Parameter optimization of tuned mass damper for three-Degree-of-freedom vibration systems

Volume 35 Number 3 3 Vietnam Journal of Mechanics, VAST, Vol. 35, No. 3 (2013), pp. 215 – 224 PARAMETER OPTIMIZATION OF TUNED MASS DAMPER FOR THREE-DEGREE-OF-FREEDOM VIBRATION SYSTEMS Nguyen Van Khang1,∗, Trieu Quoc Loc2, Nguyen Anh Tuan2 1 Hanoi University of Science and Technology, Vietnam 2 National Institute of Labour Protection, Vietnam ∗E-mail: khang.nguyenvan2@hust.edu.vn Abstract. There are problems in mechanical, structural and aerospace engineering that can be formulated as Nonlinear Programming. In this paper, the problem of parameters optimization of tuned mass damper for three-degree-of-freedom vibration systems is in- vestigated using sequential quadratic programming method. The objective is to minimize the extreme vibration amplitude of vibration models. It is shown that the constrained formulation, that includes lower and upper bounds on the updating parameters in the form of inequality constraints, is important for obtaining a correct updated model. Keywords : Vibration, tuned mass damper, optimal design, nonlinear programming. 1. INTRODUCTION Optimal design of multibody systems is characterized by a specific kind of optimiza- tion problem. Generally, an optimization problem is formulated to determine the design variable values that will minimize an objective function subject to constraints. Addition- ally, for many engineering applications, multibody analysis routine are used to calculate the kinematic and dynamic behavior of the mechanical design. As a result, most objective function and constraint values follow from the numerical analysis. Use of the tuned mass damper (TMD) as an independent means of vibration control is especially important, particularly in the case where it is almost the only or main means of vibration protection [1-6]. A tuned mass damper, also known as an active mass damper (AMD) or harmonic absorber, is a device mounted in structures to reduce the amplitude of vibrations. Its application can prevent discomfort, damage, or outright structural failure. It is frequently used in power transmission, automobiles, machine and buildings. In this paper we consider a problem of parameter optimization of tuned mass damper for three-degree-of-freedom vibration systems using sequential quadratic programming method [7-12]. 216 Nguyen Van Khang, Trieu Quoc Loc, Nguyen Anh Tuan 2. REVIEW OF SEQUENTIAL QUADRATIC PROGRAMMING METHOD The sequential quadratic programming, or called SQP, is an efficient and powerful algorithm to solve nonlinear programming problems. The method has a theoretical basis that is related to (1) the solution of a set of nonlinear equations using Newton’s method, and (2) the derivation of simultaneous nonlinear equations using Kuhn–Tu¨cker conditions to the Lagrangian of the constrained optimization problem. In this section we review some basic concepts of SQP method [7-10] for understanding the parameter optimization of the TMD installed in vibration systems. Consider a nonlinear optimization problem with equality constraints: Find x which minimizes f(x) subject to hk(x) = 0, k = 1, 2, . . . , p. (1) The Lagrange function L(x,λ), for this problem is L(x,λ) = f(x) + p∑ k=1 λkhk(x) = f(x) + λ Th(x), (2) where λk is the Lagrange multiplier for the equality constraint hk. The Kuhn–Tu¨cker necessary conditions can be stated as ∇xL = 0 ⇒ ∇f(x) + p∑ k=1 λk∇hk = 0 or ∇f(x) + λ Th(x) = 0, (3) ∇λL = 0 ⇒ hk(x) = 0, k = 1, 2, . . . , p or h(x) = 0. (4) Eqs. (3) and (4) represent a set of n + p nonlinear equations with n + p unknowns (xi, i = 1, 2, . . . , n and λk, k = 1, 2, . . . , p). These nonlinear equations can be solved using Newton’s method. For convenience, we rewrite Eqs. (3) and (4) as b(y) = 0, (5) where b = { ∇L h } (n+p)×1 , y = { x λ } (n+p)×1 , 0 = { 0 0 } (n+p)×1 . (6) According to Newton’s method, the solution of Eqs. (5) can be found iteratively as[ ∇2xL(yi) J T h (xi) Jh(xi) 0 ]{ ∆xi ∆λi } = − { ∇xL(yi) h(xi) } , (7) and xi+1 = xi +∆xi, λi+1 = λi +∆λi. (8) The first set of equations in (7) can be written separately as ∇ 2 xL(yi)∆xi + J T h (xi)∆λi = −∇xL(yi) (9) Using Eq. (8) for ∆λi and Eq. (3) for ∇xL(yi), Eq. (9) can be expessed as ∇ 2 xL(yi)∆xi + J T h (λi+1 − λi) = −∇f(xi)− J T h (xi)λi, (10) Parameter optimization of tuned mass damper for three-degree-of-freedom vibration systems 217 which can be simplified to obtain ∇ 2 xL(yi)∆xi + J T h (xi)λi+1 = −∇f(xi). (11) Eq. (11) and the second set of equations in (7) can now be combined as[ ∇2xL(yi) J T h (xi) Jh(xi) 0 ] j { ∆xi λi+1 } = − { ∇f(xi) h(xi) } . (12) Eqs. (12) can be solved to find the change in the design vector ∆xi and the new values of the Lagrange multipliers, λi+1. The iterative process indicated by Eq. (12) can be continued until convergence is achieved. Now consider the following quadratic programming problem: Find d = ∆x that minimizes the quadratic objective function Q(d) = ∇xf(xi) Td+ 1 2 dT∇2xL(xi,λi)d, (13) subject to the linear equality constraints hk(xi) +∇h T k (xi)d = 0, k = 1, 2, . . . , p ⇒ h(xi) + Jh(xi)d = 0. (14) The Lagange function L˜, corresponding to the problem of Eqs. (13) and (14) is given by L˜(d,λ) = ∇xf(xi) Td+ 1 2 dT∇2xL(xi,λi)d+ λ T [h (xi) + Jh (xi)d] . (15) The Kuhn – Tu¨cker necessary conditions can be stated as ∇xf(xi) +∇ 2 xL(xi,λi)d+ J T h (xi)λ = 0, (16) h(xi) + Jh(xi)d = 0. (17) The Eqs. (16) and (17) can be combined in the following matrix form as[ ∇2xL(yi) J T h (xi) Jh(xi) 0 ] j { di λi } = − { ∇f(xi) h(xi) } . (18) Eq. (18) can be identified to be same as Eq. (12) in matrix form. This shows that the orginal problem of Eq. (1) can be solved iteratively by solving the quadratic programming problem defined by Eq. (13). In fact, when inequality constraints are added to the original problem, the quadratic programming problem of Eqs. (13) and (14) becomes Find x which minimizes Q(d) = (∇f(xi)) Td+ 1 2 dT∇2xL(xi,λi,µi)d, (19) subject to hk(xi) + (∇hk(xi)) Td = 0, k = 1, 2, . . . , p (20) gj(xi) + (∇gj(xi)) Td ≤ 0, j = 1, 2, . . . , m (21) with the Lagrange function given by L(x,λ,µ) = f(x) + p∑ k=1 λkhk(x) + m∑ j=1 µjgj(x) = f(x) + λ Th(x) + µTg(x) (22) 218 Nguyen Van Khang, Trieu Quoc Loc, Nguyen Anh Tuan Since the minimum of the augmented Lagrange function is involved, the sequential quadratic programming method is also known as the projected Lagrangian method. 3. CALCULATING OPTIMAL PARAMETERS OF TMD FOR THE THREE-DEGREE-OF-FREEDOM VIBRATION SYSTEMS In this section we study the influence of installed position of TMD on the behaviour of three-degree-of-freedom vibration systems using the sequential quadratic programming algorithm. 3.1. Vibration equation of system with the excited harmonic force at the mass m1 Consider a damped linear vibration system of three-degree-of-freedom as shown in Fig. 1a. The vibrating system has three masses m1, m2, m3; stiffness coefficients, respec- tively, k1, k2, k3 and viscous coefficients, respectively, c1, c2, c3; the mass m1 is excited by harmonic force F (t) = F0 cos(Ωt). The motion equations of the system have the following form m1y¨1 + (c1 + c2)y˙1 − c2y˙2 + (k1 + k2)y1 − k2y2 = F0cos(Ωt) m2y¨2 − c2y˙1 + (c2 + c3)y˙2 − c3y˙3 − k2y1 + (k2 + k3)y2 − k3y3 = 0 m3y¨3 − c3y˙2 + c3y˙3 − k3y2 + k3y3 = 0 . (23) a) b) c) d) Fig. 1. The system of three-degree-of-freedom under excited force at m1 a) Primary system without TMD; b) System with TMD at m1 c) System with TMD at m2; d) System with TMD at m3 The steady-state response of the system has the form y(t) = a cos(Ωt) + b sin(Ωt) (24) with y(t) =   y1(t)y2(t) y3(t)   ; a0 =   a01a02 a03   ;b0 =   b01b02 b03   . Parameter optimization of tuned mass damper for three-degree-of-freedom vibration systems 219 From Eq. (23) and Eq. (24), comparing coefficients of cos(Ωt) and sin(Ωt), we get the system of linear algebraic equations for unknown elements of vectors a and b (k1 + k2 −m1Ω 2)a01 + (c1 + c2)Ωb01 − k2a02 − c2Ωb02 = F0 −(c1 + c2)Ωa01 + (k1 + k2 −m1Ω 2)b01 + c2Ωa02 − k2b02 = 0 −k2a01 − c2Ωb01 + (k2 + k3 −m2Ω 2)a02 + (c2 + c3)Ωb02 − k3a03 − c3Ωb03 = 0 c2Ωa01 − k2b01 − (c2 + c3)Ωa02 + (k2 + k3 −m2Ω 2)b02 + c3Ωa03 − k3b03 = 0 −k3a02 − c3Ωb02 + (k3 −m3Ω 2)a03 + c3Ωb03 = 0 c3Ωa02 − k3b02 − c3Ωa03 + (k3 −m3Ω 2)b03 = 0 . (25) By solving the system of Eqs. (25), we receive the values of elements a0i, b0i (i = 1, 2, 3) of vectors a0 and b0. For numeric calculation, the values of the coefficients are given as m1 = m2 = m3 =100 kg, k1 = k2 = k3 = 10 5 N/m, c1 = c2 = c3 = 1000 Ns/m, Ω = 47 rad/s, F (t) = 10 cos(47t). 3.2. Installation positions of TMD a) System installed TMD in m1 As the first variant to quench vibrations of the system, we installed TMD with mass mtc, spring stiffness ktc and viscous resistance ctc on mass m1 (Fig. 1b). The equation of the system oscillations m1y¨1 + (c1 + c2 + ctc)y˙1 − c2y˙2 − ctcy˙tc + (k1 + k2 + ktc)y1 − k2y2 − ktcytc = F0 cos(Ωt) m2y¨2 − c2y˙1 + (c2 + c3)y˙2 − c3y˙3 − k2y1 + (k2 + k3)y2 − k3y3 = 0 m3y¨3 − c3y˙2 + c3y˙3 − k3y2 + k3y3 = 0 mtcy¨tc − ctcy˙1 + ctcy˙tc − ktcy1 + ktcytc = 0 (26) The steady-state response of the system has the form y(t) = a cos(Ωt) + b sin(Ωt) (27) where y(t) =   y1(t) y2(t) y3(t) ytc(t)   ; a =   a1 a2 a3 atc   ;b =   b1 b2 b3 btc   From Eqs. (26)-(27), comparing coefficients of cos(Ωt) and sin(Ωt) we get the system of linear algebraic equations for unknown elements of vectors a and b (k1 + k2 + ktc −m1Ω 2)a1 + (c1 + c2 + ctc)Ωb1 − k2a2 − c2Ωb2 − ktcatc − ctcΩbtc = F0 −(c1 + c2 + ctc)Ωa1 + (k1 + k2 + ktc −m1Ω 2)b1 + c2Ωa2 − k2b2 + ctcΩatc − ktcbtc = 0 −k2a1 − c2Ωb1 + (k2 + k3 −m2Ω 2)a2 + (c2 + c3)Ωb2 − k3a3 − c3Ωb3 = 0 c2Ωa1 − k2b1 − (c2 + c3)Ωa2 + (k2 + k3 −m2Ω 2)b2 + c3Ωa3 − k3b3 = 0 −k3a2 − c3Ωb2 + (k3 −m3Ω 2)a3 + c3Ωb3 = 0 c3Ωa2 − k3b2 − c3Ωa3 + (k3 −m3Ω 2)b3 = 0 −ktca1 − ctcΩb1 + (ktc −mtcΩ 2)atc + ctcΩbtc = 0 ctcΩa1 − ktcb1 − ctcΩatc + (ktc −mtcΩ 2)btc = 0 . (28) Solving the system of Eqs. (28), we receive the elements ai, bi (i = 1, 2, 3) of vectors a and b. 220 Nguyen Van Khang, Trieu Quoc Loc, Nguyen Anh Tuan For optimization problems, there is an optimization criterion (i.e. evaluation func- tion) that has to be minimized or maximized. Here we must find the optimal values mtc, ktc, ctc of TMD in order to minimize the expression of vibration amplitude of m1 R1 = √ a21 + b 2 1, with boundary constraints 5 ≤ mtc(kg) ≤ 10; 1000 ≤ ktc(N/m) ≤ 100000; 5≤ ctc(Ns/m) ≤ 1000. Using the sequential quadratic programming algorithm in MAPLE software, we can quickly and conveniently calculate the optimal parameters for TMD R1 = 0.00000451601155 m; ktc = 22099.62597299 N/m; ctc = 5 Ns/m;mtc = 10 kg. Some calculating results are provided in Tab. 1 and in Fig. 2. Table 1. Effective vibration reduction system under excited force at m1 before and after installing TMD at m1 Location Vibration amplitude (m) Efficient vibration damping (%) Without TMD With TMD increase Reduced m1 0.0000653278 0.000004516 93.08 m2 0.0000393333 0.000002719 93.08 m3 0.0000335052 0.000002316 93.08 Fig. 2. Amplitude of three degrees of freedom system under excited force at m1 before and after installing TMD at m1 Parameter optimization of tuned mass damper for three-degree-of-freedom vibration systems 221 b) System installed TMD in m2 As second variant to quench vibrations of the system, we installed TMD with mass mtc, spring stiffness, ktc and viscous resistance, ctc on mass m1 (see Fig. 1c). The vibration equations of the system have following form m1y¨1 + (c1 + c2)y˙1 − c2y˙2 + (k1 + k2)y1 − k2y2 = F0 cos(Ωt) m2y¨2 − c2y˙1 + (c2 + c3 + ctc)y˙2 − c3y˙3 − ctcy˙tc − k2y1 + (k2 + k3 + ktc)y2 − k3y3 − ktcytc = 0 m3y¨3 − c3y˙2 + c3y˙3 − k3y2 + k3y3 = 0 mtcy¨tc − ctcy˙2 + ctcy˙tc − ktcy2 + ktcytc = 0 (29) From Eq. (27) and Eq. (29), comparing coefficients of cos(Ωt) and sin(Ωt) we get the system of linear algebraic equations for unknown elements of vectors a and b (k1 + k2 −m1Ω 2)a1 + (c1 + c2)Ωb1 − k2a2 − c2Ωb2 = F0 −(c1 + c2)Ωa1 + (k1 + k2 −m1Ω 2)b1 + c2Ωa2 − k2b2 = 0 −k2a1 − c2Ωb1 + (k2 + k3 + ktc −m2Ω 2)a2 + (c2 + c3 + ctc)Ωb2 − k3a3 − c3Ωb3 − ktcatc − ctcΩbtc = 0 c2Ωa1 − k2b1 − (c2 + c3 + ctc)Ωa2 + (k2 + k3 + ktc −m2Ω 2)b2 + c3Ωa3 − k3b3 + ctcΩatc − ktcbtc = 0 −k3a2 − c3Ωb2 + (k3 −m3Ω 2)a3 + c3Ωb3 = 0 c3Ωa2 − k3b2 − c3Ωa3 + (k3 −m3Ω 2)b3 = 0 −ktca2 − ctcΩb2 + (ktc −mtcΩ 2)atc + ctcΩbtc = 0 ctcΩa2 − ktcb2 − ctcΩatc + (ktc −mtcΩ 2)btc = 0 (30) Solving the system of Eqs. (30), we receive the elements ai, bi (i = 1, 2, 3) of vectors a and b. Thus, to minimize the vibration amplitude of m2 we must find optimal values mtc, ktc, ctc of TMD to minimize the expression R2 = √ a22 + b 2 2 with boundary constraints 5 ≤ mtc (kg) ≤ 10; 1000≤ ktc (N/m) ≤ 100000; 5≤ ctc (Ns/m) ≤ 1000. Using SQP, we find the optimal parameters for TMD R2 = 0.00000485578798 m; ktc = 22099.07992772 N/m; ctc = 5 Ns/m;mtc = 10 kg. Some calculating results are shown in Tab. 2 and in Fig. 3. Table 2. Effective vibration reduction system under excited force at m1 before and after installing TMD at m2 Location Vibration amplitude (m) Efficient vibration damping (%) Without TMD With TMD increase reduced m1 0.0000653278 0.0000992695 51.95 m2 0.0000393333 0.0000048558 87.65 m3 0.0000335052 0.0000041363 87.65 222 Nguyen Van Khang, Trieu Quoc Loc, Nguyen Anh Tuan Fig. 3. Vibration amplitude of system under excited force at m1 before and after installing TMD at m2 c) System installed TMD in m3 As third variant to quench vibrations of the system, we installed TMD with mass mtc, spring stiffness, ktc and viscous resistance, ctc on mass m3 (see Fig. 1d). The equation of the system oscillations m1y¨1 + (c1 + c2)y˙1 − c2y˙2 + (k1 + k2)y1 − k2y2 = F0 cosΩt m2y¨2 − c2y˙1 + (c2 + c3)y˙2 − c3y˙3 − k2y1 + (k2 + k3)y2 − k3y3 = 0 m3y¨3 − c3y˙2 + (c3 + ctc)y˙3 − ctcy˙tc − k3y2 + (k3 + ktc)y3 − ktcytc = 0 mtcy¨tc − ctcy˙3 + ctcy˙tc − ktcy3 + ktcytc = 0 . (31) From Eq. (27) and Eq. (31), comparing coefficients of cos(Ωt) and sin(Ωt) we get the system of linear algebraic equations for unknown elements of vectors a and b (k1 + k2 −m1Ω 2)a1 + (c1 + c2)Ωb1− k2a2 − c2Ωb2 = F0 −(c1 + c2)Ωa1 + (k1 + k2 −m1Ω 2)b1 + c2Ωa2 − k2b2 = 0 −k2a1 − c2Ωb1 + (k2 + k3 −m2Ω 2)a2 + (c2 + c3)Ωb2− k3a3 − c3Ωb3 = 0 c2Ωa1 − k2b1 − (c2 + c3)Ωa2 + (k2 + k3 −m2Ω 2)b2 + c3Ωa3 − k3b3 = 0 −k3a2 − c3Ωb2 + (k3 + ktc −m3Ω 2)a3 + (c3 + ctc)Ωb3 − ktcatc − ctcΩbtc = 0 c3Ωa2 − k3b2 − (c3 + ctc)Ωa3 + (k3 + ktc −m3Ω 2)b3 + ctcΩatc − ktcbtc = 0 −ktca3 − ctcΩb3 + (ktc −mtcΩ 2)atc + ctcΩbtc = 0 ctcΩa3 − ktcb3 − ctcΩatc + (ktc −mtcΩ 2)btc = 0 . (32) Solving the system of Eqs. (32), and identify the elements ai, bi (i = 1, 2, 3) of vectors a and b. Thus, to minimize the vibration amplitude of m3 we must find optimal values mtc, ktc, ctc of TMD to minimize the expression R3 = √ a23 + b 2 3 with boundary constraints 5 ≤ mtc (kg) ≤ 10; 1000≤ ktc (N/m) ≤ 100000; 5≤ ctc (Ns/m) ≤ 1000. Using SQP, we find the optimal parameters for TMD R3 = 0.00000266217877 m; ktc = 22106.994965140063 N/ m; ctc = 5 Ns/ m;mtc = 10 kg. Some calculating results are shown in Tab. 3 and in Fig. 4. Parameter optimization of tuned mass damper for three-degree-of-freedom vibration systems 223 Table 3. Effective vibration reduction system under excited force at m1 before and after installing TMD at m3 Location Vibration amplitude (m) Efficient vibration damping (%) Without TMD With TMD increase reduced m1 0.0000653278 0.0000471 27.88 m2 0.0000393333 0.0000514 30.64 m3 0.0000335052 0.00000266 92.05 Fig. 4. Vibration amplitude of system under excited force at m1 before and after installing TMD at m3 From the simulation results in Figs 1-4 we have the following observations: When the TMD is installed on mass m1, the vibration amplitudes of masses m1, m2, m3 are significantly reduced. When the TMD is installed on mass m2, the vibration amplitude of masses m2 and m3 are significantly reduced, and the vibration amplitudes of mass m1 decreased very little. When the TMD is installed on the mass m3, the vibration amplitude of massm3 significantly reduced, and the vibration amplitudes of massesm1, m2 decreased very little. 4. CONCLUSION In this paper, the sequential quadratic programming (SQP) method is used to cal- culating parameter optimization of the tuned mass damper (TMD) for three-degree-of- freedom vibration systems. The following concluding remarks have been reached: - If the TMD is attached to the vibration source (excited force or kinematical ex- citement), the effect of vibration reduction will be achieved globally. - If the TMD is attached to the place far away from the vibration source, the effect of vibration reduction will be achieved in the upper masses from the position of TMD. 224 Nguyen Van Khang, Trieu Quoc Loc, Nguyen Anh Tuan - The SQP method can be used in solving complex constrained optimization prob- lems for multibody systems. ACKNOWLEDGEMENT This paper was completed with the financial support by The Vietnam National Foundation for Science and Technology Development (NAFOSTED). REFERENCES [1] Cyril M. Harris, Allan G. Piersol, Harris’ shock and vibration handbook (fifth edition), McGraw-Hill (2002). [2] M. Geradin, D. Rixen, Mechanical vibrations, John Wiley & Sons, Chichester and Masson, Paris (1994). [3] Nguyen Van Khang, Engineering vibration (in Vietnamese), Science and Technical Publishing House, Hanoi (2005). [4] B.G. Korenev, L.M. Reznikov, Dynamic vibration absorbes, John Wiley & Sons, Chichester (1993). [5] Nguyen Dong Anh, La Duc Viet, Reduced vibration by energy dissipation devices (in Viet- namese), Natural Sciences and Technology Publishing House, Hanoi (2007). [6] Nguyen Huy The, The dynamic absorbes (in Vietnamese), Master of Science Thesis, Hanoi University of Science and Technology (2005). [7] Singresu S. Rao, Engineering optimization theory and practice (fourth edition), John Wiley & Sons, Inc (2009). [8] M. Asghar Bhatti, Practical optimization methods with mathematica applications, Springer (2000). [9] Jorge Nocedal, Stephen J. Wright, Numerical optimization (second edition), Springer (2006). [10] R. Fletcher, Practical methods of optimization, John Wiley & Sons, Chichester (1987). [11] D. Bestle, Analyse und optimierung von mehrko¨rpersystemen, Springer-Verlag, Berlin (1994). [12] Nguyen Nhat Le, The basic problem of optimization and optimal control (in Vietnamese), Science and Technical Publishing House, Hanoi (2009). Received February 22, 2012 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 35, N. 3, 2013 CONTENTS Pages 1. N. T. Khiem, L. K. Toan, N. T. L. Khue, Change in mode shape nodes of multiple cracked bar: I. The theoretical study. 175 2. Nguyen Viet Khoa, Monitoring a sudden crack of beam-like bridge during earthquake excitation. 189 3. Nguyen Trung Kien, Nguyen Van Luat, Pham Duc Chinh, Estimating effective conductivity of unidirectional transversely isotropic composites. 203 4. Nguyen Van Khang, Trieu Quoc Loc, Nguyen Anh Tuan, Parameter optimization of tuned mass damper for three-degree-of-freedom vibration systems. 215 5. 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