**下面为大家整理一篇优秀的****paper****代写****范文****- ****Design of an Active Suspension System for Automobiles****，供大家参考学习，这篇论文讨论了汽车主动悬架系统的设计。车辆行驶在任意形状道路上都会受到振动，这些振动被认为对乘客的舒适性和车辆的耐用性都是不利的。因此，汽车悬架系统的设计目的，就是为了将振动降到最低，保证在各种路面条件和车辆机动情况下的路面保持稳定。**

Introduction

Vehicles moving on the randomly profiled road are exposed to vibrations. These vibrations are considered to be undesirable for the passengers in terms of comfort and for the durability of the vehicles. Therefore the main task of the vehicle suspension system is to minimize the vibration and to guarantee road holding for a variety of road conditions and vehicle maneuvers. This report will discuss how to considers a quarter car models instead of full model. It will simplify the analysis with a reasonable representation of the full model. A quarter model is shown in Figure 1.

Figure 1 Quarter-car model for active suspension design

In this report, we are interested in studying the response of ms and mu, therefore the state variables are chosen to be:

(1)

in which zs is the displacement of body ms relative to the ground, zu is the displacement of body mu relative to the ground.

According to Newton's second law and free body diagram approach (the derivation of the system equations using the free body diagram approach is irrelevant to this study and thus is skipped here, the system equations are presented below), the equations of motion for the system are written as Equation (2):

(2)

in which ms is the sprung mass, mu is the unsprung mass, ks is the spring constant between ms and mu, ds is the damper between ms and mu, kt is the spring constant of the tyre, Fact is the force from the electromagnetic active suspension system, and zr is the road irregularity.

Equation (2) does not explicitly include the gravity term. Gravity can be regarded as a constant value that is added to the displacement of both the suspension spring and the tyre. Gravity will cause certain amount of initial displacement of both ms and mu. It can be treated as an offset at the initial condition. The system equations do not change when considering the gravity. Equatoin (2) is a linear time-invariant representation of the system.

Assessing Passenger Comfort and Ride Handling

ystem response of the model. Some studies have proposed evaluation methods to compare vehicle comfort characteristics based on standard mathematical formulae and frequency analyses (Nahvi, Foulad and Nor 2009). Other studies have suggested that vibrations up to 12 Hz affect all of the human organs, while those above 12 Hz have local effects. Low-frequency (4-6 Hz) cyclic motions, like those caused by tires rolling over an uneven road, can resonate the body. Just one hour of seated vibration exposure may cause muscle fatigue and make the user more susceptible to back injury (Hostens, et al. 2003).

In general, an ideal suspension system should provide both a comfortable riding experience and satisfactory handling within a reasonable range of deflection. These criteria subjectively depend on the purpose of the vehicle. Therefore, a good suspension design guarantees to reduce the disturbance to the outputs (e.g. vehicle displacement). A suspension system with proper cushioning needs to be “soft” against road irregularity and “hard” against loading disturbances. A heavily damped suspension will yield good vehicle handling, but also transfers much of the road input to the vehicle body. Therefore, a suspension design is a trade-off between these two goals. The design should work up to some extent with respect to optimized riding comfort and road holding ability, depending on the design goals.

This study accesses riding comfort and ride handling using a mathematical model. Program Equation (2) in MATLAB using the following physical parameters (this study begins with the upper limits of the parameters, future comparison will use parameter values in the varying ranges of each individual parameter):

% Physical parameters

mb = 450; % kg

mw = 50; % kg

bs = 5000; % N/m/s

ks = 30000 ; % N/m

kt = 200000; % N/m

And construct a state-space model using Equation (2)

% State matrices

A = [ 0 1 0 0; [-ks -bs ks bs]/mb ; ...

0 0 0 1; [ks bs -ks-kt -bs]/mw];

B = [ 0 0; 0 1/mb ; 0 0 ; [kt -1]/mw];

C = [1 0 0 0; 1 0 -1 0; A(2,:)];

D = [0 0; 0 0; B(2,:)];

qcar = ss(A,B,C,D);

The transfer function from actuator to body travel and acceleration has an imaginary-axis zero with natural frequency 63.2 rad/s. This is called the tire-hop frequency.

tzero(qcar({'x','acc'},'F'))

ans =

-0.0000 +63.2456i

-0.0000 -63.2456i

Similarly, the transfer function from actuator to suspension deflection has an imaginary-axis zero with natural frequency 20.0 rad/s. This is called the rattlespace frequency.

zero(qcar('zu','F'))

ans =

-0.0000 +20.0000i

-0.0000 -20.0000i

Road disturbances influence the motion of the car and suspension. Passenger comfort is associated with small body acceleration. The allowable suspension travel is constrained by limits on the actuator displacement. Plot the open-loop gain from road disturbance and actuator force to body acceleration and suspension displacement.

Figure 2 Bode plot of the open-loop gain

Because of the imaginary-axis zeros, feedback control cannot improve the response from road disturbance zr to body acceleration at the tire-hop frequency, and from zr to suspension deflection at the rattlespace frequency. Moreover, because of the relationship and the fact that the wheel position roughly follows zr at low frequency (less than 5 rad/s), there is an inherent trade-off between passenger comfort and suspension deflection: any reduction of body travel at low frequency will result in an increase of suspension deflection.

Evaluate Different Spring and Damper Values

Different spring and damper values will result in different passenger comfort and ride handling. Three groups of values are used to simulation the system response to the same road disturbance.

Case 1: Mild Springs

% Physical parameters

mb = 450; % kg

mw = 50; % kg

bs = 2500; % N/m/s

ks = 30000 ; % N/m

kt = 200000; % N/m

Figure 3 System response of a system with mild springs

Case 2: Stiff Springs

% Physical parameters

mb = 450; % kg

mw = 47.5; % kg

bs = 5000; % N/m/s

ks = 50000 ; % N/m

kt = 300000; % N/m

Figure 4 System response of a system with stiff springs

Case 3: Soft Springs

% Physical parameters

mb = 450; % kg

mw = 50; % kg

bs = 1500; % N/m/s

ks = 10000 ; % N/m

kt = 100000; % N/m

Figure 5 System response of a system with soft springs

Compare these three cases, it is found that systems with soft springs (with smaller spring constant values) tend to more oscillations. This effect can be reduced by increasing the damper values. On the other hand, system with soft springs have smaller spring forces, which means the passenger comfort is improved. The system with stiff springs (with larger spring constant values) tend to have less oscillations, therefore the car body acceleration is smaller, which means road holding is improved. Based on the simulation results, there is a trade-off between road holding (vehicle handling) and passenger comfort. If the design goal is to achieve good vehicle handling, the system should be heavily damped, but it will also transfer much of the road input to the vehicle boy. If the design goal is to achieve passenger comfort, the system should be slightly damped. However, this will reduce the stability of the vehicle, meaning the handling is sacrificed.

Feedback Controller Design

This study designs a feedback controller so that when the road disturbance (zr) is simulated by a unit step input, the output zs has a settling time less than 5 seconds and an overshoot less than 5%. For example, when the car runs onto a 10 cm high step, the bus body will oscillate within a range of +/- 5 mm and will stop oscillating within 5 seconds.

The controller design starts with the implementation of a classical PID type controller:

(3)

To begin with, the parameters are set to be, and . Simulation the system response using a step input.

Figure 6 System response of the system with a P controller

From Figure 6, the percent overshoot is 9mm, which is larger than the 5mm requirement, but the settling time is satisfied, less than 5 seconds. In this case, increasing the proportional gain will not make the system response meet the design requirement. It will reduce the settling time, but the overshooting will also increase. The solution is to add a derivative term in the controller.

To choose the proper gain that yields reasonable output from the beginning, this study starts with choosing a pole and two zeros for PID controller. A pole of this controller must be at zero and one of the zeros has to be very close to the pole at the origin, at 1. The other zero, we will put further from the first zero, at 3. Plot the root locus of the system with the pole and two zeros, as shown in Figure 7.

Figure 7 Root locus of the system

Now that the closed-loop transfer function, controlling the system is a matter of tuning the controller gains. Figure 7 shows that the system has larger damping than required, but the settling time is very short. This response still doesn't satisfy the 5% overshoot requirement. However, this can be rectified by adjusting the Kp and Kd gains to obtain a better response. Increase both Kp and Kd and run the simulation again. Figure 8 shows the system response with the updated Kp and Kd gains.

Figure 8 System response with updated gains

In Figure 8 the percent overshoot and settling time meet the requirements of the system. The percent overshoot is about 5% of the input's amplitude and settling time is 2 seconds which is less than the 5 second requirement.

The simulation results indicate that a PD controller adequately controls the system. The design requirement does not specify the steady state error, therefore integral gain Ki can be omitted. In summary, a proportional control alone will not guarantee the system response meet the design requirement. By adding a derivative gain, the design requirements are met.

References

Hostens, I., Y. Papaioannou, A. Spaepen, and H. Ramon. "A study of vibration characteristics on a luxury wheelchair." Journal of Sound and Vibration, 2003: 433-452.

Nahvi, Hassan, Mohammad Hosseini Foulad, and Mohd Jailani Mohd Nor. "Evaluation of Whole-Body Vibration and Ride Comfort in a Passenger Car." International Journal of Acoustics and Vibration, 2009: 143-149.

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