Simulating Coupled Longitudinal, Pitch and Bounce Dynamics of Trucks with Flexible Frames ()
1. Introduction
The dynamic analysis of trucks requires a mathematical model of the vehicle structure (including engine, cab, and transmission), suspensions and tires, and the road excitation. While flexural vibration of the chassis can often be neglected in smaller, relatively stiff automobiles, large trucks and buses can experience significant “beaming mode” vibration. Beaming is response of the frame at its first modal transverse bending frequency, and for nonarticulated trucks this frequency can be on the order of bounce and pitch frequencies of a rigid vehicle [1-3]. Beaming response can be sizeable at the centre of the frame midway between the steered wheels and rear axle [4].
Approaches to modeling flexible vehicles range from 1) ignoring body flexibility by using a lumped mass model [5]; 2) modeling the frame as a regular free-free beam and calculating, estimating or measuring modal masses and stiffnesses [4,6]; and 3) modeling the entire vehicle using the finite element method [1,7-10]. As in most other dynamic systems, the analyst is faced with a spectrum of possible model complexity. The simplest models are easy to implement and computationally efficient but of limited accuracy and predictive ability. The most complex models present great computational burden but are potentially more accurate if parameters can be accurately determined. Low-order models with frame flexibility are typically limited to pitch-plane dynamics (bounce and pitch motions), and assume that frame angular motions are small and motion of any point can be assumed vertical. Longitudinal effects such as propulsion and braking forces, aerodynamic drag, tire rolling, slip resistance, and road inclination, when incorporated in order to predict forward speed, gradeability, or fuel economy; are usually decoupled from pitch and bounce modes. To predict vertical and pitch motions of the passenger compartment or payload, a vertical road input is typically applied to a pitch plane model, possibly with frame flexibility. Longitudinal motion determines when the vehicle encounters bumps, and therefore when suspension and body motions in the pitch plane are excited. Vertical road undulations and the resulting pitch plane response can also affect longitudinal motion. In other words, for certain vehicle parameters and road roughness, there can be two-way coupling between longitudinal and pitch/bounce motion [11-13]. To maximize the accuracy of vehicle response prediction when such coupling is present, and to further account for the effect of frame flexibility, the typical small-vertical-motion model with frame vibration must be extended to allow the large rigid body motions that arise from longitudinal motion and change in road inclination.
This paper presents a model which includes pitch and bounce motions, longitudinal dynamics, and transverse frame vibration. Forces and velocities are resolved along coordinate axes parallel and perpendicular to the undeformed frame. Large rigid body motions are allowed, onto which small flexible vibrations are superimposed. Frame flexibility is incorporated using modal expansion of a free-free beam. The model allows for complete pitch-plane representation in which motive forces can propel the truck forward over varying terrain, including hills. The effect of frame flexibility on vehicle dynamics can then be studied. The bond graph formalism is used to generate the model. Bond graphs, in addition to facilitating integration of flexible and rigid subsystems, allow for inclusion of more complex submodels for components such as suspension and engine if desired.
The following section reviews pitch plane models and approaches to including flexibility effects. Section 3 provides an overview of the bond graph modeling language. Section 4 gives schematics and equations for the rigid and flexible portions of the vehicle model, describes how terrain undulations are incorporated, and presents the final bond graph. Section 5 contains model output, including a study of the effect of road roughness on the coupling between frame flexibility, vehicle response in the pitch plane, and longitudinal motion. Discussion, conclusions and future work comprise Section 6.
2. Literature Review
Traditionally, linear pitch plane models have been the starting point for low-order modeling of truck dynamics including frame flexibility. Margolis and Edeal [4] created a 2 degree-of-freedom, small vertical motion bus frame model to which the engine, cab, and load were added in addition to suspension forces. The bond graph approach facilitated addition of external components to the flexible substructure. Frequency response of the frame showed a dominant beaming mode at approximately 10 Hz, very near the unsprung mass frequency. Margolis and Edeal [5] extended the aforementioned model to a five-axle tractor-trailer with submodels for engine, cab, and sleeper module/fuel tank. Non-linear elements were included in the model. Dynamic motions, in which the relative velocity across the suspension remained nearly zero, were identified as a significant effect of interaction between vehicle dynamics and frame vibration.
Michelberger et al. [14] identified a discrete transfer function for a free-free beam model of a two-axis bus, and then estimated modal characteristics based on measured data. For an air suspension bus, frequencies of 6.7 and 12 Hz were associated with the first two bending modes of the frame. Interaction of frame motion with the other vehicle dynamic elements is foreseeable given the 9.1 Hz engine mount natural frequencies and 10.3 - 10.4 Hz wheel-hop frequencies of the front and rear axles. Ibrahim et al. [9] modeled a truck frame using modal superposition, with modal properties calculated using a finite element model. The linear model included truck longitudinal velocity and a cab suspended by two linear suspension systems. The model assumed constant longitudinal velocity and no aerodynamic effects or tire lift-off. Frame flexibility was found to strongly affect driver’s vertical acceleration and cab’s pitch acceleration for a truck with frame natural frequencies of 7.25, 13, and 18 Hz. Yi [15] generated a 20-node finite element model of a frame modeled as a block, with both vertical and lateral vibration degrees of freedom. Response of rigid vs. flexible models to a pulse steering input showed significant discrepancies in predictions of lateral motion and tire deformation. Cao [16] developed a modal superposition representation of a truck frame rail with five segments—a front rail, kick-down rail, mid rail, kick-up rail, and rear tail. The frame rails were assumed to be the primary contributors to beaming mode. In the model 312 Nastran elements were used. The frame was modeled in isolation rather than as part of a vehicle model, without considering the effects of engine, cab, box and cab mounts that were acknowledged to affect the beaming frequency and nodes.
Truck frames have also been represented in ways other than free-free beams and finite element models for transverse vibration. Lumped-parameter subsystems are presented in [8,17]. In the former, torsional compliance was of primary concern for predicting roll angles, load transfer and yaw stability. The model in Aurell [17], in contrast to prior models in which the frame was represented as two rigid masses joined by a roll-axis torsional spring, introduced a “warp model” in which a pitch-axis torsional spring connected two longitudinal frame rails. A transverse bending degree of freedom was added by discretizing the longitudinal elements into two rigid masses connected by a pitch-axis torsional spring. Goodarzi and Jalali [8] formulated a 13 degree of freedom model with three rigid transverse beams joined by two massless flexible beams. The flexible beams have torsional and transverse compliance. Measured data and the influence coefficient method were used to populate the mass, stiffness, and damping matrices of a linear model. Flexibility was found to have a significant effect on pitch and roll angle predictions for a bus. While discretized bending representations are easy to implement and offer more insight into vehicle response than a rigid frame, a large number of elements are typically required to give a very close approximation of the lowest natural frequencies. Modal expansion has the advantage of giving the correct natural frequencies of the first n modes that are retained. Given the uncertainties in determining natural frequencies analytically for a complex irregular beam with multiple attachment points, either a discretized or modal expansion model can be tuned to match measured natural frequencies in practical applications.
The previously cited works show a range of modeling complexities in accounting for flexibility of heavy truck frames, and verify the potential importance of that vibration in vehicle dynamics models. However, prior models assume small angular motions of the frame, and do not include longitudinal dynamics along with pitch and bounce dynamics. As stated in the introduction, this paper combines longitudinal, rigid pitch and bounce, and transverse flexural dynamics of a truck into a single, computationally efficient model implemented using the bond graph graphical modeling language. Background on bond graphs follows, after which the model details are presented.
3. Bond Graph Modeling Language
In bond graphs, generalized inertias and capacitances store energy as a function of the system state variables (momentum and displacement, respectively), sources provide inputs from the environment, and generalized resistors remove energy from the system. The time derivatives of generalized momentum p and displacement q are generalized effort e and flow f. Power is the product of effort and flow. For example, force and voltage are efforts, velocity and current are flows, linear momentum and flux linkage are generalized momenta, and translational displacement and charge are generalized displacement variables.
Power-conserving elements allow changes of state to take place. Such elements include power-continuous generalized transformer (TF) and gyrator (GY) elements that algebraically relate elements of the effort and flow vectors into and out of the element. In certain cases, such as large motion of rigid bodies in which coordinate transformations are functions of the geometric state, the constitutive laws of these power-conserving elements can be state-modulated. Generalized series and parallel connections are represented by 1 and 0 junctions. All elements bonded to a 1-junction have common flow, and their efforts sum algebraically to zero. All elements bonded to a 0-junction have common effort, and all flows algebraically sum to zero. Sources represent ports through which the system interacts with its environment. See for example Figure 1 where the effort source represents either the force source or battery, and generalized flow associated with the 1-junction is either velocity or current.
The power conserving bond graph elements—TF, GY, 1-junctions, 0-junctions, and the bonds that connect them —are collectively referred to as “junction structure”. Figure 2 defines the symbols and constitutive laws of sources, storage and dissipative elements, and powerconserving elements in scalar form. Bond graphs may also be constructed with the constitutive laws and junction structure in matrix-vector form, in which case the bond is indicated by a double-line. Power bonds contain a half-arrow that indicates the direction of algebraically positive power flow, and a causal stroke normal to the bond that indicates whether the effort or flow variable is the input or output from the constitutive law of the connected elements. See Figure 3 where the effect of causal stroke location on the constitutive law form of two generic elements A and B is illustrated.
The constitutive laws in Figure 2 are consistent with the placement of the causal strokes. Full arrows are reserved for modulating signals that represent powerless information flow such as orientation angles that determine the transformation matrix between a body-fixed and inertial reference frame. Bond graphs, because they use the same small set of symbols for energy storage, dissipate and exchange in any energy domain (electrical, mechanical, hydraulic, etc.), the assembly of submodels from various disciplines is straightforward. For example, connecting a motor model to a linkage model requires simply “bonding” the 1-junction for motor output rotational speed to the 1-junction for linkage input link rotational speed.
Causality (equation input-output structure) automatically propagates through the entire model upon assembly of submodels. The graphical representation of causality
allows for visual detection of input-output conflicts between subsystems, and immediately indicates any numerical simulation issues such as algebraic loops or differential-algebraic equations. The reader is supposed to refer to [18] for more details on bond graph modeling.
The truck model is implemented using bond graphs in this paper, and uses 20 sim [19] commercial software to enter the graph, generate and simulate equations of motion, and plot results. The implementation could be done (albeit with more tedious equation derivation) in software such as Matlab.
4. Vehicle Model
Figure 4 shows a schematic of the pitch plane vehicle model. The rigid aspects of the model are based on [20] and [13]. The model permits large angular motions of the sprung mass and uses nonlinear constitutive laws for aerodynamic drag and tire slip and rolling resistance. The aerodynamic drag constitutive law assumes that drag coefficient and frontal area are constant, and the effect of crosswinds is not considered. After the rigid model aspects are reviewed, the superposition of transverse beam vibrations onto the motion of the rigid frame will be described.