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Numerical optimization of tuned mass absorbers attached to strongly
nonlinear Du�ng oscillator
P. Brzeskia, P. Perlikowskia,∗, T. Kapitaniaka
aDivision of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
Abstract
We investigate the dynamics of the vertically forced Du�ng oscillator with suspended tuned mass absorber.Three di�erent types of tuned mass absorbers are taken into consideration, i.e., classical single pendulum,dual pendulum and pendulum-spring. We numerically adjust parameters of absorbers to obtain the bestdamping properties with the lowest mass of attached system. The modi�cation of classical case (singlependulum) gives the decrease of Du�ng system amplitude. We present strategy of parameters tuning whichcan be easily applied in a large class of systems.
Keywords: Du�ng oscillator, tuned mass absorber, energy absorption, optimization.
1. Introduction
A tuned mass damper (TMD) was patented by Frahm in 1909 [1]. His device was a linear oscillatorwhich consists of mass and linear spring with the same natural frequency as a damped system. Under thiscondition one can avoid resonance of main mass, but the decrease of amplitude is observed only aroundresonant frequency. The next important modi�cation of the TMD was the addition of a dash-pot [2] whichimplies the increase of the range of frequencies for which e�ective suppressing of oscillations is observed.There are a lot of modi�cations of the classical passive TMD, most of them have important practicalapplications, i.e., to prevent damage of buildings due to seismic excitation [3, 4], to suppress vibration oftall buildings subjected to wind [5, 6], to achieve the best properties of cutting processes [7, 8], to mitigatevibration of �oors or balconies [9, 10], to reach stable rotations of rotors [11, 12, 13], or to stabilize drillstrings [14] and many more. Despite the fact that scientists and engineers are working on designing thebest passive device, there are also lots of e�orts to improve properties of TMD by adding control (hybrid,semi-active and active systems) [15, 16, 17, 18, 19].
The linear TMD decreases oscillations of the main system only around its resonant frequency (also naturalfrequency of the TMD), but outside this range one can observe an increase of amplitude. The solution ofthis problem was proposed by Roberstson [20] and Arnold [21]. They replaced linear spring of the TMDby the nonlinear one (with linear and nonlinear parts of sti�ness). This resulted in the improvement ofdamping properties when compared with the classic design. In recent years much more attention was paidto the possibility of purely nonlinear spring implementation [22, 23, 24]. The authors show that with such aspring there is no prominent damped frequency and the TMD works in wide range of excitation frequencies.
Simultaneously with improving the TMD Hatwal et al. [25, 26, 27] proposed the device called a tunedmass absorber (TMA) where the linear (nonlinear) oscillator is replaced by a pendulum. As the naturalfrequency of a pendulum depends only on its length, it is much easier to tune it in practical applications. Thependulum is used as the TMA independently on the excitation direction, in horizontal case the pendulum
∗Corresponding authorEmail address: [email protected] (P. Perlikowski)
Preprint submitted to Elsevier June 3, 2013
is oscillating for any frequency while for vertical direction only in its parametric resonances. The dynamicsof the TMA with vertical forcing of the base mass was considered in a few papers [26, 27, 28, 29, 30, 31,32, 33, 34, 35, 36, 37]. Presented analysis allows to understand the dynamics and the response of the mainmass around primary and secondary resonances of the pendulum. In our previous paper [38] we presentedcomplete bifurcation analysis of the TMA applied to forced Du�ng oscillator in two parameters space (theamplitude and the frequency of excitation). We showed oscillatory and rotational periodic solutions (internalresonances) and their coexistence. The same phenomena were also observed for systems where main massis oscillating horizontally [39, 40, 41, 42] and for combined veritcal and horizontal excitation [43].
The recent important studies on the TMD and the TMA take into account devices that consist of manysingle systems or with more than one degree of freedom. In [42, 44, 45, 46] one can �nd an application ofmultiple TMD with natural frequencies distributed over a de�ned range of frequencies. Such a constructiondamps the motion of the primary system more e�ectively than single TMD. Another advantage of multipleTMD is the reduction of the mass of individual TMD. Alternative construction of multiple TMD is connectingthem in series: linear oscillators [47], linear and nonlinear systems [15] and purely nonlinear devices [48]. Allthree approaches give better damping properties than single TMD. There are also a lot of publications onthe multiple TMA. Starting from works of Vyas and Bajaj [49, 50], where authors increase e�ciency of TMAby di�erentiation pendulums lengths. Signi�cant advantages of this set up was con�rmed experimentallyby Ikeda [35]. As in the case of TMD one can �nd many di�erent construction of multidegree TMA, i.e.,rotational pendulums TMA [51, 52] or the TMA with rotational and translational movements [53]. Theconnection of pendulums in series (double pendulum) is e�cient [54] but causes a lot of practical problems- its dynamics is very complex and one can not be sure that desired attractor will be achieved [55, 56].
In this paper we consider three di�erent types of TMA suspended on the forced Du�ng oscillator. Thepurpose of our analysis is to study and compare energy absorption properties of each system. We show thatby careful choice of parameters one can achieve large decrease of Du�ng system amplitude.
In Section 2 we show models of systems under consideration. Section 3 is devoted to optimization ofsingle TMA parameters. In Section 4 and 5 we show how modi�cations of classical TMA in�uence dampinge�ciency. Finally, in Section 6 we conclude on our investigations.
2. Model of the system
The horizontally forced single-well Du�ng oscillator, which we consider here as a base system, is shownin Fig. 1(a). In Fig. 1(b) one can see a classical TMA with single pendulum mounted on the Du�ng system.The �rst modi�cation assumes the addition of the second pendulum, which length and mass is di�erent fromthe �rst one (Fig. 1(c)). The second change of classical case is shown in Fig. 1(d) where the mass, whichcan move along the pendulum, is mounted on the linear spring. The motion of each subsystem is describedby following generalized coordinates: the vertical position of Du�ng oscillator is described by coordinate x,the angular displacements of the �rst or the second pendulum are given by angles φ or φi (i = 1, 2) andthe position of mass µ by coordinate y.
The equations of motion are derived separately for four cases (see Fig. 1) using Lagrange equations ofthe second type. The notation of the systems parameters is the same for all of them: M is mass of Du�ngoscillator, m and l correspond to mass and length of pendulum's rod (for dual pendulum case mi and licorrespond to mass and length of i-th pendulum's rod), k1 and k2 are linear and nonlinear parts of springsti�ness. The viscous damping coe�cient of Du�ng oscillators is given by c and of pendulum by cP (cPi fordual pendulum case). System presented in Fig.1(d) requires two more parameters: µ is mass suspended onthe spring, k3 is coe�cient of linear spring sti�ness and cS is damping coe�cient of linear oscillator (damperis not shown in Fig. 1 (d)). To control damping coe�cient it is possible to attach an oil viscous damper [57]or magnetorheological damper to TMA [58, 59, 60]. Energies, dimensional equations and transformationto dimensionless equations are shown in Appendix A. The transformation to dimensionless parameters(depicted by letter D) has been done in a way that allows to hold accessibility to physical parameters. Wetake initial parameters from paper of Ikeda [35] (only the construction of damper is di�erent). Here wepresent the equations:
2
a)
Mc
F ( t)cos ω
k +k x1 2
2x
M
( )φT
c
2 2
φ1l1
m1
F ( t)cos ω
k +k x1 2
2x
( )φT1 1
φ2
l2
m2
b)
c) d)
Mc
1
lm
F ( t)cos ω
k +k x1 2
2x
Mc
lm,
F ( t)cos ω
k +k x1 2
2x
1
lS
yμ
k3
( )φT
φ
φ
( )φT
Figure 1: Model of Du�ng oscillator (a); classical TMA (b); dual-pendulum TMA (c); pendulum-spring TMA (d).
Du�ng oscillator - Fig. 1(a)
Dimensionless equation:
x+ x+ k2Dx3 + cDx = FD cos (ωτ) , (1)
The dimensionless parameters have the following values: k2D = 30.303, cD = 4.90 · 10−3, F = 0.0002.
Single pendulum TMA - Fig. 1(b)
Dimensionless equations:
(1 +mD) x+1
2mDlD
[φ cos (φ)− φ2 sin (φ)
]+ x+ k2Dx3 + cDx = FD cos (ωτ) (2)
1
2mDlDx cos (φ) +
1
3mDl2Dφ+
1
2mDlDgD sin (φ) + cPDφ = 0 (3)
The dimensionless parameters have the following values: k2D = 30.3, cD = 4.90 ·10−3, cPD = 7.38 ·10−5,lD = 0.0779, mD = 0.1, gD = 0.0540, FD = 0.0002.
Dual pendulum TMA - Fig. 1(c)
Dimensionless equations:
(1 +m1D +m2D) x+2∑
i=1
{1
2miDliD
[φi cos (φi)− φ2
i sin (φi)]}
+ x+ k2Dx3 + cDx = FD cos (ωτ) (4)
3
1
2m1Dl1Dx cos (φ1) +
1
3m1Dl21Dφ1 +
1
2m1Dl1DgD sin (φ1) + cP1Dφ1 = 0 (5)
1
2m2Dl2Dx cos (φ2) +
1
3m2Dl22Dφ2 +
1
2m2Dl2DgD sin (φ2) + cP2Dφ2 = 0 (6)
The dimensionless parameters have the following values: k2D = 30.3, cD = 4.90·10−3, cP1D = 1.845·10−5,cP2D = 1.845 · 10−5, l1D = 0.0779, m1D = 0.025, l2D = 0.0779, m2D = 0.025, gD = 0.054, FD = 0.0002.
Pendulum-spring TMA - Fig. 1(d)
Dimensionless equations:
(1 +mD + µD) x+[12mDlD + µD (lSD + y)
] [φ cos (φ)− φ2 sin (φ)
]+
+µDy sin (φ) + 2µDφy cos (φ) + x+ k2Dx3 + cDx = FD cos (ωτ)(7)
[12mDlD + µD (lSD + y)
]x cos (φ) +
[13mDl2D + µD (lSD + y)
2]φ+
+2µD (lSD + y) φy +[12mDlD + µD (lSD + y)
]gD sin (φ) + cPDφ = 0
(8)
µDx sin (φ) + µDy − µD (l2D + y) φ2 + k3Dy − µDgD cos (φ) + cSDy = 0 (9)
The dimensionless parameters have the following values: k2D = 30.303, k3D = 0.455, cD = 4.90 · 10−3,cPD = 3.69 · 10−5, cSD = 4.90 · 10−3, lD = 0.0779, mD = 0.025, lSD = 0.072, µD = 0.025, gD = 0.0540,FD = 0.0002.
3. Single pendulum TMA
A single pendulum system is the classical TMA. To achieve the highest decrease of Du�ng oscillatoramplitude in a wide range of excitation frequencies, pendulum parameters have to be carefully chosen.This problem focused attention of many researchers and many successful applications both in mechanicalengineering and in constructions (high buildings, long bridges etc.) have appeared. Nevertheless, theoptimization and appropriate selection of absorber parameters is still a challenging task. In this section weshow parameter optimization for the system presented in Fig.1 (b). We also describe undesirable behaviourthat can occur if the parameters are wrongly taken. In our calculations we assume that pendulum is a rodwith constant cross-section and density. It is characterized by three dimensionless parameters: length (lD),mass (mD) and damping coe�cient (cPD).
The length of the pendulum rod is chosen to ensure that the natural frequencies of the absorber andthe Du�ng system are equal (in linear approximation). In the next subsection we present optimization ofdamping coe�cient of TMA. We show the in�uence of this parameter on the absorption e�ciency and theresponse of Du�ng system. In our work we assume that damping properties should be preserved for a widerange of excitation frequencies and because of that we use the frequency response curves (FRC) to evaluatethe e�ectiveness of the absorber. For selected range of cPD we calculate FRCs for ω ∈ (0.65, 1.35). Then,basing on the FRCs we calculate the L2-norm taking into account displacement of Du�ng oscillator:
L2 =
(∫ 1.35
0.65
[max (x)]2dω
)1/2
,
this value corresponds to the square root of the squared area under the FRC. Such indicator let us highlightthe importance of amplitude minimization along the whole FRC. The second possible indicator could be
4
just a maximum amplitude of FRC but then optimization process is focused only on minimization of onevalue - not the whole range of FRC.
The optimization has been done as follows: �rst we assume constant value of dimensionless mass mD =0.1 (which means that mass of the absorber is ten times smaller than the mass of the Du�ng oscillator)and calculate the FRCs for di�erent values of cPD. Basing on this criteria we determine the optimal valueof cPD and then we calculate the optimum ratio rOPT between optimal cPD and mD. In many practicalapplications minimization of the absorber mass is a priority, therefore basing on ratio rOPT we investigatethe in�uence of absorber massmD (for each mass we recalculate cPD according to rOPT ). This let us presenthow the mass of the absorber a�ects the overall dynamics of considered system. All calculations are madeusing path-following toolbox Auto-07p [61].
3.1. Optimization of damping coe�cient
When damping coe�cient cPD is too small one can observe the decrease of Du�ng system amplitudeonly around principal resonance and when it is too big, pendulum oscillates with small amplitude aroundsteady state or, in case of over critical damping, stays in hanging down position. Fig. 2 shows the conse-quences of wrongly chosen damping coe�cient. The black and gray lines correspond to stable and unstableperiodic solutions respectively. To provide comparison we plot amplitude of Du�ng systems without TMA(continuous line). The FRC for cPD = 8.08 · 10−7 (dashed-dotted line) explicates some phenomena that canoccur when damping in a pivot point is too small (0.2% of critical damping). One can observe that absorberoperates in desirable manner (decreases amplitude of structure oscillations) only for ω ∈ (0.91, 1.11) and inboth resonance peaks we observe a destabilization of periodic solutions by saddle-node and Neimark-Sakerbifurcations leading to dangerous jumps. Moreover, in most practical applications we can not accept maxi-mum angular displacement of pendulum greater than 0.2 [rad] [57, 62]. Therefore, despite the fact that theabsorber is extremely e�ective in principal resonance of the structure (ω = 1.0), disadvantages mentionedabove disqualify practical usage of the system with too low damping coe�cient. This is the reason whywe claim that additional damping added to pendulum can increase the applicability of TMA. AnalyzingFRC for cPD = 4.0 · 10−4 (dashed line) one can say that if the damping coe�cient is overestimated (in thiscase 99% of critical damping) the oscillations of the pendulum are very small, hence the amplitude of basestructure is not satisfactory damped out. The second visible e�ect is shift of the resonance peak (it occursfor lower value of excitation frequency ω). The interesting phenomena that can be observed is the transitionfrom two resonances for low values of damping to one resonance for overestimated damping. When dampingcoe�cient increases the second resonance peak vanishes and only the �rst one is preserved (see Figure 3(a)).
As we mention before, we �x mass of the pendulum rod (mD = 0.1) and check the in�uence of pendulumdamping coe�cient on the amplitude of Du�ng oscillator. Our aim is to �nd an optimum value of dampingcoe�cient for which the decrease of base oscillations should be observed in a wide range of excitationfrequency - not only in principal resonance. The FRCs were calculated for 100 equally spaced values of cPD
in range (0.1 · 10−4, 4.0 · 10−4) for ω ∈ (0.7, 1.3), then we create the three dimensional surface shown in Fig.3(a). For small values of cPD one can observe a coexistence of periodic attractors, moreover the amplitudeof the base structure oscillations is relatively high. In the range cPD ∈
(0.5 · 10−4, 1.5 · 10−4
)the amplitude
of Du�ng system is close to zero, hence the energy extraction is e�ective. Further increasing of the valueof cPD leads to increase of Du�ng oscillator amplitude. Fig. 3(b) is a magni�cation of the area near theoptimum value of damping coe�cient.
Although three dimensional surface plots give a good overview of the dynamics of the system and showthe importance of this parameter, to determine the optimal value of cPD we use L2-norm of FRC shown inFig. 3(c) versus damping coe�cient cPD. The minimum occurs for the following value of cPD = 7.38 · 10−5
and we take it as the recommended value of cPD. The black line presented in Fig. 3(b) is the FRC calculatedfor cPD = 7.38 · 10−5. On the basis of the above calculations one can �nd an optimum ratio between cPD
and mD:
rOPT =cPD
mD=
7.38 · 10−5
0.1= 7.38 · 10−4 (10)
5
ω
1.00.8 1.20.9 1.1 1.30.7
ω
max
()
φ
1.00.8 1.20.9 1.1 1.30.7
NS
without TMA
c =8.08 10PD-7
c =4.0 10PD-4
NS
c =8.08 10PD-7
c =4.0 10PD-4
0.01
0.02
0.03
0.04
0
max
(x)
0.5
1.0
1.5
2.0
2.5
0.0
-0.5
(a) (b)
Figure 2: The in�uence of wrongly chosen damping coe�cient: maximum displacement of Du�ng oscillator (a), maximumangular displacement of pendulum (b). Continuous line shows the response of Du�ng oscillator without TMA, dashed-dottedline with too small and dashed line with too large damping coe�cient cPD. The black and gray lines correspond to stable andunstable periodic solutions, the NS stands for Neimark-Saker bifurcation and other changes of stability of periodic solutionsalong the FRC occur through saddle-node bifurcation.
3.2. In�uence of the pendulum mass with mass dependent damping
In previous subsection we calculate optimum ratio rOPT , now we use it to investigate the in�uenceof the absorber mass on the response of Du�ng oscillator. It is important to show how decrese of massa�ects e�ciency of the absorber because the mass is often strictly limited due to constriction reasons andits minimization is often a priority. Our calculations are performed for mD ∈ (0.005, 0.4) and presented inFig. 4. On horizontal axis we show only the value of mD but we hold the ratio rOPT between mass anddamping coe�cient of pendulum so cPD also changes .
In Fig. 4 we show three dimensional plot of FRC versus the mass of the absorber. We observe thatusing optimal ratio rOPT one can get satisfactory results for a wide range of mass mD (mitigation of basestructure oscillations). Analyzing 4 (a) one can see that with the increase of mD e�ciency of the damperimproves but as we mentioned before the priority is to decrease mD to minimum acceptable value. Blacklines in Fig. 4 (a) correspond to the FRCs for mD = 0.05, mD = 0.1 and mD = 0.2 respectively. One cansee that all three curves di�er slightly and the change in damping properties is small. To give an overviewof TMA e�ciency for di�erent mD indicator L2-norm of FRCs is present it Fig. 4(b).
Fig. 4 (b) shows L2-norm of the FRCs for di�erent values of mD. In L2-norm curve we do not observea minimum but for mD greater than 0.05 the rate of decrease of L2-Norm drops rapidly. Hence, in the nextsections we take mD = 0.05 as an optimal value of absrober mass and try to improve damping propertiesby changing the design of TMA.
In Fig. 5(a) we show FRC for system without absorber and for three di�erent values of TMA mass(damping coe�cient for each case is determined using rOPT ). One can see that although mD = 0.2 guaran-tees best damping e�ciency, reduction of mass of the absorber tomD = 0.05 does not in�uence the e�ciencysigni�cantly (compare to Du�ng oscillator without TMA). For mD = 0.05 maximum amplitude of structureoscillations represents 6% of Du�ng oscillator maximum amplitude. If mass of the TMA is increased tomD = 0.1 and mD = 0.2, maximum amplitudes constitute 4.5% and 4.28% of Du�ng system amplituderespectively.
4. Dual-pendulum TMA
Previous calculations let us �nd the optimum parameters for the single pendulum TMA and understandtheir in�uence on the response of Du�ng oscillator. In this section we investigate system with two pendulums
6
max
(x)
ωm
ax
(x)
ω
cPD
cPD
a)
b)
c)
L2 N
orm
(7.38 10 , 6.57 10 )-5
cPD0 1 10
-42 10
-43 10
-44 10
-46 10
-4
11 10-4
14 10-4
-4
Figure 3: The FRC versus damping coe�cient cPD (a) and its zoom (b) in narrow range of cPD, the L2-norm of the FRCs(c). The black line in (b) corresponds to FRC with the optimum values of parameters.
7
a) ω
mD
max
(x)
b)
mD
L2 N
orm
0 0.1 0.2 0.3 0.44 10
-4
8 10-4
12 10-4
16 10-4
Figure 4: The FRC versus mass mD with �xed ratio rOPT (a), the L2-norm of the FRCs (b). The black lines (a) correspondto the FRC calculated for mD = 0.05, mD = 0.1 and mD = 0.2.
0.01
0.02
0.03
0.04
0
max
(x)
1.00.8 1.20.9 1.1 1.30.7
(a)
1.00.8 1.20.9 1.1 1.30.7
0.0005
0
0.0010
0.0015
0.0020
0.0025
ω
max
(x)
without TMA
m =0.05D
m =0.1D
m =0.2D
m =0.05D
m =0.1D
m =0.2D
(b)
ω
Figure 5: The FRCs plots show the in�uence of di�erent TMA masses on response of Du�ng oscillator. In (a) by continuousline we plot the FRC of the undamped Du�ng system, dashed-dotted, dashed and dotted lines correspond to mD = 0.05,mD = 0.1 and mD = 0.2 respectively. In (b) we show only the FRC for Du�ng oscillator with TMA. The black and gray linescorrespond to stable and unstable periodic solutions. Changes of stability of periodic solutions along the FRC occur throughsaddle-node bifurcation.
8
instead of one (see Fig. 1(c)). Ikeda [35] examines the dynamics of system with two identical pendulumswhere their damping e�ciency is comparable to single pendulum TMA. We argue that di�erentiation of thelengths of pendulums can lead to improve TMA e�ciency. When one considers the single pendulum TMAor the TMA with two identical pendulums the length of the pendulums rods is chosen to ensure that thenatural frequencies of the absorbers and the Du�ng oscillator are equal (in linear approximation). Whenwe take into account nonlinear e�ects the resonance peak shifts and occurs for di�erent value of excitationfrequency. Therefore by changing the length of one of the pendulums we can damp out another frequency.
The single pendulum is characterized by three parameters (length lD, mass mD and damping coe�cientcPD). We replace it by two pendulums with smaller masses. The parameters of dual-pendulum TMA areindicated by following parameters: liD, miD and ciPD, where i = 1, 2. The TMA with dual-pendulumcan be also used to reduce mass of single pendulum. We assume that we want to minimize total mass ofthe TMA and use values considered as minimum in previous section (total mass of the absorber is equalmD = 0.05). Masses of pendulums are equal (m1D = m2D = 1
2mD = 0.025), the damping coe�cients arecalculated using optimum ratio rOPT (c1PD = c2PD = 1
2cPD = 1.845 · 10−5). The only di�erence betweenpendulums is their length. The length of the �rst pendulum is �xed and is the same as in the case of singlependulum TMA (l1D = l1 = 0.0779). We change the length of the second pendulum. As a bifurcationparameter we take the following ratio between the lengths of pendulums:
lf =l2Dl1D
. (11)
To present how lf in�uences the system dynamics we calculate the three dimensional FRCs surface forlf ∈ (0.5, 1.5) (Fig. 6(a)). One can see that the change of ratio lf gives the possibility to improve dampingproperties by increasing the length of the second pendulum. To determine the optimum value of lf wecalculated the L2-norm of FRCs shown in Fig. 6(b). Its minimum occurs for lf = 1.2 for which in Fig.6(a)the black line is drawn.
5. Pendulum-spring TMA
In Section 4 we showed that damping properties of TMA can be improved by splitting its mass into twoslightly di�erent pendulums. By changing length of one pendulum we introduce another frequency into thesystem. Similar e�ect can be achieved by modi�cation of system's design as presented in Fig. 1(d). Weassume that mass of the TMA (mD = 0.05) and damping coe�cient of the pendulum (cPD = 3.69 · 10−5)have the same values as in Section 3.2. The pendulum-spring TMA is characterized by four additionaldimensionless parameters: sti�ness of the spring k3D, free length of spring lSD, damping coe�cient cSD andmass at the end of spring µD. Ratio mf describes mass distribution between mD and µD and is given asfollows:
mD = mf (mD + µD) , (12)
µD = (1−mf ) (mD + µD) , (13)
We assume that ratio mf is a bifurcation parameter and varies in range (0.5, 1), the other parametershave the following values: k3D = 0.455, l2D = 0.072, cmD = 4.90 ·10−3. To obtain better damping propertiesthan for single pendulum TMA, we proceed optimization of absorber's mass distribution. First, we showhow the mass distribution mf in�uences the system's dynamics. In Fig. 7(a) we present three dimensionalsurface for mf ∈ (0.5, 1.0). For mf = 1.0 our system is identical to the system described in Section 3.2(µD = 0.0). Whenmf decreases one can observe that resonance peak which occurs for ω < 1 becomes smallerbut another resonance appears for ω > 1. Although we observe two resonances, the maximum amplitude ofDu�ng oscillator decreases. The L2-norm of FRC can be once again reduced by proper distribution of themass of the absrober. In practical applications it may be also important that we can shift maximum pickof FRC towards higher frequencies.
9
a)
b)
max
(x)
ω
lf
1.00.80.6 1.2 1.4
L2 N
orm
lf
(1.2 , 8.14 10 )-4
8 10-4
9 10-4
10 10-4
11 10-4
Figure 6: The FRC versus ratio between pendulums lengths lf (a), the L2-norm of the FRCs (b). The black line in (a)corresponds to FRC with the optimum lf value.
The damping e�ciency indicator L2-norm of FRC is plotted in Fig. 7 (b). Maximum decrease ofstructure oscillations can be achieved for mf = 0.719 (mass of the pendulum rod represents 71.9% of totalmass of the absorber) and one can say that this value guarantees optimum damping properties. Black solidline in Fig.7 (a) corresponds to FRC for mf = 0.719.
6. Conclusions
In this paper we present three di�erent types of TMA. We compare them and optimize numericallytheir parameters to obtain the best damping properties. We show that with careful tuning of dampingcoe�cient, masses and mass distribution between pendulums and pendulum/mass on the spring one canachieve large decrease of Du�ng amplitude. Now, we test our optimized parameters for systems with muchlarger excitation amplitude. We show the FRC plots for Du�ng system without TMA and with threeconsidered types of TMAs for FD = 0.0002 (Fig. 8(a,b)) and FD = 0.001 (Fig. 8(c,d)), all the otherparameters of the systems are the same in both cases. The increase of excitation makes strong nonlinearcharacter of Du�ng system much more noticeable - large curvature of FRC. The continuous line indicatesthe FRC of Du�ng oscillator without TMA, then dashed-dotted, dashed and dotted the FRCs of Du�ngoscillator with single pendulum, dual-pendulum and pendulum-spring TMA respectively. The black andgray lines correspond to stable and unstable periodic orbits respectively. In all cases the change of stabilityoccurs by saddle-node bifurcation.
Fig. 8(a) shows the summary of results obtained for FD = 0.0002 (considered in whole paper). ForDu�ng oscillator without TMA periodic solutions along the FRC become unstable in the narrow rangeof excitation frequency, but for Du�ng system with any type of TMA all periodic solutions along theFRC are stable. In classical case (the single pendulum TMA) for m1D = 0.05 the maximum amplitude ofstructure oscillations represents 6% of maximum amplitude of Du�ng system without TMA (for Du�ng
10
a)
b)
max
(x)
ω
mf
L2 N
orm
mf1.00.8 0.90.6 0.70.5
(0.719 , 7.68 10 )-4
7.5 10-4
8 10-4
9 10-4
8.5 10-4
Figure 7: The FRC versus mass distribution mf (a), the L2-norm of the FRCs (b). The black line in (a) corresponds to FRCwith the optimum mass distribution.
oscillator FRC plot see Fig. 5(a)). For dual-pendulum (with total dimensionless mass m = 0.05 ) thisvalue decreases to 4.5%. Finally, for pendulum-spring system this value is equal to 4.0%. In Fig. 8(c,d)the excitation amplitude is �ve times larger (FD = 0.001) and one can observe that Du�ng system withoutTMA becomes unstable and stabilizes after attaching TMA similarly to the case with smaller excitation.The single pendulum TMA reduces amplitude of structure oscillations to 7% of maximum amplitude ofDu�ng oscillator without TMA, then for dual-pendulum system and pendulum spring we observe the samepercentage reduction (6%). Pendulum-spring dynamic absorber gives the best damping properties but ithas more complicated design and can be hard to implement in practical applications. As mitigation e�ectis almost the same as for dual-pendulum we think that the best solution is multiple pendulums TMA.Moreover, such a design enables minimization of oscillating masses of pendulums' rods.
The optimization procedure has been performed for small amplitude of excitation, where the hardeningbehavior of Du�ng oscillator is slightly visible. Nevertheless, the optimal TMA has similar energy extractionproperties for much larger values of excitation amplitude. This allow us to claim that the optimizationprocedure is robust and the TMA tuned for low amplitude is also very e�ective for high excitations, wherethe hardening e�ect is clearly visible.
Acknowledgment
This work has been supported by the Foundation for Polish Science, Team Programme under projectTEAM/2010/5/5. We also acknowledge valuable discussions with Ekaterina E. Pavlovskaia and SerhiyYanchuk.
References
[1] H. Frahm, Device for damping vibrations of bodies, 1909.
11
0.05
0.1
0.15
0
max
(x)
1.00.8 1.20.9 1.1 1.30.7
ω
0.01
0.02
0.03
0.04
0
max
(x)
1.00.8 1.20.9 1.1 1.30.7
without TMA
dual-pendulum
pendulum-spring
single pendulum
(c)
(a)
without TMA
dual-pendulum
pendulum-spring
single pendulum
ω
0.004
0.008
0.012
max
(x)
1.00.8 1.20.9 1.1 1.30.7
0.0005
0
0.0010
0.0015
0.0020
0.0025
max
(x)
(d)
(b)
ω
1.00.8 1.20.9 1.1 1.30.70
ω
Figure 8: FRCs for Du�ng system without and with three types of considered TMA for FD = 0.0002 (a,b) and FD = 0.001(c,d). For FD = 0.0002 maximum amplitude of Du�ng oscillator equals: 0.0024 for simple one-pendulum TMA, 0.0018 fordual-pendulum TMA and 0.0016 for pendulum-spring TMA. For FD = 0.001 maximum amplitude of Du�ng oscillator equals:0.012 for simple one-pendulum TMA, 0.001 for dual-pendulum TMA and 0.0099 for pendulum-spring TMA. The black andgray lines correspond to stable and unstable periodic solutions. Changes of stability of periodic solutions along the FRC occurthrough saddle-node bifurcation.
12
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Appendix A
We show formulas for the kinetic energy T , potential energy V , Rayleigh dissipation D, general forcesQ and �nal form of dimensionless equations. All of the primes used in de�nitions of dimensionless variables
14
will be omitted hereafter in the analysis.
Base structure - Du�ng oscillator (a)
Energy of the system:T = 1
2Mx2, V = 12k1x
2 + 14k2x
4, D = 12cx
2.Equation of motion:
Mx+ k1x+ k2x3 + cx = F cos (ωt) ,
then by introducing dimensionless time τ = tω1, where ω1 =√
k1
M is the natural frequency of Du�ng
oscillator, we reach dimensionless equation:
x′ + x′ + k2Dx′3 + cDx′ = FD cos (ω′τ) ,
where: x′ = xl0, x′ = x
l0ω1, x′ = x
l0ω21, k2D =
k2l20
Mω21, cD = c
Mω1, FD = F
Ml0ω21, ω′ = ω
ω1. The dimension
parameter are equal to: l0 = 1.0[m], M = 3.63[kg], k1 = 660[Nm ], k2 = 20000[ Nm3 ], c = 0.240[Ns
m ], F =0.132[N ].
Single pendulum (b)
Energy and generalized forces of the system: T = 12 (M +m) x2+ 1
6ml2φ2+ 12mlxφ cos (φ), V = 1
2k1x2+
14k2x
4 − 12mlg cos (φ), D = 1
2cx2, Q = T (φ) ∂φ
∂φ , where T (φ) = cP φ is a damping momentum of thependulum.
Equations of motion:
(M +m) x+1
2ml
[φ cos (φ)− φ2 sin (φ)
]+ k1x+ k2x
3 + cx = F cos (ωt)
1
2mlx cos (φ) +
1
3ml2φ+
1
2mlg sin (φ) + cP φ = 0
Dimensionless equations of motion:
Introducing dimensionless time τ = tω1, where ω1 =√
k1
M is the natural frequency of Du�ng oscillator
we reach dimensionless equations:
(1 +mD) x′ +1
2mDlD
[φ′ cos (φ′)− φ′2 sin (φ′)
]+ x′ + k2Dx′3 + cDx′ = FD cos (ω′τ)
1
2mDlDx′ cos (φ′) +
1
3mDl2Dφ′ +
1
2mDl1DgD sin (φ′) + cPDφ′ = 0
where: x′ = xl0, x′ = x
l0ω1, x′ = x
l0ω21, φ′
1 = φ1, φ′1 = φ1
ω1, φ′
1 = φ1
ω21, k2D =
k2l20
Mω21, cD = c
Mω1, cPD = cP
Ml20ω1,
lD = ll0, mD = m
M , gD = gl0ω2
1, ω′ = ω
ω1and parameter values: l0 = 1.0[m], M = 3.63[kg], k1 = 660[Nm ],
k2 = 20000[Nm ], c = 0.240[Nsm ], cP = 4.00 · 10−5[Nms], l = 0.0779[m], m = 0.363[kg], F = 0.132[N ].
Dual-pendulum (c)
Energy and generalized forces of the system: T = 12 (M +m1 +m2) x
2+∑2
i=1
[16mil
2i φ
2i +
12milixφi cos (φi)
],
V = 12k1x
2 + 14k2x
4 −∑2
i=1
[12milig cos (φi)
], D = 1
2cx2, Q =
∑2i=1 Ti (φi)
∂φi
∂φi, where Ti (φi) = ciφi is a
damping momentum of i-th pendulum.Equations of motion:
(M +m1 +m2) x+2∑
i=1
{1
2mili
[φi cos (φi)− φi
2 sin (φi)]}
+ k1x+ k2x3 + cx = F cos (ωt)
15
1
2m1l1x cos (φ1) +
1
3m1l
21φ1 +
1
2m1l1g sin (φ1) + cP1φ1 = 0
1
2m2l2x cos (φ2) +
1
3m2l
22φ2 +
1
2m2l2g sin (φ2) + cP2φ2 = 0
Dimensionless equations of motion:
Introducing dimensionless time τ = tω1, where ω1 =√
k1
M is the natural frequency of Du�ng oscillator
we reach dimensionless equations:
(1 +m1D +m2D) x′ +
2∑i=1
{1
2miDliD
[φ′i cos (φ
′i)− φ′2
i sin (φ′i)]}
+ x′ + k2Dx′3 + cDx′ = FD cos (ω′τ)
1
2m1Dl1Dx′ cos (φ′
1) +1
3m1Dl21Dφ′
1 +1
2m1Dl1DgD sin (φ′
1) + cP1Dφ′1 = 0
1
2m2Dl2Dx′ cos (φ′
2) +1
3m2Dl22Dφ′
2 +1
2m2Dl2DgD sin (φ′
2) + cP2Dφ′2 = 0
where: x′ = xl0, x′ = x
l0ω1, x′ = x
l0ω21, φ′
1 = φ1, φ′1 = φ1
ω1, φ′
1 = φ1
ω21, φ′
2 = φ2, φ′2 = φ2
ω1, φ′
2 = φ2
ω21,
k2D =k2l
20
Mω21, cD = c
Mω1, cP1D = c1
Ml20ω1, cP2D = c2
Ml20ω1, l1D = l1
l0, m1D = m1
M , l2D = l2l0, m2D = m2
M ,
gD = gl0ω2
1, ω′ = ω
ω1and parameter values: l0 = 1.0[m], M = 3.63[kg], k1 = 660[Nm ], k2 = 20000[Nm ], c =
0.240[Nsm ], cP1 = 2.00 ·10−5[Nms], cP2 = 2.00 ·10−5[Nms], l1 = 0.0779[m], m1 = 0.0449[kg], l2 = 0.0779[m],
m2 = 0.0446[kg], F = 0.132[N ].
Spring-pendulum (d)
Energy and generalized forces of the system: T = 12 (M +m+ µ) x2+ 1
2µy2+
[16ml2 + 1
2µ (lS + y)2]φ2+
+ 12mlxφ cos (φ)+µx [y sin (φ) + (lS + y) φ cos (φ)], V = 1
2k1x2+ 1
4k2x4−
[12ml + µ (lS + y)
]g cos (φ)+ 1
2k3y2,
D = 12cx
2 + 12cS y
2, Q = T (φ) ∂φ∂φ , where T (φ) = cP φ is a damping momentum of the pendulum.
Equations of motion:
(M +m+ µ) x+[12ml + µ (lS + y)
] [φ cos (φ)− φ2 sin (φ)
]+
+µy sin (φ) + 2µφy cos (φ) + k1x+ k2x3 + cx = F cos (ωt)
[12ml + µ (lS + y)
]x cos (φ) +
[13ml2 + µ (lS + y)
2]φ+
+2µ (lS + y) φy +[12ml + µ (lS + y)
]g sin (φ) + cP φ = 0
µx sin (φ) + µy − µ (lS + y) φ2 + k3y − µg cos (φ) + cS y = 0
Dimensionless equations of motion:
Introducing dimensionless time τ = tω1, where ω1 =√
k1
M is the natural frequency of Du�ng oscillator
we reach dimensionless equations:
(1 +mD + µD) x′ +[12mDlD + µD (lSD + y′)
] [φ′ cos (φ′)− φ′2 sin (φ′)
]+
+µDy′ sin (φ′) + 2µDφ′y′ cos (φ′) + x′ + k2Dx′3 + cDx′ = FD cos (ω′τ)
16
[12mDlD + µD (lSD + y′)
]x′ cos (φ′) +
[13mDl2D + µD (lSD + y′)
2]φ′+
+2µD (lSD + y′) φ′y′ +[12mDlD + µD (lSD + y′)
]gD sin (φ′) + cPDφ′ = 0
µDx′ sin (φ′) + µDy′ − µD (lSD + y′) φ′2 + k3Dy′ − µDgD cos (φ′) + cSDy′ = 0
where: x′ = xl0, x′ = x
l0ω1, x′ = x
l0ω21, φ′ = φ, φ′ = φ
ω1, φ′ = φ
ω21, y′ = y
l0, y′ = y
l0ω1, y′ = y
l0ω21, k2D =
k2l20
Mω21,
k3D = k3
Mω21, cD = c
Mω1, cPD = cP
Ml20ω1, cSD = cS
Mω1, lD = l1
l0, m1D = m1
M , l2D = l2l0, µD = µ
M , gD = gl0ω2
1,
ω′ = ωω1
and parameter values: l0 = 1.0[m], M = 3.63[kg], k1 = 660[Nm ], k2 = 20000[ Nm3 ], k3 = 300.3[Nm ],
c = 0.240[Nsm ], c1 = 4.00 · 10−5[Nms], cS = 0.240[Ns
m ], l1 = 0.0779[m], m1 = 0.0899[kg], l2 = 0.072[m],µ = 0.04[kg], F = 0.132[N ].
17
