0-litre version had in a front armrest. The LE was
Sliding pillar suspension
A sliding pillar suspension is a form of independent front suspension for light cars. The stub axle and wheel assembly are attached to a vertical pillar or kingpin which slides up and down through a bush or bushes which are attached to the vehicle chassis as part of transverse outrigger assemblies, sometimes resembling a traditional beam axle, although fixed rigidly to the chassis. Steering movement is provided by allowing this same sliding pillar to rotate. Sliding pillar independent suspension was first used by Decauville in 1898, the first recorded instance of independent front suspension on a motor vehicle. In this system, the stub axle carrying the wheel was fixed to the bottom of a pillar which slid up and down through a bush in a transverse axle fixed to the front of the chassis; the top of the pillar was pivoted on a transverse semi-elliptic leaf spring. This system was copied by Sizaire-Naudin a few years later. In around 1904, the New Jersey inventor J. Walter Christie introduced a sliding pillar suspension system with vertical coil springs, which may be the inspiration for that used by Lancia on its Lambda from around 1922.
Lancia continued with sliding pillar suspension until the 1950s Appia. In turn, this was copied for a single year by Nash on its unibody 600 model. Sliding pillar suspension systems have been used by several cyclecar manufacturers, the French maker Tracta, in several prototype vehicles. In 1909 H. F. S. Morgan introduced a fundamentally similar system using a sliding stub axle on a fixed pillar, used first on Morgan Motor Company cyclecars on their cars up to the current time; the Morgan design is an inverted sliding pillar, as are most of the designs. A drawback of the sliding pillar system is that the track changes with differential suspension movement, such as when one wheel rises over an obstacle; this is an issue where the track is narrow in relation to suspension travel. The effective track is the hypotenuse AC or AD of the triangle ABC, where AB is the fixed pillar spacing. However, many types of suspension, such as the swing axle have similar issues. Track variation is considered less important than changes in wheel camber, nonexistent in a sliding pillar system.
This suspension system is rare, but was used most notably in the groundbreaking Lancia Aurelia coupe. Plunger suspension - A similar sliding suspension, used for the rear suspension of some motorcycles
Twist-beam rear suspension
The twist-beam rear suspension is a type of automobile suspension based on a large H or C shaped member. The front of the H attaches to the body via rubber bushings, the rear of the H carries each stub-axle assembly, on each side of the car; the cross beam of the H holds the two trailing arms together, provides the roll stiffness of the suspension, by twisting as the two trailing arms move vertically, relative to each other. The coil springs bear on a pad alongside the stub-axle; the shock is colinear with the spring, to form a coilover. In many cases the damper is used as a restraint strap to stop the arm descending so far that the coil spring falls out through being unloaded; this location gives them a high motion ratio compared with most suspensions, which improves their performance, reduces their weight. The longitudinal location of the cross beam controls important parameters of the suspension's behaviour, such as the roll steer curve and toe and camber compliance; the closer the cross beam to the axle stubs the more the camber and toe changes under deflection.
A key difference between the camber and toe changes of a twist beam vs independent suspension is the change in camber and toe is dependent on the position of the other wheel, not the car's chassis. In a traditional independent suspension the camber and toe are based on the position of the wheel relative to the body. With twist-beam if both wheels compress together their camber and toe will not change, thus if both wheels started perpendicular to the road and are compressed together they will stay perpendicular to the road. The camber and toe changes are the result of one wheel being compressed relative to the other; this suspension is used on a wide variety of front wheel drive cars, was ubiquitous on European superminis. Rear torsion-beam axles were introduced and popularised by Volkswagen when they changed from rear engined RR layout cars to front wheel drive FF layout cars in the 1970s; the design was applied in the Audi 50 / Volkswagen Polo, Volkswagen Golf and Scirocco, all introduced in 1974.
This suspension is described as semi-independent, meaning that the two wheels can move relative to each other, but their motion is still somewhat inter-linked, to a greater extent than in a true independent rear suspension. This can mildly compromise the ride quality of the vehicle. For this reason, some manufacturers have changed to different linkage designs; as an example, Volkswagen dropped the twist-beam in favour of a true IRS for the Volkswagen Golf Mk5 in response to the Ford Focus' Control Blade rear suspension as well as the Hyundai Elantra or newer and Hyundai i30. General Motors in Europe Vauxhall/Opel have continued to use twist- or torsion- beam suspension; this is at a cost saving of €100 per car compared to multi-link rear suspension. Their latest version as used in the 2009-on Opel Astra uses a Watts linkage at a cost of €20 to address the drawbacks and provide a competitive and cost effective rear suspension; the Renault Megane and Citroen C4 have stayed with the twist beam. Low cost Can be durable Fewer bushings than multi-link suspension that are less stressed and less prone to wear Simple Neat package, reduces clutter under floor Fairly light weight Springs and shocks can be light and low cost No need for a separate anti-roll bar - the axle itself performs that function Basic toe vs lateral force characteristic is oversteer Since toe characteristics may be unsuitable, adding toe-control bushings may be expensive.
Camber characteristics are limited. Not easy to adjust roll stiffness Welds see a lot of fatigue, may need a lot of development Not much recession compliance - can be poor for impact harshness, will cause unwelcome toe changes Wheel moves forward as it rises, can be poor for impact harshness Need to package room for exhaust and so on past the cross beam Camber compliance may be high No redress for wheel alignment. Alignment geometry is factory-set and not adjustable. Any deviation from factory specifications/tolerances could mean a bent axle or compromised mounting points. A picture of a twist beam
Multibody system is the study of the dynamic behavior of interconnected rigid or flexible bodies, each of which may undergo large translational and rotational displacements. The systematic treatment of the dynamic behavior of interconnected bodies has led to a large number of important multibody formalisms in the field of mechanics; the simplest bodies or elements of a multibody system were treated by Euler. Euler introduced reaction forces between bodies. A series of formalisms were derived, only to mention Lagrange’s formalisms based on minimal coordinates and a second formulation that introduces constraints; the motion of bodies is described by their kinematic behavior. The dynamic behavior results from the equilibrium of applied forces and the rate of change of momentum. Nowadays, the term multibody system is related to a large number of engineering fields of research in robotics and vehicle dynamics; as an important feature, multibody system formalisms offer an algorithmic, computer-aided way to model, analyze and optimize the arbitrary motion of thousands of interconnected bodies.
While single bodies or parts of a mechanical system are studied in detail with finite element methods, the behavior of the whole multibody system is studied with multibody system methods within the following areas: Aerospace engineering Biomechanics Combustion engine and transmissions, chain drive, belt drive Dynamic simulation Hoist, paper mill Military applications Particle simulation Physics engine Robotics Vehicle simulation The following example shows a typical multibody system. It is denoted as slider-crank mechanism; the mechanism is used to transform rotational motion into translational motion by means of a rotating driving beam, a connection rod and a sliding body. In the present example, a flexible body is used for the connection rod; the sliding mass is not allowed to rotate and three revolute joints are used to connect the bodies. While each body has six degrees of freedom in space, the kinematical conditions lead to one degree of freedom for the whole system; the motion of the mechanism can be viewed in the following gif animation A body is considered to be a rigid or flexible part of a mechanical system.
An example of a body is the arm of a wheel or axle in a car or the human forearm. A link is the connection of a body with the ground; the link is defined by certain constraints. Typical constraints are: Universal Joint; the degrees of freedom denote the number of independent kinematical possibilities to move. In other words, degrees of freedom are the minimum number of parameters required to define the position of an entity in space. A rigid body has six degrees of freedom in the case of general spatial motion, three of them translational degrees of freedom and three rotational degrees of freedom. In the case of planar motion, a body has only three degrees of freedom with only one rotational and two translational degrees of freedom; the degrees of freedom in planar motion can be demonstrated using a computer mouse. The degrees of freedom are: left-right, forward-backward and the rotation about the vertical axis. A constraint condition implies a restriction in the kinematical degrees of freedom of one or more bodies.
The classical constraint is an algebraic equation that defines the relative translation or rotation between two bodies. There are furthermore possibilities to constrain the relative velocity between two bodies or a body and the ground; this is for example the case of a rolling disc, where the point of the disc that contacts the ground has always zero relative velocity with respect to the ground. In the case that the velocity constraint condition cannot be integrated in time in order to form a position constraint, it is called non-holonomic; this is the case for the general rolling constraint. In addition to that there are non-classical constraints that might introduce a new unknown coordinate, such as a sliding joint, where a point of a body is allowed to move along the surface of another body. In the case of contact, the constraint condition is based on inequalities and therefore such a constraint does not permanently restrict the degrees of freedom of bodies; the equations of motion are used to describe the dynamic behavior of a multibody system.
Each multibody system formulation may lead to a different mathematical appearance of the equations of motion while the physics behind is the same. The motion of the constrained bodies is described by means of equations that result from Newton’s second law; the equations are written for general motion of the single bodies with the addition of constraint conditions. The equations of motions are derived from the Newton-Euler equations or Lagrange’s equations; the motion of rigid bodies is described by means of M q