Modern Mechanical Engineering, 2011, 1, 93-103
doi:10.4236/mme.2011.12012 Published Online November 2011 (http://www.SciRP.org/journal/mme)
Copyright © 2011 SciRes. MME
The Effect of Mass Ratio and Air Damper Characteristics
on the Resonant Response of an Air Damped Dynamic
Vibration Absorber
Ranjit G. Todkar1, Shridhar G. Joshi2
1 Department of Mechanical Engineering, P.V.P. Institute of Technology, Budhagaon, Sangli, India
2Department of Mechanical Engineering, Walchand College of Engineering, Sangli, India
E-mail: rgtodkar@gmail.com
Received October 21, 2011; revised November 25, 2011; accepted November 12, 2011
Abstract
In this paper, it is shown that, a road vehicle 2DOF air damped quarter-car suspension system can conven-
iently be transformed into a 2DOF air damped vibrating system representing an air damped dynamic vibra-
tion absorber (DVA) with an appropriate change in the ratio µ of the main mass and the absorber mass i.e.
when mass ratio µ >> 1. Also the effect of variation of the mass ratio, air damping ratio and air spring rate
ratio, on the motion transmissibility at the resonant frequency of the main mass of the DVA has been dis-
cussed. It is shown that, as the air damping ratio in the absorber system increases, there is a substantial de-
crease in the motion transmissibility of the main mass system where the air damper has been modeled as a
Maxwell type. Optimal value of the air damping ratio for the minimum motion transmissibility of the main
mass of the system has been determined. An experimental setup has been designed and developed with a
control system to vary air pressure in the damper in the absorber system. The motion transmissibility charac-
teristics of the main mass system have been obtained, and the optimal value of the air damping ratio has been
determined for minimum motion transmissibility of the main mass of the system
Keywords: Air Damped Dynamic Vibration Absorber, Motion Transmissibility, Effect of Mass Ratio, Air
Damper, Optimization
1. Introduction
Many real engineering systems such as, a road vehicle
suspension system, a dynamic vibration absorber system,
a vibration isolation system of machinery (where the floor
supporting the machine is sufficiently flexible), a double
centrifugal pendulum system etc., can be adequately re-
presented as 2DOF vibrating systems [1]. As such, in this
paper, a 2DOF air damped vibrating system representing
a 2DOF air damped dynamic vibration absorber has been
studied. In this case, the mass ratio µ i.e., ratio of the
main mass m2 to the auxiliary mass m1 is greater than
unity (µ >> 1) and is in the range of 2.5 to 5.0. Also the
effect of variation of the mass ratio, air spring rate ratio
and air damping ratio on the motion transmissibility of
main mass has been discussed. It has been shown that, as
the air damping ratio in the absorber system increases,
there is a substantial decrease in the motion transmissi-
bility of the main mass system in the neighborhood of
resonant frequency for the case where the air damper is
modeled as a Maxwell type [2,3]. Optimal value of the
air damping ratio for the minimum transmissibility of the
main mass system has been determined. An experimental
setup has been designed with an air pressure control sys-
tem for setting the appropriate value of air damping ratio
in the system.The motion transmissibility characteristics
of the main mass m2 of the dynamic vibration absorber
model have been obtained.
2. Equations of Motion
Equations of motion have been derived and are given
respectively in Tables 1, 2 and 3 for the following [2],
1) A 2DOF dynamic vibration absorber system with
system damping only and without air damper (Refer Fi-
gure 1 in Table 1), here after referred as Case 1.
2) A 2DOF air damped dynamic vibration absorber
system with system damping and, Vogit type model for
R. G. TODKAR ET AL.
94
Table 1. A general 2DOF viciously damped vibrating sys-
tem with system damping.
m
2
m
1
x
1
(t)
x
2
(t)
u (t)
c
2
k
2
c
1
k
1
A general 2DOF vibrating system (Dynamic Vibration Absorber
Model). µ >>1.
Equations of motion

2212 112 12222
mxkxxcxxkx ucxu 
 
(1)

111 121 12
mxkxxcxx 
 (2)

1
2
22
2
22 0011
2
22
42 3
420 31
X
Mt2U
AA A
BBB BB

 











(3)

1
222
2
20 1
1
22
42 3
420 31
X
Mt1 U
AA A
BBB BB

 











(4)
where
A22 = 2 ζ2, A11 = v2 + 4 ζ1 ζ2v, A00 = 2 (ζ1 v2 + ζ2 v), and
A2 = 4 ζ1 ζ2 v, A1 = 2 ζ1 v2, A1 = 2 ζ1 v2, A0 = v2 and
B4 = 1, B3 = 2 (ζ1 + (ζ1) + ζ2 v ), B2 = (1 + (1) + v2),
B1 = 2 (v ζ2 + ζ1 v2), B0 = v2
air damper (Refer Figure 2 in Table 2), hereafter re-
ferred as Case 2.
3) A 2DOF air damped dynamic vibration absorber
system with system damping and with Maxwell type
model for air damper (Refer Figure 3 in Table 3), here-
after referred as Case 3.
Motion Transmissibility
Assuming the steady state solutions in the form x1 =
Xejwt , x2 = X2ejwt and y = Yejwt the base excitation as u =
Uejwt and following the usual procedure of solution, the
equations of motion have been solved and the expres-
sions for the motion transmissibility Mt2 (for the main
mass ) and Mt1 (for the auxiliary mass) have been ob-
tained and are given respectively in Equations (3) and (4)
for Case 1 in Table 1 and in Equations (7) and (8) for Case
2 in Table 2 and in Equations (12) and (13) in Table 3. [4].
Table 2. 2DOF Air damped vibrating system using an air
damper (Vigot Model), with system damping coefficients c1,
c2 and air damper characteristics i) air ddamping. Ratio ζa
and ii) air spring rate ratio k = (ka/k1) ,where k1 = stiffness
of auxiliary spring and ka = stiffness of air spring.
Case 2
m
1
c
1
x
1
(t)
x
2
(t
)
u (t)
k
1
k
a
c
a
k
2
c
2
m
2
Air Dam
p
e
r
DOF air damped vibrating system using an air damper (Vigot
Model). (Dynamic Vibration Absorber Model). µ >>1.
Equations of motion
 

2212 112 1
22 22
aa
mxkkxx ccxx
kx ucx u
   

 
 (5)
 
1111 211 2aa
mxkkxx ccxx 
 (6)

1
222
2
22 0011
2
22
42 3
420 31
X
Mt2 U
aa a
bbb bb

 







(7)

1
222
2
20 1
1
22
42 3
420 31
X
Mt1 U
aa a
bbb bb

 





 
 
(8)
where
a22 = 2 ζ2 v,
a11 =(v2 + 4 ζ1 ζ2 v + 4 ζa k0.5 ζ2 v),
a00 = 2(ζ1 v2 + ζa k0.5 v2 + ζ2 v + k ζ2 v)
and
a2 = 4 ζ1 ζ2 v + 4 ζ2 ζa k0.5 v)
a1 = 2 (ζ1 v2 + ζa k0.5 v2 + ζ2 v + k ζ2 v)
a1 = 2 (ζ1 v2 + ζa k0.5 v2 + ζ2 v + k ζ2 v),
a0 = v2 (1 + k)
and
b4 = 1, b3 = 2 (ζ1 + (ζ1 / µ) + ζ2 v + ζa [k0.5 + (k0.5/µ)])
b2 = (1 + k +4 ζ1 v ζ2 + 4 ζa k0.5 v ζ2 + (1) + v2 + (k/µ))
b1 =2 ( v ζ2 + k v ζ2 + ζ1 v2 + ζa k0.5 v2 ),
b0 = v2 (1 + k)
Copyright © 2011 SciRes. MME
R. G. TODKAR ET AL.
Copyright © 2011 SciRes. MME
95
Table 3. 2DOF air damped vibrating system using an air damper (Maxwell Model) , with system damping coefficients c1, c2 and
air damper characteristics air damping ratio ζa and air spring ratio k = (ka/k1) , where k1 = stiffness of auxiliary spring and ka =
stiffness of air spring.
Equations of motion
 
2212112122 22 2a
mxkxxcxxkxykx ucx u
  
 (9)
12
0
aa
cyx kyx
 
 (10)
111 121121a
mxkx xcxxcxy
 
 (11)
m
2
m
1
k
a
y (t)
Air Damper
x
1
(t)
c
1
k
1
u (t)
c
a
c
2
k
2
1
22
2
42 53
44220055 33 11
2
2
642 53
6420 531
X
Mt2 U
aaa aaa
bbbb bbb
 
 
2
 



 
 
 
(12)
2DOF Air damped vibrating system using an air
damper (Maxwell Model) µ >> 1. dynamic
vibration absorber model.
1
22
2
42 3
420 31
1
2
642 53
6420 531
X
Mt1 U
aaa aa
bbbb bbb
 
 
2




 
 
 
(13)
where
δ= k0.5/(2 ζa )
a55 = 2 ζ2 v , a44 = [v2 + 4 δ ζ2 v + 4 ζ1 ζ2 v], a33 = [2δv2+ 8 δ ζ1 ζ2 v + 2ζ1 v2 + 2v δ (ζ1 + ζ2) + ζ2 v (1+2k)]
a22 = [δ2 v (v + 4 ζ1 ζ2 )+δ v (4 ζ1 v + 4 ζ2 + 2 ζ2 k )+ v2(1 + k )], a11 = δ2 (4 ζ1 v2 + 2 ζ2 v) + δ(k v2 + 2 v2), a00 = δ2 v2
and
a4 = 4 ζ1 ζ2 v, a3 = 2ζ1 v2 + 8 δ ζ1 ζ2 v + 2 ζ2 v (1 + k),
a2 = δ[4ζ1 v2 + 2 ζ2 v + k 2 ζ2 v + 2 ζ2 v] + 4δ ζ1 ζ2 v + v2 [1 + k],
a1 = δ2 2 v [ζ1 v + ζ2] + δ v2 [2 + k], a0 = δ2 v2
and
b6 = 1,b5 = (2 δ + 2 ζ1 + (2 ζ1/μ) + 2 ζ2 v), b4 = [1 + k + (1 /μ) + v2+ (k/μ ) + 4 ζ1 ζ2 v] + 4 δ [ζ1 + (ζ1/μ) + ζ2 v] + δ2
b3 = δ[2 + 8 ζ1 ζ2 v + (2/μ) + 2 v2 + k + (k/μ)] + [2 ζ2 v+2 ζ2 v k + 4 ζ1 ζ2 v2] + δ2 [2 ζ1 + (2 ζ1/μ) + 2 ζ2 v],
b2 = δ2 [1+ 4 ζ1 ζ2 v + (1 /μ) + v2] + δ [4 ζ2 v + 4 v2 ζ1 + 2 ζ2 v k ] + v2(1+k), b1 = δ2 [ 2 ζ2 v + 4 v2 ζ1] + 2 δ v2 ,
b0 = δ2 v2
m
2
m
1
k
a
y (t)
Air Damper
x
1
(t)
c
1
k
1
u (t)
c
a
c
2
k
2
m
2
m
1
x
1
(t)
x
2
(t)
u (t)
c
2
k
2
c
1
k
1
Figure 1. A general 2DOF vibrating system (dynamic
vibration absorber model). µ >>1.
m
1
c
1
x
1
(t)
x
2
(t
)
u (t)
Figure 3. 2DOF air damped vibrating system using an air
damper (maxwell model) µ >> 1. dynamic vibration absor-
ber model.
k
1
k
a
c
a
k
2
c
2
m
2
Air Dam
p
er
3. Motion Transmissibility Mt2 (μ >> 1)
For the air damped dynamic vibration absorber system,
mass ratio µ has been varied in the range of 2.5 to 5.0,
where λ is the ratio of excitation frequency w to the un-
damped natural frequency w1 of the system (m1,k1)), have
Figure 2. DOF air damped vibrating system using an air
damper (vigot model). (dynamic vibration absorber model). µ
>>1.
R. G. TODKAR ET AL.
96
been plotted for Case 1, Case 2 and Case 3. The peak
values of Mt2 (at resonance) are given in Tables 4, 5 and
6 respectively.
3.1. Effect of Variation of Mass Ratio µ
The values of µ are varied as µ = 2.5, µ = 3.3 and µ = 5.0
when ζ1 = 0.1, ζ2 = 0, k = 0.1 and ζa = 0.05 with spring
rate ratio (k2/k1) as 6.49. Table 4 gives respectively the
peak values of Mt2 at resonant frequencies obtained for
Case 1, Case 2 and Case 3. It is seen that, as the value of
µ increases, there is no substantial change in the value of
Mt2 at the first resonant frequencies for the case where
the air damper is modeled as a Maxwell type. Figure 4,
Figure 5 and Figure 6 show the corresponding Mt2 vs λ
plots.
Table 4. Peak values of Mt2 with ζ1 = 0.100, ζ2 = 0.0, k = 0.1 and ζa = 0.05 and value of mass ratio µ is varied.
µ = 2.5 µ = 3.3 µ = 5.0
Air damper
modeled as a
Air damper
modeled as a
Air damper
modeled as a
Peak Values of
Mt2 With system
damping only
Case 1 Maxwell
Model Case 3
With system
damping only
Case 1
Vigot Vigot
Model Case 2 Model Case 2
Maxwell
Model Case 3
With system
damping only
Case 1 Vigot
Model Case 2
Maxwell
Model Case 3
Mt2 10.032 2.5732 0.487 7.652 2.3208 0.61 18.63 6.008 0.4948
1st peak
λ 0.91 0.920 1.01 0.91 0.92 1.0 0.87 0.88 1.01
Mt2 4.295 7.42 4.3402 3.4503 5.8782 5.154 6.662 4.1069 1.428
2nd
Peak λ 1.76 1.79 1.64 1.76 1.78 1.64 1.3 1.35 1.35
Table 5. Peak values of Mt2 with µ=3.3, ζ1= 0.133, ζ2 =0.0 and ζa = 0.05 and value of spring rate ratio k is varied.
k = 0.075 k = 0.100 k = 0.150
Air damper
modeled as a
Air damper
modeled as a
Air damper
modeled as a
Peak Values o
f
Mt2
With system
damping only
Case 1 Maxwell
Model Case 3
With system
damping only
Case 1
Vigot Vigot
Model Case 2 Model Case 2
Maxwell
Model Case 3
With system
damping only
Case 1 Vigot
Model Case 2
Maxwell
Model Case 3
Mt2 9.2 2.836 0.5698 9.2 2.90 0.6152 9.2 3.045 0.6872
1st peak
λ 0.90 0.91 1.00 0.90 0.91 1.00 0.90 0.93 1.01
Mt2 4.357 5.244 2.535 4.357 5.09 2.87 4.357 3.49 4.81
2nd
peak λ 1.53 1.57 1.48 1.53 1.57 1.49 1.53 1.51 1.59
Table 6. Peak values of Mt2 with µ = 3.3, ζ1 = 0.133, ζ2 = 0.0 and k = 0.10 and value of air damping ratio ζa is varied.
ζa = 0.025 ζa = 0.05 ζa = 0.075
Air damper
modeled as a
Air damper
modeled as a
Air damper
modeled as a
Peak Values of
Mt2
With system
damping only
Case 1 Maxwell
Model Case 3
With system
damping only
Case 1
Vigot Vigot
Model Case 2 Model Case 2
Maxwell
Model Case 3
With system
damping only
Case 1 Vigot
Model Case 2
Maxwell
Model Case 3
Mt2 9.2 3.077 0.8848 9.2 2.9 0.6152 9.2 2.752 0.5135
1st peak
λ 0.9 0.91 1.02 0.9 0.91 1.00 0.9 0.915 0.99
Mt2 4.357 5.157 7.367 4.357 5.090 2.87 4.357 5.035 2.011
2nd
peak λ 1.53 1.58 1.53 1.53 1.57 1.49 1.53 1.57 1.47
Copyright © 2011 SciRes. MME
97
R. G. TODKAR ET AL.
Figure 4. Mt2 vs λ when ζ1 = 0.1, ζ2 = 0, k = 0.1 and ζa = 0.05.
Figure 5. Mt2 vs λ when ζ1 = 0.1, ζ2 = 0, k = 0.1 and ζa = 0.05.
Figure 6. Mt2 vs λ when ζ1 = 0.1, ζ2=0, k = 0.1 and ζa = 0.05.
3.2. Effect of Variation of Air Damper Spring
Rate Ratio k
The values of k are varied as k = 0.075, k = 0.10 and
k = 0.15 when ζ1 = 0.133, ζ2 = 0.0, µ = 3.3 and ζa = 0.05
with spring rate ratio (k2/k1) as 6.49. Table 5 gives re-
spectively the peak values of Mt2 at resonant frequencies
obtained for Case 1, Case 2 and Case 3. It is seen that, as
the value of air damper spring rate ratio k increases, there
is a small increase in the peak value of Mt2 at the resonant
frequencies for the case where the air damper is modeled
as a Maxwell type.
3.3. Effect of Variation of Air Damping Ratio ζa
The values of air damping ratio ζa are varied as ζa =
0.025, ζa = 0.050 and ζa =0.075 when µ = 3.3, ζ1 = 0.133,
ζ2 = 0 and k = 0.10 with spring rate ratio (k2/k1) as 6.49.
Table 6 gives respectively the peak values of Mt2 at
resonant frequencies obtained for Case 1, Case 2 and
Case 3. It is seen that, as the value of air damping ratio ζa
increases there is a substantial decrease in the value of
Mt2 at the resonant frequencies in the case where the air
damper is modeled as a Maxwell type.
4. Optimal Value ζaopt of Air Damping Ratio ζa
The air damping is highly effective when the air damper
was modeled as Maxwell type (Case 3). As such, a 2DOF
air damped dynamic vibration absorber system for Case
3 is taken for optimization of air damping ratio ζa The
equation of the motion transmissibility Mt2 (of the main
mass m2) is given by Equation (13) of Table 3 for Case
3 when the air damper is modeled as a Maxwell type
model [3]. The value of Mt2 is affected by the system
parameters i.e. mass ratio µ, system damping ratio ζ1 and
the air damper characteristics 1) air spring rate ratio k
and 2) air damping ratio ζa
For minimizing the value of Mt2, consider equation
(13) for Mt2 is
2
1
22
2
42 53
4 4220 0553311
22
64 253
64 531
Mt2
20
X
U
aλaλaaaλaλ
bλbλbλbbλbλbλ

 


 

(where constants a44, a33, a22, a11, a00, b6, b5, b4, b3, b2, b1
and b0 have been given in Table 3). The equation for
Mt2 is rearranged in terms of ascending powers of ζa as


 



 

2
432
4321
432
4321
X
Mt2 U
aaaa
aaaa
0
0
A
AA A
BBBB




A
B
4
a
(14)
where A4, A3, A2, A1, A0, B4, B3, B2, B1 and B0 are the
constants containing system damping ratios ζ1, ζ2, k, µ, λ
and v. The equation (14) for Mt2 is differentiated w.r.t. ζa
and set equal to zero i.e. (Mt2)/(ζa ) = 0, a polyno-
mial in terms of ζa is obtained as
765
32
7 6 5 4
3 2 1 00
aaa
aa
a
hhhh
hhhh
 



(15)
Copyright © 2011 SciRes. MME
R. G. TODKAR ET AL.
98
where his (i = 0, 1, 2, 3, 4, 5, 6 and 7) are the constant
coefficients containing µ, ζ1, ζ2 ,k and λ . The expressions
derived for this are very lengthy and have not been in-
cluded in the body of the write-up. The optimal value
ζaopt of ζa is obtained by solving the Equation (15) and
with the optimal value thus obtained the values of Mt2
have been determined .
4.1. Effect on Optimal ζaopt and on Mt2 for
Various Values of Air Spring Rate Ratio k
The values of ζaopt for the air damper modeled as a
Maxwell type model have been obtained for
1) k = 0.025, k = 0.05, k = 0.075 and k = 0.1 and the
results are given in Table 7,
2) k = 0.200, k = 0.3,k = 0.4 and k = 0.5, the results are
given in Table 8 and
3) k = 0.75, k =1, k =2 and k=3, the results are given in
Table 9 (Refer also Figure 7).
4.2. Effect of Air Spring Rate Ratio k on Optimal
Value ζaopt of ζa When µ = 0.335, ζ1 = 0.133,
ζ2 = 0.0 and λ = 1
Figure 7 shows the effect k on Optimal Value ζaopt of air
damping ratio ζa. From the results of analysis, it is seen
that, as the value of ζaopt increases with the increase in air
spring rate ratio k, the value of Mt2 increases. Figure 7
shows the variation of the value of Mt2 with ζaopt for
increasing values of k.
4.3. Effect of Variation of Mass Ratio µ on
Optimal Value ζaopt of ζa
Figure 8 shows the effect of variation of mass ratio µ on
ζaot where air damper is modeled as a Maxwell type. The
value of µ is varied as µ = 2.5 and µ = 5.0 when ζ1 =
0.133, ζ2 = 0 , k = 1 and λ = 1 with the spring ratio (k2/k1)
Table 7. Values of ζaopt when air spring rate ratio k is varied.
µ = 3.3, ζ1 = 0.133, ζ2 = 0.0, λ = 1
Air Damper
modeled as:
Maxwell type k = 0.025 k = 0.05 k = 0.075 k = 0.1
0.3804 0.3907 0.4004 0.4101
Mt2 (min)
ζaopt 0.07 0.10 0.12 0.13
Table 8. Values of ζaopt when air spring rate ratio k is varied.
µ = 3.3, ζ1 = 0.133, ζ2 = 0.0, λ = 1
Air Damper
modeled as:
Maxwell type k = 0.2 k = 0.3 k = 0.4 k = 0.5
0.4452 0.4754 0.5014 0.5236
Mt2 (min)
ζaopt 0.18 0.21, 0.23 0.24
k = 3
k = 0.025
ζ
aopt
0.8
0.7
0.6
0.5
0.4
0 0.1 0.2 0.3 0.
4
Mt2
Figure 7. Mt2 vs ζa (Effect of k) for k = 0.025 to 3.0.
Figure 8. Mt2 vs ζa (Effect of µ ).
as 6.49 . It is seen that, as the value of µ increases, there
is a significant reduction in the value of ζaopt and there is
also a substantial decrease in the minimum value of Mt2.
5. Experimental Setup
Figure 9 shows the experimental setup designed and de-
veloped for dynamic response analysis of the 2DOF air
damped dynamic vibration absorber system (refer also
Plate 1). The setup consists of a cam operated mecha-
nism to provide sinusoidal base excitation .The necessary
software has been developed to collect and process the
dynamic displacements to obtain graphical plots of the
input excitation u(t) vs time and the main mass response
motion x2(t) vs time. The system also incorporates the
facility to control the operating air pressure in the system
through a computer interfaced system as shown in Fig-
ure 9. The values of the main mass and auxiliary mass
have been selected in accordance with values reported in
the literature. The ratio of main mass m2 to auxiliary m1
is about 5 to 10. The 2DOF air damped vibrating system
data selected is as , main mass m2 = 6.0 kg , auxiliary
mass m1 = 0.815 kg ,auxiliary spring rate ratio k1= 970
N/M, the spring rate of the spring supporting the main
mass m2 is k2 = 6300 N / M and mass ratio is μ = (m2/m1 )
is 7.36 .
Copyright © 2011 SciRes. MME
R. G. TODKAR ET AL.
Copyright © 2011 SciRes. MME
99
Figure 9. Experimental setup for 2DOF dynamic vibration absorber system (µ >> 1 ).
Plate 1. Experimental detup for an air damped 2DOF vibration absorber system.
R. G. TODKAR ET AL.
100
5.1. Specifications of the Air Damper
Using the approach of R.D.Cavanaugh [3] for the design
of air damper, following relations have been developed
[4,5].
1)

2
2s
ac
knvpiN
t
(15)
2)



0.5
4
0.5
1e
Q
apipe tpip
lpiNd
where

1
Q128 sπ2
oc
vnm
1
(16)
Using these relations, a cylinder-piston and air-tank
type air damper has been developed [3]. The specifica-
tions of the developed air damper are : piston diameter dp
= 29.85 mm cylinder bore dc = 30.00 mm., piston rod
diameter dr = 10.00 mm, piston length lp = 13.0mm. and
height of piston bottom from the cylinder bottom hp =
15.00 mm In the experimental investigation, the first step
was to select the value of the air damping ratio ζa associ-
ated with the air spring rate ratio k to be set and the cor-
responding set of capillary pipe dimensions like pipe
diameter dpipe and pipe length lpipe. The ratio (pi/Nt),
where pi is the operating air pressure and Nt is the ratio
(v.t/vc) is the basis for the selection of the air damping
ratio ζa (also refer Figure 10 and Figure 11).The para-
meters dpipe, lpipe and the ratio (pi/Nt) have been varied to
change the value of damping ratio ζa in the system.
(pi / Nt)
d.
- - -
....
c
= 10 mm
d
c
= 20 mm.
___ d
c
= 30 mm.
Figure 10. k vs (pi/Nt).
(pi / Nt )
ζ
a
.... d.
d.
___ d
pipe
= 2.5 mm
- - -
pipe
= 2.0 mm
pipe
= 1.5 mm
for l
pipe
= 3.0 m
Figure 11. ζa vs (pi/Nt).
5.2. Air Pressure Control
A computer interfacing system containing the closed
loop air pressure control system with a set of two LVDTs
to sense the main mass displacement x2(t) and base exci-
tation u(t) has been developed .The ratio (pi/Nt) plays an
important role in controlling the air damping 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 the ratio Nt= (vt/vc) or keeping the
air pressure in the system at atmospheric pressure and
adjusting the value of Nt by adjusting the tank volume vt.
6. Experimental Analyses
6.1. Experimental Curves for Motion
Transmissibility Mt2 vs Frequency Ratio λ
Using the experimental setup (shown in Figure 9 and
Plate 1) and by setting the appropriate values of the air
spring rate ratio k and the air damping ratio ζa, the ex-
perimental plots of Mt2 vs λ have been obtained for the
following cases
1) With µ = 1.5, ζ1 = 0.133, ζ2 = 0 and without air
damper.(Refer Figure 12 and Table 10 ).
2) With µ = 1.0, ζ1 = 0.133, ζ2 = 0 and air damper, with
k = 0.423 and ζa = 0.1326 (Refer Figure 13 and Table 11)
3) With µ = 1.5, ζ1 = 0.133, ζ2 = 0 and air damper, with
k = 0.423 and ζa = 0.1326 (Refer Figure 14 and Table 11).
6.2. Experimental Curves Mt2 vs λ for Using
Optimal Values of Air Damping Ratio ζaopt
Table 12 shows the theoretical and experimental peak
values of motion transmissibility Mt2 at resonant fre-
quency with the air damper set for the optimal air damp-
ing ratio ζaopt: k = 0.1 with ζaopt = 0.53 and k = 0.4 with
ζaopt = 0.68. The experimental results have been shown in
Figure 15 and Figure 16.
Figure 12. Mt1 vs λ for Case 6.1 (i).
Copyright © 2011 SciRes. MME
101
R. G. TODKAR ET AL.
Table 10. Peak values of Mt2 for Case 6.1 (i).
Peak Values of Mt2 Theoretical Results Experimental Results
Mt2 6.235 5.50
1st
peak λ 0.92 1.179
Figure No. 12
Figure 13. Mt2 vs λ for 6.1 (ii), µ = 1.0, k = 0.423 and ζa =
0.1326.
Table 11. Peak values of Mt2 for the Case 6.1 (ii) and for the
Case 6.1 (iii) with ζ1 = 0.133, ζ2 = 0.0.
µ = 1.0, k = 0.423 and
ζa = 0.1326
µ = 1.5, k = 0.423 and
ζa = 0.1326
Peak Values
of Mt2 Theoretical
Values
Experimental
Values
Theoretical
Values
Experimental
Values
Mt2 5.689 3.33 6.13 3.33
1st peak
λ 0.92 1.35 0.92 1.32
Figure No. 13 14
Figure 14. Mt2 vs λ for 6.1 (iii), µ = 1.5, k = 0.423 and ζa =
0.1326.
Table 12. Peak values of Mt2 with optimal valve of optimal
air damping ratio ζaopt with µ = 1.5, ζ1= 0.133, ζ2 = 0.0.
Theoretical
Values
Experimental
Values
Theoretical
Values
Experimental
Values
Peak Values
of Mt2 k =0.1, ζaopt = 0.53 k = 0.4, ζaopt=0.68
Mt22.535 1.74 1.623 1.5
1st peak
λ 1.03 1.0 1.19 1.48
Figure No.15 16
Figure 15. Mt2 vs λ. For µ = 1.5, ζ1 = 0.133, k = 0.1, ζaopt = 0.53.
Figure 16. Mt2 vs λ. µ = 1.5, ζ1 = 0.133, k = 0.4, ζaopt = 0.68.
7. Conclusions
In this paper, the effect of mass ratio and the air damper
characteristics on the resonant response of an air damped
2DOF vibrating system representing an air damped dy-
namic vibration absorber model have been studied with
the air damper modeled as a Maxwell type . There is no
substantial change in the value of Mt2 with the increase
Copyright © 2011 SciRes. MME
R. G. TODKAR ET AL.
Copyright © 2011 SciRes. MME
102
in the value of mass ratio µ. However, with the increase
in the value of the air spring rate ratio k there is a con-
siderable increase in the value of the Mt2 at the resonant
frequency where the air damper is modeled as a Maxwell
type. It is seen that, with the increase in the value of the
air damping ratio ζa there is a considerable decrease in
the value of the Mt2 at the resonant frequency where the
air damper is modeled as a Maxwell type. Further it is
seen that as the value of the air spring rate ratio k in-
creases , the value of the optimum value ζaopt of the air
damping ratio ζa increases with increase in the value of
motion transmissibility Mt2. It is also observed that there
is a considerable reduction in the value of ζaopt with the
increase in the value of the mass ratio µ, in the range µ =
2.5 to µ = 5.0. An experimental setup has been devel-
oped with an appropriate air pressure control system. A
cylinder-piston and air-tank type air damper has been
designed and developed to obtain the desired value of the
air damping ratio ζa from the air damper. From the re-
sults of the experimental analysis shown in Figure 13
and Figure 14, it is seen that the experimental peak val-
ues of Mt2 are close to the corresponding theoretical
peak values of Mt2 obtained from the theoretical analysis
where the air damper is modeled as a Maxwell type.
From the Figure 15 and Figure 16, it is seen that the
theoretical and experimental values of Mt2 for ζaopt =
0.53 with k = 0.1 and ζaopt = 0.68 with k = 0.40 are in
good agreement.
From the theoretical and experimental investigations
carried out, it is seen that the addition of the air damping
in the absorber system (m1,k1) improves substantially the
motion transmissibility characteristics of the main mass
of the 2DOF dynamic vibration absorber model over a
range of excitation frequencies in the region of reso-
nance.
8. Acknowledgements
The first author Prof. R.G.Todkar acknowledges with
thanks the authorities of P.V.P.Institute of Technology,
Budhgaon, Dist. SANGLI (Maharashtra) INDIA 416 304
for their support and encouragement during the period of
this work.
9. 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] T. Asami and Nishihara, “Analytical and Experimental
Evaluation of an Air Damped Dynamic Vibration Ab-
sorber: Design Optimizations of the Three-Element Type
Model,” Transaction of the ASME, Vol. 121, 1999, pp.
334-342.
[3] R. D. Cavanaugh, “Hand Book of Shock and Vibration,”
Air Suspension Systems and Servo-controlled Isolation
Systems, Chapter 33, 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 Engineering College, Ujjain, 4-6 October 2007, pp.
19-28.
[5] P. Srinivasan, “Mechanical Vibration Analysis,” Tata
Mc-Hill Publishing Co., New Delhi, 1990.
103
R. G. TODKAR ET AL.
Nomenclature
k1 stiffness of spring for absorber mass
k2 stiffness of spring for main mass
m1 absorber mass
m2 main mass
µ mass ratio = (m2/m1)
w1 (k1/m1)1/2
w2 (k2/m2)1/2
v natural frequency ratio = (w2/w1)
ζ1 system damping ratio for main mass system
ζ2 system damping ratio for auxikary mass system
w applied frequency
λ frequency ratio = (w/w1)
dp piston diameter
dc cylinder bore
lp length of the piston
hp height 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 of the air damper
ζa damping ratio provided by the air spring
ζaopt optimal value of air damping ratio
u(t) base excitation
x1(t)dynamic displacement response auxiliary mass m1
x2(t)dynamic displacement response of main mass m2
Mt1 motion transmissibility of the auxiliary mass m1
Mt2 motion transmissibility of the main mass m2
Copyright © 2011 SciRes. MME