A frequency-domain pseudo-force method for dynamic structural analysis nonlinear systems and nonproportional damping

A Frequency-Domain Pseudo-Force Method for Dynamic Structural Analysis Nonlinear Systems and Nonproportional Damping

W.G. Ferreira

Department of Civil Engineering. Technological Center. Federal University of Espirito Santo. Av. Fernando Ferrari, s/n. Goiabeiras 29060-000 Vitória. ES. Brazil

[e-mail]

A.M. Claret

Department of Civil Engineering. School of Mines UFOP. Federal University of Ouro Preto. 35400-000 Ouro Preto. MG. Brazil

F. Venancio-Filho

W.J. Mansur

Department of Civil Engineering. COPPE/UFRJ and Department of Applied Mechanics and Structures School of Engineering. Federal University of Rio de Janeiro. Caixa Postal 68509 21945-970 Rio de Janeiro. RJ. Brazil

[e-mail], [e-mail]

A frequency-domain method for nonlinear analysis of structural systems with viscous, hysteretic, nonproportional and frequency-dependent damping is presented. The nonlinear effects and nonproportional damping are considered through pseudo-force terms. The modal coordinates uncoupled equations are iteratively solved. The treatment of initial conditions in the frequency domain which is necessary for the treatment of the uncoupled equations is initially adressed.

Keywords: Dynamic nonlinear analysis, frequency domain, damping

Introduction

Frequency-domain (FD) methods of are mandatory for a rigorous dynamic analysis of structural systems which present frequency-dependent properties and hysteretic damping. Interaction forces in structural systems with soil or fluid-structure interaction can be frequency-dependent and, consequently, the properties of such systems (stiffness and damping) can depend on the frequency spectrum of the excitation. On the other hand, in dynamic structural analysis, hysteretic damping can only be rigorous by considered by means of FD methods. Furthermore, these methods take adequately into account nonproportional damping in structural systems.

Although FD methods pertain to the class of superposition methods and, therefore, are essentially linear they have been applied to the analysis of nonlinear systems through appropriate linearization techniques. Kawamoto (1983) developed a hybrid frequency-time-domain method for nonlinear analysis in the FD. Aprile, Beneditti, and Trombetti (1994) extended Kawamoto’s method with the consideration of hysteretic and nonproportional damping, proposing the Generalized-Alternate Frequency Time (G-AFT) method. Venancio-Filho and Claret (1991) introduced the Step-by-step Incremental Linearization in the Frequency Domain (SILFD) method which considers the dependence of damping with frequency.

W.G. Ferreira et al. (1999a, b) developed the FD dynamic analysis of a SDOF system submitted to initial conditions and a pseudo-force method for the FD dynamic analysis of nonlinear MDOF systems with viscous, hysteretic and nonproportional damping. The consideration of initial conditions in the FD is necessary for the treatment of the linearized equations in modal coordinates which stem from the pseudo-force method.

In this paper there the treatment of initial conditions in the FD is initially presented. Thereafter the pseudo-force method is developed. The dynamic equilibrium equations in physical coordinates are transformed into equations in modal coordinates which are uncoupled through pseudo-force terms resulting from nonlinearities and nonproportional damping. These equations are then solved by a sequence of linearized segments.

Numerical examples validate the analysis of SDOF systems subjected to initial conditions. Examples of linear and nonlinear analysis of systems with hysteretic, nonproportional and frequency-dependent damping are presented.

Response of a SDOF System to Initial Conditions

The problem addressed in this Section is to solve in the frequency domain the dynamic equilibrium equation of a SDOF system,

subjected to given initial conditions: initial displacement n(0) and initial velocity . In Eq.(1) m, c and k are, respectively, the mass, damping and stiffness, and v is the displacement of the structural system. Both c and k can be frequency dependent, the usual case being when only c is frequency dependent. The FD solution of Eq.(1) for a RHS p(t) was developed by Venancio-Filho and Claret (1992, 1995) through the Implicit Fourier Transform (ImFT) formulation. Herein the solution for given initial conditions is presented.

Response to initial displacement v(0). This response is calculated, in the frequency domain, as the sum of the initial displacement v(0) plus the response to a step force –kv(0). In this way, taking into account Eq. (15) from Venancio-Filho and Claret (1992), the response due to v(0) is

In Eq. (2) v_d = {v_d(t_o), v_d(t₁), ...,v_d(t_n), ..., v_d(t _N-1)} is the vector formed by the response at the discrete times t_n = nDt (n = 0, 1, 2, ..., n, ..., N - 1) and e is the matrix defined as

where H is the (N x N) diagonal matrix formed by the complex frequency response functions H(w_m) calculated at the discrete frequencies w_m=mDw (m=0, 1, 2, ..., n, ..., N-1). For the generic frequency w_m

where l is the hysteretic damping coefficient. Yet, in Eq. (3), ^E*is the (N x N) matrix whose generic term is , E is the corresponding matrix with negative sign in the exponentials, and 1 in Eq. (2) is a (N x 1) vector with 1’s in every position.

Response to initial velocity. An initial velocity produces, in the time domain, a response given by

where h(t) is the unit-impulse response function. Bearing in mind that h(t) is the inverse Fourier Transform of H(w) expressed as

v(t) can be obtained, alternatively, in the frequency domain as

The transformation of Eq. (7) into discrete form gives

or, taking into account that

The response vector v_v = {v_v(t_o), v_v(t₁), ...,v_v(t_n), ..., v_v(t_N-1)} is then, from Eq. (9), given by

where d is a (Nx1) vector with 1 in the first position and zero elsewhere.

Total response to external loading and initial conditions. If, on top of the initial conditions, the effect of an external load p = {p(t_o), p(t₁), ..., p(t_n), ..., p(t_N-1)} is considered, one adds v_d from Eq. (2), and v_v from Eq. (10) to from Eq. (15) (Venancio-Filho and Claret, 1992) which results in

Nonlinear Analysis of SDOF Systems

The basic assumption of this method of analysis is that the nonlinear spring behavior can be well approximated by a chain of piece-wise linear segments as in Fig. (1). In this way, the nonlinear analysis is performed through a sequence of linear analysis in the FD bearing in mind that the initial conditions in the each segment are the final ones in the precedent one. This method of nonlinear analysis in the frequency domain was previously named as SILFD method (Venancio-Filho and Claret, 1991).

In Fig. (1) it is indicated that the total displacement in the current step I (step 5 in Fig. (1)) is

where the are the incremental displacements in the previous steps and is the displacement in that step. The corresponding spring force is

Introducing now into the dynamic equilibrium equation

f_s from Eq. (13) and taking into account that and leads to

On the other hand, the stiffness k_I can be expressed as k_I = k₀- Dk where Dk is the modification of the linear stiffness k₀ due to nonlinearity. Substituting for k_I in Eq. (15) leads to

noting the presence of two pseudo-force terms in the RHS. Eq. (16) is solved through an iterative process. Taking into account Eq. (11), the k^(th) step of the process is expressed by the following equation:

where v0Iand are, respectively, the initial displacement and initial velocity of the current linearized segment (final values in the previous segment).

Analysis of MDOF Systems with Nonlinearity and Nonproportional Damping

The dynamic equilibrium equation of a MDOF system with nonlinear behavior is

where v, and are, respectively, the vectors of displacements, velocities, and accelerations; m and c are, respectively, the mass and damping matrices; _fsis the vector of resisting forces taking into account the nonlinear behavior; and p(t) is the vector of applied loading.

The generalization of Eqs. (12) and (13) for a MDOF system leads to the following expressions for the displacement and resisting force vectors:

and

In these equations k_i and Dv_i are, respectively, the stiffness matrix and the vector of incremental displacements; kI and are, respectively, the stiffness matrix and the vector of current displacements in the current (I^th) linearized segment. Substituting for f_s in Eq. (18) and taking into account that and yields

The stiffness matrix k_I in the current segment can be expressed as

where k₀ is the initial stiffness and Dk_Iis the modification of k₀due to nonlinearity. Introducing now Eq. (22) into Eq. (21) one obtains

noting the pseudo-force terms in the RHS.

The LHS of Eq. (23) is linear whereas the nonlinearity is considered by the pseudo-force terms in the RHS. As the LHS is linear Eq. (23) can be transformed into an equation in modal coordinates through the modal transformation . Therefore, the transformed equation in modal coordinates is

In this equation F and L are, respectively, the normal mode matrix and the diagonal matrix containing the natural frequencies squared, bearing in mind that F and L are derived from the linear stiffness matrix k₀and the normal modes are normalized according to F^tm F = I. Additionally, in Eq. (24) DK= F^tDkF, P = F^tp(t) and D = F^t dF in which d is the damping matrix incorporating viscous and hysteretic damping. Finally, it is worthwhile to bear in mind the analogy between Eqs. (24) and (16).

Noting that damping is hysteretic and nonproportional, matrix D can be split as D = D_d + D_f (Claret and Venancio-Filho, 1991), D_d being a diagonal matrix which contains the diagonal elements of D and D_f containing zero elements in the diagonal and the off diagonal elements of D. Introducing D into the LHS of Eq. (24) and transferring the term D_f to the RHS yields.

The LHS of Eq. (25) is linear and the coupling due to nonlinearity and nonproportional damping is present by the pseudo-force terms in the RHS. The j^th equation of Eq. (25) is

where MC stands for number of modal coordinates. The solution of this equation (in total coordinates) is obtained through an iterative process. By analogy with Eq. (17) the k^(th) iterative step of this process is

In Eq. (27) and are the initial conditions of Y_j in the beginning of the current linearized segment (final conditions in the previous one) and e_jis the matrix corresponding to e in Eq. (17) with where D_vjan D_Hjare the terms of D_dj, Eq. (26), corresponding, respectively, to viscous and hysteretic damping and L_j is the frequency squared of mode j. The iterative steps are performed until a convergence threshold is attained.

Numerical Examples

The first examples concern the treatment of initial conditions. In order to validate the analysis for initial conditions Eq. (1) is solved by means of Eqs. (2) and (10) for v(0)=0.03m and

=0.60m/s, respectively. The system properties are m=17500kg, k=17.5 MN/m, and x=5%. The results are shown, respectively in Figs. (2) and ⁽³⁾. The FD solutions indicate a very good agreement with the known solutions in the time-domain (Clough and Penzien, 1993). In order to take into account frequency-dependent damping the dependencies of Figs. (4) and ⁽⁵⁾ are considered. The responses for v(0)=0.03m are displayed, respectively, in Figs. (6) and ⁽⁷⁾.

A SDOF system with m=17.5t, k=875kN/m is submitted to a triangular load with a peak of 35kN at t=0 and duration of 0.6sec. The damping constant is c=21kNsec/m which corresponds to a damping ratio x=8.5%. The FD response agrees very well with the time-domain one obtained by Newmark-b method, Fig. (8). The solutions obtained with constant damping (c=21kNs/m) and with frequency-dependent damping, Fig. (9), are presented in Fig. (10). These results indicate the influence of the damping dependence in the response.

The shear building studied extensively by Clough and Penzien (1993) is now considered with a discrete damper, Fig. (10). This damper causes a situation of high nonproportionality. The properties and the load are indicated in Fig. (11). The responses were obtained in the FD domain by an exact method in physical coordinates Venancio-Filho (1999) and, in modal coordinates, by the pseudo-force method with convergence thresholds of 5% and 1%. The results are presented in Figs. 12a, ^b and ^c. The result with 1% agrees very well with the exact one.

The last example is the two DOF system of Fig. (13) which is a simplified model of a nuclear reactor contaiment. The foundation and superstructure masses are mf=3,0E+11kg and ms=1,0E+8kg, respectively. The superstructure stiffness is ks=6.0E+11N/m. The soil has a nonlinear behaviour with stiffness represented by the bilinear model of Fig. (14). The damping constant of the soil is 3.79E+09Ns/m, calculated according to the expressions given by Richart et al. (1970). The system is submitted to an impact load given in Fig. (15) which represents an aircraft crash in the reactor. The results of linear and nonlinear analyses are displayed, for comparison, in Fig. (16).

Concluding Remarks

A FD pseudo-force method for nonlinear dynamic structural systems with viscous, hysteretic, nonproportional, and frequency dependent damping has been presented. The treatment of initial conditions in the frequency domain, which is necessary for the treatment of the uncoupled linearized equations was initially addressed. Numerical solutions in the frequency domain show a very good agreement with known ones in the time domain. Solutions of systems with frequency-dependent damping indicate the feasibility of the treatment of such systems and exhibit the influence of that type of damping in the response. The method of nonlinear analysis of MDOF systems is a pseudo-force method in which the uncoupled linearized equations in modal coordinates are solved in the frequency domain. Numerical examples of small MDOF systems with high nonproportional damping and with nonlinearities reflects the quite good potential of the presented approach.

Presented at DINAME 99 – 8th International Conference on Dynamics Problems in Mechanics, 4-8 January 1999, Rio de Janeiro. RJ. Brazil. Technical Editor: Hans Ingo Weber.

• Aprile, A., Beneditti A., and Trombeti, M., 1994. “On Nonlinear Dynamic Analysis in the Frequency Domain”, Earthquake Engineering and Structural Dynamics, Vol. 23, pp. 363-388.
• Claret, A.M. and Venancio-Filho, F., 1991, “A Modal Superposition Pseudo-force Method for Dynamic Analysis of Structural Systems with Nonproportional Damping”, Earthquake Engineering and Structural Dynamics, Vol. 20, pp. 303-315.
• Clough, R. W. and Penzien, J., 1993. “Dynamics of Structures”, 2nd edition, McGraw-Hill, New York, USA, 738p.
• Ferreira, W.G., Claret, A.M., and Venancio-Filho, F., 1999a. “Dynamic Response due to Initial Conditions by Frequency Domain Method”, Applied Mechanics in the Americas, Proceedings of the Sixth Pan-american Congress of Applied Mechanics (PACAM VI) and Eighth International Conference on Dynamic Problems in Mechanics (DINAME99) Vol. 8, Rio de Janeiro, Brasil, pp. 1339-1342.
• Ferreira, W.G., Venancio-Filho, F., Mansur, W.J. and Claret, A.M., 1999b. “A Frequency Domain Method for Nonlinear Dynamic Structural Analysis”, Applied Mechanics in the Americas, Proceedings of the Sixth Pan-american Congress of Applied Mechanics (PACAM VI) and Eighth International Conference on Dynamic Problems in Mechanics (DINAME99) Vol. 8, Rio de Janeiro, Brasil, pp. 1343-1346.
• Ferreira, W.G., 1998. “Nonlinear Dynamic Analysis in the Frequency Domain of Structural Systems with Nonproportional Damping” (In Portuguese), D.Sc. Thesis, COPPE/Federal University of Rio de Janeiro, Rio de Janeiro, Brazil, 93p.
• Kawamoto, J.D., 1983. “Solution of Nonlinear Dynamic Structural Analysis by a Hybrid Frequency Time Domain Approach”, Department of Civil Engineering, MIT, Cambridge, Massachussets, USA, 320 p.
• Richart, F.E., Woods, R.D., and Hall, J.R., 1970. “Vibrations of Soil and Structures”, Prentice-Hall, New Jersey, USA, 414p.
• Venancio Filho, F. and Claret, A.M., 1991. “An Implicit Fourier Transform Method for Nonlinear Dynamic Analysis with Frequency-dependent Damping”, Proceedings of the DAMPING’91 Conference, San Diego, California, USA, pp. BCB1-BCB16.
• Venancio Filho, F. and Claret, A.M., 1992. “Matrix Formulation of the Dynamic Analysis of SDOF Systems in the Frequency Domain”. Computers and Structures, Vol. 42, pp. 853-855.
• Venancio Filho, F. and Claret, A.M., 1995. “Frequency Domain Analysis of SDOF Systems: Nodal and Modal Coordinates Formulation”. Computers and Structures, Vol. 56, pp. 189-191.
• Venancio Filho, F., 1999. " Frequency-Domain Dynamic Analysis of Systems with Viscous and Hysteretic Damping", Nonlinear Dynamics, Chaos, Control and their Applications to Engineering Sciences, published by ABCM, SBMAC and SIAM, Vol. 2, pp. 136-151.

Publication Dates

• Publication in this collection
  04 Oct 2001
• Date of issue
  2000