The performance of the proposed optimal DVA is compared with that obtained by existing optimal solution in literature. It is shown that the proposed optimal parameters are possible to obtain superior vibration suppression compared to existing optimal formula. Extended simulations are carried out to examine the performance of the optimally designed DVA and the sensitivity of the optimum parameters. The simulation results show that the improvement of the vibration performance on damped rotary system can be as much as 90% by using DVA.
Vietnam Journal of Mechanics, Vietnam Academy of Science and Technology DOI: https://doi.org/10.15625/0866-7136/14897 OPTIMAL PARAMETERS OF DYNAMIC VIBRATION ABSORBER FOR LINEAR DAMPED ROTARY SYSTEMS SUBJECTED TO HARMONIC EXCITATION Vu Duc Phuc1,∗ , Tong Van Canh2 , Pham Van Lieu3 Hung Yen University of Technology and Education, Vietnam Korea Institute of Machinery and Materials, South Korea Hanoi University of Industry, Vietnam ∗ E-mail: ketquancs@gmail.com Received 18 March 2020 / Published online: 16 July 2020 Abstract Dynamic vibration absorber (DVA) is a simple and effective device for vibration absorption used in many practical applications Determination of suitable parameters for DVA is of significant importance to achieve high vibration reduction effectiveness This paper presents a method to find the optimal parameters of a DVA attached to a linear damped rotary system excited by harmonic torque To this end, a closed-form formula for the optimum tuning parameter is derived using the fixed-point theory based on an assumption that the damped rotary systems are lightly or moderately damped The optimal damping ratio of DVA is found by solving a set of non-linear equations established by the Chebyshev’s min-max criterion The performance of the proposed optimal DVA is compared with that obtained by existing optimal solution in literature It is shown that the proposed optimal parameters are possible to obtain superior vibration suppression compared to existing optimal formula Extended simulations are carried out to examine the performance of the optimally designed DVA and the sensitivity of the optimum parameters The simulation results show that the improvement of the vibration performance on damped rotary system can be as much as 90% by using DVA Keywords: dynamic vibration absorber, torsional excitation, optimazed parameters, rotary systems INTRODUCTION Vibration control is essential in many engineering fields Among vibration control methods, the dynamic vibration absorbers (DVAs) are widely applied because of its efficiency, reliability, and rather low expense [1] The early study on DVA was conducted by Frahm [2] Ormondroyd and Den Hartog [3] first introduced the concept of the DVA with spring and viscous damper arranged in parallel Den Hartog proposed in his book, the fixed-points theory, which helps find out the closed-form optimal parameters of DVA c 2020 Vietnam Academy of Science and Technology Vu Duc Phuc, Tong Van Canh, Pham Van Lieu attached to undamped structures [4] That approach mainly aims at reducing the maximum amplitude magnification factor of the primary system, which is still widely used nowadays [5–7] Since then, a number of optimization criteria have been proposed for optimal design of DVA, in which the H∞ and H2 optimizations were employed by many authors [1, 8, 9] Nishihara and Asami [10] proposed an analytical solution for the optimal parameters of DVA using H∞ optimization, which minimized the maximum displacement of the primary mass Shen et al [11] studied the optimal design of DVA with negative stiffness based on the H∞ optimization The H2 optimization was used to minimize the mean square displacement of the main mass [12], and the power dissipated by the absorber [13] Yamaguchi [14] found the optimal parameters of DVA using a stability maximization criterion for minimizing the transient vibration of the system Argentini et al [15] proposed a closed-form optimal tuning of TMD coupled with an undamped single DOF system forced by a rotating unbalance Bisegna and Caruso [16] took the exponential time-decay rate of the system transient response as an optimality condition Then, the closed-form expressions of the optimal exponential time-decay rate were proposed for undamped systems The other optimization approaches, such as the frequency locus method [17], the min-max criteria [18], and the numerical optimization scheme [19–21], and averaging technique [22] were also proposed The above-mentioned studies have provided a comprehensive background to the design optimization of DVAs However, there have been few studies on DVA for rotary systems with torsional vibration The torsional vibrations usually result in significant harmful effects on rotating systems For example, torsional vibration causes the fluctuation in rotational speed of electric motor leading a severe perturbation of the electro-magnetic flux and thus additional oscillation of the electric currents in the motor [13] Torsional vibration is one of the greatest danger factors for the shaft line and the crankshaft of the marine power transmission system [23] Minimization of torsional vibration helps to increase the fatigue durability and the efficient functioning of a large turbo-generator [24] Recently, several attempts have been made to find out the closed-form optimal parameters of DVA used to reduce torsional vibration of undamped rotary system [25] For the damped rotary system, Phuc et al [26] have focused on approximating the damped rotary system by an equivalent undamped rotary system by using the least square criterion [27], from which the optimization problem was solved by using the fixed-points theory In this paper, the optimal parameters for a DVA attached on damped rotary system under torsional excitation are determined Approximation approach for lightly damped systems is used to derive approximated solution for the optimum tuning parameter of DVA Then, the Chebyshev equioscillation theorem is used to find out the optimal damping ratio This paper is organized as follows In Section 2, the model of damped rotary system coupled with a DVA is introduced and the system equations of motion are presented In Section 3, the optimization problems are solved for the optimal parameters of DVA In Section presents the numerical results The present method is compared to an existing method Extended numerical results are also presented to examine the performance of the optimal DVA and investigate the effect of mass and damping ratio of damped rotary system on the optimal results Finally, Section concludes the paper Optimal parameters of dynamic vibration absorber for linear damped rotary systems subjected to harmonic excitation MODEL OF DAMPED ROTARY SYSTEM ATTACH DYNAMIC VIBRATION ABSORBER UNDER TORSIONAL EXCITATION 2.1 Vibration equations of damped rotary system with dynamic vibration absorber Fig shows the model of the damped rotary system attached DVA In which the damped rotary system is the one DOF shaft attached a main disk which has the characteristic parameters are mass (ms ), inertial momentum ( Js ), torsional stiffness (k s ) and internal coefficient of torsional viscous damper (cs ) The DVA consists of a passive disk having inertial momentum ( Ja ) is connected to the damped rotary system by springs and dampers with stiffness (k j ) and coefficient of viscous damper (c j ), respectively The springs and dampers are arranged in parallel and they are distributed on the circles with radius e1 and e2 , respectively To create symmetrical, the springs (k j ) and dampers (c j ) are the same Because a damper (cs ) is imposed in the model, the system is damped To change the damped rotary system into undamped rotary system, damping cs can be set Vu Duc Phuc, Tong Van Canh, Pham Van Lieu to zero Model of damped rotary systemattached attached DVA Fig Fig Model of damped rotary system DVA By applying the second order Lagrange equation, the differential equations of motion for the systemByin applying Figure can obtainedorder as: Lagrange equation, the differential equations of mothebesecond tion for the system in Fig ! ì( 1J scan + J abe + J aq!!a + cas )q!!sobtained sq s + k sq s = M t ù n (1) J!!a ) ăs + Ja ă2a !+ csn s + 2k s s = Mt , í J(a J(sq!!+ + q ) + c å r a nje2q a + å k j en1 q a = ï (1) ˙ j =1 ợ ă ă j =1 Ja (θr + θ a ) + ∑ c j e2 θ a + ∑ k j e1 θ a = j =1 In the equations (1), q s and q a are thej=relative torsional angles of main disk and passive disk, respectively harmonic moment given by: In Eqs M (1), θ a aretorsional the relative torsional angles of main disk and passive disk, s and t is θthe respectively Mt is the harmonic torsional moment given by M t = M sin Wt (2) Mt = M0 sin Ωt, (2) W is the excitation frequency, e1 and e2 indicate the radial positions of springs and dampers, where Ω is the excitation frequency, e1 and e2 indicate the radial positions of springs and respectively dampers, respectively 2.2 Amplitude magnification factor (AMF) By describing the harmonic excitation torque in a complex form as: M t = M eiWt (3) The solution for equation (1) can be determined as follows: ìïq s (t ) = qˆs (W)eiWt í (4) Vu Duc Phuc, Tong Van Canh, Pham Van Lieu 2.2 Amplitude magnification factor (AMF) By describing the harmonic excitation torque in a complex form as Mt = M0 eiΩt (3) The solution for Eqs (1) can be determined as follows θs (t) = θˆs (Ω)eiΩt , θ a (t) = θˆa (Ω)eiΩt (4) Substituting Eqs (4) and its derivation into Eqs (1), and solving the obtained equation, the relative torsional angle of main disk θˆs can be obtained as M0 θˆs = ( α j , ζ j ), ks where (5) (α j , ζ j ) is the transfer function of the system described by (α j , ζ j ) = − (1 + µη ) β2 µ2 η β4 + 2ζ s βi − (6) n −µη β2 + ∑ γ2 µα2j + 2λ2 µβα j ζ j i j =1 The other parameters in Eq (6) are given as follows µ= ma , ms η= cj ζj = , 2m a ω j ρa , ρs ωs = γ= ks , Js e1 , ρs λ= e2 , ρs cs ζs = , 2Js ωs ωj = kj , ma Ω β= , ωs αj = ωj , ωs (7) where µ is the mass ratio; γ, λ and η respectively represent the ratio between radial position of springs, radial position of dampers and the gyration radius of passive disk to the gyration radius of main disk; ωs indicates the natural frequency of rotary system; α and ζ are the tuning and damping ratios, respectively; β is the frequency ratio; the indexes j and s stand for the DVA and main disk, respectively The amplitude magnification factor (AMF) is defined as the magnitude of the complex transfer function as H = | (α, ζ )| = Aζ + B Cζ + Dζ + E (8) Optimal parameters of dynamic vibration absorber for linear damped rotary systems subjected to harmonic excitation In Eq (8), all the springs and dampers of the DVA are assumed identical, that means α j = α and ζ j = ζ ( j = 1, 2, , n) The other parameters in Eq (8) are described as A = 4λ4 n2 β2 α2 , B = (η β2 − nα2 γ2 )2 , C = 4n2 α2 λ4 β2 µη β2 + β2 − + 4β2 ζ s2 , D = 8ζ s β2 λ2 nα (η β2 − nα2 γ2 )(µη β2 + β2 − 1)+ η β2 (nµα2 γ2 − β2 + 1)+ nα2 γ2 ( β2 − 1) , E = nµα2 η γ2 β2 + (1 − β2 )(η β2 − nα2 γ2 ) + 4β2 ζ s2 (η β2 − nα2 γ2 )2 (9) To reduce torsional vibration of damped rotary system, the parameters α and ζ are needed to define for minimum torsional angle θs (Eq (4)) at resonant frequency These parameters are called optimal parameters of DVAs (signed αopt and ζ opt ) that will be determined in the below section OPTIMAL PARAMETER OF DYNAMIC VIBRATION ABSORBER When the rotary system is coupled with a damper, the fixed-points feature is diminished However, the amplitude magnification curves roughly pass two points when the rotary system is lightly or moderately damped, and the mass ratio between the rotary system and the DVA is small To satisfying the above-mentioned conditions, it is assumed that the fixed-point theory is approximately maintained [28] Based on this assumption, an approximate solution for the optimum tuning parameter αopt for the damped model can be realized The two approximated fixed points (signed as S and T) are found by finding the intersections of the amplitude magnification curves The two AMF curves defined at ζ equals to 0, and ζ approaches ∞ are chosen H | ζ =0 = B = E (η β2 − nα2 γ2 )2 , [nµα2 η γ2 β2 + (1 − β2 )(η β2 − nα2 γ2 )]2 + 4β2 ζ s2 (η β2 − nα2 γ2 )2 (10) H |ζ →∞ = lim ζ →∞ Aζ + B = Cζ + Dζ + E A = C (µη β2 + β2 − 1)2 + 4β2 ζ s2 (11) Equating H in Eqs (10) and (11) results in (η β2 − nα2 γ2 )2 [nµα2 η γ2 β2 +(1 − β2 )(η β2 − nα2 γ2 )] + 4β2 ζ s2 (η β2 − nα2 γ2 )2 = (µη β2 + β2 − 1) + 4β2 ζ s2 (12) Solving Eq (12) gives the frequency ratios at S and T as β2S,T = nα2 γ2 (µη + 1) + η ∓ n2 α4 γ4 (µη + 1)2 + η (η − 2nα2 γ2 ) η ( η µ + 2) (13) Vu Duc Phuc, Tong Van Canh, Pham Van Lieu To find the optimal tuning ratio, let the ordinates of points S and T be equal resulting in µη β2S + β2S −1 + 4β2S ζ s2 = µη β2T + β2T −1 + 4β2T ζ s2 (14) By substituting β S , β T in Eq (13) into Eq (14), and then solving this equation, αopt is found as follows η γ n (1 + η µ ) √ αopt = − 2ζ s2 − 2ζ s2 ( η µ + 1) (15) From Eq (15), the optimal tuning parameter of undamped system can be calculated by setting the damping ratio of the main system ζ s = Then resulting αopt for undamped system is η (16) αopt = √ γ n (1 + η µ ) It is seen that αopt in Eq (16) is the same as the one derived for undamped system by using the fixed-point theory [25, 26] Phuc et al [26] found the optimal tuning parameter of DVA for damped rotary system using an equivalent undamped model as following αopt = 4ζ s2 2ζ s +1− π π η γ n (1 + η µ ) √ (17) The comparison between the optimal parameter proposed in this paper and [26] will be presented in the next section Fig shows the AMF curves versus β with varying damping ratios of DVA (ζ ) To reduce the maximum peaks of AMF, we determine the value of ζ so that the AMF function has two equal peak values with a minimal distance from a straight line L as Vu Duc Phuc, Tong Van Canh, Pham Van Lieu shown in Fig 10 S z = 0.03 z = 0.04 z = 0.05 T D H L 0.75 b1 0.8 0.85 0.9 0.95 b2 b3 1.05 1.1 1.15 1.2 1.25 b Fig Demonstration of parameters in equation (18) Fig Demonstration of parameters in Eq (18) In this way, the optimum solution can be found by using the Chebyshev equioscillation theorem [18, 19] To this end, the following equations will be solved: ì dH dH dH = 0; = 0; =0 ï d b b = b2 d b b = b3 í d b b = b1 ï ỵ H ( b1 ) - H ( b ) = 0; L - [ H ( b1 ) + H ( b ) ] = 0; 2D - [ H ( b1 ) - H ( b ) ] = (18) Optimal parameters of dynamic vibration absorber for linear damped rotary systems subjected to harmonic excitation In this way, the optimum solution can be found by using the Chebyshev equioscillation theorem [18, 19] To this end, the following equations will be solved dH dH dH = 0, = 0, = 0, dβ β= β1 dβ β= β2 dβ β= β3 (18) H ( β ) − H ( β ) = 0, 2L − [ H ( β ) + H ( β )] = 0, 2∆ − [ H ( β ) − H ( β )] = where ∆ indicates the maximum peak value of the AMF curve determined from the ordinate H = L (see Fig 2) β , β and β are the frequency ratios at which the AMF curve reaches maximum and minimum Unlike the result of Ghosh and Basu [28], which only found the optimal tuning parameter, this study allows finding the optimal damping parameter through solving the system of nonlinear equations (18) for unknowns (i.e β , β , β , L, ∆ and ζ ) Compared to the system of equations of Liu and Coppola [19], the unknown α is eliminated in our system of equations (18) This is because the expression of the optimal tuning parameter (15) has been substituted into the amplitude magnification factor ( H ) before the equation system equations (18) were established The fsolve function provided by Matlab is used to solve system equations (18) With respect to the nonlinear equations system, the fsolve solver requires good initial values of the roots for a quick convergence The initial parameters were set as: β = 0.85, β = 1, β = 1.05, L = 0, ∆ = The initial value of ζ is selected as the optimal value for undamped rotary system, that is ζ= µη γ2 nλ4 (1 + µη ) (19) The optimal damping ratio of undamped in Eq (19) is determined by using the fixedpoint theory [25, 26] NUMERICAL RESULTS AND DISCUSSION To demonstrate the proposed method, this section presents numerical simulation for a sample damped DVA system with the parameters given in Tab The parameters of this system are taken from reference [26] First, a comparison is performed to compare Table Input parameters of damped rotary system and DVA Parameters Unit Value Mass of main disk (ms ) Gyration radius of main disk (ρs ) Amplitude of excitation moment ( M0 ) Stiffness of main spring (k s ) Gyration radius of passive disk (ρ a ) Radial position of dampers of DVA (e2 ) Radial position of springs of DVA (e1 ) Number of springs and dampers of DVA (n) kg m Nm Nm/rad m m m - 0.12 8.0 12,000 0.12 0.08 0.05 Vu Duc Phuc, Tong Van Canh, Pham Van Lieu the current method with the existing method Then, further simulations are carried out to examine the performance of the optimal DVA and investigate the effect of several important parameters of DVA system on the optimal parameters 4.1 Model comparison The present method is compared with the method proposed by Phuc et al [26] In their method [26], the authors obtained the optimal parameters of damped DVA system via two steps In the first step, the damped rotary system is converted into an equivalent undamped rotary system using the least squares estimation of equivalent linearization method, which was developed by Anh et al [27] The second step determines the optimal parameters of DVA of equivalent undamped rotary system using the traditional fixed point theory Tab and Fig show the effect of damping ratio of rotary system ζ s on the optimal parameters determined by present method and the method proposed in [26] The optimal parameters are determined for two values of mass ratio, µ = 0.033 and µ = 0.05 Fig 3(a) shows that increasing ζ s leads to the reduction of optimal tuning parameter of DVA Higher µ value requires a smaller αopt value for both methods On contrary, ζ opt calculated by the present method increases with increasing either ζ s or µ The value of ζ opt found by [26] is almost constant in entire range of ζ s Fig shows that the optimal parameters of the present method and [26] are only approximated at low ζ s Table Optimum tuning ratio and damping ratio of DVA for different rotary system damping ratios Rotary system damping ratios (ζ s ) 0.010 0.015 0.020 0.025 0.030 0.035 0.040 Optimum tuning ratio (αopt ) Phuc et al [26] Present study Optimum damping ratio (ζ opt ) Phuc et al [26] Present study µ = 0.033 µ = 0.05 µ = 0.033 µ = 0.05 µ = 0.033 µ = 0.05 µ = 0.033 µ = 0.05 1.1539 1.1503 1.1466 1.1430 1.1393 1.1357 1.1321 1.1356 1.1320 1.1284 1.1248 1.1212 1.1177 1.1141 1.1611 1.1608 1.1604 1.1599 1.1592 1.1585 1.1576 1.1426 1.1424 1.1420 1.1415 1.1408 1.1401 1.1393 0.0516 0.0516 0.0516 0.0516 0.0516 0.0516 0.0516 0.0626 0.0626 0.0626 0.0626 0.0626 0.0626 0.0626 0.0527 0.0530 0.0532 0.0539 0.0546 0.0559 0.0580 0.0641 0.0644 0.0648 0.0654 0.0658 0.0662 0.0672 Fig shows the amplitude amplification factor determined by the present method and [26] for several values of ζ s There are two peaks of the AMF, which occur around the resonance frequency It can be seen from Fig 4(a) that at low damping ratio such as ζ s = 0.01, the AMF calculated by the two methods are almost the same This is because the values of optimum parameters obtained by the two methods at low damping ratios are approximated as shown in Fig When the damping ratio increases, the maximal peak value of AMF determined by [26] is higher than that of the present method (show in Fig 4(d)) Tab shows the AFM calculated at the resonant frequency ( H( β=1) ) by the present method and [26] with varying the damping ratio This table also shows the maximum Vu Duc Phuc, Tong Canh, Pham 8 Vu Duc Phuc, Tong VanVan Canh, Pham VanVan LieuLieu 8Optimal parameters of dynamic vibration Vu Duc Phuc,for Tong Vandamped Canh, rotary Pham Van Lieusubjected to harmonic excitation absorber linear systems 88 Vu Duc Phuc, Tong Van Canh, Pham Van Lieu VuDuc DucPhuc, Phuc,Tong TongVan VanCanh, Canh, Pham Van Lieu Vu Van Lieu Phuc etPham al.[27] Present study Phuc et al.[27] Present study 1.2 1.2 1.2 Phuc et al.[27] 1.2 a opta opt a opt a opt a opt opt 1.2 1.2 1.151.15 1.15 1.15 1.15 1.15 1.1 1.1 1.1 1.1 1.1 1.1 Phuc et al.[27] Phuc al.[27] Phuc et et al.[27] µ=0.05 µ=0.05 µ=0.05 µ=0.05 µ=0.05 µ=0.05 Present study Present study Present study Present study µ=0.033 µ=0.033 µ=0.033 µ=0.033 µ=0.033 µ=0.033 0.0050.010.010.015 0.0150.020.020.025 0.0250.030.030.035 0.0350.040.04 0 0.005 0.025 0.03 0.03 0.035 0.035 0.04 0.04 a) a) 0 0.005 0.005 0.01 0.01 0.015 0.015 0.02 0.02 0.025 Phuc et0.025 al.[27] 0.03 Present study Phuc et al.[27] Present study a) a) 00 0.005 0.01 0.015 0.02 0.025 0.035 0.04 0.005 0.01 0.015 0.02 0.03 0.035 0.04 0.070.07 µ=0.05 µ=0.05 (a) Phuc et al.[27] al.[27] Present study Phuc et Present study a) 0.07 zoptzopt z z zzopt opt opt opt 0.07 µ=0.05 µ=0.05 Phuc et et al.[27] Present study 0.065 Phuc al.[27] Present study 0.065 0.07 µ=0.05 0.07 µ=0.05 0.065 0.065 0.06 0.06 0.065 0.065 µ=0.033 µ=0.033 0.060.06 µ=0.033 0.055 0.055 µ=0.033 0.06 0.06 0.055 µ=0.033 0.055 µ=0.033 0.05 0.05 0.055 0.055 0.05 0.01 0.015 0.015 0.020.02 0.025 0.025 0.030.03 0.035 0.035 0.040.04 0.01 0.05 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.05 Main system damping ratio 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Main system damping ratio z z 0.05 b) s b) b) 0.01 s Main system damping ratio 0.015 0.02 0.025 0.03 0.035 0.04 z s 0.01 0.015 Main 0.02 0.025 ratio0.03 0.035 0.04 system damping z Fig Effect of main system damping ratio and mass ratio on optimal parameters b) Fig.Fig s Effect of main system damping ratio and mass ratio on optimal parameters Main system damping ratio zs zon optimal parameters Effect of main system damping ratio and mass ratio Main system damping ratio b) s Effect of main system damping ratio and on optimal parameters Figure 44Fig shows the amplitude amplification factor determined by the present method and Fig Effect of main system damping ratio andmass massratio ratio on optimal parameters Figure shows the3the amplification factor determined by the present method [27][27] Figure shows amplitude amplification factor determined by the method andand [27] (b) Fig 3amplitude Effect of main system damping ratio and mass ratio on present optimal parameters several of There are two peaks of the AMF, which occur around the resonance zamplitude for for several values of zthe There are two peaks of the AMF, which occur around the resonance values shows amplification factor determined by the present method and [27] forFigure several values of There are two peaks of the AMF, which occur around the resonance s s zthe Figure shows amplitude amplification factor determined by the present method and [27] s Figure 4ofshows the amplitude amplification factor determined by theoptimal present method and [27] forFig several values ofmain are two peaks of the AMF, which occur around the resonance zof Effect system damping ratio and mass ratio on parameters frequency frequency frequency for3.several values There are two peaks of the AMF, which occur around the resonance s z There for several values of zs s There are two peaks of the AMF, which occur around the resonance frequency frequency frequency (a) (b) Fig Comparison of the AMF by present method and [27] with varying damping ratios (µ = 0.033) Comparison of AMF the AMF by method and [27] with varying damping ratios µ= 0.033) 44Comparison of the AMF bypresent present method and [27] with varying damping ratios = 0.033) Fig.Fig 4Fig Comparison of the by present method and [27] with varying damping ratios ((µ =(µ 0.033) Fig Comparison of the AMF by present method and [27] with varying damping ratios ( µ = 0.033) Fig Comparison of the AMF by present method and [27] with varying damping ratios (µ = 0.033) (c) (d) Fig Comparison of the AMF by present method and [26] with varying damping ratios (µ = 0.033) Vu Duc Phuc, Tong Van Canh, Pham Van Lieu It can be seen from Figure (a) that at low damping ratio such as z s = 0.01 , the AMF calculated by the two methods are almost the same This is because the values of optimum parameters obtained by the two methods at low damping ratios are approximated as shown in Figure When the damping ratio increases, the maximal peak value of AMF determined by [27] is higher than that of the present 10 method (show in Figure 4(d)) Vu Duc Phuc, Tong Van Canh, Pham Van Lieu Table shows the AFM calculated at the resonant frequency ( H ( b =1) ) by the present method and AFM ) calculated by the two Fromthethis table, AFM the maximum reduction [27]( H with the damping ratio Thismethods table also shows maximum ( H max ) calculated by maxvarying percentages of H estimated at βthe = maximum 1, and Hreduction and of 5.24%, respectively the two methods From this table, percentages at b = , Thereand max are 1.89% H estimated fore,Hpresent method clearly shows better vibration suppression compared to [26] max are 1.89 % and 5.24 %, respectively Therefore, present method clearly shows better vibration suppression compared to [27] Table Comparison of AFM estimated at resonance frequency (H( β=1) ) and the maximal value Table 3: Comparison of AFM estimated at resonance frequency (H(b = 1)) and the maximal value of AFM of AFM ((H Hmax ) with varying the damping ratio max ) with varying the damping ratio resonancefrequency frequency HHmax ininresonance frequency region HH in in resonance resonance frequency region Rotary max Rotary system system Phuc et Present Improvement Improvement from Improvement from Phuc’s Phuc et et Present Improvement fromfrom Phuc’s Phuc et Present Phuc Present damping damping al [26] study result in percentage Phuc’s result in Phuc’s result in terms al.al.[27] [26] studystudy result in percentage terms al.[27] study ratios ratios percentage terms percentage terms µ = 0.033 µ = 0.033 ((ζzs ) ) s 0.010 0.010 0.015 0.020 0.015 0.025 0.020 0.030 0.035 0.025 0.040 0.030 6.227 6.200 5.9266.2275.867 5.657 5.561 5.926 5.414 5.318 5.1945.6575.096 4.993 4.920 5.414 4.810 4.790 5.194 µ = 0.033 6.2 5.867 5.561 5.318 5.096 0.43 1.00 1.70 1.77 1.89 1.46 0.42 0.43 1.00 1.70 1.77 1.89 7.074 7.074 6.740 6.443 6.74 6.145 6.443 5.880 5.631 6.145 5.399 5.88 µ = 0.033 7.008 6.6537.008 6.327 6.653 6.029 5.5726.327 5.501 6.029 5.272 5.572 0.93 0.93 1.29 1.80 1.29 1.89 1.80 5.24 2.31 1.89 2.35 5.24 0.0355 compares 4.993 the maximum 4.92 1.46 determined 5.631 by proposed 5.501 2.31 and [26] Fig of AMF formulae with 0.040 varying the damping ratio of the0.42 damped rotary It can be seen 4.81 4.79 5.399system 5.272 2.35 that the increase rate of difference in maximal peaks of AMF is increased as ζ is greatly increased s Figure compares the maximum of AMF determined by proposed formulae and [27] with varying the Therefore, optimal a better mitigation than of [26] damping the ratioproposed of the damped rotaryparameters system It cangive be seen that the increase rate of that difference in in terms of maximum magnification maximal peaks of AMF is increasedfactor as z s is greatly increased Therefore, the proposed optimal parameters give a better mitigation than that of [27] in terms of maximum magnification factor Fig The effect of damping ratio of main structure on maximum of AMF (µ = 0.033) The sensitivity of optimal parameters on the amplitude magnification factor is presented in Fig In which, the vertical axes show the deviation percentage of AMF, which are calculated by Hζ,αopt − Hαopt ,ζ opt ∆Hζ = × 100, (20) Hαopt ,ζ opt 10 Vu Duc Phuc, Tong Van Canh, Pham Van Lieu Fig The effect of damping ratio of main structure on maximum of AMF (µ = 0.033) The sensitivity of optimal parameters on the amplitude magnification factor is presented in Figure In which, the vertical axes show the deviation percentage of AMF, which are calculated by: Ha opt ,z optrotary systems subjected to harmonic excitation Optimal parameters of dynamic vibration absorber forHlinear damped z ,a opt DH z = and And ∆Hα = H a opt ,z opt ´ 100 (20) Hα,ζ opt − Hαopt ,ζ opt H a ,z - Ha ,z × 100, DH a =Hαoptopt,ζ opt opt opt ´ 100 H a opt ,z opt (21) 11 (21) where Hα,ζ opt , Hζ,αopt and Hαopt ,ζ opt are the maximum values of AMF determined at the Ha tuning where Hdamping maximum values AMF determinedindexes at the correspondent a ,z , H z ,a and ,z are the correspondent and ratios given inofthe subscript opt opt opt opt damping and tuning ratios given in the subscript indexes Fig Effect of error in optimal parameters on AMF (zs = 0.1, µ = 0.033) Fig Effect of error in optimal parameters on AMF (ζ s = 0.1, µ = 0.033) Figure demonstrates that the tuning parameter is more sensitive to the variation of AMF than the damping ratio When the damping ratio varies from -10% to 40% around z opt , the maximum Fig demonstrates that the tuning parameter is more sensitive to the variation of deviation percentage of AMF ( DH z ) is below 1%, whereas DH z is higher than 40% in the case of AMF than the damping ratio When the damping ratio varies from −10% to 40% around error in a opt In addition, the actual optimal damping ratio should be about 15% higher than the ζ opt , the maximum deviation percentage of AMF (∆Hζ ) is below 1%, whereas ∆Hζ is calculated value with equation (19) In the case of tuning ratio, the actual optimal tuning ratio should higher than 40% in the case of error in αopt In addition, the actual optimal damping ratio should be about 15% higher than the calculated value with Eq (19) In the case of tuning ratio, the actual optimal tuning ratio should be about 1.5% smaller than the value found by proposed equation (15) However, the differences between ∆Hζ and ∆Hα computed by actual optimal parameters and proposed formulae (Eqs (20) and (21)) are relatively small (about 2%) Therefore, the proposed optimal parameters are approximated solutions, although they can still be used with satisfactory accuracy Fig also shows that the actual optimal damping and tuning ratios should be around 20% higher and 3% lower than those proposed in [26], respectively Therefore, the proposed optimal parameters in this paper are closer to actual optimality point than those in [26] Vu Duc Duc Phuc, Phuc, Tong Tong Van Van Canh, Canh, Pham Pham Van Van Lieu Lieu Vu 11 11 be than the the value value found found by by proposed proposed equation equation (15) (15) However, However, the the differences differences be about about 1.5% 1.5% smaller smaller than between and computed by actual optimal parameters and proposed formulae [equations D H D H between DHz and DH a computed by actual optimal parameters and proposed formulae [equations z a (20) relatively small small (about (about 2%) 2%) Therefore, Therefore, the the proposed proposed optimal optimal parameters parameters are are (20) and and (21)] (21)] are are relatively approximated solutions, although they can still be used with satisfactory accuracy Figure also shows approximated solutions, although they can still be used with satisfactory accuracy Figure also shows that damping and and tuning tuning ratios ratios should should be be around around 20% 20% higher higher and and 3% 3% lower lower than than that the the actual actual optimal optimal damping those proposed in [27], respectively Therefore, the proposed optimal parameters in this paper are 12 Vu DucTherefore, Phuc, Tong Van Pham Van Lieu those proposed in [27], respectively theCanh, proposed optimal parameters in this paper are closer to actual optimality point than those in [27] closer to actual optimality point than those in [27] 4.2 investigation 4.2 Numerical investigation 4.2 Numerical Numerical investigation In this section, numerical simulation is performed to examine the effect optimal DVA In numerical simulation simulation is is performed performed to to examine examine the the effect optimal optimal DVAon onthe the In this this section, section, numerical on the system vibration Moreover, the effects of damping ratio of effect rotary systemDVA and mass system vibration Moreover, the effects of damping ratio of rotary system and mass ratio on the system vibration Moreover, the effects of damping ratio of rotary and mass ratio on the ratio on the optimal parameters of DVA are investigated The system parameters of the damped optimal parameters of DVA DVA are are investigated investigated The The parameters parameters of of the the damped damped rotary rotary system system are are the the optimal parameters of rotary system are the same insame Table as those depicted in Tab same as as those those depicted depicted in Table Fig shows the AMF and vibration response in time domain of the damped rotary Figure 77 shows theand AMF andoptimally vibration response response in DVA time domain domain of of the the damped damped rotary rotary system system Figure shows the AMF and vibration in time system without DVA with designed without DVA and with optimally designed DVA without DVA and with optimally designed DVA Fig Performance without DVA DVA and and with with optimal optimal DVA, DVA,(a) (a)AMF, AMF,(b) (b)vibration vibrationresponse response (a) of the system without (b) (W == w wss,, zzss == 0.01, 0.01, µµ == 0.0333, 0.0333, n=4) n=4) in time domain (W 7(a) shows DVA remarkably reducesDVA, the peak peak of AMF AMF around the Fig 7.Figure Performance of the withoutDVA DVAremarkably and with optimal (a) of AMF, (b)around vibration thatsystem using optimal reduces the the response time domain (Ω =the ωsresonance, ,resonance, ζ s = 0.01,such µ = as 0.0333, = 4)the resonance frequency from the such as at bb ==n0.85, 0.85, theAMF AMFmay maynot not At thein frequencies far from at be reduced In general, vibration control using using passive passive DVAs DVAs isis normally normally very very effective effective for foraacertain certain frequency-bandwidth vibration of remarkably the system system isis usuallythe large around the around resonant vibration of the usually large around the resonant Fig 7(a) shows However, that usingthe optimal DVA reduces peak of AMF frequency (see Figure effectiveness of optimal DVA isis still stillsuch wellachieved achieved (a)) Hence, effectiveness optimal DVA well the resonance frequency At the the frequencies farof from the resonance, as at β = 0.85, the AMFThe mayvibration not be reduced In general, vibration control using passive DVAs is ratio normally response in time domain in in Figure Figure 7(b) 7(b) is is simulated simulated at at the the frequency frequency ratio bb==11 very effective for a certain frequency-bandwidth However, the vibration of the system the (resonance condition) It is clear that the rotary rotary system system attached attached with with optimal optimal DVA DVAquickly quicklyobtains obtains the is usually large around the resonant frequency (see Fig 7(a)) Hence, the effectiveness of steady state vibration with vibration amplitude amplitude significant significant lower lower than than that that without without DVA, DVA, the theeffective effective optimal DVA is still well achieved in resonance condition of vibration control to 87,6% condition [Figure [Figure 7(b)] 7(b)] The vibration response in time domain in Fig 7(b) is simulated at the frequency Figure shows the deviation percentage percentage of of AMF AMF calculated calculated by by equations equations (20) (20) and and (21) (21) for for ratio β = (resonance condition) It is clear that the rotary in system attached with optimal several values of z s Figure 8(a) shows that increasing increasing zz results results in aa larger larger difference difference between between the the DVA quickly obtains the steady state vibration with vibration amplitude significant lower optimal damping ratio (signed ) Such as, is about 2.7% DDzzopt DDzzopt actual and without predictedDVA, (signed as ascontrol Such as, in 2.7% for for opt ) to opt is about than that the effective of vibration 87.6% resonance condition , and Dz opt is approximate 17% for z ss == 0.1 z(Fig = 0.02 0.1 7(b)) ss Fig shows the deviation percentage of AMF calculated by Eqs (20) and (21) for theshows actualthat and increasing predicted tuning (signed as )) isis also Thevalues difference predicted optimal optimal tuning ratio (signed asDDaaoptopt also several of ζ s between Fig 8(a) ζ results in aratio larger difference between increased with thepredicted increase ofoptimal in However, of isis relatively DDaa z ss as shown the actual and damping ratio8(b) (signed as ∆ζthe value Such ∆ζ is about in Figure Figure 8(b) However, the)value ofas, relatively opt opt optopt 2.7% for = 0.02, is approximate 17% for ζ s ratio = 0.1.((zz ==0.1 small (lessζ sthan 2%) and even∆ζinoptthe case of highest damping highest damping ratio From Figure Figure 8,8, the the 0.1).) From s The difference between the actual and predicted optimals tuning ratio (signed as ∆αopt ) is also increased with the increase of ζ s as shown in Fig 8(b) However, the value of ∆αopt is relatively small (less than 2%) even in the case of highest damping ratio (ζ s = 0.1) From Fig 8, the maximum differences of ∆Hζ and ∆Hα determined at actual optimal parameters and proposed formulae are around 0.453% and 3.622%, respectively 12 Vu Duc Phuc, Tong Van Canh, Pham Van Lieu 12 Vu Duc Phuc, Tong Van Canh, Pham Van Lieu maximum differences ofvibration actual optimal parameters proposed DH z and DHfor a determined Optimal parameters of dynamic absorber linear dampedatrotary systems subjected to harmonic and excitation 13 formulae are around 0.453% and 3.622%, respectively maximum differences of DH z and DH a determined at actual optimal parameters and proposed 10 formulae are around 0.453% and 3.622%, respectively zs = 0.02 10 zs = 0.04 zs = 0.02 zs = 0.06 zs = 0.04 zs = 0.08 zs = 0.06 zs = 0.1 zs = 0.08 DHz 46 DHz 68 24 02 -2 -10 -2 -10 Proposed 0.1 z =minimal Actual minimal Proposed minimal Actual minimal s -0.453 -5 -0.453 -5 10 15 20 25 Error percentage of zopt (%) 10 15 20 25 (a) Error percentage of zopt (%) 25 30 30 35 40 40 zs = 0.02 25 20 z = 0.04 zss = 0.02 z = 0.06 zss = 0.04 z = 0.08 zss = 0.06 = 0.1 zzs = 0.08 20 15 DHa DHa 35 15 10 s zs = 0.1 10 5 0 -5 -4 -5 -4 -3.622 -3.622 -3 -2 -3 -2 -1 Error percentage of a opt (%) -1 Error percentage of a opt (%) 2 3 4 Fig Effect of error in optimal parameters on AMF for different zs (µ = 0.033) Fig 8the Effect of error inpercentage optimal(b) parameters on with AMF for different zs (µit= illustrates 0.033) Figure shows deviation of AMF varying that the µ and sensitivity of optimal is dependent on µ particularly, the sensitivity of a opt isthat reduced Figure showsparameters the deviation percentage of In AMF with varying the µ and it illustrates Fig Effect of error in optimal parameters on AMF for different ζ s (µ = 0.033) sensitivity of optimal is dependent on µ 10% In particularly, of a optfrom is reduced a optsensitivity with the increase of µ parameters For example, with the same erroneous inthe , DH a reduces 59.21% the increase Forincreased example, with same in a optthat , DH DH and towith 46.43% when ofµµ is from the 0.02 to 10% 0.06.erroneous It is seen the values from of 59.21% a reduces Fig shows the deviation percentage of AMF with varying µ and it illustratesathat D determined parameters varytowith varying ; however, the changes are D minor µ is optimal H a and toH z46.43% whenat actual increased from 0.02 0.06 It isµ seen that the values of the sensitivity of optimal parameters is dependent on µ In particularly, the sensitivity of DH z determined at actual optimal parameters vary with varying µ ; however, the changes are minor αopt is reduced with the increase of µ For example, with the same 10% erroneous in αopt , ∆Hα reduces from 59.21% to 46.43% when µ is increased from 0.02 to 0.06 It is seen that the values of ∆Hα and ∆Hζ determined at actual optimal parameters vary with varying µ; however, the changes are minor Fig 10 shows the comparison of the AMF of a damped rotary system determined by the method in [26] and the proposed method in this study In this figure, the solid green line indicates the AMF determined by [26] and the solid blue line presents the AMF calculated with optimal tuning and damping ratios determined by the present method It can be seen that the maximum peak value of AMF curve obtained by the proposed Vu Duc Phuc, Tong Van Canh, Pham Van Lieu 13 1.2 14 Phuc, Van Canh, Pham Vu Vu DucDuc Phuc, TongTong Van Canh, Pham Van LieuVan Lieu DHz 0.8 1.2 0.6 0.4 DHz 0.2 µ = 0.02 µ = 0.04 µ = 0.06 0.8 0.6 0.4 0.2 -0.2 -10 -8 -6 -0.2 -10 -8 -4 -6 60 50 40 -2 Error percentage of zopt (%) -4 -2 Error percentage of zopt (%) 6 D Ha 10 10 µ = 0.02 µ = 0.04 µ = 0.02 = 0.04 µ µ = 0.06 60 50 µ = 0.06 40 D Ha 8 30 20 13 µ = 0.02 µ = 0.04 µ = 0.06 30 20 10 10 0 -10 -10 -10 -10 -8 -8 -6 -6 00 22 4 -4-4 -2-2 Error percentage of a opt (%) Error percentage of a opt (%) 6 8 10 10 Effect error in parameters optimal parameters onAMF the AMF differentmass mass ratios Fig Effect Fig of error in of optimal on the forfor different ratios(z(s ζ=s 0.03) = 0.03) Fig Effect of error in optimal parameters on the AMF for different mass ratios (zs = 0.03) 6.5 6.5 5.756 5.5 5.756 H H 5.5 5.752 5.752 5.88 5.88 5 4.5 4.5 3.5 3.5 0.8 Phuc et al [27] Propsed method with a opt and z calculated by equation 19 Phuc et al.method [27] with Propsed a opt and zopt Propsed method with a opt and z calculated by equation 19 0.85 0.9 0.95 1.05 1.1 1.15 b and Propsed method with a opt zopt Fig 10 Comparison 0.8 of the AMF determined proposed1method1.05 and [27] with µ = 0.033, zs = 0.03) 0.85 0.9 by the 0.95 1.1 (n=4,1.15 b Figure 10 shows the comparison of the AMF of a damped rotary system determined by the method in [27] and the proposed method in this study In this figure, the solid green line indicates the Fig 10 Comparison of the AMF determined bydetermined the proposedbymethod and [27]method with (n=4, = 0.033, zs = 0.03) Fig 10 Comparison AMF the proposed andµ[26] AMF determined by [27] andof thethe solid blue line presents the AMF calculated with optimal tuning and with (n = 4, µ = 0.033, ζ = 0.03) s damping ratios determined by the present method It can be seen that the maximum peak value of by the Figure 10 shows the comparison of the AMF of a damped rotary system determined AMF curve obtained by the proposed method is smaller than that by [27] Figure 10 further shows the method in [27] and the proposed method in this study In this figure, the solid green line indicates the AMF estimated with the optimal tuning ratio proposed in this study, while a non-optimal damping AMF determined by [27] and the solid blue line presents the AMF calculated with optimal tuning and damping ratios determined by the present method It can be seen that the maximum peak value of AMF curve obtained by the proposed method is smaller than that by [27] Figure 10 further shows the AMF estimated with the optimal tuning ratio proposed in this study, while a non-optimal damping Optimal parameters of dynamic vibration absorber for linear damped rotary systems subjected to harmonic excitation 15 method is smaller than that by [26] Fig 10 further shows the AMF estimated with the optimal tuning ratio proposed in this study, while a non-optimal damping ratio is used as calculated by Eq (19) The AFM curve for this case is indicated by the red-dashed line in Fig 10 It is observed that even using only the proposed optimal tuning ratio, the maximum value of the AMF curve is still smaller than that of [26] Hence, the AMF by the proposed method is a generally superior compared to [26] CONCLUSIONS In this study, the optimum parameters of a DVA attached to a damped rotary system are proposed The aim of optimization is to minimize the maximum amplitude magnification factor of the damped rotary system A closed-form formula for the optimum tuning ratio is obtained using the fixed-point theory with assuming low-to-moderate damping in the damped rotary system A semi-analytical process for calculating the optimal damping ratio of DVA is presented The numerical results prove that the proposed optimal parameters have better suppression of the resonant vibration amplitude than the existing method and the control performance of DVA can be up to 90% A time history analysis is performed to demonstrate the efficiency of the proposed formulae It is shown that there is a slight difference between actual and proposed optimal tuning and damping ratios of DVA Therefore, the proposed optimal formulation is an approximate solution Nevertheless, the error caused by approximated solution is very small as shown by 3.62% difference in deviation percentage of magnification factor (Fig 8) ACKNOWLEDGMENT This research is funded by Hung Yen University of Technology and 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https://doi.org/10.1002/stc.446 [28] A Ghosh and B Basu A closed-form optimal tuning criterion for TMD in damped structures Structural Control and Health Monitoring, 14, (4), (2007), pp 681–692 https://doi.org/10.1002/stc.176 ... dynamic vibration absorber for linear damped rotary systems subjected to harmonic excitation MODEL OF DAMPED ROTARY SYSTEM ATTACH DYNAMIC VIBRATION ABSORBER UNDER TORSIONAL EXCITATION 2.1 Vibration. .. differences ofvibration actual optimal parameters proposed DH z and DHfor a determined Optimal parameters of dynamic absorber linear dampedatrotary systems subjected to harmonic and excitation 13 formulae... deviation percentage of AMF, which are calculated by: Ha opt ,z optrotary systems subjected to harmonic excitation Optimal parameters of dynamic vibration absorber forHlinear damped z ,a opt DH