The origin of MSC.ADAMS can be traced back to a program of research initiated by Chace at the University of Michigan in 1967. By 1969 Chace (1969, 1970) and Korybalski (Chace and Korybalski, 1970) had completed the original version of DAMN (Dynamic Analysis of Mechanical Networks). This was historically the first general program to solve time histories for systems undergoing large displacement dynamic motion. This work led in 1971 to a new program DRAM (Dynamic Response of Articulated Machinery) that was further enhanced by Angel (Chace and Angel, 1977). The first program forming the basis of MSC.ADAMS was completed by Orlandea in 1973 and published in a series of two ASME papers (Orlandea et al., 1976a, b). This was a development of the earlier two-dimensional programs to a three-dimensional code but without some of the impact capability contained in DRAM at that time. Blundell (1991) describes how the MSC.ADAMS software is used to study the behaviour of systems consisting of rigid or flexible parts connected by joints and undergoing large displacement motion and in particular the application of the software in vehicle dynamics. The paper also lists a num- ber of other systems based on MSC.ADAMS that had at that time been developed specifically for automotive vehicle modelling applications. Several of the larger vehicle manufacturers have at some time integrated MSC.ADAMS into their own in-house vehicle design systems. Early examples of these were the AMIGO system at Audi (Hudi, 1988), and MOGESSA at Volkswagen (Terlinden et al., 1987). The WOODS system based on user defined worksheets was another system at that time in this case developed by German consultants for Ford in the UK (Kaminski, 1990). Ford’s global vehicle modelling activities have since focused on in-house generated linear models and the ADAMS/Chassis™ (formerly known as ADAMS/Pre™) package, a layer over the top of the standard MSC.ADAMS pre- and post-processor that is strongly tailored towards productivity and consistency in vehicle analysis. Another customized application developed by the automotive industry is described in Scapaticci et al. (1992). In this paper the authors describe how MSC.ADAMS has been integrated into a system known as SARAH (Suspension Analyses Reduced ADAMS Handling). This in-house system for the automotive industry was developed by the Fiat Research Centre Handling Group and used a suspension modelling technique that ignored suspension layout but focused on the final effects of wheel centre trajectory and orientation. At Leeds University a vehicle-specific system was developed under the supervision of Crolla. In this case all the commonly required vehicle dynamics studies have been embodied in their own set of programs (Crolla et al., 1994) known as VDAS (Vehicle Dynamics Analysis Software). Examples of the applications incorporated in this system included ride/handling, suspensions, natural frequencies, mode shapes, frequency response and steady state handling diagrams. The system included a range of models and further new models could be added using a pre-processor. Crolla et al. (1994) also define two fundamental types of MBS program, the first of which is that such as MSC.ADAMS where the equations are 18 Multibody Systems Approach to Vehicle Dynamics generated in numerical format and are solved directly using numerical inte- gration routines embedded in the package. The second and more recent type of MBS program identified formulates the equations in symbolic form and often uses an independent solver. The authors also describe toolkits as collections of routines that generate models, formulate and solve equa- tions, and present results. The VDAS system is identified as falling into this category of computer software used for vehicle dynamics. Other examples of more recently developed codes formulate the equations algebraically and use a symbolic approach. Examples of these programs include MESA VERDE (Wittenburg and Wolz, 1985), AUTOSIM (Sayers, 1990), and RASNA Applied Motion Software (Austin and Hollars, 1992). Crolla (Crolla et al., 1992) provides a summary comparison of the differ- ences between numeric and symbolic code. As stated MBS programs will usually automatically formulate and solve the equations of motion although in some cases such as with the work described by Costa (1991) and Holt (Holt and Cornish, 1992; Holt, 1994) a program SDFAST has been used to formulate the equations of motion in symbolic form and another program ACSL (Automatic Continuous Simulation Language) has been used to generate a solution. Special-purpose programs are designed and developed with the objective of solving only a specific set of problems. As such they are aimed at a specific group of problems. A typical example of this type of program would be AUTOSIM described by Sayers (1990, 1992), Sharp (1997) and Mousseau et al. (1992) which is intended for vehicle handling and has been developed as a symbolic code in order to produce very fast simulations. Programs such as this can be considered to be special purpose as they are specifically developed for a given type of simulation but do, however, allow flexibility as to the choice and complexity of the model. An exten- sion of this is where the equations of motion for a fixed vehicle modelling approach are programmed and cannot be changed by the user such as the HVOSM (Highway-Vehicle-Object Simulation Model) developed at the University of Michigan Transport Research Institute (UMTRI) (Sayers, 1992). The program includes tyre and suspension models and can be used for impact studies in addition to the normal ride and handling simulations. Crolla et al. (1992) indicate that the University of Missouri has also devel- oped a light vehicle dynamics simulation (LVDS) program that runs on a PC and can produce animated outputs. In the mid-1980s Systems Technology Inc. developed a program for vehicle dynamics analysis non- linear (VDANL) simulation. This program is based on a 13 degree of free- dom, lumped parameter model (Allen et al., 1987) and has been used by researchers at Ohio State University for sensitivity analysis studies (Tandy et al., 1992). The modelling of the tyre forces and moments at the tyre to road contact patch is one of the most complex issues in vehicle handling simulation. The models used are not predictive but are used to represent the tyre force and moment curves typically found through laboratory or road-based rig test- ing of a tyre. Examples of tyre models used for vehicle handling discussed in this book include: (i) A sophisticated tyre model known as the ‘Magic Formula’. This tyre model has been developed by Pacejka and his associates (Bakker et al., Introduction 19 1986, 1989; Pacejka and Bakker, 1993) and is known to give an accurate representation of measured tyre characteristics. The model uses modified trigonometric functions to represent the shape of curves that plot tyre forces and moments as functions of longitudinal slip or slip angle. In recent years the work of Pacejka has become well known throughout the vehicle dynamics community. The result of this is a tyre model that is now widely used both by industry and academic institutions and is undergoing contin- ual improvement and development. The complexity of the model does, however, mean that well over 50 or more parameters may be needed to define a tyre model and that software must be obtained or developed to derive the parameters from measured test data. (ii) The second model considered here is known as the Fiala tyre model (Fiala, 1954) and is provided as the default tyre model in MSC.ADAMS. This is a much simpler model that also uses mathematical equations to rep- resent the tyre force and moment characteristics. Although not so widely recognized as Pacejka’s model the fact that this model is the default in MSC.ADAMS and is simpler to use led to its inclusion. The advantage of this model is that it only requires 10 parameters and that the physical sig- nificance of each of these is easy to comprehend making this a good start- ing point for students and newcomers to the discipline. The parameters can also be quickly and easily derived from measured test data without recourse to special software. It should also be noted, however, that this model unlike Pacejka’s is not suitable for combined braking and cornering and can only be used under pure slip conditions. The Fiala formulation also has some limitations at high slip angles and is thus unsuitable for limit manoeuvres even when pure slip conditions are included. (iii) The fourth modelling approach is to use a straightforward interpolation model. This was the original tyre modelling method used in MSC.ADAMS (Ryan, 1990). This methodology is still used by some companies but has, to a large extent, been superseded by more recent parameter-based models. The method is included here as a useful benchmark for the comparison of other tyre models in Chapter 5. Another tyre model is provided for readers as a source listing in Appendix B. This model (Blundell, 2003) has been developed by Harty and as with the Fiala model has the advantage of requiring only a limited number of input parameters. The implementation is more complete, however, than the Fiala model and includes representation of the following: ● Comprehensive slip ● Load dependency ● Camber thrust ● Post limit It has been found that the Harty model is robust when modelling limit behav- iour including, for example, problems involving low grip or prolonged wheelspin. 20 Multibody Systems Approach to Vehicle Dynamics 1.9 Benchmarking exercises In addition to the software discussed so far other multibody systems analysis programs such as DADS, SIMPACK, and MADYMO are com- mercially available and used by the engineering community. A detailed description of each of these is beyond the scope of this text and in any case the rapid development of commercial software means any such assessment here would rapidly become outdated. In broad terms DADS and SIMPACK appear to be comparable with MSC.ADAMS although at the time of writ- ing MSC.ADAMS is the most widely used, particularly by the vehicle dynamics community. MADYMO is a program recognized as having a multibody foundation with an embedded non-linear finite element cap- ability. This program has been developed by TNO in the Netherlands and complements their established crash test work with dummies. Recent developments in MADYMO have included the development of biofidelic humanoid models to extend the simulation of crash test dummies to ‘real world’ pedestrian impact scenarios. A detailed comparison between the various codes is beyond the capability of most companies when selecting an MBS program. In many ways the use of multibody systems has followed on from the earlier use of finite element analysis, the latter being approximately 10 years more mature as applied commercial software. Finite element codes were subject to a rigorous and successful series of benchmarks under the auspices of NAFEMS (National Agency for Finite Elements and Standards) during the 1980s. The pub- lished results provided analysts with useful comparisons between major finite element programs such as NASTRAN and ANSYS. The tests per- formed compared results obtained for a range of analysis methods with various finite elements. For the vehicle dynamics community Kortum et al. (1991) recognized that with the rapid growth in available multibody systems analysis programs a similar benchmarking exercise was needed. This exercise was organized through the International Association for Vehicle System Dynamics (IAVSD). In this study the various commercially available MBS programs were used to benchmark two problems. The first was to model the Iltis mili- tary vehicle and the second a five-link suspension system. A review of the exercise is provided by Sharp (1994) where some of the difficulties involved with such a wide-ranging study are discussed. An example of the problems involved would be the comparison of results. With different investigators using the various programs at widespread locations a simple problem occurred when the results were sent in plotted form using different size plots and inconsistent axes making direct comparisons between the codes extremely difficult. It was also very difficult to ensure that a consistent mod- elling approach was used by the various investigators so that the comparison was based strictly on the differences between the programs and not the models used. An example of this with the Iltis vehicle would be modelling a leaf spring for which in many programs there were at the time no standard elements within the main code. Although not entirely successful the exercise was useful in being the only known attempt to provide a comparison between all the main multibody programs at the time. It should also be recognized that in the period since the exercise programs such as MSC.ADAMS have been extensively developed to add a wide range of capability. Introduction 21 Anderson and Hanna (1989) have carried out an interesting study where they have used two vehicles to make a comparison of three different vehicle simulation methodologies. They have also made use of the Iltis, a vehicle of German design, which at that time was the current small utility vehicle used by the Canadian military. The Iltis was a vehicle that was considered to have performed well and had very different characteristics to the M-151 Jeep that was the other vehicle in this study. The authors state that the M- 151 vehicle, also used by the Canadian military, had been declared unsafe due to a propensity for rolling over. Work has been carried out at the University of Bath (Ross-Martin et al., 1992) where the authors have compared MSC.ADAMS with their own hydraulic and simulation package. The results for both programs are com- pared with measured vehicle test data provided in this case by Ford. The Bath model is similar to the Roll Stiffness Model described later in this book but is based on a force roll centre as described by Dixon (1987). This requires the vehicle to actually exist so that the model can use measured inputs obtained through static rig measurements, using equipment of the type described by Whitehead (1995). The roll-centre model described in this book is based on a kinematic roll centre derived using a geometric construction as described in Chapter 4, though there is little to preclude a force-based prediction by modelling the test rig on which the real vehicle is measured. As a guide to the complexity of the models discussed in Ross-Martin et al. (1992), the Bath model required 91 pieces of information and the MSC.ADAMS model, although not described in detail, needed 380 pieces of information. It is also stated in this paper that the MSC.ADAMS model used 150 sets of non-linear data pairs that suggests detailed modelling of all the non-linear properties of individual bushes throughout the vehicle. 22 Multibody Systems Approach to Vehicle Dynamics Kinematics and dynamics of rigid bodies 2 2.1 Introduction The application of a modern multibody systems computer program requires a good understanding of the underlying theory involved in the formulation and solution of the equations of motion. Due to the three-dimensional nature of the problem the theory is best described using vector algebra. In this chapter the starting point will be the basic definition of a vector and an explanation of the notation that will be used throughout this text. The vector theory will be developed to demonstrate, using examples based on suspen- sion systems, the calculation of new geometry and changes in body orienta- tion, such as the steer change in a road wheel during vertical motion relative to the vehicle body. This will be extended to show how velocities and accel- erations may be determined throughout a linked three-dimensional system of rigid bodies. The definition of forces and moments will lead through to the definition of the full dynamic formulations typically used in a multibody systems analysis code. 2.2 Theory of vectors 2.2.1 Position and relative position vectors Consider the initial definition of the position vector that defines the loca- tion of point P in Figure 2.1. In this case the vector that defines the position of P relative to the reference frame O 1 may be completely described in terms of its components with magnitude Px, Py and Pz. The directions of the components are defined by P {R P } 1/1 Z 1 Px X 1 Y 1 Pz O 1 Py Fig. 2.1 Position vector attaching the appropriate sign to their magnitudes: (2.1) The use of brackets { } here is a shorthand representation of a column matrix and hence a vector. Note that it does not follow any quantity that can be expressed as the terms in a column matrix is also a vector. In writing the vector {R P } 1/1 the upper suffix indicates that the vector is measured relative to the axes of reference frame O 1 . In order to measure a vector it is necessary to determine its magnitude and direction relative to the given axes, in this case O 1 . It is then necessary to resolve it into com- ponents parallel to the axes of some reference frame that may be different from that used for measurement as shown in Figure 2.2. In this case we would write {R P } 1/2 where the lower suffix appended to {R P } 1/2 indicates the frame O 2 in which the components are resolved. We can also say that in this case the vector is referred to O 2 . Note that in most cases the two reference frames are the same and we would abbreviate {R P } 1/1 to {R P } 1 . It is now possible in Figure 2.3 to introduce the concept of a relative pos- ition vector {R PQ } 1 . The vector {R PQ } 1 is the vector from Q to P. It can also be described as the vector that describes the position of P relative to Q. {} / R Px Py Pz P 11 ϭ 24 Multibody Systems Approach to Vehicle Dynamics P Py 2 Px 2 {R P } 1/2 Z 1 X 1 Y 1 O 1 Pz 2 X 2 Y 2 Z 2 O 2 Fig. 2.2 Resolution of position vector components P Z 1 X 1 Y 1 O 1 Q {R Q } 1 {R PQ } 1 {R P } 1 Fig. 2.3 Relative position vector These vectors obey the triangle law for the addition and subtraction of vec- tors, which means that (2.2) It also follows that we can write (2.3) Application of Pythagoras’ theorem will yield the magnitude |R P | of the vector {R P } 1 as follows: (2.4) Similarly the magnitude |R PQ | of the relative position vector {R PQ } 1 can be obtained using (2.5) Consider now the angles X , Y and Z which the vector {R P } 1 makes with each of the X, Y and Z axes of frame O 1 as shown in Figure 2.4. This gives the direction cosines lx, ly and lz of vector {R P } 1 where These direction cosines are components of the vector {l P } 1 where (2.7) It can be seen that {l P } 1 has unit magnitude and is therefore a unit vector. { {} || l R R p P P } 1 ϭ lx x Px R ly y Py R lz z Pz R P P P cos cos cos ϭϭ ϭϭ ϭϭ || || || R PxQx PyQy PzQz PQ ϭϪϩϪϩϪ () () () 2 2 2 RPxPyPx P 2 ϭϩϩ 22 {} {} RRR RRR QP Q P QPQP 11 1 {} {} or { } { } 1 11 ϭϪ ϭϩ {} {} RRR RRR PQ P Q PQPQ 1 1 {} {} or { } { } 11 11 ϭϪ ϭϩ Kinematics and dynamics of rigid bodies 25 P {R P } 1 Z 1 Px X 1 Y 1 Pz O 1 Py θz θy θx Fig. 2.4 Direction cosines (2.6) 2.2.2 The dot (scalar) product The dot, or scalar, product {A} 1 • {B} 1 of the vectors {A} 1 and {B} 1 yields a scalar C with magnitude equal to the product of the magnitude of each vec- tor and the cosine of the angle between them. Thus: {A} 1 • {B} 1 ϭ |C| ϭ |A| |B| cos (2.8) The calculation of {A} 1 • {B} 1 requires the solution of {A} 1 • {B} 1 ϭ {A} 1 T {B} 1 ϭ AxBx ϩ AyBy ϩ AzBz (2.9) (2.10) The T superscript in {A} T 1 indicates that the vector is transposed. Clearly {A} 1 • {B} 1 ϭ {B} 1 • {A} 1 and the dot product is a commutative operation. The physical significance of the dot product will become appar- ent later but at this stage it can be seen that the angle between two vectors {A} 1 and {B} 1 can be obtained from (2.11) A particular case which is useful in the formulation of constraints repre- senting joints and the like is the situation when {A} 1 and {B} 1 are perpen- dicular making cos ϭ0. As can be seen in Figure 2.6 the equation that enforces the perpendicular- ity of the two spindles in the universal joint can be obtained from {A} 1 • {B} 1 ϭ 0 (2.12) 2.2.3 The cross (vector) product The cross, or vector, product of two vectors, {A} 1 and {B} 1 , is another vec- tor {C} 1 given by {C} 1 ϭ {A} 1 ؋ {B} 1 (2.13) The vector {C} 1 is perpendicular to the plane containing {A} 1 and {B} 1 as shown in Figure 2.7. cos { } 1 ϭ A B AB • {} |||| 1 where and [ ]{} {}B Bx By Bz AAxAyAz T 11 ϭϭ 26 Multibody Systems Approach to Vehicle Dynamics {B} 1 {A} 1 θ Fig. 2.5 Vector dot product The magnitude of {C} 1 is defined as |C| ϭ |A| |B| sin (2.14) The direction of {C} 1 is defined by a positive rotation about {C} 1 rotating {A} 1 into line with {B} 1 . The calculation of {A} 1 ؋ {B} 1 requires {A} 1 to be arranged in skew-symmetric form as follows: (2.15) Multiplying this out would give the vector {C} 1 : (2.16) Exchange of {A} 1 and {B} 1 will show that the cross product operation is not commutative and that {A} 1 ؋ {B} 1 ϭϪ{B} 1 ؋ {A} 1 (2.17) {}C AzBy AyBz AzBx AxBz AyBx AxBy 1 ϭ Ϫϩ Ϫ Ϫϩ {} {}CABAB Az Ay Az Ax Ay Ax Bx By Bz 11111 0 0 0 { } { } [ ] ϭϭϭ Ϫ Ϫ Ϫ ؋ Kinematics and dynamics of rigid bodies 27 {B} 1 {A} 1 Fig. 2.6 Application of the dot product to enforce perpendicularity {B} 1 {A} 1 {C} 1 θ Fig. 2.7 Vector cross product . bushes throughout the vehicle. 22 Multibody Systems Approach to Vehicle Dynamics Kinematics and dynamics of rigid bodies 2 2.1 Introduction The application of a modern multibody systems computer program. grip or prolonged wheelspin. 20 Multibody Systems Approach to Vehicle Dynamics 1.9 Benchmarking exercises In addition to the software discussed so far other multibody systems analysis programs such. also be described as the vector that describes the position of P relative to Q. {} / R Px Py Pz P 11 ϭ 24 Multibody Systems Approach to Vehicle Dynamics P Py 2 Px 2 {R P } 1/2 Z 1 X 1 Y 1 O 1 Pz 2 X 2 Y 2 Z 2 O 2 Fig.