Vehicle Suspension Systems Based On Ride Quality Engineering Essay

Published: Last Edited:

This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.

Abstract - This paper reviews different types of suspension system configurations used in the current automotive industry. After a brief introduction on each of the configurations, paper explores the role of the suspension systems in the context of vehicle ride quality, vehicle handling & durability. Discussions and comments are based on the reviewed simulation & experimental results previously achieved.

Keywords - Passive suspension, Active suspension, semi-active suspension, ride quality, vehicle handling & durability.


Suspension system is one of the important components of a vehicle, which plays a crucial role in handling performance and ride comfort characteristics of a vehicle. A suspension system acts as a bridge between the occupants of a vehicle and the road it rides on. It has two main functionalities; one is to isolate the vehicle body with its passengers from external disturbance inputs such as road surface irregularities, aerodynamics forces, vibrations of the engine and driveline, and non-uniformity of the tire/wheel assembly & the other is to maintain a ¬rm contact between the road and the tyres to provide guidance along the track (handling performance). Usually, road surface irregularities, ranging from potholes to random variations of the surface elevation profile, acts as a major source that excites the vibration of the vehicle body through the tire/wheel assembly and the suspension system. [1] In a conventional passive suspension system which comprises of only springs and dampers, a trade-off is needed to resolve the con¬‚icted requirements of ride comfort and good handling performance. The reason is that stiff suspension is required to keep the road wheel in contact with the road surface as much as possible, because all the forces acting on the vehicle do so through the contact patches of the tires; on the other hand, soft suspension is needed to isolate the disturbance from the road.

Multi-body dynamics has been used extensively by automotive industry to model and design vehicle suspension. Before modern optimization methods were introduced, design engineers used to follow the iterative approach of testing various input parameters for vehicle suspension performance. The whole analysis will be continued until the predefined performance measures were achieved. Design optimization, parametric studies and sensitivity analyses were difficult, if not impossible to perform. This traditional optimization process usually accompanied by prototype testing, could be difficult and time-consuming for complete complex systems. With the advent of various optimization methods along with developments in computational technology, the design process has been speeded up to reach optimal values and also facilitated the studies on influence of design parameters in order to get the minimum/maximum of an objective function subjected to the constraints. These constraints incorporate the practical considerations into the design process [2].

Various researches have been carried out using optimization methods, genetic algorithms, 2DOF models, fuzzy adaptive control models, quarter-car models, etc. Analyses of prior researches show that the suspension parameters are optimally designed to attain the best compromise between ride quality and suspension deflections. However, inadequate investigations had been done to apply optimization technique at design stage itself so that suspension parameters satisfy the comfort for whole-body vibration assessment. The present work aims at the investigation & research into various automotive suspension system configurations and their role in the context of vehicle ride quality, vehicle handling & vehicle durability.

The design of a vehicle suspension system includes ride comfort, body motion, road handling and suspension travel. Ride comfort directly relates to the acceleration sensed by passengers; body motion means bounce, pitch and roll of sprung mass are created by cornering, acceleration or deceleration; road handling is associated with the contact forces of tyres and the road surface; suspension travel refers to the displacement between a sprung mass and an unsprung mass.

Fig. 1. Six degree-of-freedom vehicle model [4]


A vehicle body is generally a rigid body with six degree-of-freedom (DOF) motions shown in Fig.1 [4], it consists of longitudinal, lateral and heave motions and roll, pitch and yaw motions. These motions are restricted by suspension geometries in vehicles and are more or less coupled with one another. Moreover, as the suspensions have a mechanical structure with unsprung mass, coupling also occurs between the sprung and unsprung masses. Regardless of such coupling problems, the reduced-order mathematical model is useful for designing an active suspension control system. Therefore a quarter-vehicle model or a half-vehicle model is often used for theoretical analysis and design of active suspension systems [4], [5]. In this section, a linear quarter-vehicle model and a linear half-vehicle model of an active suspension system are introduced. Their linear quadratic (LQ) controllers are designed based on the models, practical active suspension system models are also analysed in terms of non-linear properties and uncertain dynamic disturbances.


Suspension systems are classified in to three groups: Passive, Semi Active and Active suspension systems.


Passive suspension system consists of an energy dissipating element, which is the damper, and an energy-storing element, which is the spring. Since these two elements cannot add energy to the system this kind of suspension systems are called passive. No external energy is directly supplied to the suspension.


Self-levelling suspensions are the variations of passive suspension in which the primary lift component (usually air springs) can adjust for the changes in load. Air springs, which are self-adjusting, are usually used on heavy load trucks and some premium luxury cars. A height control valve monitors the deflection of the suspension, and when its mean position has varied from normal ride height for a designated period of time, the air pressure in the spring is adjusted to bring the deflection within the desired range. The most notable feature of an air suspension is that as the pressure changes with load, the spring stiffness changes correspondingly causing the natural frequency of the suspension to remain constant. [3]


The Active Suspension Technology is a suspension system which can modify its settings in real time to control body motion in response to any road abnormality or during cornering, braking or acceleration. These types of systems usually respond to inputs from either the road or the driver using different sensors. A vehicle equipped with an active suspension can provide both a comfortable and firm ride, thus keeping a perfect balance between smoothness and good road handling. Citroen's Active Wheel incorporates an in-wheel electrical suspension motor that controls torque distribution, traction, turning manoeuvres, pitch, roll and suspension damping for that wheel, in addition to an in-wheel electric traction motor.

Various actuators are used in the active suspension in order to actuate the passive suspension to work accordingly. They are as follows:

HYDRAULIC ACTUATORS: Hydraulically actuated suspensions are controlled with the use of hydraulic servomechanisms. The hydraulic pressure to the servos is supplied by a high pressure radial piston hydraulic pump. Sensors continually monitor body movement and vehicle ride level, constantly supplying the computer with new data. As the computer receives and processes data, it operates the hydraulic servos, mounted beside each wheel. Almost instantly, the servo regulated suspension generates counter forces to body lean, dive, and squat during various driving manoeuvres. In practice, the system has always incorporated the desirable self-levelling suspension and height adjustable suspension features, with the latter now tied to vehicle speed for improved aerodynamic performance, as the vehicle lowers itself at high speed.

Colin Chapman - the inventor and automotive engineer who founded Lotus Cars and the Lotus Formula One racing team - developed the original concept of computer management of hydraulic suspension in the 1980s, as a means to improve cornering in racing cars. Lotus developed a version of its 1985 Excel model with electro-hydraulic active suspension, but this was never offered to the public.

Computer Active Technology Suspension (CATS) co-ordinates the best possible balance between ride and handling by analysing road conditions and making up to 3,000 adjustments every second to the suspension settings via electronically controlled dampers. These suspensions are incorporated on Jaguar XJ (1997), XK8 (1996), S-type (2002) [9]

ELECTROMAGNETIC RECUPERATIVE: This type of active suspension uses linear electromagnetic motors attached to each wheel independently allowing for extremely fast response and allowing for regeneration of power used through utilizing the motors as generators. This comes close to surmounting the issues with hydraulic systems with their slow response times and high power consumption. It has only recently come to light as a proof of concept model from the Bose company, the founder of which has been working on exotic suspensions for many years while he worked as an MIT professor. Electronically controlled active suspension system (ECASS) technology was patented by the University of Texas Centre for Electro mechanics in the 1990s and has been developed by L-3 Electronic Systems for use on military vehicles. The ECASS-equipped HMMWV exceeded the performance specifications for all performance evaluations in terms of absorbed power to the vehicle operator, stability and handling.

Vehicles used this suspension were: Infiniti Q45 (1990), Toyota Supra (1990), Mitsubishi GTO (1991), Lexus GS (2007) [9]

Quarter Vehicle Active Suspension System modelling:

The quarter-vehicle model was initially developed to explore active suspension capabilities and gave birth to the concepts of skyhook damping and fast load levelling which are now being developed toward actual, large-scale production applications.

Fig. 2. Two degree-freedom quarter-vehicle model [3]

Where, mb: quarter body mass (or sprung mass) (Kg);

mw: wheel mass (or unsprung mass) (Kg);

Ks: suspension spring stiffness (N/m)

Kt: tyre stiffness (N/m);

c: damping coef¬cient (Ns/m);

G0 : road roughness coef¬cient (m3/cycle);

U0: vehicle original forward velocity (m/s);

f0: low cut-off frequency (Hz);

z0: road displacement (m);

zw: wheel displacement (m);

zb: body displacement (m);

fa: actuator force (N);

The quarter vehicle model is shown in Fig.2. The dynamic differential equations of this suspension system can be represented as:

mbz¨b = fa + c(Ë™zw −zË™b)+Ks(zw −zb) (1)

mwz¨w = −fa −c(Ë™zw −zË™b)−Ks(zw −zb)−Kt(zw −z0 ) (2)

The road surface is a natural changing condition for vehicle. For better riding comfort, a perfect road surface model is necessary to design vehicle active suspension control system. There are many possible ways to analytically describe the road inputs, which can be classi¬ed as shock or vibration [5]. Shocks are the discrete events of relatively short duration and high intensity, e.g. a pronounced bump or pothole on an otherwise smooth road. Vibrations, on the other hand, are characterised by prolonged and consistent excitations that are called "rough" roads. In this section, the rough road is considered. The International Organization for Standardization (ISO) has proposed a series of standards of road roughness classi¬cation using the Power Spectral Density (PSD) values (ISO 1982), as shown in Table II. Due to the ISO, the road displacement PSD can be described as

G(n) = G(n0)(n/n0)−w

Here, n is the space frequency (m−1) and time frequency f is f = nv (v is the vehicle speed), n0 is the reference space frequency, G(n) is the road displacement PSD, G(n0) is road roughness coefficient shown in Table II, w is the linear fitting coefficient, always w = 2. Then based on the standard road surface description, the road surface input model has been built through an inform filter by Gaussian White noise and successfully used in many presented works [6], [7]. The equation of road surface input is:

z˙0 = −2πf0z0 + 2π (GoUowo)0.5

Where, f0 is low cut-off frequency,

G0 is road roughness co-efficient,

w0 is a Gaussian white noise.


Semi-active systems can only change the viscous damping coefficient of the shock absorber, and do not add energy to the suspension system. Though limited in their intervention (for example, the control force can never have different direction than that of the current speed of the suspension), semi-active suspensions are less expensive to design and consume far less energy. In recent times, research in semi-active suspensions has continued to advance with respect to their capabilities, narrowing the gap between semi-active and fully active suspension systems.

SOLENOID/VALVE ACTUATED: This type is the most economic and basic type of semi-active suspensions. They consist of a solenoid valve which alters the flow of the hydraulic medium inside the shock absorber, therefore changing the dampening characteristics of the suspension setup. The solenoids are wired to the controlling computer, which sends them commands depending on the control algorithm (usually the so called "Sky-Hook" technique).

Vehicles using this suspension: 1994 Citroën Xantia Activa, 1992 Citroën Xantia VSX, 2001 Citroën C5

MAGNETO RHEOLOGICAL DAMPER: Another fairly recently-developed method incorporates magneto rheological dampers with a brand name MagneRide. It was initially developed by Delphi Corporation for GM and was standard, as many other new technologies, for Cadillac Seville STS (from model 2002), and on some other GM models from 2003. This was an upgrade for semi-active systems ("automatic road-sensing suspensions") used in upscale GM vehicles for decades, and it allows, together with faster modern computers, changing the stiffness of all wheel suspensions independently on every road inch on highway speed. These dampers are finding increased usage in the USA and already leases to some foreign brands, mostly in more expensive vehicles. In this system, being in development for 25 years, the damper fluid contains metallic particles. Through the onboard computer, the dampers' compliance characteristics are controlled by an electromagnet. Essentially, increasing the current flow into the damper raises the compression/rebound rates, while a decrease softens the effect of the dampers. Information from wheel sensors (about suspension extension), steering, acceleration sensors and some others is used to calculate the optimized stiffness. Very fast reaction of the total system allows, for instance, make softer passing by a single wheel above a bump or a rock on the road.

Vehicles using these suspensions: 2002 Cadillac Seville STS, 2003 Chevrolet Corvette, 2008 + Audi TT, 2008 + Alfa Romeo Mito


Ride quality refers to the sensation of a passenger in the environment of a moving vehicle. Ride comfort problem is mainly caused by the vibrations of the vehicle body, which may be induced by a variety of sources, such as road surface irregularities, aerodynamics forces, vibrations of the engine and driveline, and non-uniformity of the tire/wheel assembly. Usually, road surface irregularities, ranging from potholes to random variations of the surface elevation profile, act as a major source that excites the vibration of the vehicle body through the tire/wheel assembly and the suspension system [1].

Human beings feel uncomfortable when exposed to vibrations with frequencies in motion sickness regime: 0.1-1Hz. And, ride quality is considered to be improved as the magnitude of the seat acceleration and displacement is reduced [2].


ROAD ROUGHNESS: Usually the roads tend to have surface irregularities due to changes in the ground profile due to which troughs and crest are often found on the surface rather than having a smooth profile. These irregularities fit into the "broad-band random signals" this can be decomposed into Fourier's transform process described in terms of sine waves of different amplitudes and phases; with wave number varying from 0.01 cycles/foot to 0.1 cycles/foot for a Portland concrete road to a smooth continuous surface of bituminous asphalt surface road. This causes the deviation in the elevation as the vehicle passes on the road. This roughness causes the vertical displacement input to the vehicle, thus exciting ride vibrations which are mainly responsible for pitch and bounce excitations. At low wave number, the roll frequency is low in it relative magnitude as compared to its vertical excitation. Thus bounce plays a dominant part as compared to pitch excitation. Hence, the wheelbase is preferably kept nearly equal to the mean wavelength of the road as bounce is accepted over pitch which causes "sea-sickness".[3]

TIRE / WHEEL ASSEMBLY: Ideally tire/wheel assembly is designed to be soft and compliant to absorb some part of shocks experienced from the road. But due to the manufacturing defects in the tires, hub, brake assembly, variations in the force / moments are experienced in the axle at the ground as it rolls thus causing excitation in the ride vibration. Also improper wheel balancing causes vibration excitation which is transmitted through the steering system to the rider. Deformities in tires also contribute to the vibrations affecting the ride quality and handling. [3]

DRIVELINE EXCITATION: Generally driveline is considered to be an integral part of power train package but in large wheelbase, front engine vehicles with rear wheel drive often experience considerable amount of vibration through the drive shaft. Mainly due to the unsupported span of the driveshaft which tends to sag after a certain period of time due to it's self-weight. The major cause of excitation being the uneven mass distribution over the driveshaft along its circumference. [3]

ENGINE / TRANSMISSION: Engine is considered to be the main source of vibration due to many rotating parts in it. Due to improper mounting, these vibrations are passed on to the entire vehicle thus degrading the ride quality of vehicle. [3]


Analysis of the suspension system generally implies the time response of the system. The following optimization methods and procedure is adopted for analysis.


The performance characteristics which are of most interest when designing the vehicle suspension are passenger ride comfort, road holding and suspension travel. The passenger ride comfort is related to passenger acceleration, suspension travel is related to relative distance between the unsprung mass and sprung mass and road handling is related to the tyre displacement.

Among the above three characteristics ride comfort is chosen to be the most important characteristic and is expressed in an objective function as

As per ISO2631 standards the passenger feels highly comfortable if the weighted RMS acceleration is below 0.315 m/s2 [1], [8][ ISO: 2631-1-1997]. So, it is considered as constraint.

At least 5 inches of suspension travel must be available in order to absorb a bump acceleration of one-half "g" without hitting the suspension stops and also an upper bound to maximum acceleration should be kept so that at any time suspension will not hit suspension stops [2] and [3]. Both these are taken as constraints

Dynamic tyre force will increases with increase in tyre deflection so an upper bound to maximum tyre deflection is placed and it is considered as one more constraint [2] and [3]

The other performance characteristic viz. road holding is included as constraints and is restricted by [2].

Human being feel comfortable within a frequency zone of 0.8 Hz and 1.5 Hz and also another criterion for good suspension system often considered is the maximum allowable jerk experienced by the passengers. Both these are added as two more constraints. [8] and [3]

In order to make pitch motion die faster, natural frequency of front suspension should be greater than the rear suspension and it is considered as constraint. [3]


A sinusoidal shape of the road profile as shown in Fig.3 consisting of two successive depressions of depth h = 0.05 m, length = 20 m and vehicle velocity V = 20 m/s is used for analysis [2].

 C:\Users\Sachin Sawant\Desktop\Untitled.png

 As a function of time, the road conditions are given by

Where tau and w are the time lag between front and rear wheels and the forcing frequency respectively and are given by

Fig.4 Road Profile

In this study, the right and left sides have same amplitude road profile but there is a time delay of 0.2 sec and also the rear wheel will follows the same trajectory as the front wheels with a time delay of tau as shown in fig.3 This road input will help to introduce bounce, pitch and roll motion simultaneously.


The handling characteristics of an automobile are concentrated on the characteristics of the tyres and the suspension geometry. Tyres are the vehicle's reaction point with the roadway. They manage the input of forces and disturbances from the road, and they are the final link in the driver's chain of output commands. Tyre characteristics are therefore a key factor in the effect the road has on the vehicle, and in the effectiveness of the output forces that control vehicle stability and cornering characteristics. The tyre's basic characteristics are managed by the system of springs, dampers, and linkages that control the way in which tyres move and react to disturbances and control inputs. Tyres play a significant part in vehicle handling characteristics. Low profile sidewalls improve steering response but also stiffen the ride, so for optimum handling, the tyres, springs and shock absorbers must all work together as a package. For example, rollover of vehicles is one of the major causes of highway accidents. Rollover generally occurs when a vehicle is subjected to extreme steering and braking inputs. To prevent rollover, there is a need to optimize those vehicle and suspension parameters that will make the design less susceptible to rollover. Increasing safety requirements on the handling performance of vehicles demand tools, which allow predicting the tyre influences on the vehicle dynamics. Computer simulation of handling performance of vehicles facilitates evaluation of the influence of tyre design changes on handling properties. The vehicle motions are caused primarily by tyre forces and moments that result from power train, braking, and steering inputs. Tyre forces represent a significant part of the vehicle dynamic behaviour.


HALF TRACK CHANGE: It is the lateral movement of the steering wheels while cornering. It is the measure of how much the contact patch moves in and out of the vehicle body in corners. It helps in controlling the centripetal force acting on the vehicle thus maintaining the stability of vehicle during cornering. [10]

CAMBER ANGLE: Camber angle alters the handling qualities of a particular suspension design; in particular, negative camber improves grip when cornering. This is because it places the tire at a more optimal angle to the road, transmitting the forces through the vertical plane of the tire, rather than through a shear force across it. Another reason for negative camber is that a rubber tire tends to roll on itself while cornering. If the tire had zero camber, the inside edge of the contact patch would begin to lift off of the ground, thereby reducing the area of the contact patch. By applying negative camber, this effect is reduced, thereby maximizing the contact patch area. Note that this is only true for the outside tire during the turn; the inside tire would benefit most from positive camber. [10]

STEER ANGLE: Steer angle is measured in the top elevation between the longitudinal axis of the vehicle and the line of intersection of the wheel plane and road surface. Steer angle is taken as positive if the front of the wheel toes towards the vehicle.

CASTOR ANGLE: The castor angle is the angle, measured in degrees, formed between the axis of the kingpin and the perpendicular to the ground, looking at the vehicle from the side,As this angle is formed longitudinally relative to the vehicle, it's more exact definition is: longitudinal castor angle, In practical usage it is known more simply as the castor angle. It has been established by convention that, if the extension of the kingpin axis falls in front of the point where the wheel stands on the ground, the castor angle is defined as positive, and if it falls behind this point the castor angle is defined as negative, the castor angle is zero when the kingpin is perfectly vertical. The castor angle given to the kingpin creates two very important phenomenon for the ride of the vehicle, the first is related to stability, in maintaining the straight line travel of the vehicle, with the relative return of the steering after steering round a bend, and the second is the tilt of the wheel which occurs during steering, and which is observed by the inclination of the wheel being turned,


This is created on the basis of the distance between the point at which the kingpin axis extension falls (in relation to the direction of travel) and the point of contact between the tyre and the ground. Two wheels with positive castor angle (the kingpin extension falls before the point of contact with the ground) using two different systems, one is to incline the kingpin, and the other is to move the kingpin position in relation to the wheel axis, the straight line travel stability exists in both cases. In fact, in the case of the positive castor angle, the wheel is pulled, as it is the line of application of the force applied to the axis that passes in front of the wheels mid-point, without taking the direction of travel into consideration, each attempt made by the wheel to deviate from straight line travel will be counteracted by the straightening couple generated by the force and by the wheel rolling resistance. However, in the case of negative castor angle, the wheel is pushed, as it is the line of application of the force applied to the axis passes behind the mid-point of the wheel, without taking the direction of travel into consideration. Each attempt made by the wheel to deviate from straight line travel will be helped and increased by the couple generated by the force and by the wheel rolling resistance. Consequently, the best stability condition for straight line travel is obtained with a positive castor angle and with the wheel being pulled, in fact, in this case, the phenomenon of wheel-wobble and the consequent effects on the steering are avoided.


Product design in today's environment requires careful attention to all aspects of the design, including such issues as sustainability, recyclability and durability of the product. Durability of ground vehicles can be particularly challenging because of the way the vehicle is used and the roads on which the vehicle is driven and the product design can change significantly during both the design phase and the product lifecycle. Testing the product is the most common way to verify durability, but testing is also expensive and time-consuming. Virtual durability testing methods in today's computing environment offer rapid, repeatable predictions of the loads a vehicle will experience and the resulting life that the product will have when put into service.

Simulation of overall durability process is typically defined by an organization's access to the necessary data and simulation tools. This article will focus on the various components of durability analysis, with respect to vehicles that travel our roads, and how the challenge of vehicle durability is being addressed.



A modern passenger vehicle is expected to last at least 100,000 miles in the U.S. market and may see loads repeated 1 million times or more during its life. An extreme load, such as driving through a deep pothole with the brakes locked, will be repeated only 10 or 20 times during the life of the vehicle. A typical field-loads measurement group will instrument and measure customer-owned vehicles going through standard-use events. This data will be analyzed using multiple methods to break it into small customer-use samples that will be extrapolated so they represent the entire population of customers. The engineering team decides the percentile customer to which they wish to design or test, and targets for different subsystems will be developed using the data.

The proving grounds team will use the customer data to develop correlated events to repeat the customer-use cases in a compressed schedule at the proving grounds. International teams will correlate schedules at various proving grounds so they have the option of testing at any location. The proving grounds schedule will add events, such as car washes and tire changes, to reproduce the customer experience and also remove non-damaging drive time from the schedule to significantly compress the time required to test the vehicle.

Damage plots (Figure 1) are commonly used to track the damage on a vehicle. The damage plot shown in the chart below compares the damage in two proving ground schedules for busses. The damage shows that the bus will see different loads when run at the two different schedules. Typically a bus will be developed to pass both schedules.

Damage PlotDamage plot from "Design Life Validation Testing of Heavy Duty Transit Buses" ORTECH report #98-1-05-03-01-10362 ORTECH Corp, Mississauga,ON,CA


Early in a vehicle program the weight of the vehicle is set and the resulting loads are predicted. Early load predictions are either historically based or estimated with g-loads. Loads also can be predicted using road surfaces, tire models and early design multi-body dynamic models. During the vehicle development cycle, load predictions must be constantly updated to respond to design changes, consumer-use changes, and other program-related decisions. When high loads are found, it is desirable to use the load prediction process to reduce the loads. There is a significant amount of variability in any load, both measured and predicted, because of the effects of the operator.


The complete vehicle durability problem can be analyzed using Durability Director within the HyperWorks suite. Durability director employs the HyperWorks tools to manage all of the different stages of durability. These include:

Multi-body solution: To predict early loads and chassis loads employing Ftire and a road surface, or measured loads from wheel force transducer systems.

Finite element pre-processing and solver solution: To create finite element models and predict stress or fatigue-life of a component.

Post-processing solution: To visualize and streamline automation of results analysis for multiple CAE solvers.

Open architecture: To enables fast and easy export of stress and load results to fatigue codes for advanced fatigue life prediction.

Process automation solution: To automate the durability prediction process and makes it simple to perform multiple iterations of the durability process to fine-tune a model.

There are many challenges involved in ensuring that a vehicle can withstand the expected loading requirements over its lifetime. HyperWorks and Durability Director represent a comprehensive and streamlined solution that guides organizations through the complex process of predicting fatigue and improving vehicle design to meet today's durability targets.


Development in suspension systems in various approaches are required in order to enhance stability of the vehicle thus leading to safety of the passengers. This journal gives a brief review of various suspension configurations. The optimization and analysis of quarter vehicle model has been given and the effect of road profile has been discussed. Various roles of suspension configurations in context of vehicle ride quality, handling and durability have been described alongwith the suspension responses towards these factors.