Upload
umer
View
8
Download
0
Embed Size (px)
Citation preview
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 1/13
International Journal of Traffic and Transportation Engineering 2012, 1(2): 19-31
DOI: 10.5923/j.ijtte.20120102.03
Matlabr and Simulink Use in Response Analysis of
Automobile Suspension System in Design
Oluwole O. O
Mechanical Engineering Department, University of Ibadan, Ibadan
Abtract In designing automobile suspension, response analysis is an important tool. In this paper, the response analysis
of the auto suspension to various road conditions is studied using MATLABR tools and SIMULINK. The suspension
system was modeled as a combination of dashpot and spring system in parallel attached to the auto body. Laplace
transforms were used in obtaining transfer functions from the ordinary differential equations which described the system.
The system, a second order open loop with a unity fee d back ‘type 1’system was subjected to different inputs and the
response was studied using Matlab inbuilt commands and SIMULINK. The system was observed to be stable to frequency
input using the Nyquist diagram. Very fast settling time was observed for the responses . It was observed that it was mucheasier to design compensators for the system using the Matlab commands. Using MATLAB root locus plot, the system
could be re-designed by choosing new locus points and the new gains and damping ratios could be obtained. With the
rlocfind command, the gains in the graphics window and the damping ratio could be found as well. With this a better
design of the system could be obtained by compensating the system. This brings a dynamism into the des ign system.
Keywords Response, Auto Suspension, Transfer Function, Design
1. Introduction
Automobile suspension system containing the damper andspring system serve the purpose of absorbing shocks due to
road incongruities thereby making our ride pleasant. Other
method of studying the system include the Finite Element
Method(FEM). However, this is very cumbersome and
weeks of programming and are needed to fully s tudy system
of response of system to various inputs. With the advent of
control engineering software like Matlab, the process of
design has been greatly s implified and s timulating.
This work studies the use of Matlab and Simulink in
studying auto body’s transient response to various types of
road incongruities and how this is important in Engineering
Education.
2. Methodology
The automobile suspension system was modeled as a
combination of dashpot and spring in pa rallel attached to the
auto body. Mathematical equations for motion of the system
was formulated and the transfer functions derived using
Laplace transforms. The transfer function was analysed to
obtained its various parameters.The transfer function was
* Corresponding author:
[email protected] (Oluwole O. O)
Published online at http://journal.sapub.org/ijtte
Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved
then subjected to various probable road inputs. These are,
step input, impulse, ramp and sinusoidal.
2.1. Mathematical Derivation of system Equations
Figure 1 shows a simplified schematic diagram of the auto
suspension system. During motion, the vertical displacement
of the tires put the auto suspension system into motion. Th e
motion Xi at point P is the road input to the system while the
vertical motion of the auto- body, XO is the output. This
vertical motion of the auto-body is what the driver and the
occupants feel.
Figure 1. Simplified auto-suspension system
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 2/13
20 Oluwole O .O et al.: Matlabr and Simulink Use in Response Analysis of
Automobile Suspension Syst em in Design
The equat ion for the system motion is
ii kx xbkx xb xm 000 (1)
Taking Laplace transforms of (1) and inputting zero
initial conditions gives
2
0 0 0 0 0 0( ) (0) (0) ( ) (0) ( )
( ) (0) ( ) I i I
m s X s sx x b sX s x kX s
b sX s x kX s
(2)
k bs s X k bsms s X I )()( 2
0
functiontransfer overall k bsms
k bs
s X
s X
i
2
0
)(
)(
This overall transfer function describes a linear second
order ‘type 1’ unity feedback system which can be
represented in block diagrams as shown in Fig.2.
Figure 2. Block diagram of the auto – suspension system
The forward transfer function is (bs+k)/(ms2) while the
feedback is unity.
Values used for the simulation were; m= 1000Kg; b=
20KN-s/m; k=500KN/m2.
2.2. Using Matlab in Response Analysis
Various Matlab commands were used in response analyses
of the auto-suspension system. Inputs were step input,
impulse, ramp and sinusoidal.
2.2.1. Unit Step Response
For unit step response i.e. R(s)=1/s at t>0 ,the use of the
command step(num,den) gave the des ired result.
2.2.2. Unit Impulse Response
The impulse co mmand has R(s) =1;
Using impulse(num,den), the graph was plotted.
2.2.3. Unit Ramp Response
Ramp response is represented by R(s)=1/s2
Since Matlab has no direct ramp command, the step
command can be used
The step command of G(s)/s is obtained where G(s) is the
overall transfer function for the system. Altenatively, the
command lsim(num,den,r,t) OR lsim(A,B,C,D,u,t) could be
used
Where r and u are the input t ime funct ions. The command
goes thus:
num=[0 20 500];
den=[1 20 500];
t=0:0.005:0.4;
r=t;
y=lsim(num,den,r,t);plot(t,r,'-',t,y,'o')
Fig.5 shows the result of the unit ramp response.
2.2.4. The response of the steady-state of the system to
sinusoidal Input
The response of the system to s inusoidal input was studied
using the Bode plot. The Nyquist and Nichols plots gave
same results. Use was made of the command Bode(num,den)
to get the Bode diagram.
2.3. Using SIMULINK R
in Response Analysis
In using Simulink, it was important to obtain the poles first
using the rootlocus plot. This plot gives the open loop zero
and open loop poles for the system. These values are input
into the transfer function property forms.
2.3.1. Root locus plot of (20s+100)/(s2+ 20s+500)
The rootlocus plot(Fig.3) was obtained by using the
following commands :
num=[0 20 500];
den=[1 20 500];
rlocus(num,den)
The root locus plot showed the open loop zero and open
loop poles for the system.
These could be observed to tally with the complex
conjugate open loop poles(roots of s 2+20s+500) which were
obtained by using the following procedure.
b=[1 20 500];
roots(b)
ans =
-10.0000 +20.0000i
-10.0000 -20.0000i
>> a=[20 500];
>> roots(a)
ans = -25
Values obtained were:
Open loop zero: s= -25
Open loop poles: s=-10 j20Poles (-10+20j and -10-20j ) were input into the transfer
function property form. The simulink flow diagram was set
up(Fig.4) and the simulation started. Results of different
input signals were obtained.
3. Results and Discussion
3.1. Results
Results of the system responses to different inputs using
Matlab commands are presented in Figs. 5 -11. The response
to step input(Fig.5) showed a rise_time of 0.043s, peak_time
= 0.11s, a maximum_overshoot of 0.3305, settling_time of
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 3/13
International Journal of Traffic and Transportation Engineering 2012, 1(2) : 19-31 21
0.34s, final value of 1 and a peak amplitude of 1.34. The
response to an impulse input (Fig.6)showed a peak
amplitude of 20.4 at time of 0.011s and a settling time of
0.383s. The response to a unit ramp input(Fig.7) showed the
a good response output. Figs. 8 and 11 showed good
response to sinusoidal inputs while Figs. 9 and 10 showed
very sharp initial rise t ime for sudden steep inputs.
Fig. 12 -14 shows the frequency response plots using Bode,
Nyquist and Nichols diagrams. Delay margins to frequency
response was as small as 0.057s. Closed loop was stable. The
values obtained showed the system stability to different
sinusoidal inputs .
The system responses to different inputs using
SIMULINK are presented in Figs. 15-22. Fig.23 presents the
use of the signal generator in building a signal. The results
followed the same pattern obtained using Matlab commands .
3.2. Discussion
3.2.1. Response to various inputs
Response to the various inputs (Figs.5-11)showed a fast
rise time and settling time. This showed that there is good
compensator in the system.
Figure 3 . Root locus plot of (20s+500)/(s2+20s+500)
Figure 4 . Simulink flow diagram with a signal generator, the transfer funct ion and the scope
-80 -70 -60 -50 -40 -30 -20 -10 0-25
-20
-15
-10
-5
0
5
10
15
20
250.250.50.680.80.880.94
0.975
0.994
0.250.50.680.80.880.94
0.975
0.994
10203040506070
Root Locus
Real Axis
I m a g i n a r y A x i s
20(s+25)
poles(s)
Zero-PoleSignal
GeneratorScope
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 4/13
22 Oluwole O .O et al.: Matlabr and Simulink Use in Response Analysis of
Automobile Suspension Syst em in Design
Figure 5 . Response curve to a step input
Rise_time = 0.043s; P eak_time = 0.11s; max_overshoot =0.3305;
settling_time = 0.34s; Final Value=1;Peak Amplitude=1.34
Figure 6. Response to impulse input
0 0.1 0.2 0.3 0.4 0.5 0.6-5
0
5
10
15
20
25
Impulse Response
Time (sec)
A m p l i t u d e
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 5/13
International Journal of Traffic and Transportation Engineering 2012, 1(2) : 19-31 23
3.2.2. Response of the s teady-state to sinusoidal input
The response of the steady-state of the system to
sinusoidal input was studied here. The frequency of the input
signal was varied and the response to this input by the
system at s teady state was studied here. The Bode, Nichols
and Nyquist diagrams reveal the same response to sinusoidalinputs with varying frequencies. The s ystem studied showed
stability. The Nyquist plot is a polar plot of the frequency
response . We could observe the same values obtained in the
Bode plot are replicated here: Peak Response: Peak
gain=3.82Db at frequency of 18.9rad/sec
Minimum stability Margins:Phase margin=103deg. At a
frequency of 31.6 rad/sec with a delay margin of 0.057sec
showed that the closed loop is stable.
Peak amplitude= 20.4 at time 0.011s
Settl ing Time=0.383s
Figure 7 . Response to a unit ramp input
Figure 8 . Response to a sinusoidal input sin3 t
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Unit-Ramp Response
Time(sec)
U n i t - R a m p I n p u t a n d S y s
t e m O
u t p u t
Unit Ramp Input
System Output
Unit Ramp Input
Output
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time(sec)
Response of 20s+100/(s2+20+100) to sinusoidal imput
r ( t )
c ( t )
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 6/13
24 Oluwole O .O et al.: Matlabr and Simulink Use in Response Analysis of
Automobile Suspension Syst em in Design
Figure 9 . Response to cos3t input
Figure 10. Response to cos3t input
3.2.3. Modeling, s imulating and Designing With Matlab
In the root locus plot (Fig.3), the system could be
re-designed by choosing new locus points and the new gains
and dampiong rat ios could be obtained. With the rlocfind
command, the gains in the graphics window and the da mping
ratio could be found as well. With this a better design of the
system could be obtained by co mpensating the system. This
brings a dynamism into the design system.
3.2.4. Modeling, s imulating and Designing With Simulink
With Simulink, the user has to obtain the poles first us ing
the rootlocus plot. This plot gives the open loop zero and
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5
-1
-0.5
0
0.5
1
1.5Response to cos(3t)
t
r ( t )
c ( t )
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5
-1
-0.5
0
0.5
1
1.5Response to -cos3t
t
r ( t )
c ( t )
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 7/13
International Journal of Traffic and Transportation Engineering 2012, 1(2) : 19-31 25
open loop poles for the system. These values are then input
into the transfer function property forms. Thus, it is
important when using Simulink to work with both the
rootlocus plot and the Simulink trans fer forms. It is the
input into the transfer funct ions that brings out the responses
of the system to various input signals(Figs.15-22). Input
signals can be obtained using the signal buider (Fig.23).
Figure 11. Response to -sin3t input
Figure 12. Bode diagram for (20s+500)/(s2+20s+500)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0 Response to r=-sin3t
r ( t )
c ( t )
t
Bode Diagram of G(s)=20s+500/(s2+20s+500)
Frequency (rad/sec)
-80
-60
-40
-20
0
20
System: sys
Peak gain (dB): 3.82
At frequency (rad/sec): 18.9
M a g n i t u d e ( d B )
100
101
102
103
-135
-90
-45
0
P h a s e
( d e g )
System: sys
Phase Margin (deg): 103
Delay Margin (sec): 0.057
At frequency (rad/sec): 31.6
Closed Loop Stable? Yes
System: sys
Phase Margin (deg): 103
Delay Margin (sec): 0.057
At frequency (rad/sec): 31.6
Closed Loop Stable? Yes
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 8/13
26 Oluwole O .O et al.: Matlabr and Simulink Use in Response Analysis of
Automobile Suspension Syst em in Design
Figure 13. Nyquist plot for for (20s+500)/(s2+20s+500)
Figure 14. Nichols plot for for (20s+500)/(s2+20s+500)
Nyquist Diagram
Real Axis
I m a g i n a r y A x i s
-1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
0 dB
-20 dB
-10 dB
-6 dB
-4 dB
-2 dB
20 dB
10 dB
6 dB
4 dB
2 dB
System: sys
Peak gain (dB): 3.83
Frequency (rad/sec)
System: sys
Phase Margin (deg): 103
Delay Margin (sec): 0.057
At frequency (rad/sec): 31.6
Closed Loop Stable? Yes
System: sys
Phase Margin (deg): -180
Delay Margin (sec): Inf
At frequency (rad/sec): 0
Closed Loop Stable? Yes
Nichols Chart
Open-Loop Phase (deg)
O p e n - L o o p G a i n ( d B )
-360 -315 -270 -225 -180 -135 -90 -45 0-40
-30
-20
-10
0
10
20
30
40
6 dB
3 dB
1 dB
0.5 dB
0.25 dB
0 dB
-1 dB
-3 dB
-6 dB
-12 dB
-20 dB
-40 dB
System: sys
Peak gain (dB): 3.83
Frequency (rad/sec): 19.3
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 9/13
International Journal of Traffic and Transportation Engineering 2012, 1(2) : 19-31 27
Figure 15. Response to step input response at a step time of 1 sec
Figure 16. Response to pulse inputs( undulating layers on road surfaces)
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 10/13
28 Oluwole O .O et al.: Matlabr and Simulink Use in Response Analysis of
Automobile Suspension Syst em in Design
Figure 17. Response to ramp input of slope 1
Figure 18. Response to sinusoidal input(bumps and depths on road surfaces)
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 11/13
International Journal of Traffic and Transportation Engineering 2012, 1(2) : 19-31 29
Figure 19. Response to random inputs
Figure 20. Response to square wave inputs
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 12/13
30 Oluwole O .O et al.: Matlabr and Simulink Use in Response Analysis of
Automobile Suspension Syst em in Design
Figure 21. Output of Sharp square input and descent
Figure 22. Response to no input (smooth road)
7/21/2019 Transient response suspension system10.5923.j.ijtte.20120102.03.pdf
http://slidepdf.com/reader/full/transient-response-suspension-system105923jijtte2012010203pdf 13/13
International Journal of Traffic and Transportation Engineering 2012, 1(2) : 19-31 31
Figure 23. Use of signal builder for input signals
4. Conclusions
Auto-suspension system for compensators using Matlab
commands and simulink has been done in this work. Using
MATLAB root locus plot, the system could be re-designed
by choos ing new locus points and the new gains and
damping ratios could be obtained. With the rlocfind
command, the gains in the graphics window and the dampingratio could be found as well. With this a better design of the
system could be obtained by co mpensating the system. This
brings a dynamism into the design system.
With Simulink, the user has to obtain the poles first us ing
the rootlocus plot. This plot gives the open loop zero and
open loop poles for the system. These values are then input
into the transfer function property forms. Thus, it is
important when using Simulink to work with both the
rootlocus plot and the Simulink transfer forms. It is the input
into the transfer functions that brings out the responses of the
system to various input signals. Input signals can be obtained
using the signal buider.
REFERENCES
[1]
Ogata.K (2002) ‘Modern Control Theory’ Prentice Hall India p.131
[2] Bandyopadhyay.M.N(2003) ‘Control Engineering’ Prentice – Hall of India.p.84
[3] WIKEPEDIA(2012) ‘unsp rung mass’ en.wikepedia.org/wiki/Unsprung_mass
[4] NPL(2012) ‘Optimal road hump for comfortable speedreduction’ www.npl.co.uk/
[5]
ieee (2012) ‘Analysis of vehicle rotation during passage overspeed control road’ ieeexplore.ieee.org/
[6]
Arrb(2012) ‘Roughometer –II with GPS’ www.arrb.com.au
[7] Popcenter(2012) Problem guides’ www.popcenter.org
[8] Ehow(2012) ‘Bad Car struts behavior over bumps’
www.ehow.com
[9] Ite(2012) ‘Comparative study of speed humps’ www.ite.org
[10] SCCS (2012) Speed bumps and auto suspension analysis’www.sccs.swarthmore.edu
[11] Managemylife(2012) ‘How to make a car comfortable and
quiet’ www.managemylife.com
[12] Pikeresearch(2009) ‘Bumps and Toyota’s Gree auto
Dominance’ www.pikeresearch.com
[13] SAGEPUB(2012) ‘Effect of obstacle in the road to dynamic
response of a vehicle passing over the obstacle’www.sagepub.com
[14] CARTECHAUTOPARTS(2012) ‘Car coil springs’www.cartechautoparts.com
[15] NACOMM(2012) ‘Vehicle chassis analysis’www.nacomm03.ammindia.org/articles