th
Anoshirvan Farshidianfar and Neda Nickmehr
Department of mechanical engineering, Ferdowsi University of Mashhad
Mashhad, Iran
Farshid@um.ac.ir
Abstract
Impact dampers in various forms are used to control the response of vibratory systems subjected to either forced or self-excitations.
Reducing the amplitude of vibration of van der Pol oscillator using an impact damper is investigated in this paper. The Incremental Harmonic Balance (IHB) method is used to obtain analytical solution which is verified by numerical integration.
Impact vibration exists in many parts of technology. Vibration dampers, shakers and impact machines are different cases that this phenomenon performs a very useful role, so mechanical systems whose elements impacts on one another during operation, have been extensively investigated by researchers.
Impact dampers have been successfully reduced the high amplitude of various systems such as turbine blades [1], tall flexible structures [2] and machine tools [3]. Masri [4] in 1970 studied general motion and stability of impact dampers. Bepat et al.
[5] have considered single impact damper and provided easy to use design charts for the former. The use of bean-bags as the secondary mass, instead of metallic shots has been proposed by Popplewell and Semercigi [6] for noiseless operation of the impact dampers. Ema and Marui [7] investigated analytically and experimentally the characteristics of an impact damper in free damped vibration in detail to improve the damping capability of cutting tools such as long drills. A.K.Mallik et al.
[8,9] proposed the performance of an impact damper for controlling the amplitude of non-linear vibrating systems such as Duffing oscillator and self-excited van der Pol oscillator using harmonic balance and piecewise equivalent linearization approach, respectively. He presented, with the optimum design, two symmetric collisions per cycle are always seen to occur.
H. Luo and S. Hanagud [10] discussed the dynamics of a class of vibration absorbers with elastic stops that operate as an impact damper, they could reduce the amplitude of vibration near resonance region by using an auxiliary mass and motion limiting stop. In this paper their model is considered to reduce the amplitude of van der Pol oscillator.
An impact damper consists of an auxiliary mass which undergoes repeated collisions with the main system; the vibration of the main mass is reduced mainly by the transfer of the momentum and loss of energy during impacts.
The motion of the system affects the excitation force in a self-excited system, the most usual mathematical model is given by the autonomous van der Pol system in which the position
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ICSV15 • 6-10 July 2008 • Daejeon • Korea dependent damping force changes sign once in every cycle and performs like the source of energy to provide the vibration.
In the present paper an attempt is made to use an impact damper for controlling the amplitude of van der Pol oscillator and analyse the system by IHB method.
The IHB method as a powerful approach with unique advantages is capable of dealing with strongly non-linear systems. In this method non-linear differential equations are transformed into a set of linearized incremental algebric equations in terms of Fourier coefficients and so only linear equations have to be solved iteratively in each incremental step. With a variable parameter this method is ideally suited to parametric studies for obtaining solution diagrams, after seeking the solution for a particular value of the parameter, this parameter will then be incremented or decremented. The IHB method was originally presented for analyzing periodic structural vibrations [11] and has been successfully applied to various types of non-linear dynamic problems such as dry friction damper, cubic non-linear system, and wheel shimmy system combined coulomb and quadratic damping.
Finally the results that are obtained from IHB method have been compared with those calculated by numerical integration.
The considered model of a two-degree-of-freedom vibration absorber is shown in figure 1, the system is modelled as a van der Pol oscillator and an auxiliary mass m
2
, existence of the motion limiting stop s plays the role of impact damper and introduces nonlinearities because of the contact and noncontact motion states during the vibration, y =x
1
- x
2 indicates the relative motion between the main mass m
1 and the auxiliary mass m
2
.
In the models described by Galhoud et al . [12], it was assumed that as long as y >= d , there always exists a contact force f ( t ) that was described by: f ( t )
 k
4
( y
 d )
 c
4
, y
 d
(1a) f ( t )

0 , y
 d (1b)
So the equations of the system can be expressed by:
1) In a noncontact state m
1
 x 
1
 e ( x
2
1 m
2
 x 
2
 c
2 x 
2
 
)
 k
2

1
 k
1 x
2
 x
1

0
F
2 sin (
 t )
(1c)
(1d)
2) In a contact state m
1
 x 
1
 e ( x
2
1
 
) x 
1
 k
1 x
1
 f ( t )

0
(1e) m
2
 x 
2
 c
2 x 
2
 k
2 x
2
 f ( t )

F
2 sin (
 t )
(1f)
Here the dot indicates differentiation with respect to time t .
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ICSV15 • 6-10 July 2008 • Daejeon • Korea
3.
The non-linear equations of n-DOF autonomous system have the following form:

2
M q   
C q  
K q
 f ( q , q  ,

,

)

0 (2)
Here the dots denote derivatives with respect to the dimensionless time
   t and ω is the oscillation frequency. The parameter λ is added to the equations for the aim of parametric studies. q

[ q
1
, q
2
,...., q n
]
T
is the unknown vector of the system, and f is a non-linear function of

,

, q , q . M , C
and matrices, respectively.
K are mass, linear viscous damping and linear stiffness
For obtaining periodic solution by IHB method two steps exist, first is a Netwon–Raphson procedure. Assuming q
0
,

0
,

0
express a state of vibration of the system, by adding the corresponding increments to them, neighboring state can be found: q
 q
0
  q ,
  
0
  
,
  
0
  
Where
 q

[
 q
1
, q
0
 q
2
,....,
 q n
]
T

[ q
10
, q
20
,...., q n 0
]
T
(3)
(4)
Substituting expressions (3) into Eq. (2) and neglecting small terms of higher order, yields the following linearized incremental relation:

0
2
M
 q  
(

0
C

C
N

R

( 2

0
M
)
 
( K q 
0

C 
0

Q )
 

K
N

P
)
 q
 
(5)
The elements of the matrices K
N
, C
N
, Q , P and R are given by:
K
N

(
 f /
 q )
0
, C
N

(
 f /
 q  )
0
, Q

(
 f /
 
)
0
, P

(
 f /
 
)
0
(6)
R
 
(

0
2
M  q 
0
 
0
C 
0

K q
0
 f
0
) (7)
If the system is odd, the steady state response can be expanded into the sum of N odd harmonic terms as follows: q j 0

K
N 

1
[ a jK co s ( 2 K

1 )
  b jK sin ( 2 K

1 )

]

C
S
A j
(8)
 q j 0


[
 a jK cos ( 2 K

1 )
   b jK sin ( 2 K

1 )

]

C
S

A j
(9)
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ICSV15 • 6-10 July 2008 • Daejeon • Korea
Where
C
S

{ cos

, cos
A j

{ a j 1
, a j 2
, .
.
.
, a jN
, b j 1
, b j 2
, .
.
,.
b jN
}
T
3

.
.
.
, cos ( 2 N

1 )

, sin

, sin 3

., .
,.
sin ( 2 N

1 )

}
(10)
(11)
The unknowns’ vectors and their increments can be found as follows: q
0

SA ,
 q
0

S

A (12)
A

{ A
1
, A
2
,...
A n
}
T
,

A

{

A
1
,

A
2
,...,

A n
}
T
, S








C
S
C
S
C
S
C
S







Substituting expression (12) into Eq. (5) and performing the Galerkin procedure yields:
2


0
2


0


(
 q )
T
[

0
2
M
 q  
(

0
(
 q )
T
[ R

( 2

0
M  q 
0
C

C
N

C 
0
)
 q  
( K

Q )
  

K
N
)
 q ] d

P
 
] d


Or simply
K mc

A

R

R mc
  
P
 
(13)
(14)
(15)
In which
K mc
 
0
2
M
 
0
C

C
N

K

K
N
R
R mc
M



(

0

2
( 2

0
M
M



0
C
C ) A


K
Q
) A

2
0


S
T
M S
 d

, C

2

0

S f
0
T
C
 d

K
K
N


2
0


S
T
2
0


S
T
K Sd

,
K
N
Sd

,
C
N f
0


2
0


S
T
C
N
2

0

S
T
Sd
 f
0 d

(16)
Q

2

0

S
T
Q d

, P

2
0


S
T
P d

To seek the solution for equation (15), one solves it iteratively such that the norm of the R is smaller than a certain convergence criterion, updating K mc
, R mc
, R and P at each step. It is the
Newton-Raphson procedure of seeking solutions, and therefore the increments are not necessarily infinitesimally small.
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ICSV15 • 6-10 July 2008 • Daejeon • Korea
4.1 steady state solutions
In this section we present periodic solution and frequency of a van der Pol oscillator (Eq. (1c)) using IHB method and steady state response for Eq. (1d).
The dimensionless time
τ
is expressed by
τ
=
ω t, in which t is the time variable and
ω
is the oscillation frequency, rewriting equation (1c) with respect to
τ
and using the parameters in
Table 1, yields:

2
 
1
 e ( X
2
1

1 )

1

X
1

0 (17)
Where x
0
 
, X
1
 x
1
/ x
0
Clearly enough equation (17) is a special case of equation (2) with n =1. As the oscillator is symmetrical, one can consider a solution with m odd harmonic terms up to cos(2m-1)τ and sin(2m-1)τ , as follows:
X
1
(

)
 n

1 ,

1 2 m 
[
3 ,...
a n cos n
  b n sin n

] (18) with reference to equation (15) , K mc
will be a 2 m *2 m coefficient matrix which corresponds to the Fourier coefficient vector: A=[a
1
, a
3
,. . . , a
2m-1
, b
1
, b
3
,. . . , b
2m-1
]
T
.
The vector R mc
, R and P can be calculated using equation (16). It should be noted that in Eq.
(15) the number of incremental unknowns is greater than the number of equations available, because of variables
Δω
and
Δλ
. In this investigation,
λ
is constant (
λ
=1) so it taken to be the control parameter (i.e.
Δλ
=0) while
Δω
should be regarded as an unknown. Therefore, one of the Fourier coefficients has to be fixed (e.g., a i
=0 or b j
=0), to have the number of equations equal to that of the unknowns. This will not cause any effects for autonomous system but it only represents a shift of one of dependent variable on the time axis.
With respect to equation (18), e=0.25 is assigned and 5 odd harmonic terms are employed in the computation; the highest harmonic terms are cos9τ and sin9τ .
The IHB periodic solution is shown in figure 2 and is expressed as follows: x
1

( t )

2
0 .
0620
.
0010 sin ( cos(
3 * 0 .
0 .
9961
9961 * t
*
) t )


0 .
0058
0 .
0006 sin cos(
( 5
3 * 0 .
9961 * t
* 0 .
9961 * t )

)

0 .
0032
0 .
00025 cos( sin
5 * 0 .
9961 * t )
( 7 * 0 .
9961 * t )
(19)
For comparison the results computed by the Numerical Integration method (NI method) are also presented in the figure 2, the IHB method are exactly the same as the NI results.
Also the steady state response for Eq. (1d) using the parameters in Table 1 is found to be: x
2
( t )

( 90 sin t

100 cos t ) / 181 (20)
4.2 Numerical simulation for contact state
By substituting Eqs. (19) and (20) into the following criterion the first contact time t
1 is determined:
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ICSV15 • 6-10 July 2008 • Daejeon • Korea x
1
 x
2
 d , d
0
 d / x
0
Now a computer code is written to analyse the equations (1e) and (1f) using initial conditions x
1
( t
1
), x
2
( t
1
), x’
1
( t
1
) and x’
2
( t
1
) that are obtained from expressions (19) and (20).
In this section the role of impact due to limiting stop, for reducing the limit cycle amplitude of a van der Pol oscillator is reported. Here the efforts are devoted to reduce the amplitude by properly choosing the parameter h=c
4
/k
4
. Two different cases are studied; system with h=0.0001 (c
4
=0.01 N.S/M, k
4
=100 N/M ) and h=0.01 (c
4
=0. 1 N.S/M, k
4
=10 N/M). Figures 3, 4 show the amplitude of limit cycle for cases 1 and 2, respectively.
Figures 3, 4 indicate that, within an equal value of d
0
=1 for two situations, the limit cycle amplitude will be reduced by increasing the value of h .
The performance of an elastic limiting stop as an impact damper, attached to a van der Pol oscillator has been presented. The incremental harmonic balance as an analytical method has been used to determine the steady state solution for nonlinear self-excited van der Pol oscillator and the numerical integration has been applied to simulate the contact situation. Because of unique advantages of IHB method for strongly nonlinear system, the procedure that has been approached in this paper can be used for greater value of e .
For efficient operation the value of h should choose to be higher.
The next stage of research should concentrate upon study contact and non contact state together and also forced excitations.
Table 1. System parameters
Case1:
Parameter
Main mass, m
1
(kg)
Auxiliary mass, m
2
(kg)
Main stiffness, k
1
(N/M) e
K
4
(N/M)
C
4
(N.S/M) value
1
0.1
1
0.25
100
0.01
Case 2:
C
K
4
4
(N/M)
(N.S/M)
Auxiliary stiffness, k
2
(N/M) and damping, c
2
(N.S/M)
Amplitude & frequency of auxiliary mass's excitation force, F
2
(m) and
Ω
(rad/sec)
10
0.1
1
1
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ICSV15 • 6-10 July 2008 • Daejeon • Korea x
1
(t) k
1 s k
4 x
2
(t) k
2 e x
2
(
1
 
) m
1 m
2 d c
4 x
3
(t)
F
2
(t) c
2
Figure 1. Model of a vibration absorber with an elastic stop
Limit cycle
Amplitude=2.0019
t (sec)
Figure 2. Periodic solution of van der Pol oscillator before impact for e=0.25
;
The IHB method ; o the NI method
Limit cycle
Amplitude=1.0185
t (sec)
Figure 3. Time response of van der Pol oscillator after impact, Case 1: h=0.0001
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ICSV15 • 6-10 July 2008 • Daejeon • Korea
Limit cycle
Amplitude=0.942
t (sec)
Figure 4. Time response of van der Pol oscillator after impact, Case 2: h=0.01
[1] A. L. PAGET, “Vibration in steam turbine buckets and damping by impacts”, Engineering 143 , 305-307
(1937).
[2] H. H. REED, “Hanging chain impact dampers-a simple method of damping tall flexible structures”, Wind
Effects on Buildings and Structures, Proceedings of Internal Research Seminar, Ottawa.
1967.
[3] M. M. SADEK, “Impact dampers for controlling vibration in machine tools”, Machinery 120 , 152-161
(1972).
[4] S. F. MASRI, “General motion of impact dampers”, Journal of the Acoustical Society of America 47 ,
229-237 (1970).
[5] C. N. BAPAT, S. SANKAR and N.POPPLEWELL, “Experimental investigation of controlling vibrations using multi unit impact dampers’’, 53 rd Shock and Vibration Bulletin 4, 1-12 (1983).
[6] N. POPPLEWELL and S. E. SEMERCIGIL “Performance of the bean bag impact damper for a sinusoidal external force”, Journal of Sound and Vibration 133 , 193-223 (1989).
[7] S. Ema, E. Marui, “Damping characteristics of an impact damper and its application”, International Journal of Machine Tools and Manufacturers 36(3) , 293–306 (1997).
[8] S. CHATTERJEE, A. K. MALLIK and A. GHOSH “On impact dampers for nonlinear vibrating systems”,
Journal of Sound and Vibration 187(3) , 403-420 (1995).
[9] S. CHATTERJEE, A. K. MALLIK and A. GHOSH, “Impact dampers for controlling self-excited oscillation”, Journal of Sound and Vibration 193(5) , 1003-1014 (1996).
[10] H. Luo and S. Hanagud, “On the Dynamic of Vibration Absorbers with Motion-Limiting Stops”,
Journal of Applied Mechanics 65 , 223-233 (1998).
[11] S.L. Lau, Y.K. Cheung, “Amplitude incremental variational principle for nonlinear vibration of elastic systems”, American Society of Mechanical Engineers Journal of Applied Mechanics 48 , 959–964 (1981).
[12] L. E. Gelhoud, S. F. Masri and J. C. Auderson, “Transfer Function of a Class of Nonlinear Multidegree of Freedom Oscillators”, Journal of Applied Mechanics 54 , 215-225 (1987).
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