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Optimal design of a damped dynamic vibration absorber for vibration control of structure excited by ground motion
Engineering Structures 30 (2008) 282–286 www.elsevier.com/locate/engstruct
Short communication
Optimal design of a damped dynamic vibration absorber for vibration control of structure excited by ground motion W.O. Wong ∗ , Y.L. Cheung Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong Received 15 June 2006; received in revised form 12 October 2006; accepted 12 March 2007 Available online 17 April 2007
Abstract Optimum parameters of a dynamic vibration absorber of non-traditional form have been derived for suppressing vibration of a single degreeof-freedom system due to ground motion. The reduction of transmission of motion from the support to the mass of the structure is compared for the cases of using the traditional and the proposed dynamic absorbers. Under the optimum tuning condition of the absorbers, it is proved analytically that the proposed absorber provides a larger suppression of resonant vibration amplitude of the primary system excited by ground motion than the traditional absorber. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Vibration absorber; Tuned mass damper; Damping
1. Introduction Dynamic vibration absorbers can be used for passive control of narrowband vibration. As described in the classical textbooks on dynamic vibration absorber [1–3], an undamped dynamic vibration absorber is an auxiliary mass–spring system which, when correctly tuned and attached to a vibrating body subject to a harmonic excitation, eliminates steady-state motion of the point to which it is attached. The traditional damped vibration absorber has a damper added between the absorber mass m and the primary mass M as shown in Fig. 1(a) to limit the vibration amplitude when the lower resonance is experienced during system start-up and stopping. However, it is not possible to eliminate steady-state vibrations of the original mass after damping is added in the auxiliary mass–spring system. Optimization of the frequency and damping parameters of the traditional damped vibration absorber for minimization of the resonant vibration amplitudes based on the fixed-points theory is well documented in the textbook of Den Hartog [1]. Recent advances of absorber designs involve addition of active controlled elements [4–6], adaptive elements that can be used to change the tuned condition of the absorber [7,8], ∗ Corresponding author. Tel.: +852 2766 6667; fax: +852 2365 4703.
E-mail address: [email protected] (W.O. Wong). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.03.007
and passive device with non-traditional elements for broadband vibration control [9–11]. These advanced absorbers are useful in applications requiring the change of tuning parameters as the main system changes in time. On the other hand, the traditional dynamic vibration absorber may provide a cheaper and convenient solution for vibration suppression and isolation of vibrating systems with harmonic excitation. In this paper, the optimum tuning frequency and damping of a damped dynamic vibration absorber of non-traditional form as shown in Fig. 1(b) have been derived for suppressing vibration of a single-degreeof-freedom system due to ground motion. The derivation of the formulae for the optimum tuning frequency and damping of the absorber was based on the fixed-points theory of Den Hartog [1]. It is proved that the proposed absorber provides a larger suppression of resonant vibration amplitude of the primary system excited by ground motion than the traditional absorber. 2. The traditional damped dynamic vibration absorber A schematic diagram of a traditional damped dynamic vibration absorber attached to an undamped mass–spring system is shown in Fig. 1(a). This vibration model is called model A in the following discussion. The amplitude ratio
W.O. Wong, Y.L. Cheung / Engineering Structures 30 (2008) 282–286
283
Fig. 1. A damped dynamic vibration absorber as an auxiliary mass–spring–damper (m–k–c) system attached to a primary system (M–K ) (a) traditional design of the absorber [1], (b) the proposed design of the absorber for suppressing the vibration of the mass M due to ground motion.
|X 1 /Y |A is: X1 Y A (k − mω2 + jcω)K = [(K − Mω2 )(k − mω2 ) − mω2 k] + jcω(K − Mω2 − mω2 ) s (γ 2 − λ2 )2 + (2ζ γ λ)2 = [(1 − λ2 )(γ 2 − λ2 ) − µγ 2 λ2 ]2 + (2ζ γ λ)2 (1 − µλ2 − λ2 )2
(1) √ √ where √ λ = ω/ K /M, γ = k/m/ K /M, µ = m/M, ζ = c/2 mk. Based on the result of Liu and Liu [12], the optimum tuning condition and the optimum damping using the fixed-points theory [1] leading to minimum vibration amplitude at resonance are
Assuming a harmonic disturbing input y = Y e− jωt , and the responses may be written as x1 = X 1 e− jωt
and
x2 = X 2 e− jωt .
(7)
Therefore Eqs. (5) and (6) become −Mω2 X 1 = −K (X 1 − Y ) − k(X 1 − X 2 )
(8)
−mω2 X 2 = −k(X 2 − X 1 ) − jcω(X 2 − Y ).
(9)
Solving these equations yields
√
1 γopt A = , 1+µ s 3µ ζopt A = . 8(1 + µ)
(2) (3)
An approximate value of the amplitude ratio at resonance derived by Den Hartog is s X1 2+µ = . (4) Y µ max A Eq. (4) above shows that the theoretical limit of the amplitude ratio at resonance is one when the mass ratio µ approaches infinity. 3. A variant form of the damped dynamic vibration absorber A variant form of the damped dynamic vibration absorber as shown in Fig. 1(b) is called model B in the following discussion. In Fig. 1(b), the motions of the primary system and the dynamic vibration absorber are governed by the following equations: M x¨1 = −K (x1 − y) − k(x1 − x2 ) m x¨2 = −k(x2 − x1 ) − c(x˙2 − y˙ ).
(5) (6)
X1 K (k − mω2 + jcω) + jkcω . = Y (K − Mω2 + k)(k − mω2 + jcω) − k 2
(10)
Eq. (10) may be rewritten with the non-dimensional variables as used in Eq. (1) as X1 (γ 2 − λ2 + 2 jζ γ λ) + 2 jµγ 3 ζ λ = . Y (1 − λ2 + µγ 2 )(γ 2 − λ2 + 2 jζ γ λ) − µγ 4
(11)
The ratio between the vibration amplitude of the mass M and that of the support is X G = 1 Y s
(γ 2 − λ2 )2 + (1 + µγ 2 )2 (2γ ζ λ)2 [(1 − λ2 )(γ 2 − λ2 ) − µγ 2 λ2 ]2 + (1 + µγ 2 − λ2 )2 (2γ λζ )2
= s =
A + Bζ 2 C + Dζ 2
(12)
where A = (γ 2 − λ2 )2 , B = (1 + µγ 2 )2 (2γ λ)2 , C = [(1−λ2 )(γ 2 −λ2 )−µγ 2 λ2 ]2 and D = (2γ λ)2 (1+µγ 2 −λ2 )2 . Eq. (12) is calculated with mass ratio µ = 0.2 at four different damping ratios and the results are plotted in Fig. 2. It can be observed that there are four stationary points (P, Q, R and S) at which the response |X 1 /Y | is independent of damping of the absorber. By the fix-points theory, the stationary points can be found under this condition, r X1 A = (13) Y C ς =0 r X1 B = . (14) Y D ς =∞
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To find λ P and λ Q , we consider γ 2 − λ2 1 + µγ 2 = − . (1 + µγ 2 − λ2 )(γ 2 − λ2 ) − µγ 4 1 + µγ 2 − λ2
(17)
Manipulating Eq. (17) yields λ4 − (1 + γ 2 + µγ 2 )λ2 + 2γ 2 (1 + µγ 2 )/(2 + µγ 2 ) = 0. (18) Suppose that the roots of Eq. (18) are λ P and λ Q . Eq. (18) may be rewritten as (λ2 − λ2P )(λ2 − λ2Q ) = λ4 − (λ2P + λ2Q )λ2 + λ2P λ2Q = 0. (19) Comparing Eqs. (18) and (19) yields Fig. 2. Vibration amplitude ratio between the primary system and the ground motion. µ = 0.2 and γ = 1.1. ζ = 0.1 (- - - -), ζ = 0.3 (—), ζ = 0.5 (– – –), ζ = 0.7 (-·-·-·-).
λ2P + λ2Q = (1 + γ 2 + µγ 2 ).
(20)
In order to reach the optimum tuning, the responses of mass M at λ = λ P and λ = λ Q should be the same [1], therefore 1 + µγ 2 1 + µγ 2 = − , 1 + µγ 2 − λ2P 1 + µγ 2 − λ2Q
λ Q > λ P > 0.
(21)
Manipulating Eq. (21) yields λ2P + λ2Q = 2(1 + µγ 2 ).
(22)
From Eqs. (20) and (22) the optimum tuning frequency is γopt = √
Fig. 3. Displacement ratio between the primary system and the ground motion. µ = 0.2 and γ = 1.1. (X 1 /Y )ζ =0 (—), (X 1 /Y )ζ =∞ (- - - -).
At the stationary or fixed points, P, Q, R and S, we may write A B = C D ∴
(15)
(γ 2 − λ2 )2 (1 + µγ 2 )2 = [(1 − λ2 )(γ 2 − λ2 ) − µγ 2 λ2 ]2 (1 + µγ 2 − λ2 )2
or γ 2 − λ2 1 + µγ 2 = ± . (1 − λ2 )(γ 2 − λ2 ) − µγ 2 λ2 1 + µγ 2 − λ2
(16)
To have a better understanding of the phase relationship between the displacement ratio (X 1 /Y )ζ =0 and (X 1 /Y )ζ =∞ , they are calculated using Eq. (11) with mass ratio µ = 0.2 and frequency ratio γ = 1.1. The two displacement ratios are plotted in Fig. 3. (X 1 /Y )ζ =0 and (X 1 /Y )ζ =∞ at λ P and λ Q are in opposite phase while (X 1 /Y )ζ =0 and (X 1 /Y )ζ =∞ at λ R and λ S are in phase.
1 . 1−µ
(23)
The frequency ratios of the stationary points, λ P and λ Q can be solved from Eqs. (20), (22) and (23) as r µ 1 2 1− (24) λP = 1−µ 2−µ r µ 1 λ2Q = 1+ . (25) 1−µ 2−µ Substitute Eq. (24) or (25) into Eq. (12), the response |X 1 /Y | at the stationary points is s 2−µ G P,Q = . (26) µ To determine the optimum damping in order to make points P and Q to be the maximum points on the response curve. It requires zero slope at the two stationary points, P and Q. We may therefore write 2 ∂ X1 =0 (27) 2 Y ∂λ ∂p ∂q q− 2p=0 ∂λ2 ∂λ where p = (γ 2 − λ2 )2 + (2ζ γ λ + 2µγ 3 ζ λ)2 q = [(1 + µγ 2 − λ2 )(γ 2 − λ2 ) − µγ 4 ]2 + (2ζ γ λ)2 (1 + µγ 2 − λ2 )2 .
(28)
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W.O. Wong, Y.L. Cheung / Engineering Structures 30 (2008) 282–286
Differentiating p and q with respect to λ2 , we have ∂p = 2(λ2 − γ 2 ) + (2ζ γ )2 (1 + µγ 2 )2 ∂λ2 ∂q = −2[(1 + µγ 2 − λ2 )(γ 2 − λ2 ) − µγ 4 ] ∂λ2 × [1 + (1 + µ)γ 2 − 2λ2 ] + (2ζ γ )2 (1 + µγ 2 − λ2 )(1 + µγ 2 − 3λ2 ). Eq. (28) may be rewritten as p ∂p ∂q = . q ∂λ2 ∂λ2 Under the optimum tuning condition 2 p X1 2−µ = = . Y q µ
(29)
(30)
(31)
(32)
Fig. 4. Comparison of the amplitude ratio between the vibration of the primary system to the ground motion for Model A (– – –) and Model B (solid line) at mass ratio m/M = 0.2.
Substituting Eqs. (29), (30) and (32) into Eq. (31), we have 2(λ2 − γ 2 ) + (2ζ γ )2 (1 + µγ 2 )2 2−µ = {−2[(1 + µγ 2 − λ2 )(γ 2 − λ2 ) − µγ 4 ] µ × [1 + (1 + µ)γ 2 − 2λ2 ] + (2ζ γ )2 (1 + µγ 2 − λ2 )(1 + µγ 2 − 3λ2 )}.
(33)
Eq. (33) may be rewritten as ζ2 = 2µ(λ2 − γ 2 ) + 2(2 − µ)[(1 + µγ 2 − λ2 )(γ 2 − λ2 ) − µγ 4 ][(1 + (1 + µ)γ 2 − 2λ2 )] (2γ )2 (2 − µ)(1 + µγ 2 − λ2 )(1 + µγ 2 − 3λ2 ) − µ(2γ )2 (1 + µγ 2 )2
.
(34) Substituting Eqs. (23)–(25) into the about equation results in the optimum damping as 2 ζ P,Q =−
µ(µ − 3)(1 − µ) . q µ 4(2 − µ) −1 ± 2−µ
(35)
2 Taking an average of ζ P,Q produces
ζa2 + ζb2 µ(3 − µ) = . (36) 2 8 From Eq. (26), the resonant amplitude ratio of model B under optimum tuning and damping may be written as s X1 2−µ = . (37) Y µ max B 2 ζopt =
The amplitude ratios between the vibration of the primary system to the ground motion for models A and B under optimum tuning and damping with mass ratio µ = 0.2 were calculated using Eqs. (1)–(3), (12), (23) and (36), and the calculation result is plotted in Fig. 4. The non-dimensional resonant amplitude of mass M of model B is smaller than that of Model A. The additional reduction of vibration amplitude in using model B is 7%. To prove that the proposed absorber is in general superior to the traditional absorber for vibration suppression, the additional
Fig. 5. Percentage reduction of vibration amplitude of M of the proposed absorber relative to the traditional absorber at different mass ratio.
reduction of vibration amplitude of the primary system using the proposed absorber relative to the case of using the traditional absorber is written as X1 X1 X1 − Y Y Y max B max A max A s s ! s 2+µ 2−µ 2+µ = − µ µ µ =
2µ p > 0, 2 + µ + 4 − µ2
for 0 > µ ≥ 2.
(38)
The above equation shows that the proposed absorber provides smaller resonant vibration amplitude of the primary mass M excited by ground motion than the traditional absorber if the absorbers are optimally tuned. Eq. (38) is plotted in Fig. 5 for showing the percentage reduction of vibration of the mass M of the proposed absorber relative to the traditional absorber at different mass ratio.
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4. Conclusion Optimum tuning condition including the frequency and damping ratios of the proposed absorber has been derived based on the fixed-points theory. Under the optimum tuning condition of the absorbers, it is proved analytically that the proposed absorber provide a larger suppression of resonant vibration amplitude of the primary system excited by ground motion than the traditional absorber. The comparison revealed that though model B requires a larger amount of damping than model A, the resonant vibration amplitude under the optimized condition of model B is always less than that of model A. It provides an alternative design for the traditional damped dynamic vibration absorber. References [1] Den Hartog JP. Mechanical vibrations. Dover Publications Inc; 1985. [2] Hunt JB. Dynamic vibration absorbers. Mechanical Engineering Publications Ltd; 1979. [3] Korenev BG, Reznikov LM. Dynamic vibration absorbers, theory and technical applications. John Wiley & Sons; 1993. [4] Jalili N, Knowles DW. Structural vibration control using an active resonator absorber: Modeling and control implementation. Smart
Materials and Structures 2004;13(5):998–1005. [5] Chen YD, Fuh CC, Tung PC. Application of voice coil motors in active dynamic vibration absorbers. IEEE Transactions on Magnetics 2005; 41(3):1149–54. [6] Lin J. An active vibration absorber of smart panel by using a decomposed parallel fuzzy control structure. Engineering Applications of Artificial Intelligence 2005;18(8):985–98. [7] Bonello P, Brennan MJ, Elliott SJ, et al. Designs for an adaptive tuned vibration absorber with variable shape stiffness element. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences 2005;461:3955–76. [8] Williams KA, Chiu GTC, Bernhard RJ. Nonlinear control of a shape memory alloy adaptive tuned vibration absorber. Journal of Sound and Vibration 2005;288:1131–55. [9] Taflanidis AA, Angelides DC, Manos GC. Optimal design and performance of liquid column mass dampers for rotational vibration control of structures under white noise excitation. Engineering Structures 2005;27(4):524–34. [10] Webste AL, Semke WH. Broad-band viscoelastic rotational vibration control for remote sensing applications. Journal of Vibration and Control 2005;11(11):1339–56. [11] Rudinger F. Optimal vibration absorber with nonlinear viscous power law damping and white noise excitation. Journal of Engineering MechanicsASCE 2006;132(1):46–53. [12] Liu K, Liu J. The damped dynamic vibration absorbers: Revisited and new result. Journal of Sound and Vibration 2005;284:1181–9.
Short communication
Optimal design of a damped dynamic vibration absorber for vibration control of structure excited by ground motion W.O. Wong ∗ , Y.L. Cheung Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong Received 15 June 2006; received in revised form 12 October 2006; accepted 12 March 2007 Available online 17 April 2007
Abstract Optimum parameters of a dynamic vibration absorber of non-traditional form have been derived for suppressing vibration of a single degreeof-freedom system due to ground motion. The reduction of transmission of motion from the support to the mass of the structure is compared for the cases of using the traditional and the proposed dynamic absorbers. Under the optimum tuning condition of the absorbers, it is proved analytically that the proposed absorber provides a larger suppression of resonant vibration amplitude of the primary system excited by ground motion than the traditional absorber. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Vibration absorber; Tuned mass damper; Damping
1. Introduction Dynamic vibration absorbers can be used for passive control of narrowband vibration. As described in the classical textbooks on dynamic vibration absorber [1–3], an undamped dynamic vibration absorber is an auxiliary mass–spring system which, when correctly tuned and attached to a vibrating body subject to a harmonic excitation, eliminates steady-state motion of the point to which it is attached. The traditional damped vibration absorber has a damper added between the absorber mass m and the primary mass M as shown in Fig. 1(a) to limit the vibration amplitude when the lower resonance is experienced during system start-up and stopping. However, it is not possible to eliminate steady-state vibrations of the original mass after damping is added in the auxiliary mass–spring system. Optimization of the frequency and damping parameters of the traditional damped vibration absorber for minimization of the resonant vibration amplitudes based on the fixed-points theory is well documented in the textbook of Den Hartog [1]. Recent advances of absorber designs involve addition of active controlled elements [4–6], adaptive elements that can be used to change the tuned condition of the absorber [7,8], ∗ Corresponding author. Tel.: +852 2766 6667; fax: +852 2365 4703.
E-mail address: [email protected] (W.O. Wong). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.03.007
and passive device with non-traditional elements for broadband vibration control [9–11]. These advanced absorbers are useful in applications requiring the change of tuning parameters as the main system changes in time. On the other hand, the traditional dynamic vibration absorber may provide a cheaper and convenient solution for vibration suppression and isolation of vibrating systems with harmonic excitation. In this paper, the optimum tuning frequency and damping of a damped dynamic vibration absorber of non-traditional form as shown in Fig. 1(b) have been derived for suppressing vibration of a single-degreeof-freedom system due to ground motion. The derivation of the formulae for the optimum tuning frequency and damping of the absorber was based on the fixed-points theory of Den Hartog [1]. It is proved that the proposed absorber provides a larger suppression of resonant vibration amplitude of the primary system excited by ground motion than the traditional absorber. 2. The traditional damped dynamic vibration absorber A schematic diagram of a traditional damped dynamic vibration absorber attached to an undamped mass–spring system is shown in Fig. 1(a). This vibration model is called model A in the following discussion. The amplitude ratio
W.O. Wong, Y.L. Cheung / Engineering Structures 30 (2008) 282–286
283
Fig. 1. A damped dynamic vibration absorber as an auxiliary mass–spring–damper (m–k–c) system attached to a primary system (M–K ) (a) traditional design of the absorber [1], (b) the proposed design of the absorber for suppressing the vibration of the mass M due to ground motion.
|X 1 /Y |A is: X1 Y A (k − mω2 + jcω)K = [(K − Mω2 )(k − mω2 ) − mω2 k] + jcω(K − Mω2 − mω2 ) s (γ 2 − λ2 )2 + (2ζ γ λ)2 = [(1 − λ2 )(γ 2 − λ2 ) − µγ 2 λ2 ]2 + (2ζ γ λ)2 (1 − µλ2 − λ2 )2
(1) √ √ where √ λ = ω/ K /M, γ = k/m/ K /M, µ = m/M, ζ = c/2 mk. Based on the result of Liu and Liu [12], the optimum tuning condition and the optimum damping using the fixed-points theory [1] leading to minimum vibration amplitude at resonance are
Assuming a harmonic disturbing input y = Y e− jωt , and the responses may be written as x1 = X 1 e− jωt
and
x2 = X 2 e− jωt .
(7)
Therefore Eqs. (5) and (6) become −Mω2 X 1 = −K (X 1 − Y ) − k(X 1 − X 2 )
(8)
−mω2 X 2 = −k(X 2 − X 1 ) − jcω(X 2 − Y ).
(9)
Solving these equations yields
√
1 γopt A = , 1+µ s 3µ ζopt A = . 8(1 + µ)
(2) (3)
An approximate value of the amplitude ratio at resonance derived by Den Hartog is s X1 2+µ = . (4) Y µ max A Eq. (4) above shows that the theoretical limit of the amplitude ratio at resonance is one when the mass ratio µ approaches infinity. 3. A variant form of the damped dynamic vibration absorber A variant form of the damped dynamic vibration absorber as shown in Fig. 1(b) is called model B in the following discussion. In Fig. 1(b), the motions of the primary system and the dynamic vibration absorber are governed by the following equations: M x¨1 = −K (x1 − y) − k(x1 − x2 ) m x¨2 = −k(x2 − x1 ) − c(x˙2 − y˙ ).
(5) (6)
X1 K (k − mω2 + jcω) + jkcω . = Y (K − Mω2 + k)(k − mω2 + jcω) − k 2
(10)
Eq. (10) may be rewritten with the non-dimensional variables as used in Eq. (1) as X1 (γ 2 − λ2 + 2 jζ γ λ) + 2 jµγ 3 ζ λ = . Y (1 − λ2 + µγ 2 )(γ 2 − λ2 + 2 jζ γ λ) − µγ 4
(11)
The ratio between the vibration amplitude of the mass M and that of the support is X G = 1 Y s
(γ 2 − λ2 )2 + (1 + µγ 2 )2 (2γ ζ λ)2 [(1 − λ2 )(γ 2 − λ2 ) − µγ 2 λ2 ]2 + (1 + µγ 2 − λ2 )2 (2γ λζ )2
= s =
A + Bζ 2 C + Dζ 2
(12)
where A = (γ 2 − λ2 )2 , B = (1 + µγ 2 )2 (2γ λ)2 , C = [(1−λ2 )(γ 2 −λ2 )−µγ 2 λ2 ]2 and D = (2γ λ)2 (1+µγ 2 −λ2 )2 . Eq. (12) is calculated with mass ratio µ = 0.2 at four different damping ratios and the results are plotted in Fig. 2. It can be observed that there are four stationary points (P, Q, R and S) at which the response |X 1 /Y | is independent of damping of the absorber. By the fix-points theory, the stationary points can be found under this condition, r X1 A = (13) Y C ς =0 r X1 B = . (14) Y D ς =∞
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To find λ P and λ Q , we consider γ 2 − λ2 1 + µγ 2 = − . (1 + µγ 2 − λ2 )(γ 2 − λ2 ) − µγ 4 1 + µγ 2 − λ2
(17)
Manipulating Eq. (17) yields λ4 − (1 + γ 2 + µγ 2 )λ2 + 2γ 2 (1 + µγ 2 )/(2 + µγ 2 ) = 0. (18) Suppose that the roots of Eq. (18) are λ P and λ Q . Eq. (18) may be rewritten as (λ2 − λ2P )(λ2 − λ2Q ) = λ4 − (λ2P + λ2Q )λ2 + λ2P λ2Q = 0. (19) Comparing Eqs. (18) and (19) yields Fig. 2. Vibration amplitude ratio between the primary system and the ground motion. µ = 0.2 and γ = 1.1. ζ = 0.1 (- - - -), ζ = 0.3 (—), ζ = 0.5 (– – –), ζ = 0.7 (-·-·-·-).
λ2P + λ2Q = (1 + γ 2 + µγ 2 ).
(20)
In order to reach the optimum tuning, the responses of mass M at λ = λ P and λ = λ Q should be the same [1], therefore 1 + µγ 2 1 + µγ 2 = − , 1 + µγ 2 − λ2P 1 + µγ 2 − λ2Q
λ Q > λ P > 0.
(21)
Manipulating Eq. (21) yields λ2P + λ2Q = 2(1 + µγ 2 ).
(22)
From Eqs. (20) and (22) the optimum tuning frequency is γopt = √
Fig. 3. Displacement ratio between the primary system and the ground motion. µ = 0.2 and γ = 1.1. (X 1 /Y )ζ =0 (—), (X 1 /Y )ζ =∞ (- - - -).
At the stationary or fixed points, P, Q, R and S, we may write A B = C D ∴
(15)
(γ 2 − λ2 )2 (1 + µγ 2 )2 = [(1 − λ2 )(γ 2 − λ2 ) − µγ 2 λ2 ]2 (1 + µγ 2 − λ2 )2
or γ 2 − λ2 1 + µγ 2 = ± . (1 − λ2 )(γ 2 − λ2 ) − µγ 2 λ2 1 + µγ 2 − λ2
(16)
To have a better understanding of the phase relationship between the displacement ratio (X 1 /Y )ζ =0 and (X 1 /Y )ζ =∞ , they are calculated using Eq. (11) with mass ratio µ = 0.2 and frequency ratio γ = 1.1. The two displacement ratios are plotted in Fig. 3. (X 1 /Y )ζ =0 and (X 1 /Y )ζ =∞ at λ P and λ Q are in opposite phase while (X 1 /Y )ζ =0 and (X 1 /Y )ζ =∞ at λ R and λ S are in phase.
1 . 1−µ
(23)
The frequency ratios of the stationary points, λ P and λ Q can be solved from Eqs. (20), (22) and (23) as r µ 1 2 1− (24) λP = 1−µ 2−µ r µ 1 λ2Q = 1+ . (25) 1−µ 2−µ Substitute Eq. (24) or (25) into Eq. (12), the response |X 1 /Y | at the stationary points is s 2−µ G P,Q = . (26) µ To determine the optimum damping in order to make points P and Q to be the maximum points on the response curve. It requires zero slope at the two stationary points, P and Q. We may therefore write 2 ∂ X1 =0 (27) 2 Y ∂λ ∂p ∂q q− 2p=0 ∂λ2 ∂λ where p = (γ 2 − λ2 )2 + (2ζ γ λ + 2µγ 3 ζ λ)2 q = [(1 + µγ 2 − λ2 )(γ 2 − λ2 ) − µγ 4 ]2 + (2ζ γ λ)2 (1 + µγ 2 − λ2 )2 .
(28)
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W.O. Wong, Y.L. Cheung / Engineering Structures 30 (2008) 282–286
Differentiating p and q with respect to λ2 , we have ∂p = 2(λ2 − γ 2 ) + (2ζ γ )2 (1 + µγ 2 )2 ∂λ2 ∂q = −2[(1 + µγ 2 − λ2 )(γ 2 − λ2 ) − µγ 4 ] ∂λ2 × [1 + (1 + µ)γ 2 − 2λ2 ] + (2ζ γ )2 (1 + µγ 2 − λ2 )(1 + µγ 2 − 3λ2 ). Eq. (28) may be rewritten as p ∂p ∂q = . q ∂λ2 ∂λ2 Under the optimum tuning condition 2 p X1 2−µ = = . Y q µ
(29)
(30)
(31)
(32)
Fig. 4. Comparison of the amplitude ratio between the vibration of the primary system to the ground motion for Model A (– – –) and Model B (solid line) at mass ratio m/M = 0.2.
Substituting Eqs. (29), (30) and (32) into Eq. (31), we have 2(λ2 − γ 2 ) + (2ζ γ )2 (1 + µγ 2 )2 2−µ = {−2[(1 + µγ 2 − λ2 )(γ 2 − λ2 ) − µγ 4 ] µ × [1 + (1 + µ)γ 2 − 2λ2 ] + (2ζ γ )2 (1 + µγ 2 − λ2 )(1 + µγ 2 − 3λ2 )}.
(33)
Eq. (33) may be rewritten as ζ2 = 2µ(λ2 − γ 2 ) + 2(2 − µ)[(1 + µγ 2 − λ2 )(γ 2 − λ2 ) − µγ 4 ][(1 + (1 + µ)γ 2 − 2λ2 )] (2γ )2 (2 − µ)(1 + µγ 2 − λ2 )(1 + µγ 2 − 3λ2 ) − µ(2γ )2 (1 + µγ 2 )2
.
(34) Substituting Eqs. (23)–(25) into the about equation results in the optimum damping as 2 ζ P,Q =−
µ(µ − 3)(1 − µ) . q µ 4(2 − µ) −1 ± 2−µ
(35)
2 Taking an average of ζ P,Q produces
ζa2 + ζb2 µ(3 − µ) = . (36) 2 8 From Eq. (26), the resonant amplitude ratio of model B under optimum tuning and damping may be written as s X1 2−µ = . (37) Y µ max B 2 ζopt =
The amplitude ratios between the vibration of the primary system to the ground motion for models A and B under optimum tuning and damping with mass ratio µ = 0.2 were calculated using Eqs. (1)–(3), (12), (23) and (36), and the calculation result is plotted in Fig. 4. The non-dimensional resonant amplitude of mass M of model B is smaller than that of Model A. The additional reduction of vibration amplitude in using model B is 7%. To prove that the proposed absorber is in general superior to the traditional absorber for vibration suppression, the additional
Fig. 5. Percentage reduction of vibration amplitude of M of the proposed absorber relative to the traditional absorber at different mass ratio.
reduction of vibration amplitude of the primary system using the proposed absorber relative to the case of using the traditional absorber is written as X1 X1 X1 − Y Y Y max B max A max A s s ! s 2+µ 2−µ 2+µ = − µ µ µ =
2µ p > 0, 2 + µ + 4 − µ2
for 0 > µ ≥ 2.
(38)
The above equation shows that the proposed absorber provides smaller resonant vibration amplitude of the primary mass M excited by ground motion than the traditional absorber if the absorbers are optimally tuned. Eq. (38) is plotted in Fig. 5 for showing the percentage reduction of vibration of the mass M of the proposed absorber relative to the traditional absorber at different mass ratio.
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4. Conclusion Optimum tuning condition including the frequency and damping ratios of the proposed absorber has been derived based on the fixed-points theory. Under the optimum tuning condition of the absorbers, it is proved analytically that the proposed absorber provide a larger suppression of resonant vibration amplitude of the primary system excited by ground motion than the traditional absorber. The comparison revealed that though model B requires a larger amount of damping than model A, the resonant vibration amplitude under the optimized condition of model B is always less than that of model A. It provides an alternative design for the traditional damped dynamic vibration absorber. References [1] Den Hartog JP. Mechanical vibrations. Dover Publications Inc; 1985. [2] Hunt JB. Dynamic vibration absorbers. Mechanical Engineering Publications Ltd; 1979. [3] Korenev BG, Reznikov LM. Dynamic vibration absorbers, theory and technical applications. John Wiley & Sons; 1993. [4] Jalili N, Knowles DW. Structural vibration control using an active resonator absorber: Modeling and control implementation. Smart
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