Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 3, pp. 1081-1089, Warsaw 2017
DOI: 10.15632/jtam-pl.55.3.1081

ON THE OPTIMUM ABSORBER PARAMETERS: REVISING THE
CLASSICAL RESULTS

Volodymyr Puzyrov

Donetsk National University, Department of Higher Mathematics and Methodology of Teaching, Vinnytsya, Ukraine

e-mail: doe.seldon@gmail.com

Jan Awrejcewicz

Lodz University of Technology, Department of Automation, Biomechanics and Mechatronics, Łódź, Poland

e-mail: awrejcew@p.lodz.pl

The dynamic vibration absorber is a kind of mechanical device with inertia, stiffness, and
damping.Onceconnectedtoagivenstructureormachine, it is capableofabsorbingvibratory
energy. As a result, the primary system can be protected from excessively high vibration
levels. In this paper, we deal with classical DenHartog’smodel to clarify the known results
and improve the mathematical component of this approach.We suggest the optimal choice
of absorber parameters, which is slightly different and more general analytical approach.
The comparison of twomethods of optimization is carried out, and the corresponding error
of calculus is estimated.

Keywords: dynamic vibration absorber, Den Hartog’s model, frequency-amplitude curve,
optimization

1. Introduction

The problem of elimination or reduction of undesired vibration in various technical systems
has long history and great achievements. The concept of vibration control is widely accepted
nowadays and has been applied in many different areas, such as civil, mechanical, and aero-
nautical engineering. Passive vibration control is the most widespread, and effective methods
are available. The typical simplest and most reliable device is the dynamic vibration absorber
(DVA) or tuned-mass damper (TMD). A simpleDVA consists of amass and a spring.When the
primary system is excited by a harmonic force, its vibration can be suppressed by attaching a
DVA. Themain purpose of adding the secondary oscillator is tomove the resonant frequency of
themechanical system away from the operating frequency of the vibratory force. So, the system
becomes a 2-DOF (degree of freedom)mechanical system, but with coincidence of the exciting
frequency with one of the two natural frequencies it will be again at resonance. To eliminate
this effect, a damper is added to the DVA (Fig. 1).

The idea of vibration control was originally proposed by Frahm (1911), and many various
DVA configurationswere designed during the past century (Ormondroyd andDenHartog, 1928;
Brock, 1946; Mead, 2000; Hunt, 1979; Korenev andReznikov, 1993; Viet et al., 2011;Marano et
al., 2007).Good surveys on the subjectwere presented in (Johnson, 1995; Sun et al., 1995).With
reference to the DVA optimal design, the first criterion was offered by Ormondroyd and Den
Hartog (1928). This criterion concerned the minimization of the system response with respect
to the stationary harmonic excitation with themost “dangerous” frequency value which results
in the largest increase of the amplitude.



1082 V. Puzyrov, J. Awrejcewicz

Fig. 1. Damped DVA connected to a primary system

2. Formulation of the problem

The equations of motion of the mechanical system under consideration are

mpẍp+ ca(ẋp− ẋa)+kpxp+ka(xp−xa)= F0eiωt

maẍa+ ca(ẋa− ẋp)+ka(xa−xp)= 0
(2.1)

In this formulation,weuse the notions according to (Johnson, 1995), F(t)= F0e
iωt is a harmonic

excitation force acting on the primary system.

The amplitudes of steady state harmonic responses are

Xp = F0
ka−maω2+iωca

(kp−mpω2)(ka−maω2)−makaω2+iωca(kp−mpω2−maω2)

Xa =−F0
ka+iωca

(kp−mpω2)(ka−maω2)−makaω2+iωca(kp−mpω2−maω2)

(2.2)

In terms of dimensionless parameters, we can rewrite

|Xp|
(Xp)st

=

√

(2ζg)2+(g2−η2)2
(2ζg)2(g2−1+µg2)+ [µη2g2+(g2−1)(g2−η2)]

(2.3)

where µ = ma/mp is the mass ratio; ωa =
√

ka/ma – undamped natural frequency of the

DVA considered separately; ωp =
√

kp/mp – undamped natural frequency of the primary sys-

tem considered separately; η = ωa/ωp – tuning factor; g = ω/ωp – forcing frequency ratio;
ζ = ca/(2maωp) – damping ratio; (Xp)st = F0/kp – static displacement of the primarymass.

Many methods of optimization have been developed to opportunely design this vibration
control technique. In the classical textbook onmechanical vibrations,DenHartog (1940) pointed
out a remarkable feature: for any fixed values of η and µ, curves (2.3) intersect in two points P
and Q (named “invariant points“), as shown in Fig. 2, independently of the value of ζ. These
points are situated close enough to the peaks of the frequency-amplitude curve. Den Hartog
suggested to choose the parameter η to equalize ordinates of P and Q. Secondly, ζ was taken to
satisfy the condition of “almost horizontal” tangents in the invariant points. Thus the values

ηopt =
1

1+µ
ζopt =

√

3µ

8(1+µ)3
(2.4)

were obtained.



On the optimum absorber parameters: revising the classical results 1083

Fig. 2. Invariant points for η =0.9, µ =0.1. Solid line ζ =0.027, long dash line ζ =0.16, dot-dash line
ζ =0.5

3. Mathematical analysis of the problem

Theapproach ofDenHartog has undoubted advantages: simplicity, which is essential for applied
researchers, and rather high accuracy, as was shown in numerical simulations. At the same time,
it is possible to note two drawbacks of this approach. The first one is the absence of analytical
assessment of the difference between themaximum response value proposed and the true value.
The second is that the scheme of determining the optimal values is based on the existence of
invariant points, which is an exception rather than the rule. For example, this approach does not
fit in the case when themain body is damped itself (Warbrton andAyorinde, 1980), or for sky-
-hooked (Griffin et al., 2002; Liu and Liu, 2005) DVA. So, in such cases, one have to rely upon
numerical methods for optimization. It should be added that from the theoretical viewpoint,
the question on the existence of the exact solution of the problem in an algebraic form is not
closed.

Considering the above-mentioned, our aims are: 1) to provide amore general approachwhich
does not rely on exclusive properties of (2.3) (invariant points existence); 2) to evaluate the error
of formulas (2.4).

At the beginning, let us consider a function

f(µ,δ,h,γ) =
hγ +(γ − δ)2

hγ(γ +µγ −1)2+[µδγ − (γ −1)(γ − δ)]2
(3.1)

which we believe is more suitable for the analysis. Here

δ = η2 h =4ζ2 γ = g2 f =
( |Xp|
(Xp)st

)2
(3.2)

Obviously, the optimization of function (2.3) is the same as function (3.1). The optimal
values from (2.4) in the new notions are

δ =
1

(1+µ)2
h =

3µ

2(1+µ)3
(3.3)

The typical shapes of the surface f(γ,µ) with fixed values δ, h are presented in Fig. 3.



1084 V. Puzyrov, J. Awrejcewicz

Fig. 3. Typical character of surface (3.1): (a) δ =1−2.5µ, h =0.5, (b) according to Eqs. (3.3)

It is remarkable that for any given set of (δ,h,µ), zeros of the derivative df/dγ lead to the
following equation

γ5−γ4[4δ−2h+µ(δ−h)− 1
2
hµ2]+γ3[6δ2+4δ −4hδ +h2−2h+2µ(2δ2 −3hδ +h2)

−hµ2(2δ −h)]−γ2[4δ3−2hδ2 +6δ2−4hδ +h2+µ(5δ3−4hδ2+ δ2−3hδ+h2)]
+γδ2[δ2+4δ −2h+2µ(δ2+ δ−h)+µ2δ2]− δ4(1+µ)= 0

(3.4)

This equation of the 5-th order cannot be solved in an explicit form and, therefore, there is
noway tomake a conclusion on themaximumvalue of f. Because of this, we require an indirect
method to achieve our goal.1

Let f0 =1/κ be some fixed number. Then the equation f = f0 is equivalent to the following
polynomial equation

u(γ) = γ4+(h+hµ2+2hµ−2−2δ −2µδ)γ3 +(µ2δ2+4δ−2h+2µδ−2hµ+1
+ δ2−κ+2µδ2)γ2+(h−2δ2−2µδ2+2δκ−2δ−hκ)γ + δ2− δ2κ =0

(3.5)

In the case when curve (3.1) has two peaks, for some values of κ the line

f =
1

κ
(3.6)

intersects this curve four times, and equation (3.5) has four real positive roots. Otherwise, when
line (3.6) is above curve (3.1), equation (3.5) has no positive roots. On the assumption that
both peaks have the same height, we conclude that this is a borderline between two cases. In
other words, the discriminant of the polynomial u(γ) is equal to zero, and (3.5) has two pairs of
multiple roots. Thereby, there exist such expressions M(µ,δ,h), N(µ,δ,h) that

u(γ)= (γ2+Mγ +N)2 M2−4N > 0 (3.7)

Then, we conclude from (3.7) that

(−2δ +2µh+µ2h−2µδ −2M −2+h)γ3

+(1+2µδ2+2µδ−2h−κ+4δ + δ2−2δN −2µh+µ2δ2−M2)γ2

+(−2δ2−hκ+2δκ−2δMN −2δ +h−2µδ2)γ + δ2(1−κ)−N2 =0
1An approach based on investigation of equation (3.4) with Taylor expansions representation was

presented in (Pozdniakovich and Puzyrov, 2009). It allowed one to achieve some progress, comparatively
with (3.3), but again, no explicit form and error estimation were gained.



On the optimum absorber parameters: revising the classical results 1085

Nowwehavea systemof four equationswithfivevariablesκ, δ,h,M,N, andcan consistently
find

κ =1−
(N

δ

)2
h =
2[(1+µ)δ +M +1]

(1+µ)2

M =−δ
2(1+µ)3+µδN2(1+µ)−N2

N[δ(1+µ)2−N]

(3.8)

The last equation is

(δ −N)[δ9(1+µ)6+ δ8N(1+µ)6−4δ7N(1+µ)4−4δ6N2(1+µ)4

+2δ5N2(1+µ)2(3+µ)+2δ4N3(1+µ)2(3+µ)−4δ3N3(1+µ)
−2δ2N4(2+2µ+µ2)+ δN4+N5] = 0

(3.9)

Condition (3.9) gives an equation of the 9-th order on δ (and 5-th order on N), butwith the
substitution

δ =
δ1

(1+µ)2
N =

N1δ
2
1

(1+µ)2
(3.10)

it may be rewritten as

N21δ
2
1[N

4
1(1+µ)

2−2N31(2+2µ+µ2)+2N21(3+µ)−4N1+1]+2N41µ2δ1
− [N41(1+µ)2−4N31(1+µ)+2N21(3+µ)−4N1+1]=0

The last one being factorized as

(N1δ1−1)
{

δ1[N
4
1(1+µ)

2−2N31(2+2µ+µ2)+2N21(3+µ)−4N1+1]
+[N41(1+µ)

2−4N31(1+µ)+2N21(3+µ)−4N1+1
}

=0

finally leads to

(1−N1δ1)
{

−N1δ1[µ
√

N312(
√
2−
√

N1)− (1−N1)2][µ
√

N312(
√
2+
√

N1)+(1−N1)2]

+ [µN21 +(1−N1)2]2
}

=0

(3.11)

Evidently,

µ
√

N31(
√
2+
√

N1)+(1−N1)2 > 0 [µN21 +(1−N1)2]2 > 0

and, due to the condition κ > 0, the expression in the first square brackets must be positive to
fulfill (3.11). With this, we have the following restrictions on N1

N1δ1 < 1 µ
√

N31(
√
2−
√

N1) > (1−N1)2 (3.12)

So, we have

δ1(N1)=
2µ2N21

2µ2N31 − [µN21 +(1−N1)2]2
− 1
N1

(3.13)

and nowwe can find the maximum of κ, i.e. the minimum of ϕ(N1)= N/δ = N1δ1(N1)

dϕ

dN1
=2µ2N21

(1+µ)2N41 −2(3+µ)N21 +8N1−3
{2µ2N31 − [µN21 +(1−N1)2]2}2



1086 V. Puzyrov, J. Awrejcewicz

The numerator has two real and two complex roots

√
4+3µ−1
1+µ

−
√
4+3µ+1

1+µ

1+i
√
µ

1+µ

1− i√µ
1+µ

Only the first of them is positive and for any given value of µ provides the absolute minimum
of the function ϕ(N1).
Substituting N1 =(

√
4+3µ−1)/(1+µ) in (3.13), after simplifications we get

δ⋆1 =
8

3

16+23µ+9µ2+2(2+µ)
√
4+3µ

64+80µ+27µ2
(3.14)

The last expression is obviously positive, so the second inequality in (3.12) holds. Now we
verify the first one. It leads to

N =
64

3

64+112µ+61µ2+9µ3+(64+136µ+103µ2+27µ3)
√
4+3µ

(1+µ)2(64+80µ+27µ2)
< δ (3.15)

Eliminating the square root, one may see that

729µ6+5778µ5+19279µ4 +32758µ3+27809µ2 +9344µ > 0

So, for any µ > 0 we have N < δ, and (3.12) are fulfilled. Finally, we have

δ⋆ =
8

3

16+23µ+9µ2+2(2+µ)
√
4+3µ

(1+µ)2(64+80µ+27µ2)

h⋆ =
2

3

64+248µ+255µ2+81µ3−2(16+20µ+9µ2)
√
4+3µ

(64+80µ+27µ2)(1+µ)3

(3.16)

and respectively

η⋆ =2

√

2

3

16+23µ+9µ2+2(2+µ)
√
4+3µ

(1+µ)2(64+80µ+27µ2)

ζ⋆ =

√

64+248µ+255µ2+81µ3−2(16+20µ+9µ2)
√
4+3µ

6(64+80µ+27µ2)(1+µ)3

(3.17)

whichdetermine theoptimal values of stiffness anddamping forDVA.Forµ =0.1, the curvef(γ)
in the vicinity of peaks is presented in Figs. 4a and 4b for both cases – according to (3.3) and
(3.16).

Fig. 4. Frequency-amplitude curve in the neighbourhood of peaks. Long dash line – according to (3.3),
solid line – according to (3.16)



On the optimum absorber parameters: revising the classical results 1087

Also, substituting δ and N into (3.8)2, we get

M =−
414µ2+1104µ+768+(54µ2+168µ+192)

√
4+3µ

9(1+µ)(64+80µ+27µ2)
(3.18)

The rigorous mathematical proof of the fact that any pair of δ, h which differs from (3.16)
is worse (gives larger maximum of f) is too cumbersome to be given here. But, at least, this
verification versus pair (3.3) is rather simple. Indeed, substituting (3.3) and (3.16) one after the
other in (3.4), we determine the extremal values of γ for each case separately with the help of
Taylor expansions by

√
µ

γ
upon(3.3)
1 =1−

√
2

2

√
µ−
7

8
µ−
139
√
2

256
(
√
µ)3+

1797

2058
µ2+ . . .

γ
upon(3.3)
2 =1+

√
2

2

√
µ−
7

8
µ+
139
√
2

256
(
√
µ)3+

1797

2058
µ2+ . . .

γ
upon(3.16)
1 =1−

√
2

2

√
µ− 7
8
µ− 37

√
2

64
(
√
µ)3+

107

128
µ2+ . . .

γ
upon(3.16)
1 =1+

√
2

2

√
µ−
7

8
µ+
37
√
2

64
(
√
µ)3+

107

128
µ2+ . . .

And then, we may estimate the gain of using the values by (3.16)

maxfupon(3.3)−maxfupon(3.16) = f(γupon(3.3)2 )−f(γ
upon(3.16)
2 )

=
5
√
2

256

√
µ+
101

8192
µ+
9859
√
2

131072
µ2+ . . .

(3.19)

Numerically, such verification is even easier –wemay just combine (3.1) with condition (3.4)
for both cases separately and plot the curve

ψ(µ) =maxfupon(3.3)−maxfupon(3.16)

This curve is presented in Fig. 5, the dash line corresponds to γ
upon(3.3)
1 , so the distance between

the two branches is the height difference between the two peaks for case (3.3).

Fig. 5. Difference in f
max
values for two cases



1088 V. Puzyrov, J. Awrejcewicz

As the last step, let uswrite down the analytical expression for |Xp|/(Xp)st. Substituting δ⋆,
h⋆ according to (3.16) and γ2 =(−M +

√
M2−4N)/2 in f, we have

( |Xp|
(Xp)st

)2
= f⋆ =

A+BR

C +DR

where

A =2(A0+A1r) B =3(1+µ)σ(B0+B1r) C =2(C0+C1r)

D =3(1+µ)σ(D0+D1r) R =
√
M2−4N r =

√
4+3µ

σ =64+80µ+27µ2

Here, the polynomial coefficients are given by formulas

A0 =8192+57856µ+145536µ
2 +181000µ3 +122048µ4 +43764µ5+7083µ6+243µ7

A1 =−4096−8960µ−1152µ2+13252µ3+15443µ4 +7149µ5+1242µ6

B0 =64+376µ+564µ
2+331µ3+69µ4 B1 =−32+4µ+63µ2+46µ3+9µ4

C0 = µ
2(1437696+5792256µ+9830592µ2 +8976744µ3 +4645404µ4 +1291545µ5

+150903µ6)

C1 = µ
2(387072+1278720µ+1804896µ2 +1411020µ3 +652725µ4 +170586µ5 +19683µ6)

D0 = µ
2(155713536+906854400µ+2321104896µ2 +3410021376µ3 +3146726016µ4

+1871659224µ5 +705324672µ6 +157181148µ7 +17380089µ8 +531441µ9)

D1 = µ
2(35389440+177389568µ+397799424µ2 +529362432µ3 +463738176µ4

+275945292µ5 +108628209µ6 +25572591µ7 +2716254µ8)

Eliminating the root R from the denominator, we have

(AC −BDR2)+(BC −AD)R

in thenumerator, andmultiplier beforeRmustbezero, because of f(γ
upon(3.16)
2 )= f(γ

upon(3.16)
1 ),

i.e. f⋆ is an even function of R (direct calculation confirms this fact). To simplify the fraction
obtained, we also need to eliminate the “small‘” root r from the denominator. The final expres-
sion

|Xp|
(Xp)st

=

√

(8+9µ)2(16+9µ)−128
√

(4+3µ)3

27µ2(32+27µ)
(3.20)

is a quite remarkable recognition because reduction of the fraction is possible only at the final
stage, and the denominator before this reduction is the polynomial of the 48-th(!) order in µ.

4. Conclusion

We have discussed the problem of selection of optimal parameters of a DVA according to the
classicalDenHartogmodel (DenHartog, 1940).Wehave shownthat the solutionmaybegiven in
an accurate algebraic formwhich updates somewhat the known result. The analytical approach
presented here may be more applicable for solving problems connected with the use of passive
dampingdevices.We have also compared the results of two approaches anddetermined the error
estimation which has been illustrated by the corresponding frequency-amplitude curves.



On the optimum absorber parameters: revising the classical results 1089

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Manuscript received December 21, 2016; accepted for print April 18, 2017