A Procedure to Identify the Modal and Physical Parameters of a Classically Damped System under Seismic Motions

Abstract

The purpose of this paper is to discuss a methodology for determining some modal parameters (frequencies, damping ratios, and seismic eigenvectors) and, under certain hypotheses, the physical matrices of a general structure with proportional damping and subjected to seismic loads. The procedure is based on a time-domain state space formulation from which the modal parameters, including the seismic eigenvectors, and the complex eigenvectors of the system can be derived also in the case of a limited set of instrumentations. The conditions to normalize the eigenvectors are then illustrated and, finally, applied to derive the second-order matrices of the system. The proposed procedure is applied to some numerical examples also in the case of noise-polluted measurements and to an experimental investigation performed on a four-story steel frame subjected to earthquake excitations. An optimization procedure to improve the prediction of the first-order modal parameters of the system is also discussed and experimentally proved.

1. Introduction

System identification (SI) has shown a growing development in civil engineering during the last decades. A central issue of SI, especially within the field of civil engineering constructions, concerns the set of input/output experimental data with a particular emphasis on two main aspects: the number and the types of measurements (displacements, velocities, and accelerations) and the category of external actions to use in the process. Both the aspects are, in fact, related to the large-scale dimensions of civil engineering systems. The first aspect is widely discussed in [1, 2] where the minimum number of sensors and/or actuators required for a complete identification of a system is defined, in the case of input forces. Concerning the second aspect, Brownjohn [3] provides a clear discussion of the problem with specific reference to the external actions used in the dynamic tests on civil structures. In particular, these actions can be artificial forces applied to the structural system, such as those induced by a shaker or a vibrodine, and ambient vibrations, such as those due to traffic, earthquakes, and wind.

The use of forced vibration test data allows to identify the system’s modal parameters and, also, retrieve the normalized eigenvectors, the mass, stiffness, and damping matrices of the system. The main drawback of the forced vibration test is that it requires large machineries which are often very expensive and cumbersome especially for large-scale structures such as civil constructions. A way to overcome this shortcoming is to adopt ambient vibration tests which use ambient actions always acting on the structures. The use of ambient vibration test data allows to identify the system’s modal parameters as well. A comprehensive state-of-the-art review on fundamentals of system identification in structural dynamics in the case of environmental loads is reported in Imai et al. [4]. One common type of ambient vibrations in civil constructions is the earthquake excitations. In this field, some contributions, although the list is not exhaustive, are Ghanem and Shinozouka [5] and Shinozouka and Ghanem [6], which review and apply some structural identification algorithms for systems subjected to earthquake excitations, Beck and Jennings [7] and Mc Verry [8], which present an output-error method, respectively, in the time and frequency domain, and [9–11] which concern the identification of frequencies and dampings. In the case of the assumption of shear type buildings, some interesting contributions are the work by Yuan et al. [12] which provides a procedure to retrieve the normalized eigenvectors and the mass, stiffness, and damping matrices of the system, and Takewaki and Nakamura [13] which proposes a method of stiffness-damping simultaneous identification of the system using limited earthquake records. One of the more recent approaches in SI is based on a time-domain first-order formulation which provides the complex modal parameters of the system and a state space model capable of reproducing the input/output mapping of the system. These approaches are mostly used in control theory but notable contributions are also found in the field of identification [1, 2, 14, 15]. Some of these studies specifically concerning systems subjected to earthquake loadings are Luş et al. [16] which identifies the modal parameters of the analyzed systems including the modal shapes and Fraraccio et al. [17] which presents an experimental study of different algorithms for the health monitoring of a shear-type system.

This paper discusses a methodology for determining some modal parameters (frequencies, damping ratios, and seismic eigenvectors) and, under certain hypotheses, the physical matrices of a general structure (not a shear-type one as usually analyzed in available literature) with proportional damping and subjected to seismic loads. In particular, the methodology is based on a time-domain first-order state representation using the well-known algorithm ERA/OKID (Eigensystem Realization Algorithm/Observer Kalman filter Identification) on the basis of the procedure discussed in [16, 18]. It allows, first, to identify the complex eigenvectors and the modal parameters of the system including, beyond the frequencies and the damping ratios, the seismic eigenvectors [19, 20], sometimes also named effective participation factors [7, 8]. Then, the procedure is able to provide the conditions to normalizing the eigenvectors and, hence, deriving the second-order matrices of the system as reported in De Angelis et al. [1]. The significance of the proposed methodology consists, first, in the fact that the seismic eigenvectors are independent of the normalization of the mode shapes, and, hence, they can be determined whatever the number of sensors is and with no information and restrictions about the physical model of the system. This means that the seismic eigenvectors can represent a useful tool for health monitoring and of significance in updating models where they provide, jointly with the frequencies and dampings, additional information for validating the model. The other point of interest of the proposed procedure is that, in the case of full set of instrumentations, it provides the conditions to normalizing the eigenvectors and, hence, deriving the second-order matrices of the system without any restrictions about the system behaviour (such as the shear-type assumption). The methodology is discussed theoretically and, then, applied to some numerical examples which have shown the efficacy of the procedure also in the case of noise-polluted measurements. Finally, the procedure is also applied to an experimental test performed on a four-story steel frame subjected to earthquake excitations.

2. Some Background

Let us consider a seismically loaded 𝑁 degree of freedom structural system whose second-order equations of motion can be written as follows: 𝐌𝐱̈(𝑡)+𝐋𝐱̇(𝑡)+𝐊𝐱(𝑡)=−𝐌𝐔𝑥̈_𝑔(𝑡),𝐲(𝑡)=𝐜_𝑝𝐱(𝑡), (1) where 𝐱(𝑡) is the 𝑁×1 vector of generalized nodal displacements relative to the ground and the notation (⋅) indicates the differentiation with respect to time; 𝐌, 𝐋, and 𝐊 are the 𝑁×𝑁 mass, damping and stiffness matrices of the system, 𝐔 is the 𝑁×1 influence vector, with all terms equal to 𝟏, and 𝑥̈_𝑔(𝑡) is the ground acceleration. The 𝑚×1 output vector 𝐲(𝑡) contains the 𝑚 time histories of the displacement output measurements and 𝐜_𝑝 is the matrix which selects the degrees of freedom where the displacements are known. In the case of accelerations output measurements, equations similar to the second of (1) can be written with a matrix 𝐜_𝑎 used in place of the matrix 𝐜_𝑝 to select the degrees of freedom where the accelerations are known.

By using the modal transformation 𝐱(𝑡)=𝝓𝐪(𝑡), being 𝝓=[𝝓₁𝝓₂…𝝓_𝑁] the eigenvector matrix and 𝐪(𝑡) the modal coordinates, (1), in modal coordinates, result in the following: 𝐪̈(𝑡)+𝜺𝐪̇(𝑡)+𝛀𝐪(𝑡)=−𝝓^𝑇𝐌𝐔𝑥̈_𝑔(𝑡),𝐲(𝑡)=𝐜_𝑝𝜙𝐪(𝑡), (2) with 𝜙^𝑇𝐌𝝓=𝐈, 𝝓^𝑇𝐊𝝓=Ω=diag(𝜔²_𝑘).

If the system is classically damped, the matrix 𝜺 is diagonal, that is, 𝜺=diag(𝜀_𝑘) and 𝜀_𝑘=𝝓^𝑇_𝑘𝐋𝝓_𝑘=2𝜁_𝑘𝜔_𝑘 with 𝜁_𝑘 the 𝑘th damping ratio. In this case, the Laplace transform of (2) provides the following, for the 𝑗th dof: 𝑦_𝑗(𝑠)=−^𝑁_𝑘=1𝜙_𝑗𝑘𝑃_𝑘𝑠²+𝜀_𝑘𝑠+Ω_𝑘𝑥̈_𝑔(𝑠)=−^𝑁_𝑘=1𝜃_𝑗𝑘𝑠²+𝜀_𝑘𝑠+Ω_𝑘𝑥̈_𝑔(𝑠), (3) where 𝑃_𝑘=𝝓^𝑇_𝑘𝐌𝐔/𝝓^𝑇_𝑘𝐌𝝓_𝑘 is the 𝑘th modal participation factor and 𝜃_𝑗𝑘=𝜙_𝑗𝑘𝑃_𝑘 is the effective modal participation factor at position 𝑗 of mode 𝑘 [7, 8] or also named the seismic eigenvector 𝜽_𝑘 [19, 20]. The components 𝜃_𝑗𝑘 satisfy the following relationship: 𝜽𝐔=𝐔. (4)

By introducing the state vector 𝜼(𝑡)={𝐱(𝑡)^𝑇𝐱̇(𝑡)^𝑇}^𝑇, (1) can be transformed to the symmetrical first-order form: 𝐀𝜼̇(𝑡)+𝐁𝜼(𝑡)=𝐄𝑥̈_𝑔(𝑡),𝐲(𝑡)=𝐂𝜼(𝑡)+𝐃𝑥̈_𝑔(𝑡), (5) where the matrices 𝐀, 𝐁, 𝐄, 𝐂, and 𝐃 are given by:𝐎;𝐜𝐀=𝐋𝐌𝐌𝐎;𝐁=𝐊𝐎𝐎−𝐌;𝐄=−𝐌𝐔𝐂=_𝑝𝐎;𝐃=[𝐎], (6) being 𝐎 the zero vector or matrix.

The eigenvalue problem associated to (5) provides the complex eigenvalues 𝜆_𝑖 and eigenvectors 𝝍_𝑖(𝑖=1,2,…,2𝑁) which can be arranged respectively, in the square matrix Λ=diag(𝜆₁𝜆₂⋯𝜆_2𝑁), of dimension 2𝑁×2𝑁, and in the eigenvector matrix 𝝍=[𝝍₁𝝍₂⋯𝝍_2𝑁] of dimension 𝑁×2𝑁. It results in 𝜆^∗_𝑘=𝜆_𝑁+𝑘 and 𝝍^∗_𝑘=𝝍_𝑁+𝑘 (the superscript (*) indicates the complex conjugate) with 𝑘=1,2,…𝑁.

Considering the following matrix Ψ=[𝝍^𝑇(𝝍Λ)^𝑇]^𝑇 the complex eigenvector can be normalized according to the following relations: 𝚿^𝑇𝐀𝚿=𝝂;𝚿^𝑇𝐁𝚿=−𝝂𝚲, (7) where the matrix 𝝂 can be defined alternatively as follows:

𝝂=Λ−Λ^∗; in this case the matrix 𝝂 can be related to the natural modes 𝝓 previously defined and, in particular, equal to them when the system is classically damped.

𝝂=𝐈; in this case, when the damping is proportional, the eigenvalues have the same real and imaginary parts.

In this paper it is assumed 𝝂=𝐈.

By using the modal transformation 𝜼(𝑡)=Ψ𝝃(𝑡), with 𝝃(𝑡) expressing the modal coordinates, (5), in modal coordinates, result in the following:𝝃̇(𝑡)=𝚲𝝃(𝑡)−𝚿^𝑇𝐌𝐔𝑥̈_𝑔(𝑡)=𝚲𝝃(𝑡)−𝐐𝑥̈_𝑔(𝑡),𝐲(𝑡)=𝐜_𝑝𝚿𝝃(𝑡), (8) where the vector 𝐐 is defined as 𝐐=Ψ^𝑇𝐌𝐔, being 𝑄_𝑘=𝝍^𝑇_𝑘𝐌𝐔=𝑄^∗_𝑁+𝑘.

The Laplace transform of (8) provides the following, for the 𝑗th dof:𝑦_𝑗(𝑠)=−^𝑁_𝑘=1𝜓_𝑗𝑘𝑄_𝑘+𝜓^∗_𝑗𝑘𝑄^∗_𝑘𝑠+𝜓_𝑗𝑘𝜆^∗_𝑘𝑄_𝑘+𝜓^∗_𝑗𝑘𝜆_𝑘𝑄^∗_𝑘𝑠²−𝜆_𝑘+𝜆^∗_𝑘𝑠+𝜆_𝑘𝜆^∗_𝑘𝑥̈_𝑔(𝑠), (9) which is valid for both classically and nonclassically damped systems.

The frequencies Ω_𝑘 and damping ratios 𝜁_𝑘 can be evaluated through the following expressions: Ω_𝑘=𝛼²_𝑘+𝛽²_𝑘, 𝜁_𝑘=−𝛼_𝑘/Ω_𝑘 where the 𝑘th eigenvalue, 𝜆_𝑘, and its conjugate, 𝜆^∗_𝑘=𝜆_𝑁+𝑘, are given by 𝜆_{𝑘,𝑁+𝑘}=𝛼_𝑘±𝑖𝛽_𝑘 (𝑖=√−1 is the imaginary number).

It is worthy to mention that, if the eigenvectors are normalized through (7), then the mass, stiffness, and damping matrices can be expressed as functions of the complex eigenvalues and eigenvectors, De Angelis et al. [1], as follows:𝐌=𝝍𝚲𝝍^𝑇−1;𝐊=−𝝍𝚲⁻¹𝝍^𝑇−1;𝐋=−𝐌𝝍𝚲²𝝍^𝑇𝐌, (10) with the eigenvectors satisfying the following conditions:𝝍𝝍^𝑇=𝟎. (11)

For general damping assumptions the relationship between the modal parameters of the second- and the first-order models are quite difficult to derive, Chen et al. [11], while in the case of classical damping, the two representations are related through simple relationships. In fact, by comparing (3) and (9) it results in the following:0=𝜓_𝑗𝑘𝑄_𝑘+𝜓^∗_𝑗𝑘𝑄^∗_𝑘,𝜃_𝑗𝑘=𝜓_𝑗𝑘𝜆^∗_𝑘𝑄_𝑘+𝜓^∗_𝑗𝑘𝜆_𝑘𝑄^∗_𝑘,𝑠²+𝜀_𝑘𝑠+Ω_𝑘=𝑠²−𝜆_𝑘+𝜆^∗_𝑘𝑠+𝜆_𝑘𝜆^∗_𝑘. (12)

The quantities 𝜖_𝑗𝑘 and 𝜖^∗_𝑗𝑘, given by the second expression of (12), can be rewritten as follows:𝝐=𝝍𝚲^∗diag𝑄_𝑘,𝑄^∗_𝑘. (13) Similarly to the matrix 𝜽, also the matrix 𝝐 is independent on the transformation and, thus, satisfies a relationship analogous to (4), that is,𝝐𝐔=𝐔, (14) which is valid for general damping assumptions.

3. Identification of Modal and Physical Parameters

This section is devoted to discuss a methodology for identifying the natural frequencies, the modal damping ratios, the seismic eigenvectors, and, in some cases, the physical matrices of a linear classically damped structure subjected to seismic ground motion. The methodology is based on a time-domain first-order state representation and, in detail, identifies the following quantities: (i) first-order model; (ii) frequencies, dampings, and seismic eigenvectors; (iii) physical parameters (stiffness, damping, and mass matrices).

(i) Identification of the First-Order Model. The first phase of this methodology consists of identifying the first-order state space model of the system from general input/output data using an Eigensystem Realization Algorithm/Observer Kalman filter Identification- (ERA/OKID-) based approach [16, 21]. The identified modal model is expressed as follows (De Angelis et al. [1]):𝝃̇₁(𝑡)=𝚲𝝃₁(𝑡)+𝐁₁𝑥̈_𝑔(𝑡),𝐲(𝑡)=𝐂_1𝑝𝝃₁(𝑡), (15) where Λ, 𝐁₁ and 𝐂_1𝑝 are complex matrices given by𝚲=𝐓₀⁻¹𝐀₀𝐓₀,𝐁₁=𝐓₀⁻¹𝐁₀,𝐂_1𝑝=𝐂_0𝑝𝐓₀. (16) In (16), 𝐀₀, 𝐁₀, and 𝐂_0𝑝 are the identified continuous state matrices, and 𝐓₀ is a transformation matrix which relates the two representations of the first- and second-order of the model, De Angelis et al. [1]. The matrix Λ contains the complex eigenvalues, 𝜆_𝑘 and 𝜆^∗_𝑘, and the matrix 𝐂_1𝑝 contains the complex eigenvectors of the system.

It is worthy to remember that the relationship between the identified matrix related to the absolute accelerations, 𝐂_1𝑎, and that related to the displacements, 𝐂_1𝑝, is given by𝐂_1𝑝=𝐂_1𝑎𝚲⁻². (17)

(ii) Identification of Frequencies, Dampings, and Seismic Eigenvectors. Once the complex eigenvalues 𝜆_𝑘 and 𝜆^∗_𝑘 are identified, the frequencies and dampings parameters of the system can be derived from (12), that is,Ω_𝑘=𝜆_𝑘𝜆^∗_𝑘=𝜔²_𝑘;−𝜆_𝑘+𝜆^∗_𝑘=2𝜁_𝑘𝜔_𝑘. (18)

Once the matrices 𝐂_1𝑝 and 𝐁₁ are identified through (15), the seismic eigenvectors can be derived from (12) that is,𝜃_𝑗𝑘=𝐶_{1𝑝𝑗𝑘}𝜆^∗_𝑘𝐵_1𝑘+𝐶^∗_{1𝑝𝑗𝑘}𝜆_𝑘𝐵^∗_1𝑘. (19)

It is worthy to remind that the seismic eigenvectors identified at this stage has a double advantage. First they are independent of the normalization of the normal modes and, thus, can be determined also in case of limited set of instrumentations and without any information about the physical model of the system. In addition, they constitute a set of modal parameters which are of primary interest in many system identification applications such as finite element models or health monitoring problems. In fact, in the first case, they provide further information for the model validation which can be added to frequencies and damping; in health monitoring, they can be efficiently used in detecting damage and its location.

(iii) Identification of the Physical Parameters. The physical parameters of the system can be obtained by (10) if the complex eigenvectors contained in the matrix 𝐂_1𝑝 are normalized through (7). To this purpose the conditions, in terms of number of sensors and information about the mass of the system, to normalizing the eigenvectors and deriving the second-order matrices are discussed.

Let us assume that the complex eigenvectors contained in the matrix 𝐂_1𝑝 and the normalized complex eigenvector 𝝍 are related through a scaling matrix 𝝉 such that,𝝍=𝐂_1𝑝𝝉, (20) where 𝝉 is a complex diagonal matrix with terms such that 𝜏_𝑘=𝜏^∗_𝑁+𝑘 which contains the unknown scaling factors.

Using the scaling condition of (20), written as follows:𝐌⁻¹=𝝍𝚲𝝍^𝑇, (21) and in the common assumption of a diagonal mass matrix, (21) provides 𝑛𝑒₁=𝑚(𝑚+1)/2−𝑚 equations expressing the zero off-diagonal terms. Equation (11) provides 𝑛𝑒₂=𝑚(𝑚+1)/2 equations, where 𝑚 is the number of measurement points. The total number of equations is 𝑛𝑒_𝑇 given by 𝑛𝑒_𝑇=𝑚². Then, the eigenvectors can be normalized only when𝑛𝑒_𝑇+𝑛_𝑚≥2𝑁, (22) where 𝑛_𝑚 (≥1) is the number of known masses connected to the measured dofs. If the condition is satisfied, it is possible to determine the following quantities:

(1) the transformation matrix 𝝉, (2) the normalized eigenvectors 𝝍, and (3) the physical matrices 𝐌, 𝐋, and 𝐊.

Using (20), (11) and (21) become:^𝑁_𝑘=1𝑐_𝑖𝑘𝑐_𝑗𝑘𝜏²_𝑘+𝑐^∗_𝑖𝑘𝑐^∗_𝑗𝑘𝜏_𝑘^∗2=0,_𝑁𝑘=1𝑐_𝑖𝑘𝑐_𝑗𝑘𝜆_𝑘𝜏²_𝑘+𝑐^∗_𝑖𝑘𝑐^∗_𝑗𝑘𝜆^∗_𝑘𝜏_𝑘^∗2=𝑚⁻¹_𝑖𝑗, (23) with 𝑖,𝑗=1,2,….,𝑚. When 𝑖=𝑗, the second set of (23) is utilized only if the mass 𝑚_𝑖𝑖 is known. The system of equations (23) can be rewritten in the following form:𝐙𝐯=𝐭, (24) where 𝐯 is the vector containing the unknown terms 𝜏²_𝑘 and 𝜏_𝑘^∗2 (𝑣_𝑘=𝜏²_𝑘 and 𝑣^∗_𝑘=𝜏_𝑘^∗2), 𝐙 is the coefficients matrix, 𝐭 is a vector that contains all the known quantities. If all masses are unknown, 𝐭 is the zero vector and the system is homogeneous. Then the solution depends on a single parameter which can be determined only if one mass of the measured dofs is known. Generally the system (24) is characterized by a number of equations greater than the number of unknowns and the vector 𝐯 is given by 𝐯=𝐙⁺𝐭, where 𝐙⁺ is the pseudoinverse of the matrix 𝐙. Once the vector 𝐯 is known, the matrix 𝝉 can be derived.

In the case of a complete set of measurements, that is 𝑚=𝑁, the physical matrices of the system can be derived by (10) with 𝝍=𝐂_1𝑝𝝉.

In the case of incomplete measurements, that is, 𝑚<𝑁, it is still possible to normalize the eigenvectors if the condition (22) is satisfied. However, only reduced forms of the physical matrices are possible to be derived, Luş et al. [2]. The approximation of these reduced forms depends on the damping system assumptions.

In the case of classically damped systems, the modes of vibration 𝝓 of the system can be derived from the complex eigenvectors 𝝍. Then, the 𝑚×𝑚 “reduced” form of the mass, damping, and stiffness matrices are as follows:𝐌=𝝓^+𝑇𝝓⁺;𝐋=𝝓^+𝑇𝜺𝝓⁺;𝐊=𝝓^+𝑇𝛀𝝓⁺, (25) where 𝝓 is the 𝑚×𝑁 matrix containing the identified eigenvectors and 𝝓⁺ is its pseudoinverse. The matrices (25) are the partitions of the “full-order” physical matrices referred to the measured degrees of freedom.

In the case of nonclassically damped systems, the reduced forms of the physical matrices are expressed in terms of the complex eigenvectors 𝝍, of dimension 𝑚×2𝑁, as follows:𝐌=𝝍𝚲𝝍^𝑇−1,𝐋=−𝐌𝝍𝚲²𝝍^𝑇𝐋𝐌𝐌,𝐊≃⁻¹𝐋−𝐌𝝍𝚲³𝝍^𝑇𝐌. (26)

The “reduced” forms given by the first two equations (26) are exact and provide the known partitions of the “full-order” mass and damping matrices of the system. On the contrary the “reduced” form given by the third equation (26) provides only an approximation of the exact partition of the “full-order” stiffness matrix, Luş et al. [2].

3.1. The Case of All Unknown Masses

The assumption of having at least one known mass at the measured degrees of freedom to identify the physical properties of the system can be easily verified if the procedure is applied to buildings where the evaluation of the floor mass is quite simple.

However, the hypothesis of a known mass can be also removed in the case of damage detection problems and health monitoring models. In fact, if all the masses are unknown, it is possible to determine a transformation matrix 𝝉^′ such that𝜏²_𝑘=𝑚_𝑙⁻¹𝜏_𝑘^′2, (27) where 𝑚_𝑙 is the unknown mass related to the 𝑙th dof. Then (23) becomes^𝑁_𝑘=1𝑐_𝑖𝑘𝑐_𝑗𝑘𝜏_𝑘^′2+𝑐^∗_𝑖𝑘𝑐^∗_𝑗𝑘𝜏_𝑘^′∗2=0,_𝑁𝑘=1𝑐_𝑖𝑘𝑐_𝑗𝑘𝜆_𝑘𝜏_𝑘^′2+𝑐^∗_𝑖𝑘𝑐^∗_𝑗𝑘𝜆^∗_𝑘𝜏_𝑘^′∗2=𝛿_𝑖𝑗, (28) with the condition that the second set of equations (28) includes only the term with 𝑖=𝑗=𝑙 in addition to the terms with 𝑖≠𝑗. The solution of the system (28) provides the unknown matrix 𝝉^′.

Then, using (20) and (27), the eigenvectors are given by𝝍=𝑚_𝑙^−0.5𝐂_1𝑝𝝉^′=𝑚_𝑙^−0.5𝝍^′, (29) and the “full-order” physical matrices (𝑚=𝑁) are expressed through the following expressions:𝐌=𝑚_𝑙𝐂_1𝑝𝝉^′𝚲𝝉^′𝑇𝐂^𝑇_1𝑝=𝑚_𝑙𝐌^′,𝐊=−𝑚_𝑙𝐂_1𝑝𝝉^′𝚲⁻¹𝝉^′𝑇𝐂^𝑇_1𝑝=𝑚_𝑙𝐊^′,𝐋=−𝑚_𝑙𝐌^′𝐂_1𝑝𝝉^′𝚲²𝝉^′𝑇𝐂^𝑇_1𝑝𝐌^′=𝑚_𝑙𝐋^′. (30)

In the case of incomplete measurements, that is, 𝑚<𝑁, it is still possible to derive expressions analogous to (25) and (26) and containing the unknown mass 𝑚_𝑙.

It is interesting to observe that the identified matrices (30) depend linearly on the unknown mass 𝑚_𝑙. This could be useful in the case of finite element updating models as well as in damage detection processes. In this last case the comparison between the system’s state before and after damage would be independent on the unknown mass 𝑚_𝑙 if it is assumed constant.

4. Optimization of Modal Parameters

The identification process is generally affected by errors which depend either on modelling errors or noise pollution. The prediction of the modal parameters can be improved by minimizing the following objective function:𝑓(𝐩,𝑡)=[𝐲(𝑡)−𝐲(𝐩,𝑡)]^𝑇[𝐲(𝑡)−𝐲(𝐩,𝑡)], (31) where 𝐲(𝑡) is the measured data, and 𝐲(𝐩,𝑡) is the predicted data, being 𝐩 a vector containing the modal parameters to be optimized. The parameters 𝑝_𝑖 already identified and referred to the eigenvectors, can be constrained through (11) and (14) and the condition of diagonal mass (21); if the system is classically damped the parameters 𝑝_𝑖 can also be constrained through (12).

The predicted data 𝐲(𝐩,𝑡) included in (31) can be obtained by considering the identified first-order model:𝝃̇(𝑡)=𝚲𝝃(𝑡)+𝐐𝑥̈_𝑔(𝑡),𝐲(𝑡)=𝝍𝚲²𝝃(𝑡). (32) The McMillan transformation, Luş [18], applied to (32) provides a first-order model with real matrices and vectors given by:𝝃̇(𝑡)=𝐀_𝑀𝑐𝝃(𝑡)+𝐁_𝑀𝑐𝑥̈_𝑔(𝑡),𝐲(𝑡)=𝐂_𝑀𝑐𝝃(𝑡), (33) where the matrix 𝐀_𝑀𝑐 has the following expression:𝐀_𝑀𝑐=Ω𝐎𝐈−diag_𝑘diag𝜀_𝑘, (34) with 𝐎 and 𝐈 the zero and unity 𝑁×𝑁 matrices, 𝐁_𝑀𝑐 an 𝑁×1 vector and 𝐂_𝑀𝑐 an 𝑚×𝑁 matrix.

To perform the optimization, the system (33) is transformed from the continuous to the discrete time:𝐱(𝑘+1)=𝚽_𝑀𝑐𝐱(𝑘)+𝚪_𝑀𝑐𝑥̈_𝑔(𝑘),𝐲(𝑘)=𝐂_𝑀𝑐𝐱(𝑘), (35) where 𝑘 is a time index, Φ_𝑀𝑐 is the discrete state transition matrix, Γ_𝑀𝑐 is the discrete time input vector. It is worthy to underline that the system (35) in the discrete time is equivalent to the system (33) in the continuous time and it is used in the computational procedure. Using (35), (31) can be rewritten as follows:𝑓(𝐩,𝑘)=[𝐲(𝑘)−𝐲(𝐩,𝑘)]^𝑇[𝐲(𝑘)−𝐲(𝐩,𝑘)]. (36)

5. Applications

5.1. Numerical Applications

The identification procedure is applied, De Angelis et al. [22], to the five degrees of freedom system illustrated in Figure 1. The data used in the identification are provided by numerical analyses.

Figure 1 

Five dofs frame.

The response of the system is expressed in terms of absolute accelerations at each degree of freedom and is evaluated through the finite element model subjected to the north-south component of the El Centro Earthquake (18/5/1940). It is assumed 𝑑𝑡=0.02 sec and a number of points equal to 1560. The time history accelerations are, then, polluted with Gaussian, zero-mean, white noise sequences whose root mean square is equal to 10% of the unpolluted data.

5.1.1. Registrations at All Floors

In this first case the accelerations at each floor and the ground acceleration are measured.

The 𝐸𝑅𝐴/𝑂𝐾𝐼𝐷 algorithm, Luş et al. [16], provides the identified matrices 𝐀₀, 𝐁₀, and 𝐂_0𝑎; next, the matrices Λ, 𝐁₁, and 𝐂_1𝑝=𝐂_1𝑎Λ⁻² are derived through (16) and reported in Table 1, 2, and 3. Using the eigenvalues of the first-order system and applying (18), it is possible to identify the frequencies and damping coefficients of the system (Table 4) which are coincident with the exact ones. The complex eigenvectors 𝝍 obtained by (20) provide, through (13), the vector 𝝐. Finally, the seismic eigenvectors 𝜽_𝑘 are evaluated through (19) and the results are reported in Table 5. Figure 2 shows the first two identified seismic eigenvectors 𝜽_𝑘.

Table 1 

Eigenvalues (×10²).

Table 2 

Matrix 𝐁₁(×10⁻²).

Table 3 

Matrix 𝐂_1𝑝(×10⁻³).

Table 4 

Identified frequencies and damping coefficients.

Table 5 

Identified vectors 𝜽_𝑖=𝜖_𝑖+𝜖^∗_𝑖.

Figure 2 

First and second identified eigenvectors.

The exact values of the modal parameters have been obtained and conditions (4) and (14) are satisfied for 𝜃_𝑗𝑘 and 𝜖_𝑗𝑘.

Once the modal parameters are obtained, the physical matrices of the system can be obtained only if the complex eigenvectors are normalized. In this case, under the assumption of full measurements (i.e., 𝑚=5) and the knowledge of the mass at the first degree of freedom (i.e., 𝑛_𝑚=1), the condition (22) is satisfied and provides 26 available equations. The solution of the system (24), then, provides the vector 𝐯^′ reported in Table 6.

Table 6 

Vector 𝐯^′𝑇.

The normalized and nonscaled eigenvectors 𝝍^′ (Table 7) are derived from (29).

Table 7 

Matrix 𝝍^′.

From 𝝍^′, using the value of the mass 𝑚₁=50𝑡, it is possible to evaluate the transformation matrix and, then, the eigenvector 𝝍.

The real and imaginary parts of the eigenvectors being equal, the system is classically damped.

The vectors 𝝓_𝑘 (Table 8) are then evaluated from the eigenvectors 𝝍. In Figure 2 the first two identified eigenvectors 𝝓_𝑘 are also reported (to note the proportionality between the two eigenvectors).

Table 8 

Identified normalized eigenvectors 𝝓_𝑘.

The mass and stiffness matrices are, finally, derived from (10) and are given by:𝐌=diag50.0000;40.0000;40.0000;40.0000;30.0000, (37)𝐊=10⁵+2.3120−1.3790+0.4300−0.0860+0.0140−1.3790+1.9130−1.2820+0.3860−0.0550+0.4300−1.2820+1.8440−1.1550+0.2530−0.0860+0.3860−1.1550+1.4400−0.6050+0.0140−0.0550+0.2530−0.6050+0.3970. (38)

It is clear, then, that the proposed identification procedure provides, in the case of noise-free measurements, the exact values of the modal and physical parameters.

5.1.2. Registrations at 1st, 3rd, 4th, and 5th Floors

The identification of the first-order model and modal parameters is not affected by the number of measurements and, thus, the frequencies and the damping coefficients are exactly the same obtained in the previous case (Table 4). Also the eigenvector 𝜽_𝑘 is the same except for the component related to the 2nd dof where the data are missed.

In this case, of incomplete measurements (𝑚=4) and still one known mass at the first degree of freedom (𝑛_𝑚=1), the condition (22) is still satisfied and the eigenvectors can still be normalized. The eigenvectors 𝝓_𝑘 are, then, evaluated from the complex eigenvectors and are the same as the previous case except for the component related to the 2nd dof where the data are missed. Also, only reduced form of the physical matrices can be derived from (25). The results are the 𝑚×𝑚 reduced form obtained by removing from each of (37) and (38) the second column and the second row. The reduced stiffness 𝐊 matrix can be also derived by using (26), valid in the case of general damping. It results in the following:𝐊=10⁵−2.3115−0.42950.0860−0.0140−0.4295−1.84351.1550−0.2530+0.08601.1550−1.44000.6050−0.0140−0.25300.6050−0.3970. (39)

The comparison with the matrix (38) shows that the reduced stiffness matrix is slightly different from the exact one with differences among the correspondent diagonal terms less than 1% and those among the off-diagonal terms less than 3%. This means that the reduced stiffness matrix (39) can be considered a good approximation of the exact one.

It is worthy to compare the results of the identification process without and with the optimization procedure described in the paragraph 4, when the time history accelerations are polluted with a Gaussian, zero-mean, white noise (10%). The procedure has been adopted to improve the estimates of the modal parameters (frequencies, dampings, and eigenvectors). In the procedure to minimize the objective function (31) a Levenberg-Marquardt algorithm is used.

Table 9 shows the frequencies and damping coefficients when evaluated using the optimization procedure (𝑓^opt e 𝜁^opt) and without it (𝑓 e 𝜁). It is evident that the identification is satisfactory also without the optimization procedure which, however, reduces the differences with respect to reference values. The efficacy of the optimization procedure is also evidenced in the results reported in Tables 10 and 11 in terms of the 𝜽_𝑘 and 𝝓_𝑘 eigenvectors.

Table 9 

Identified frequencies (Hz) and dampings (%) without and with optimization.

Table 10 

Identified seismic eigenvectors 𝜽_𝑘 with optimization.

Table 11 

Identified normalized eigenvectors 𝝓_𝑘 with optimization.

5.2. Experimental Application

The proposed methodology has also been used to identify the modal and physical parameters of the structure shown in Figure 3. The test structure represents a 1 : 5 scaled adjacent structures model used in a shaking table experimentation on both coupled and uncoupled configurations, Cimellaro et al. [23]. In this paper the attention is focused to the four dofs structure in the uncoupled configuration.

Figure 3 

Experimental model.

The structure has plan dimension of 60 × 60 cm and interstory height of 60 cm, so that the structure is 240 cm high. The vertical elements have been realized with commercial steel profiles L 40 × 40 × 4 mm and all the connections are bolted. Four steel blocks, of 19 kg mass each, have been installed at each floor to simulate the floor mass. The total mass of the structure is about 500 kg.

In order to measure the structural response, the test structure has been instrumented with accelerometers at each floor and on the shaking table, and with laser displacement transducers, Figure 3. In the characterization tests several monodirectional excitations (random white noise and natural seismic records), with different intensities have been used. The data acquisition has been made by using an MTS 468D system, with sampling rate of 200 Hz.

The mass matrix is assumed to be diagonal, with 𝑚_𝑖=125 kg. In Table 12 the identified frequencies and damping coefficients are reported. The identified damping and stiffness matrices are presented in the following matrices:𝐋=10²,7.3899−0.9820−1.48200.8828−0.98205.5690−0.5126−0.5928−1.4820−0.51266.3829−1.76430.8829−0.5928−1.76435.1474𝐊=10⁶.8.2009−4.32740.36210.3665−4.32748.1550−4.79310.35450.3621−4.79318.2536−3.87630.36650.3545−3.87633.3340 (40)

Table 12 

Identified frequencies and damping coefficients.

Finally, Figure 4 shows, for the El Centro input, the experimental and numerical accelerations at each floor. The figure proves that the identified numerical model reproduces the experimental data very well.

(a)

(b)

(c)

(d)

Figure 4 

Time histories absolute accelerations: experimental 𝑣𝑠 numerical.

6. Conclusions

In this paper a procedure to identify the modal and physical parameters of a linear system subjected to ground motion has been discussed. In particular, the paper has proposed a methodology which is based on a time-domain first-order state representation and allows, first, to identify the modal parameters of the system including the seismic eigenvectors which do not require any normalization. Then, the procedure is able to provide the conditions to normalizing the eigenvectors and, hence, deriving the second-order matrices of the system. The significance of the proposed methodology can be summarized in the following points.(a)The procedure identifies the seismic eigenvectors of a classically damped system instead of the normal modes (as usually found in the literature) which, in contrast with the normal modes, do not require any normalization and, thus, they can be determined whatever the number of sensors is and with no information and restrictions about the physical model of the system. This means that the seismic eigenvectors can represent a useful tool for health monitoring and of significance in updating models where they provide, jointly with the frequencies and dampings, additional information for validating the model.(b) In the case of a full set of instrumentations, it provides the conditions to normalizing the eigenvectors and, hence, deriving the second-order matrices of the system without any restrictions about the system behaviour. This represents a new finding with respect to the available literature which instead proposes methodologies valid under the assumption of shear-type behaviour.(c)In the case of incomplete set of measurements, it is still possible to normalize the eigenvectors with the unique assumption of having at least one mass of the measured dofs known and under the satisfaction of a certain condition (22). In this case, it is also possible to obtain reduced forms of the physical matrices.(d)The assumption of having at least one known mass at the measured degrees of freedom to identify the physical properties of the system can be easily verified if the procedure is applied to buildings where the evaluation of the floor mass is quite simple. However, it is interesting to observe that the identified matrices 𝐌, 𝐊, and 𝐂 (see (30)) depend linearly on one unknown mass. This could be useful in the case of finite element updating models as well as in damage detection processes because, in this last case, the comparison between the system’s state before damage would be independent on the unknown mass if it is assumed constant.

The procedure has been discussed theoretically and, then, applied to some numerical examples which have shown the efficacy of the procedure. Finally, the procedure has been validated also by an experimental test performed on a four-story steel frame subjected to earthquake excitations. The authors are actually analyzing the extension of the proposed procedure to nonclassically damped systems.