Approximate complex eigensolution of proportionally damped linear systems supplemented with a passive damper

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X X International International Conference Conference on on Structural Structural Dynamics, Dynamics, EURODYN EURODYN 2017 2017

Approximate Approximate complex complex eigensolution eigensolution of of proportionally proportionally damped damped linear linear systems systems supplemented supplemented with with aa passive passive damper damper a,∗ a S. S. Hraˇ Hraˇccov ova,∗,, Jiˇ Jiˇrr´´ıı N´ N´aaprstek prsteka

a Institute a Institute

of Theoretical and Applied Mechanics AS CR, v.v.i. of Theoretical and Applied Mechanics AS CR, v.v.i. Proseck´a 76, CZ-19000, Prague, Czech Republic Proseck´a 76, CZ-19000, Prague, Czech Republic

Abstract Abstract An approximate and numerically efficient method is developed to complex eigenproblem associated with the discrete proportionally An approximate and numerically efficient method is developed to complex eigenproblem associated with the discrete proportionally damped systems equipped with a passive damper. The presence of the damper changes the dissipative character of the complete damped systems equipped with a passive damper. The presence of the damper changes the dissipative character of the complete system and makes it non-proportionally damped. The eigenanalysis of such systems is conventionally performed in a space of system and makes it non-proportionally damped. The eigenanalysis of such systems is conventionally performed in a space of twice the system's dimension. This makes analysis costly, particularly for large systems. The proposed method avoids using twice the system's dimension. This makes analysis costly, particularly for large systems. The proposed method avoids using this numerically demanding state-space formulation. The determination of complex eigenvalues is based on the approximate this numerically demanding state-space formulation. The determination of complex eigenvalues is based on the approximate solution of the characteristic equation that is derived in the modal space. The perturbation approach is adopted to reflect the solution of the characteristic equation that is derived in the modal space. The perturbation approach is adopted to reflect the differences in the eigenvalues of proportionally and non-proportionally damped systems. The complex eigenvectors are calculated differences in the eigenvalues of proportionally and non-proportionally damped systems. The complex eigenvectors are calculated afterwards with the use of a significantly reduced modal system and obtained eigenvalues. The proposed procedure is easily afterwards with the use of a significantly reduced modal system and obtained eigenvalues. The proposed procedure is easily programmable and enables the calculation of the individual complex eigenvalues and eigenmodes separately, which significantly programmable and enables the calculation of the individual complex eigenvalues and eigenmodes separately, which significantly reduces computational time. reduces computational time. © 2017 The Authors. Published by Elsevier Ltd. © 2017 2017 The TheAuthors. Authors.Published Publishedby byElsevier ElsevierLtd. Ltd. © Peer-review under responsibility of the organizing committee of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN EURODYN 2017. 2017. Keywords: non-proportional damping; complex eigensolution; perturbation method; passive damper. Keywords: non-proportional damping; complex eigensolution; perturbation method; passive damper.

1. Introduction 1. Introduction The numerical demands on complex eigensolution of the linear viscously damped systems strongly depend on the The numerical demands on complex eigensolution of the linear viscously damped systems strongly depend on the damping model used. In case of proportional damping [1] the eigenanalysis is relatively inexpensive. On the other damping model used. In case of proportional damping [1] the eigenanalysis is relatively inexpensive. On the other hand, for non-proportionally damped systems the eigensolution is performed in a space of twice the dimension of the hand, for non-proportionally damped systems the eigensolution is performed in a space of twice the dimension of the original problem, which significantly increases both, computational time and cost. To reduce these high numerical original problem, which significantly increases both, computational time and cost. To reduce these high numerical requirements, procedures based on the modification of traditional methods, such as Lanczos [2] or subspace iteration requirements, procedures based on the modification of traditional methods, such as Lanczos [2] or subspace iteration methods [3], were developed. The various forms of the iterative pseudo-force approach were also successfully applied methods [3], were developed. The various forms of the iterative pseudo-force approach were also successfully applied for the solution of the structural response of non-proportionally damped systems on different dynamic loads, see for the solution of the structural response of non-proportionally damped systems on different dynamic loads, see e.g. [4–6]. The similar iterative procedure was applied for the calculation of the complex eigensolution of such systems e.g. [4–6]. The similar iterative procedure was applied for the calculation of the complex eigensolution of such systems ∗ ∗

Stanislav Hraˇcov. Tel.: +420-286-882-121 ; fax: +420-286-884-634. Stanislav Hraˇcov. Tel.: +420-286-882-121 ; fax: +420-286-884-634. E-mail address: [email protected] E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. 1877-7058 2017responsibility The Authors. Published by committee Elsevier Ltd. Peer-review©under of the organizing of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN 2017. 10.1016/j.proeng.2017.09.360

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using classical normal modes [7]. Finally also the widely used perturbation theory was applied to eigensolution of the non-proportionally damped structures, see e.g. [8–10]. In this paper, the complex eigenvalues of the linear proportionally damped system supplemented with a viscous damper are evaluated using the first-order perturbation method. The eigenvalues of the proportionally damped system are assumed as the reference values of the perturbation approach. The complex eigenvectors are subsequently determined using obtained eigenvalues from substantially reduced modal system. The accuracy of the proposed method is demonstrated and verified using a numerical model of a shear frame. 2. Theoretical background The governing system of equations for a generally viscously damped linear system with n degrees of freedom can be written in a form M¨x(t) + C˙x(t) + Kx(t) = 0, (1)

where M, C and K are the (n × n) positive-definite mass, damping and stiffness matrices of the system. Symbols x¨ , x˙ , and x are the (n × 1) vectors of nodal accelerations, velocities and displacements. Assuming the eigensolution of Eq. (1) in the form leads to the (n × n) eigenvalue problem

x(t) = X eλt ,

(λ2 M + λC + K)X = 0,

(2) (3)

where X is the complex eigenvector of size (n × 1) and λ is the complex eigenvalue. Since the dimension of the problem, n, is expected to be high, and character of the problem is complex, the standard method for solving these modal parameters can be very time consuming. For the sake of the proposed method, it is convenience to transform the eigenproblem (3) using the modal transformation X = ΦQ

into modal space

˜ + Ω2 )Q = 0. (λ2 I + λC

(4) (5)

where following relations utilizing mass normalized eigenmodes Φ of undamped system were applied ˜ ΦT KΦ = Ω2 . ΦT MΦ = I, ΦT CΦ = C,

(6)

2.1. Proposed approximate method for complex eigensolution

Existing civil engineering structures are approximately considered as proportionally damped. However, this assumption is acceptable only if no specialized damping device is incorporated into the structure. Installation of such an element could introduce significant non-proportionality of the damping. Let us assume the damping matrix of the structure with damping device has the form C = C p + ddT . (7)

Matrix C p represents the proportional part of damping matrix, while the product consisting of a column vector d and its transpose stands for the non-proportional part arising from the presence of the damping device. The vector d reflects the damping property as well as the position of the device in the structure. For the characteristic equation of the system (1) with damping matrix given by Eq. (7), it holds [11]

where following relations were applied

2 1 n d˜i = 0, +∑ 2 λ i=1 λ + λC˜ pi,i + ωi 2

˜ p , d˜ = ΦT d. ΦT C p Φ = C

(8)

(9)



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Symbol ωi in Eq. (8) stands for the i-th real undamped angular eigen-frequency and C˜ pi,i is the i-th diagonal element ˜ p. of matrix C Adopting the first-order perturbation technique, the solution of j-th eigenvalue λ j can be written in the form λ j = λ p, j + ∆λ j ,

(10)

where λ p, j is the j-th eigenvalue of the classically damped system λ p, j =

−C˜ p j, j +

√ C˜ 2p j, j − 4 ω2j

(11)

2

and ∆λ j represents its deviation from the exact solution λ j . Substituting the solution (10) into Eq. (8) leads to 2 n d˜i 1 +∑ = 0, λ p, j + ∆λ j i=1 αi, j + βi, j ∆λ j + ∆λ2j

where

αi, j = λ2p, j + λ p, jC˜ pi,i + ωi 2 ,

(12)

βi, j = 2λ p, j + C˜ pi,i .

Eq. (12) can be converted to a common denominator

(13)

2 ∏ (αi, j + βi, j ∆λ j + ∆λ2j ) + (λ p, j + ∆λ j ) ∑ d˜l ∏ (αi, j + βi, j ∆λ j + ∆λ2j ) n

i=1

n

n

l=1

i=1 i≠l

(λ p, j + ∆λ j ) ∏ (αi, j + βi, j ∆λ j + ∆λ2j ) n

i=1

= 0.

(14)

Assuming the numerator of Eq. (14) is equal to zero and keeping only the first and the second power of the perturbation, ∆λ j , Eq. (14) transforms into the following relation ∏ αk, j + ∆λ j ∑ βi, j ∏ αk, j + ∆λ2j ( ∑ βi, j ∑ βl, j ∏ αk, j + ∑ ∏ αk, j )+ n

n

n

n−1

n

n

k=1

i=1

k=1 k≠i

i=1

l=i+1

k=1 k≠i,l

n

n

i=1 k=1 k≠i

(15) ⎡ ⎤ n n n n−1 n n n n ⎥ 2⎢ 2 ⎥ + ∑d˜l ⎢ β β β α ) + ∆λ )( α + ∆λ α ) + ∆λ λ ( α + (λ = 0. ∏ k, j ∑ i, j ∑ m, j ∏ k, j ⎥ j j ∑ i, j ∏ k, j ⎢ p, j j p, j ∑ ∏ k, j ⎢ ⎥ m=i+1 i=1 i=1 k=1 i=1 k=1 k=1 k=1 l=1 ⎣ ⎦ m≠l i≠l i≠l k≠i,l i≠l k≠i,l,m k≠l k≠i,l n

This equation can be simplified considering zero coefficient α j, j and reformulated as

where

a2, j ∆λ2j + a1, j ∆λ j + a0, j = 0,

(16)

⎡ ⎤ n n ⎥ 1 ⎢ ⎢βi, j β j, j + d˜j 2 βi, j + d˜i 2 β j, j + λ p, j (d˜j 2 (1 + βi, j ∑ βl, j ) + d˜i 2 (1 + β j, j ∑ βl, j ))⎥, ⎢ ⎥ ⎥ l=1 αl, j l=i+1 αl, j i=1 αi, j ⎢ ⎣ ⎦ l≠i, j l≠i, j i≠ j 2 2 n d˜j βi, j + d˜i β j, j 2 2 , a0, j = d˜j λ p, j . a1, j = β j, j + d˜j + λ p, j ∑ αi, j i=1 a2, j = 1 +

n

∑

(17)

i≠ j

The solution of the perturbation, ∆λ j , is the smaller absolute value from the roots of quadratic Eq. (16) ∆λ j,(1),(2) =

−a1, j ±

√ a21, j − 4a2, j a0, j 2a2, j

,

(18)

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More rough estimation can be obtained by neglecting the second power of ∆λ j in Eq. (16), which leads to unique solution a0, j . (19) ∆λ j = − a1, j

The calculation of complex eigenvector X j corresponding to obtained λ j is based on the determination of the eigenvector Q j of the reduced modal eigenproblem (5) having size (m × m), m ≪ n. The m real eigenmodes of Φ, ˜ identical to those used for the calculation of λ j , are selected for the construction of (m × m) matrices I, Ω2 and C according to Eq. (6). The eigenvector X j is subsequently calculated with the use of the obtained eigenvector Q j of size (m × 1) and by means of the transformation (4) from modal into original space. 3. Numerical example

The accuracy of approximate method was investigated for a linear discrete numerical model of 6-storey shear frame, see Figure 1. The lateral stiffness and mass of each storey do no vary along the height. The stiffness of the vertical elements connecting adjacent floors is equal to k = 4 ×107 Nm−1 and the floor mass is equal to m = 0.8 ×105 kg. A discrete 6-DOF numerical model of the frame was built in and analysed using procedures created in programmable language of software MATLAB. The first four natural modes and angular eigenfrequencies ωi of the frame are given in Figure 1. The proportional part of the damping matrix C p in Eq. (7) was set to be proportional to the stiffness matrix K. The proportional value of viscous coefficient of the dampers c p = 7.42 × 104 N sm−1 was calculated from low structural damping ratio ζ = 0, 005 for the first angular eigenfrequency ω1 . The vector d, which forms the non-proportional part of the damping matrix, contained an increment ∆c in value of viscous coefficient of the top damper, see Figure 1. The accuracy of the proposed method was examined for a set of various increments ∆c, while proportional part c p of viscous coefficient of all dampers was kept constant. The numerical simulation and accuracy control were also performed for different positions of the damper with higher internal damping in different storeys of structure. cp + ∆c

m m

cp m

cp cp cp cp

m m m

k k k k k k ω1 =5.391 rad s−1

ω2 =15.858 rad s−1

ω3 =25.405 rad s−1

ω4 =33.474 rad s−1

Fig. 1: The scheme of analysed 6-storey shear frame and the first four eigenmodes with corresponding angular eigenfrequencies ωi .

The calculation of the coefficients (17) were carried out with the use of all six real eigenmodes of undamped system. The perturbations (18) and (19) were determined and corresponding approximate complex eigenvalues were subsequently calculated according to Eq. (10). In order to quantify their accuracy, the eigenvalues were finally compared with the exact solution obtained using the complex subspace iteration method [3] and with results of a commonly used ˜ The j-th complex approach [12] based on the neglecting the off-diagonal elements of the modal damping matrix C. eigenvalue determined by this approximate method has a form √ −C˜ j, j + C˜ 2j, j − 4 ω2j λ˜ j = . (20) 2



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The exact and the calculated approximate eigenvalues as the functions of the increment ∆c are depicted in the complex plane in Figure 2. The presented results are related to the damping model presented in Figure 1. The identical markers on all curves correspond to the identical increment ∆c. In case of the first and second eigenvalues the curves corresponding to both proposed solutions almost coincidence with black solid line of exact solution. Changes of eigenvalue positions given by the exact solution indicate, that adding damping increment ∆c into damper placed in top storey strongly influences the fourth eigenvalue. The rest of eigenvalues including the fifth and sixth eigenvalues is affected less significantly. To quantify an accuracy of the proposed method and compare it with commonly used

Fig. 2: The first four eigenvalues λ j as functions of increment ∆c (solid black line - exact solution; dashed black line - solution using Eq. (20); dashed blue line - solution using perturbation (19); dashed red line - solution using perturbation (18))( ∗ - ∆c = 0; ○ - ∆c = 5 × c p ; ◇ - ∆c = 10 × c p ; ◻ - ∆c = 20 × c p ).

solution according to Eq. (20), the relative error in the determination of eigen-value λ j was calculated as follows √ (Re(λ j(ap) ) − Re(λ j(ex) ))2 + (Im(λ j(ap) ) − Im(λ j(ex) ))2 δA (λ j ) = ⋅ 100 [%], (21) ∣λ j(ex) ∣

where λ j(ex) and λ j(ap) are the exact and approximate solutions, respectively. This error expresses the ratio of the distance between corresponding points of the exact and the approximate eigenvalues in the complex plane and the absolute value of the exact solution. The selected values of error δA as the function of damping increment ∆c are shown in Table 1. Table 1: The error δA of eigenvalues λ j obtained by different approximate methods as the function of damping increment ∆c (δA,P - λ j obtained using Eq.(20); δA,R - λ j obtained using Eqs.(10),(19); δA,F - λ j obtained using Eqs.(10),(18))

∆c 5 × cp 10 × c p 20 × c p

δA,P

λ1 δA,R

δA,F

δA,P

λ2 δA,R

δA,F

δA,P

λ3 δA,R

δA,F

δA,P

λ4 δA,R

δA,F

0.0014 0.0073 0.0320

1.3e-06 9.9e-06 1.0e-04

1.8e-09 2.0e-08 1.8e-07

0.1073 0.5292 2.1136

0.0029 0.0225 0.1280

0.0002 0.0020 0.0118

0.5227 2.7127 9.9693

0.0808 0.6410 2.6198

0.0209 0.2113 0.4039

0.5465 5.7770 29.161

0.4127 5.5248 31.880

0.2475 6.7283 45.968

The proposed method provided higher and acceptable accuracy (< 10%) with respect to δA than the commonly used approximate approach for all eigenvalues with the exception of the fourth eigenvalue.The higher accuracy was achieved as expected for the solution using the perturbation (18). The error δA of the first to third, fifth and sixth eigenvalues

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was given by values lower than 5%. The more accurate solution of the fourth eigenvalue determined by proposed method than by the solution (20) was reached only for ∆c < 6 c p . However, almost similarly for both methods the acceptable error lower than 10% was obtained up to ∆c = 12 c p . 4. Conclusions

The paper deals with the approximate complex eigensolution of the linear non-classically damped system, which represents the proportionally damped basic structure equipped with one viscous damper. The proposed and numerically efficient procedure operates only in the original dimension of the physical problem, thus significantly reduces computational costs. The core of the method represents the first-order perturbation technique. As the reference values for the perturbation strategy, the eigenvalues of proportionally damped system are used. The applicability of the developed approach is limited to the systems having only the simple complex eigenvalues. In case of the systems with well separated eigenvalues, usually only several real eigenmodes are needed to reach acceptable accuracy. For the systems with clusters of eigenvalues, more careful selection of larger number of eigenmodes is required. The selection can be done e.g., according to the elements of the modal damping matrix. The accuracy of the suggested approach was analysed for the numerical model of proportionally damped shear frame with added viscous damper. The method provided acceptable accuracy of the eigensolution in the limited interval in the vicinity of the equivalent classical damping ratio of the damper. Nevertheless, in comparison with the traditional approximate solution, that is based on the neglecting the off-diagonal elements of the modal damping matrix, it provides a higher accuracy in this interval. To reach a tolerable accuracy even for higher damping of the damper, authors recommend to combine the presented perturbation procedure with incremental method or employ another more sophisticated approach. Acknowledgements The kind support of the Czech Science Foundation project No. 15-01035S and of the RVO 68378297 institutional support are gratefully acknowledged. References [1] A. K. Chopra, Dynamics of Structures: Theory and Applications to Earthquake Engineering, 4th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2012. [2] C. Rajakumar, Lanczos Algorithm for the Quadratic Eigenvalue Problem in Engineering Applications, International Jornal of Numerical Methods in Engineering, 105 (1993), 1-22. [3] P. Fischer, Eigensolution of nonclassically damped structures by complex subspace iteration, Computer Methods in Applied Mechanics and Engineering, 189(1) (2000), 149-166. [4] A. Ibrahimbegovic, E. L. Wilson, Simple numerical algorithms for the mode superposition analysis of linear structural systems with nonproportional damping, Computers and Structures, 33(2) (1989), 523-531. [5] F. E. Udwadia, R. Kumar, Iterative methods for non-classically damped dynamic system, Earthquake Engineering and Structural Dynamics, 23 (1994), 137-159. [6] F. B. Lin, Y. K. Wang, Y.S. Cho, A pseudo-force iterative method with separate scale factors for dynamic analysis of structures with nonproportional damping. Earthquake Engineering and Structural Dynamics 32 (2003) 329-337. [7] S. Adhikari, An iterative approach for nonproportionally damped systems. Mechanics Research Communications 38 (2011) 226-230. [8] F. Perotti, Analytical and numerical techniques for the dynamic analysis of non-classically damped linear systems, Soil Dynamics and Earthquake Engineering, 13 (1994), 197-212. [9] J. Tang, W. L. Wang, Perturbation method for determining eigensolutions of weakly damped systems, Journal of Sound and Vibration, 187(4) (1995), 671-681. [10] P. D. Cha, Approximate eigensolutions for arbitrarily damped nearly proportional systems, Journal of Sound and Vibration, 288 (2005), 813827. [11] M. G¨urg¨oze, Proportionally damped systems subjected to damping modifications by several viscous dampers, Journal of Sound and Vibration, 255(2) (2002), 407-412. [12] J. W. S. Rayleigh, The Theory of Sound, vol. 1, Dover Publication, New York, 1945.