Effects of damping matrix in the response of structures with added linear viscous dampers

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1 Effects of damping matrix in the response of structures with added linear viscous dampers J.R. Arroyo', J. Marte2 l Department of General Engineering, University of Puerto Rico, Mayagiiez Campus 'Department of Civil Engineering, University of Puerto Rico, Mayagiiez Campus Abstract In the last decade the use of some devices had been proposed and used to reduce the response in structures due to earthquakes effects by increasing the damping of the system. These devices are activated whenever the structure vibrates and have no effect otherwise, so it can be called passive damper. In this paper, the response of a structure with added linear viscous dampers is presented by using two damping matrix model of the system. The first damping matrix used is one with arbitrary constant producing a nonclassical damping matrix. That is, the eigensolution cannot be used to decouple the equations of motion. To solve the problem, the system of second order equations is transformed into a system of first order equations. The second damping model used is a classical damping matrix. In this case. the damping ratio of each mode can be greater than one. This force us to solve the problem using the convolution integral developed for overdamped systems. Finally, a comparison of the response using the two damping matrices models will be presented. It will be shown that the new formulation of the convolution integral for overdamped systems is a reasonable way to calculate the dynamic response of structures with added dampers.

2 12 Earthquake Resistant Eng~rwerlng Srrucrzwes III 1 Introduction In the last decade, the use of linear viscous dampers to reduce the response of structures ctue to a dynamic load is a technique

2 2 12 Earthquake Resistant Eng~rwerlng Srrucrzwes III 1 Introduction In the last decade, the use of linear viscous dampers to reduce the response of structures ctue to a dynamic load is a technique well accepted. The buildings with viscous dampers arrangements have an increase of the damping ratio of the whole structure. The dynamic analysis of these structures is more elaborated because these equations of motion are coupled due to the inclusion of dampers. In practice, engineers have to perform simplified analysis of new structures with this mechanism to do a preliminary design. Or in a simple way, the response due to dynamic loads of existing structures to which dampers will be added is required. Two methodologies are presented here to obtained the response of a shear building with added dampers. The first method is the state equations approach, which is the exact solution of the problem. The second method is the elimination of the off main diagonal terms of the global damping matrix. This method provided a solution without the complex eigenvalue problem of the state equations approach. This simple method introduced the necessity of determine the convolution integral for damping ratio greater than one. After the eigensolution is calculated, the damping ratio for some modes could be defined as overdamped Then, the convolution integral for undamped systems cannot be used to solve the problem. These models are used in this paper, in which the convolution integral for overdamped cases is employed to determine the seismic response of a building with added linear viscous dampers. For comparison purposes, the response is calculated using two methods: the state equations method (SEM) and the elimination of the off diagonal terms (EODT) of the global damping matrix. Finally, a numerical example is presented to confirm that the use of the EODT method for overdamped cases is an acceptable mathematical twl to calculate the response of structures with added dampers. 2 Solution of systems with non-proportional damping using the state equations method The equations of motion of a MDOF system subjected to a seismic effect at the base are [MM) + [C, M)+ [KE~) = -[MPg 6) (1) where [M] is the mass matrix, [K] is the stiffhess matrix, [C, ] is the damping matrix of the system, x(t) is the relative acceleration vector, i(t) is the - - relative velocity vector, ~(t) is the relative displacement vector and Zg (t) is the - earthquake acceleration time history. Each value x,(t) represents the lateral

3 deformation of each story of the model. Earthquake Resrstant Ei~grneerrng Structures Matrix [C,] mrrspands to the summation of the effects of the classical damping matrix of the structural system and the damping matrix due to the addition of the dampers. This matrix be calculated as Let l = + 'd 1 where C,!, can be obtained by using typical values of the damping ratio for each mode in the traditional modal analysis resulting in a diagonal matrix in which each term has the form 2wj<,. Matrix Cd can be calculated in a similar fashion as the sti&ess matrix as C,+C, -C2 0 (3) This matrix transforms the system in one with nonproportional damping. To solve the problem, eqn (l) has to be transformed into where the matrices A, B and D are formed by the global stiffens, mass and damping matices as and where vector Z corresponds to the relative velocity for the first n terms and to the relative displacement for the last n terms. Solving the eigenproblem associated to eqn (4), one obtain [@I' [AI[@] = ['l (6) PIT [B1[@l = PI (7) The new equation of motion, eqn (4), can be decouple by premultiplying by [@lt and using the following cmrdinats transformation {W1 = [@I(W (8) A decoupled system of 2n first order differential equations can be obtained as L1 (t)+a./yj = 91 ('1 j=l,..., 2n (9) qj (t) = -[@lt [M] ( I] Using the Laplace transform, the solution of eqn (9) is jig (t) = [@lt F* (10)

4 2 14 Enrtlzqunke Resistant Engineering Structtlves 111 where the first term is the response due to initial conditions and the second term is due to the external loads. Using modal superposition, the total response of each degree of freedom can be calculated as where 3 Response calculation by the elimination of the off main diagonal terms of the global damping matrix Using eqns (l) and (2), the equations of motions of the system are rji, (t) (14) and with the following transformation of coordinates = [Q] v@) (15) eqn (14) transform to [M][@] [M] x(t) + [C, + C,,] x(t) + [K] iiw + [Cd + cc,o][@] W) T Remultiplying eqn (16) by [Q] + [K] one obtains [m] x(t) = -[M] = -[M] r xbr 0) (16) [Q]' [M][@] 60) + [@lr[g + cch][@]iw+ [a]'[k][@] 70) = -[Q]' [M] r (l 7, in which the second term can be expressed as [Q]' [C, + cla] [Q] = [Q]' [cd][m] i(t) + [Q]' [Cd][Q] i(t) (18) Using the orthogonality properties [@]'[M][@]= 0 0 a,

5 Earthquake Reslsta~t Engrneevltlg Structures where <c,i, is the classical damping ratio, which is constant for each mode of vibration. The problem presented by eqn (22) is that eqn (17) cannot be decouple. To produce a simple methodology the terms outside the main diagonal will be neglected. Then, an equivalent system is created and the equations are dewupled. The solution of this system is obtained without the use of the complex eigenvalue problem. Eliminating these terms, the non-classical damping problem is transformed in a system with classical damping. Then, the damping matrix has the form 24*@, + c,, 0 0 Y [ ~ ~ U ] ] + S [ ~ ~ ] [ ~ ] = [ in which each diagonal term is now where A 0 25,,@" +Cm a ~ C i a P+c, j = hjcj =cc, (24) is a modified damping ratio expressed as Now eqn (17) can be decouple, producing the following set of equations G (l) + cj G (l) + ('1 = &yj% [( f) ~r j=1,2,3... n (26) with a participation factor defined as yj = f [ ~ ] r 3.1 Response to unit impulse function for overdamped systems Due to the inclusion of the viscous dampers, the damping ratio can be more than one. Then, the traditional convolution integral cannot be used because it was developed for underdamped systems. Let begin assuming the following solution for eqn (14) as X([) = Ale+' + 4eS2' = Ale (-{o+&)t + qe(-t@-q (27) where = Taking the exponential as the common factor

6 216 Eurthpake Resrsrant Engrneerrng Structz~res III and transfming the constant to eqn (27) becomes then, it can be shown that x(t) = e-w (B, cosh rjt + B, sinh ht) (31) An impulsive force can be represented by the force in the integrand of the impulse hction = IF (t)dt (32) It can be shown that the initial conditions due to the application of an impulsive force to a system initially at rest is Then, eqn (31) can be used to calculate the unit impulse response function, 1 using as initial condition X, = 0 y X, = - m 3.2 Convolution integral for overdamped systems The incremental response for each impulse can be expressed as dy= F(z)h(t-z)dz (35) Using the superposition method to obtained the total response due to F (t), the well-known convolution integral arrives For an overdamped system, substituting eqn (34) into eqn (36) produce It can be shown that the solution of the integral in eqn (37) for a constant force in each time interval is

7 Earthquake Res~stant Engmeering Srrucrures where ad =q'l-p. Evaluating eqn (31) and its time derivative at time equal to zero, the response to initial conditions is f d ) Einf, 09, Finally, the total solution is the response to initial conditions as presented in eqn (39) and the forced response as presented in the convolution integral. That is, using a constant force in each time interval, the solution is Then, back to the solution of eqn (26), the solution of the modal coordinates is 1 sinh G, (t - r )dr + - j: N, (r)e-w(t-r) 4, where VJ (0) = et [M] ~ (0) and il, (0) = (bt[m]&(o). Finally, for A 4; = 5, the response of the modal coordinates are obtained as - The solution in physical coordinates can be obtained using eqn (1 5). 4 Numerical examples To show the implementation of the Convolution integral for damping ratio greater than one using the elimination of the off main diagonal values and to compare it against the State Equations method the structure shown in Figure 1 will be analyzed. It consists of a building with linear viscous dampers at each story and the properties associated to it are shown in Table 1. The base excitation is a time history of El Centro earthquake.

8 2 18 Earthquake Resistant Engineering Strtlctures III 3 Table 1. Properties of the shear building model per coefficient "cn of 250,OW lb.sec/dn and the state lhg n&d &equaq and se are presented in Table 2 response was also calculated using the elimination of the off diagonal terms in ping matrix. First, the natural hquency and the darnping ratio. The most important values are the new equivalent damping king equivalent values are over to calculate g the Convolutim integral fo fbr the H& mode of vl'bration. These values are presented In Table 3 md the response is presented in Figure 3. Table 2. Dynamic properties of the shear building mdel using State Equations approach the shear building model using a diagonal damping matrix

9 loj Earrizqziake Resistant Engineering Strzrctzwes Figure 2: Response time history using the State Equation approach Figure 3: Response time history by eliminating the off diagonal terms of [C,] and overdamped Convolution integral 5 Conclusions The addition of viscous dampers in structures produces a full damping matrix, resulting in a non-classical damping matrix. This produce that the differential

220 Earthquake Resistant Engineering Str.uctwes 111 equations of motions cannot be decoupled, and the conventional modal analysis cannot be used.

10 220 Earthquake Resistant Engineering Str.uctwes 111 equations of motions cannot be decoupled, and the conventional modal analysis cannot be used. The convolution integral for overdamped cases was developed. The response of structure with added viscous dampers can be calculated with the convolution integral for underdamped and overdamped cases fot each mode of vibration. This can be done after the elimination of the off diagonal terms in the global damping matrix. It can be demonstrated as presented in this paper, that the response of structures with overdamped modes using the exact solution produces a very similar results as the method of the elimination of the off diagonal terms in the global damping matrix and the convolution integral for underdamped and overdamped modes. Acknowledgements We acknowledge the financial support provided by the Federal Emergency Management Administration of the United States under the HAZARD MITIGATION GRANT PROGRAM, to conduct the fust stage of the project Improved Structural Analysis Of Buildings With Added Dampers. References Caughey, T. K. Classical Normal Modes in Damped Linear Dynamic Systems, Journal ofapplied Mechanics, 27, pp , Singh M. y Ghafory-Ashtiany, M. Modal Time History Analysis of Nonclassically Damped Structures for Seismic Motions. Journal of Earthquake Engineering and Structural Dynamics, 14 pp., , 1986 Singh, Mahendra P. Seismic Response Combination of High Frequency Modes. Proc. 7th Eur. ConE Earthquake Engineering., Athens, Greece, Singh, Mahendra P. Seismic Response by SRSS for Nonproportional Damping. Journal of the engineering Mechanics Division, ASCE, 106(EM6), pp , 1980 Velestos A. S., Ventura C.E. Modal Analysis of Non-Classically Damped Linear Systems. Journal of Earthquake Engineering and Structural Dynamics, 14 pp , Warbuton, G. B. y Soni, S. R Errors in Response Calculations for Non- Classically Damped Structures. Journal of Earthquake Engineering and Structural Dynamics, 5, pp , 1977.

Design of Structures for Earthquake Resistance

NATIONAL TECHNICAL UNIVERSITY OF ATHENS Design of Structures for Earthquake Resistance Basic principles Ioannis N. Psycharis Lecture 3 MDOF systems Equation of motion M u + C u + K u = M r x g(t) where: