This section discusses the dynamic characteristics of the fundamental quarter car model and the significance of the model in understanding the dynamics of real vehicles. The quarter car is a 2 Degree-of-Freedom (2DoF) system commonly used in the simulation of vehicle dynamics. The model is designed to represent one quarter of a vehicle and consists of six independent parameters; the sprung and unsprung masses (𝑚𝑚𝑠𝑠 and 𝑚𝑚𝑢𝑢) and the stiffness and damping of the suspension system (𝑘𝑘𝑠𝑠 and 𝑐𝑐𝑠𝑠) and tyre (𝑘𝑘𝑡𝑡 and 𝑐𝑐𝑡𝑡). The excitation to the quarter car occurs via a vertical input to the tyre, usually a road elevation profile (where the displacement is 𝑢𝑢 and the slope of the displacement is 𝑢𝑢̇). An illustration of the 2DoF quarter car model is presented in Figure 3-1.
Figure 3-1: Schematic of the quarter car model (without tyre damping), reproduced from Gillespie (1992a).
The equations of motion for the sprung and unsprung masses (including the damping of the tyre) are given in Equations 3-1 and 3-2, respectively.
(
)
(
)
= − − − − s s s s u s s u m z k z z c z z (3-1)(
)
(
)
(
)
(
)
= − + − − − − − u u s s u s s u t u t u m z k z z c z z k z u c z u (3-2)where the acceleration, velocity and displacement of the sprung mass are represented as 𝑧𝑧̈𝑠𝑠, 𝑧𝑧̇𝑠𝑠 and 𝑧𝑧𝑠𝑠 and 𝑧𝑧̈𝑢𝑢, 𝑧𝑧̇𝑢𝑢 and 𝑧𝑧𝑢𝑢 for the unsprung mass. The sprung and unsprung masses are 𝑚𝑚𝑠𝑠
and 𝑚𝑚𝑢𝑢, the stiffness and damping of the suspension system are 𝑘𝑘𝑠𝑠 and 𝑐𝑐𝑠𝑠 and the stiffness and damping of the tyre are 𝑘𝑘𝑡𝑡 and 𝑐𝑐𝑡𝑡.
There are some important assumptions that are often made when using the quarter car model. One is that the damping in the system (shock absorbers) is viscous (and does not take into account friction or other damping within the shock absorber). Also, in most models it is assumed that the tyre is always in contact with the base (or road); however in practice and particularly at high frequencies this is not always the case (Jazar 2009, p. 933). For a 2DoF system the modes are coupled, however in many cases the dynamic forces generated by the unsprung mass mode are insignificant in comparison to the sprung mass mode. It is common in vehicle dynamics for the sprung mass mode to be the main focus and approximated as a Single Degree-of-Freedom (SDoF) system, with the simplified equations outlined herein. From the quarter car model several important relationships can be used to estimate the dynamic characteristics. Firstly, the natural frequency of the sprung mass, 𝑓𝑓𝑠𝑠𝑛𝑛, is established via the following relationship:
π
=
1
2
s sn sk
f
m
(3-3)The critical damping coefficient of the suspension system, 𝑐𝑐𝑐𝑐, is dependent upon the sprung mass and the stiffness of the suspension, and is defined as:
=2
c s s
c
k m
(3-4)The damping ratio of the suspension system, 𝜁𝜁𝑠𝑠, is the dimensionless ratio of the damping coefficient and the critical damping and defined as:
ζ =
s sc
c
c
(3-5)The damped natural frequency of the sprung mass, 𝑓𝑓𝑠𝑠𝑠𝑠, can then be established using the natural frequency and the damping ratio by the following relationship:
=
− ζ
21
sd sn s
f
f
(3-6)From Equation 3-6, it is evident that as the damping of the suspension system is increased the damped natural frequency of the sprung mass is reduced. In practice, the six parameters of the
quarter car model are rarely known and it is considerably more practical to estimate the sprung mass natural frequency and damping ratio using experimental methods rather than estimating each of the individual parameters. To establish the FRF of a quarter car, the irregular pavement elevation profile and the acceleration response of the sprung or unsprung mass must be measured (or known). The FRF is used to describe the dynamic characteristics of the vehicle, or how the sprung (or unsprung) mass mode responds with respect to frequency.
When the excitation and response are measured with the same motion (e.g. acceleration), a special case FRF is obtained known as the transmissibility FRF. The magnitude of the transmissibility is the dimensionless “ratio of response amplitude to excitation amplitude” of a system and is also known as the transfer function or gain (Gillespie 1985). The dynamic characteristics may be established from the transmissibility through various methods. Furthermore, if the dynamic characteristics and the sprung and unsprung masses are known, the remaining parameters of the model may be established. For many applications in vehicle dynamics the response of the sprung mass is of primary interest. For a vehicle with linear characteristics (idealised), the response PSD function, 𝐺𝐺𝑅𝑅𝑅𝑅(𝑓𝑓), can be obtained by weighting the road excitation PSD function, 𝐺𝐺𝐸𝐸𝑥𝑥(𝑓𝑓), with the vehicle’s transmissibility squared, 𝑇𝑇2(𝑓𝑓), shown in Equation 3-7 and illustrated in Figure 3-2.
( )
= 2( )
( )
Re Ex
G f T f G f (3-7)
Figure 3-2: A pavement excitation spectrum (left) combined with the transmissibility squared (middle) produces the response spectrum of the vehicle-road combination (right).
0.1 1 10 10-3 10-2 10-1 100 101 102 Frequency [Hz] E x c ita ti o n P S D [( m /s 2) 2/Hz ] Pavement Excitation 0.1 1 10 10-3 10-2 10-1 100 101 102 Frequency [Hz] T ra n s m is s ib ilit y [ -] Vehicle Transmissibility 0.1 1 10 10-3 10-2 10-1 100 101 102 Frequency [Hz] R es pons e P S D [ (m /s 2) 2/Hz ] Vehicle Response 22
Depending on the values selected for the parameters of the quarter car model, it is able to represent a variety of different vehicle types from passenger cars to light transport vehicles and even heavy vehicles. For example, Prem (1987) outlined parameters to create a “quarter truck model,” while Cebon (1999, pp. 67-68) described a quarter car model that is “broadly representative of single axle truck suspensions in current highway use.” One limitation of the quarter car model is that it is only useful for the investigation of the vertical vibratory motion of vehicles. To simulate other degrees of freedom of vehicles, such as pitch and roll, more intricate models are required. Complete passenger vehicles, heavy vehicles and even truck-trailer combinations can be created using elaborate numerical models (Cebon 1999, pp. 91-106).
3.1.1 Parameter Influence on the Dynamic Characteristics
Despite the quarter car’s inability to provide a true representation of a complete vehicle, the model is important to explain how each parameter influences the dynamic characteristics and is applicable to more complex vehicles. The sprung mass consists of the rigid body of the vehicle, including any passengers or cargo. The sprung mass is where the ride quality is assessed and the damped frequency is measured to evaluate whether heavy vehicles qualify as road-friendly. Equation 3-3 reveals that any increase in the sprung mass will result in a decrease of its natural frequency. The sprung mass also has an influence on the damping ratio via the level of critical damping in the suspension system. Equation 3-4 and Equation 3-5 illustrate that any increase in the sprung mass will raise the critical damping level; effectively reducing the damping ratio of the suspension system. The unsprung mass consists of anything not supported by the suspension system, such as the axle and other components for the wheels (Gillespie 1985). At the resonant frequency of the unsprung mass a phenomenon known as axle hop occurs. When axle hop occurs, the tyre begins to bounce and if the damping of the suspension system is not sufficiently great then the tyre may lose contact with the pavement (Jazar 2009, p. 933).
The suspension system, in its simplest form, consists of a spring and a hydraulic damper, or shock absorber, and connects the wheels to the body to allow for relative motion (Jazar 2009, p. 455). The primary function of the suspension system is to perform two conflicting roles: to support the body of the vehicle at a constant height to minimise vibration while also maintaining a constant load to the pavement through the tyre without any delay (Popp & Schiehlen 2010, p. 291). These conflicting roles pose a challenge to the design of an effective suspension system; it must somehow overcome the compromise between good ride quality and minimising pavement damage (Woodrooffe 1995).
The spring of the suspension system allows vibration to occur in the vehicle’s sprung mass, preventing the vehicle occupants or products being directly subjected to the irregular pavement. As shown in Equation 3-3, the stiffness of the suspension system will affect the sprung mass natural frequency, where an increase in stiffness will increase the sprung mass natural frequency. The damping of the suspension system acts to reduce vibration induced via the irregular pavement. The damping ratio is also dependent upon the sprung mass and stiffness of the suspension system (via the critical damping). It is important to note that these effects are only valid provided the suspension system is linear; it is widely noted that vehicle suspension systems are often nonlinear. The various issues and difficulties associated with estimating the dynamic characteristics of nonlinear vehicles are discussed in the following section on experimental techniques to estimate the dynamic characteristics of road vehicles.
The tyre is integral for a vehicle as it is the only component that is in contact with the road. A vehicle is able to “maneuver only by longitudinal, vertical and lateral force systems generated under the tyres” by Jazar (2009, p. 95). The tyre acts as both a spring and a damper; however the damping of the tyre is often neglected in quarter car models as the damping coefficient is relatively small in comparison to the damping of the suspension system (Jazar 2009, p. 932). Furthermore, in numerical simulations the pavement elevation is differentiated which introduces further errors into the simulation and is another factor to support the omission of the tyre’s damping.
Gillespie and Karamihas (1994) noted that “variations in tyre inflation pressure affect pavement damage by changing the size of the contact patch and the vertical tyre stiffness.” Correct inflation of the tyre is essential for “optimum performance, safety and fuel economy” and provides optimum surface contact with the pavement (Jazar 2009, p. 124). Over-inflation of the tyre increases the pressure, which in turn increases the stiffness and reduces the surface contact area to the pavement, diminishing the “ride comfort and generates vibration” (Jazar 2009, pp. 124-125). While under-inflation also reduces the tyre-road contact area, it leads to “an overloaded tyre that operates at high deflection with a low fuel economy, and low handling” (Jazar 2009, p. 124).
Cebon (1999, p. 24) described the effect of tyres enveloping the short wavelengths of road roughness “of the order of the tyre contact length.” The envelopment acts to filter the effect of the short wavelengths of the road. This phenomenon is not relevant at “normal highway speeds” as the short wavelengths correspond to high frequencies that do not influence the suspension system (Cebon 1999, p. 24). Imperfections in the manufacturing of the tyres and wheel assembly may transmit additional vibration to the axle, introducing another source of excitation
in the vehicle (Gillespie 1985). Some typical examples of imperfections in the tyre and wheel assembly include dimensional variations of the tyre, mass imbalance and non-uniform tyre stiffness (Gillespie 1985).
Wheelbase filtering, while irrelevant to the quarter car, is an important occurrence in normal (multi-wheeled) vehicles travelling over uneven pavements. Due to two or more wheels travelling along the same longitudinal path, nodes occur in the vehicle’s frequency response (Gillespie 1985). These nodes occur at frequencies equal to the speed divided by half the wheelbase length and odd multiples thereafter (Gillespie 1985). The effect of this on the vertical vibration (bounce) depends on the wheelbase length of the vehicle; for short to medium wheelbases the response is attenuated at high speed (Gillespie 1985). For vehicles with long wheelbases, the nodes are “proportionally higher” and the excitation (via the road) decreases by a “speed-squared relationship” due to wheelbase filtering (Gillespie 1985).