Modern Mechanical Engineering, 2011, 1, 84-92
doi:10.4236/mme.2011.12011 Published Online November 2011 (http://www.SciRP.org/journal/mme)
Copyright © 2011 SciRes. MME
Design, Development and Testing of an Air Damper to
Control the Resonant Response of a SDOF Quarter-Car
Suspension System
Ranjit G. Todkar
Department of Mechanical Engineering, P.V.P. Institute of Technology, Sangli, India
E-mail: rgtodkar@gmail.com
Received October 21, 2011; revised November 3, 2011; accepted November 10, 2011
Abstract
An air damper possesses the advantages that there are no long term changes in the damping properties, there
is no dependence on working temperature and additionally, it has less manufacturing and maintenance costs.
As such, an air damper has been designed and developed based on the Maxwell type model concept in the
approach of Nishihara and Asami and Nishihara [1]. The cylinder-piston and air-tank type damper character-
istics such as air damping ratio and air spring rate have been studied by changing the length and diameter of
the capillary pipe between the air cylinder and the air tank, operating air pressure and the air tank volume. A
SDOF quarter-car vehicle suspension system using the developed air enclosed cylinder-piston and air-tank
type damper has been analyzed for its motion transmissibility characteristics. Optimal values of the air
damping ratio at various values of air spring rate have been determined for minimum motion transmissibility
of the sprung mass. An experimental setup has been developed for SDOF quarter-car suspension system
model using the developed air enclosed cylinder-piston and air-tank type damper to determine the motion
transmissibility characteristics of the sprung mass. An attendant air pressure control system has been de-
signed to vary air damping in the developed air damper. The results of the theoretical analysis have been
compared with the experimental analysis.
Keywords: Ride Comfort, Quarter-Car Suspension Model, Cylinder-Piston and Air-Tank Type Air Damper,
Motion Transmissibility, Optimal Air Damping Ratio
1. Introduction
The control of response of the sprung mass of a SDOF
quarter-car suspension system subjected to the road ex-
citation is necessary in the neighborhood of the reso-
nance for the better ride comfort, road holding and sta-
bility. Various damping mechanisms such as, hydraulic,
electromagnetic, ER and MR fluid and air dampers have
been reported in the literature [1-3]. In this paper, an air
enclosed cylinder-piston and air-tank type air damper con-
figuration has been selected for design and development
because in these dampers there are no long term changes
in the damping properties, no dependence on working
temperature. Air dampers have less manufacturing and
maintenance costs. A SDOF quarter-car vehicle suspen-
sion model using such a developed air damper has been
analyzed for its motion transmissibility characteristics.
The air damper has been designed and developed as a
Maxwell type model in the approach of Nishihara and
Asami and Nishihara [1]. The air damper characteristics
such as air damping ratio and air spring rate have been
studied. Optimal values of the air damping ratio at vari-
ous values of air spring rate have been determined to ob-
tain minimum motion transmissibility of the sprung mass.
An experimental setup for SDOF quarter-car suspension
system using the developed air enclosed cylinder-piston
and air-tank type damper has been developed with an
attendant air pressure control system.
2. Development of an Air Enclosed
Cylinder-Piston and Air-Tank Type
Damper
A cylinder-piston and air-tank type damper, both the sides
of which are connected to two surge tanks through capil-
lary pipes has been developed. The arrangement is used
85
R. G. TODKAR
to set the desired damping properties by allowing the
changes in 1) Tank volume to cylinder volume ratio Nt, 2)
Operating air pressure pi, and 3) Capillary pipe length
lpipe and diameter dpipe. Figure 1(a) shows a cylinder-
piston and air-tank type air damper. Figure 1(b) shows
the mathematical models for the air Damper [4].
2.1. Air Spring Rate ka [3]
22
22
a
tc
npisns pi
kvv







Nt
where (1)
tc
vvNt
where n is the index of expansion of air, pi is the operat-
ing air pressure in the system, s is the cross sectional area
of the piston, vt is the air tank volume and vc is the air
cylinder volume.
Let air spring rate ratio
1
a
k
kk
(2)
where k1 is the suspension spring rate.
Substituting the value of ka from Equation (1) in Equa-
tion (2), we obtain
2
1
2
c
ns pi
kvk Nt





 where
Figure 1. (a) Cylinder-piston and air tank type system; (b)
Mathematical models.

22
and
4
p
rcp
s
dd vsh
  (3)
where dp is the piston diameter and dr is the piston rod
diameter and hp is the height of the cylinder volume.
Defining the terms p1, p2 and p3 as
12 3
,and
p
p
r
cc
dh
d
pp p
dd
 
c
d
and substituting the val-
ues of p1, p2, p3, s and vc in Equation (3), we obtain

22
12
31
2c
np
kppd
pk Nt






i
(4)
Assuming the values p1 = 0.985, p2 = 0.333, p3 = 0.5
and k1 = 970 N/m, the Equation (4) becomes.

0.00403146 c
pi
kd
Nt



(5)
2.1.2. Ai r Damping Ratio ζa [3]
2
at
a
r
wv
nc pi
where
12
1
a
a
k
wm



(6)
The capillary flow coefficient cr is given as

4
π
128
r
pipe
opipe
d
cl
(Refer [3]) (7)
in which d
pipe and lpipe are the capillary pipe diameter
and length respectively and µo is the viscosity of air at
atmospheric temperature. Taking the value of ka from
Equation (1) and substituting it in Equation (6), we get
12
2
11
22
a
cc
npisn pi
ws
vNtmvm Nt






 (8)
Substituting wa from Equation (8) and cr from Equa-
tion (7) in Equation (6)
we obtain


1
4
2
π
2128
c
c
a
pipe
opipe
npi
vNt
vm Nt
d
np
l
i







or

14
1
1
1
128
where π2
pipe
a
pipe
oc
l
Qpi
d
Nt
sv
Qnm











(9)
Similarly using expressions (4) and (9) respectively
Copyright © 2011 SciRes. MME
R. G. TODKAR
86
for k and ζa one can write
2
1
aQk
(10)
where



2
22
1
24
1128
4
p
rp
pipe
wdd l
Q
d



ipe
2.2. Air Damper Characteristics
From Equation (10), it is seen that ζa will be large for
small values of k, i.e. for small values of ka for given
value of k. To provide variable damping ratio, the value
of k can be varied. For the application of this device as a
variable damping unit smaller values of k (0.05 to 0.125)
are preferred. Also k depends on the ratio (pi/Nt), (refer
Equation (4)) i.e. for small values of k, ratio (pi/Nt)
should be kept small.
2.2.1. Effect of the Cylinder Diameter dc on Air
Spring Rate Ratio k (Refer Figure 2)
Here dc is varied as dc = 10 mm, dc = 20.0 mm and dc =
30.0 mm. The values of dr = (0.333) (dc) = (0.333) (30.0)
= 10mm and hp =(0.5) (dc )= (0.5) (30.0) = 15 mm have
been obtained. Considering a sliding fit between the pis-
ton and cylinder the piston diameter dp is taken as 29.95
mm for a cylinder diameter dc = 30 mm. Figure 2 shows
the effect of variation of cylinder bore dc on spring rate
ratio k .
2.2.2. Effect of dpipe on ζa (Refer Figure 3)
Using the Equation (10), the effect of variation of ratio
on air damping ratio ζa has been obtained for the values
of dpipe = 2.5, 2.0 and 1.5 mm. with lpipe = 3.0 m. Figure
4 shows the effect of air spring rate ratio k on the air
damping ratio ζa. for the values of dpipe = 2.5, 2.0 and 1.5
mm. with lpipe = 3.0 m .
Figure 2. k vs (pi/Nt ) for different values of dc = a) 10 mm,
b) 20 mm, c) 30 mm.
Figure 3. ζa vs (pi/Nt) with lpipe = 3.0 m dpipe = a) 2.5 mm, b)
2.0 mm, c) 1.5mm.
Figure 4. ζa vs k with lpipe=3.0 m and dpip = a) 2.5 mm, b) 2.0
mm, c) 1.5mm.
From the curves of Figures 2, 3 and 4, it is seen that
the developed air damper can provide appreciable increase
in the damping ratio for values of the ratio (pi/Nt) in the
range 500 to 6000 N/sq.m. per unit volume ratio (vt/vc).
2.2.3. Develop e d Air D ampe r Specifi cations
Plate 1 shows the details of the air damper cylinder and
slider assembly and air damper piston rod fitted to the
sprung mass assembly .The air damper has been devel-
oped with the physical dimensions given in Table 1.
A double acting air cylinder configuration has been
selected with the piston travel of ±15 mm amplitude. The
base excitation of ±1.5mm amplitude is provided. The
material used for the entire assembly is steel with EN8
series, properly ground and finished to the selected di-
mensions. The sprung mass is in the form of a circular
plate made up of C.I.
2.2.4. SDOF Quarter Car Vehicle Suspe nsi on
System Model [5]
Thus the developed cylinder-piston and air tank-type air
damper is capable of providing variable damping ratio.
Copyright © 2011 SciRes. MME
87
R. G. TODKAR
Plate 1. Air damper cyli nder-piston and air-tank system.
Table 1. Air damper dimensions.
dc d
p d
r h
p l
p
30 29.85 10 15 13
Figures 5(a) and (b) show a SDOF quarter-car vehicle
suspension system model with system damping only and
using the developed air damper respectively described in
Section 2, respectively.
2.3. Equations of Motion
The equations of motion are given below
1) For Case 1 the equation of motion is

111 111
mxkx ucxu 
 (11)
2) For Case 2 the equations of motion are


11 11111a
mxkx ucx ucxy 
  
(12)

10
aa
cyx kyu 
 (13)
2.4. Motion Transmissibility of the Sprung Mass
Assume the steady state solutions of the Equations (11),
(12) and (13) in the form x1 = X1ejwt, x2 = X2ejwt and the
base excitation as u = Uejwt, and following the usual pro-
cedure of solution, the expression for the motion trans-
missibility Mt1 (for the sprung mass) has been obtained
for Case 1 as



22
01
1
22
2
02 1
Mt1 AA
X
UBB B





(14)
where A1 = 2 ζ1 , A0 = 1 ,B2 = 1 ,B1 = 2 ζ1 and B0 = 1 and
for Case 2 as
(a)
(b)
Figure 5. SDOF quarter-car vehicle suspension system mo-
del. (a) SDOF quarter-car vehicle suspension System with
system damping; (b) SDOF quarter-car vehicle suspension
system with system damping and air damper with maxwell
type model.

22
2
201
1
22
23
023 1
Mt1 aa a
X
Ubbb b






 

 
(15)
where a2 = 2(ζ1 + kζa), a1 = (2 ζ1 δ + 1+ k), a0 = δ, b3
= 1, b2 = (δ+2ζ1), b1 = (2ζ1 δ + 1 + k) and b0 = δ where δ
= (k/ζa)
Figure 6 and Figure 7 respectively show the curves of
Mt1 vs λ (where λ is the ratio of excitation frequency w
to the undamped natural frequency w1 of the system
(m1,k1) for Case 1 and Case 2 .
2.5. Effect of Variation of Air Damper Spring
Rate Ratio k
The peak values of Mt1 (at resonance) for increasing
values of the air spring rate ratio k and the air damping
ratio ζa are given in Table 2 (also refer Figure 6 and
Figure 7). It is seen that as the value of air damper
spring rate ratio k and air damping ratio ζa increase, there
is an appreciable reduction in the value of Mt1 at the
resonant frequency for the case where the air damper is
modeled as a Maxwell type.
Copyright © 2011 SciRes. MME
R. G. TODKAR
88
Figure 6. Mt1 vs λ for k =0.2, 0.4 and 0.6.
Figure 7. Mt1 vs λ for ζa = 0.2, 0.3 and 0.4.
Table 2. Values of spring rate ratio k and damping ratio ζa
varied with air damper modeled as a Maxwell Mode l.
ζ1 = 0.133 , ζa = 0.3 ζ1 = 0.133, k = 0.3
k ζa
Peak Values of Mt1
0.2 0.4 0.6 0.2 0.3 0.4
Mt1 3.48 2.89 2.48 2.0 1.55 2.92
λ 0.92 0.96 0.98 0.98 1.08 1.15
Figure No. 6 7
3. Optimal Value ζaopt of Air Damping Ratio ζa
The air damping is highly effective when the air damper
is modeled as Maxwell type (Case 2 of Section 2). As
such, Case 2 is taken for optimization of air damping
ratio ζa The value of Mt1 given by Equation (15) is af-
fected by system damping ratio ζ1 and the air damper
characteristics i) air spring rate ratio k and ii) air damp-
ing ratio ζa.
Rearranging the equation as a function of ζa, we obtain
2
1
2
21
Mt1 21
aa
aa
0
0
A
AA
X
UBB



 B
(16)
where A2, A1, A0, B2, B1,B0 are the constants containing
frequency ratio λ, air spring rate ratio k and system
damping ratio ζ1. Differentiating the rearranged Equation
(16) for Mt1 w.r.t. ζa and setting it equal to zero i.e.
(Mt1)/(ζa) = 0, we obtain a polynomial in terms of
descending powers of ζa as
32
C32 1 00
aaa
CCC


(17)
where Cis are the constant coefficients containing ζ1, k
and λ (i = 0, 1, 2 and 3). The expressions derived for Cis
are lengthy and have not been included in the body of the
write-up. The optimal value ζaopt of ζa is obtained by
solving the Equation (17) and with the optimal value thus
obtained, the values of Mt1 have been determined.
Effect of Air Damping Ratio ζa on Amplitude Ratio
Mt1 for Various Values of Air Spring Rate Ratio k
The values of ζaopt for the air damper with a Maxwell
type model have been obtained for ζ1 = 0.133 and λ = 1
for different values of k. The minimum values of Mt1 (at
resonance) for increasing values of the air spring rate
ratio k respectively are given in Table 3 (Also refer Fig-
ures 8 to 11).
4. Experimental Setup
Figure 13 shows the experimental setup designed and
developed for dynamic response analysis of the SDOF
quarter car suspension system model (refer also plate 2).
The set up consists of a cam operated mechanism to pro-
vide sinusoidal base excitation of the desired amplitude
and excitation frequency. The time dependant motion of
both the base excitation u(t) and the sprung mass re-
sponse x1(t) are sensed and processed by the sensors
consisting of LVDTs interfaced with a computer system.
The software has been developed to process the input
base excitation motion u(t) vs time and the sprung mass
response motion x1(t) vs time. The system also incorpo-
rates the facility to control the operating air pressure in
the damper system through a computer interfaced system
as shown in Figure 14. (Also refer Plate 2). Plate 2 shows
all the details regarding the laboratory experimental mo-
del of a SDOF air damped SDOF quarter-car suspension
system. The values of the sprung mass and suspension
spring rate are taken respectively as 4.0 kg and 970 N/m.
Air Pressure Control
A computer interfacing system containing the closed loop
air pressure control system associated with a set of two
LVDTs to sense the suspension mass displacement x1(t)
Copyright © 2011 SciRes. MME
89
R. G. TODKAR
Table 3. Values of the air spring rate ratio k varied with ζ1
= 0.133 and λ = 1.
k
0.025 0.050 0.0750.100 0.200 0.300
Mt1(min) 3.763 3.642 3.529 3.422 3.054 2.765
ζaopt 0.06 0.09 0.11 0.13 0.21 0.27
Figure No. 8 9
k
0.400 0.500 0.6000.700 0.800 0.900
Mt1(min) 2.538 2.356 2.209 2.089 1.989 1.905
ζaopt 0.33 0.38 0.44 0.48 0.52 0.57
Figure No. 10 11
Figure 8. Mt1 vs ζa for k = 0.025, 0.05 and 0.075.
Figure 9. Mt1 vs ζa for k =0.1, 0.2 and 0.3..
Figure 10. Mt1 vs ζa for k = 0.4, 0.5 and 0.6.
Figure 11. Mt1 vs ζa for k = 0.7, 0.8 and 0.9.
Figure 12. Mt1 vs ζa. for k = 0.025 to 0.9 .
Copyright © 2011 SciRes. MME
R. G. TODKAR
Copyright © 2011 SciRes. MME
90
Figure 13. Experimental setup for an air damped SFOF suspension system model.
Plate 2. Experimental setup for a SFOF suspension system model with air damper.
91
R. G. TODKAR
Figure 14. Mt1 vs λ case (i) for ζ1 = 0 .133.
and base excitation u(t) has been developed .The ratio
(pi/Nt) plays an important role in controlling the air damp-
ing ratio ζa in the system. The appropriate value of the
ratio (pi/Nt), depending on the value of ζa desired in the
system, can be set by controlling the value of operating
air pressure pi for a given value of ratio Nt = (vt/vc) or
keeping the air pressure in the system at the atmospheric
pressure and adjusting the value of the term Nt by ad-
justing the tank volume vt.
5. Experimental Analysis
5.1. Experimental Curves for Motion
Transmissibility Mt1 vs Frequency Ratio λ
Using the experimental setup shown in Figure13 and
Plate 2 and by setting the appropriate values of the air
spring rate ratio k and the air damping ratio ζa, the ex-
perimental plots of Mt1 vs λ have been obtained for the
SDOF system as i) With system damping only and with-
out air damper (Refer Figure 14 and Table 4) ii) With
system damping and air damper, with k = 0.2 (Refer Fig-
ure 15 and Tab le 5).
5.2. Experimental Motion Transmissibility
Curves Mt1 vs λ, Using Optimal Values of
Air Damping Ratio ζaopt
Table 6 shows the theoretical and experimental mini-
mum values of motion transmissibility Mt1 at resonant
frequency (with the air damper set for the optimal air
damping ratio ζaopt at the value of ζaopt = 0.33 with air
spring rate ratio k = 0.4.
Table 4. Theoretical and Experimental Peak Values of Mt1(max)
for the Case (i) for ζ1 = 0.133.
Peak value ofTheoretical Experimental
Mt1(max) 3.798 3.70
λ 0.98 1.25
Figure 14
Figure 15. Mt1 vs λ for ζ1 = 0.133, k = 0.2 and ζa = 0.2.
Table 5. Theoretical and experimental peak values of Mt1(max)
for the case (ii) for ζ1 = 0.133, k = 0.2 and ζa = 0.2.
Peak value of Mt1(max) Theoretical Experimental
Mt1(max) 1.89 1.80
λ 0.98 1.29
Figure 15
Table 6. Theoretical and experimental peak values of Mt1(mim)
for ζ1 = 0 .133, λ = 1, k = 0.4 and ζaopt =0.33.
Mt1(min) Theoretical Experimental
Mt1 2.55 2.10
λ 1.0 1.0
Figure 16
Figure 16. Mt1 vs λ for ζ1 = 0.133, λ = 1, k = 0.4 and ζa opt =
0.33.
Copyright © 2011 SciRes. MME
R. G. TODKAR
92
6. Conclusions
In this paper, a cylinder-piston and air-tank type air damp-
er has been developed to provide variable air damping
for a SDOF quarter car vehicle suspension system. The
air damper has been based on the Maxwell type model.
The effect of the air damper characteristics i.e. air damp-
ing ratio ζa and air spring rate ratio k on the resonant re-
sponse of an air damped SDOF vehicle suspension sys-
tem has been analyzed. It is seen that as the value of the
air spring rate ratio k increases, the optimal value ζaopt
increases with decrease in the value of motion transmis-
sibility Mt1. An experimental setup has been developed
with an attendant air pressure control system. The values
of k and ζa for the air damper can be adjusted with the
appropriate changes in dimensions of pipe length lpipe, pipe
diameter dpipe of capillary pipe between the air damper
and the air tank and change in the ratio (pi/Nt). From the
results of the experimental analysis shown in Figure 14
and Figure 15, it is seen that the experimental values of
Mt1 are close to the corresponding theoretical values of
Mt1. From Figure 16, it is seen that the theoretical and
experimental minimum values of Mt1 for ζaopt = 0.33
with k = 0.4 are in good agreement. The addition of the
air damping improves substantially the motion transmis-
sibility characteristics of the sprung mass of the SDOF
quarter-car suspension model in the region of resonance.
7. References
[1] R.A. Williams, “Electronically Controlled Automotive
Suspension Systems,” Computing and Control Engineer-
ing Journal, Vol. 5, No. 3, 1994, pp. 143-148.
doi:10.1049/cce:19940310
[2] Toshihiko Asami and Nishihara, “Analytical and Ex-
perimental Evaluation of an Air Damped Dynamic Vi-
bration Absorber: Design Optimizations of the Three-Ele-
ment Type Model”, Transaction of the ASME, Vol. 121,
1999, pp. 334-342.
[3] R. D. Cavanaugh, “Air Suspension Systems and Servo-
Controlled Isolation Systems,” Hand Book of Shock and
Vibration, 2nd Edition, McGraw-Hill, New York, 1961,
pp. 1-26
[4] R. G. Todkar and S. G. Joshi, “Some Studies on Trans-
missibility Characteristics of a 2DOF Pneumatic Semi-
Active Suspension System,” Proceedings of International
Conference on Recent Trends in Mechanical Engineering,
Ujjain, 4-6 October 2007, pp. 19-28.
[5] P. Srinivasan, “Mechanical Vibration Analysis,” Tata Mc-
Hill Publishing Co., New Delhi, 1990.
Nomenclature
k1 stiffness of spring supporting sprung mass
m1 sprung mass
w1 (k1/m1)1/2
ζ1 system damping ratio
w applied frequency
λ frequency ratio = (w/w1)
dp piston diameter
dc cylinder bore
lp length of the piston
hp height of bottom of piston from bottom of the
cylinder
dpipe inside diameter of the capillary pipe
lpipe length of the capillary pipe
μo viscosity of air
n index of expansion of the air
ka stiffness of air spring
k spring rate ratio = (ka/k1)
wa (ka/m1)1/2
ca coefficient of viscous damping provided by the
air damper
ζa air damping ratio of air spring
ζaopt optimal value of air damping ratio.
u(t)base excitation
x1(t)dynamic displacement response of sprung mass
m1
Mt1motion transmissibility of the sprung mass m1
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