Approximation of structural damping and input excitation force

Abstract

Structural dynamic characteristics are the most significant parameters that play a decisive role in structural damage assessment. The more sensitive parameter to the damage is the damping behavior of the structure. The complexity of structural damping mechanisms has made this parameter to be one of the ongoing research topics. Despite all the difficulties in the modeling of damping, there are some approaches like as linear and nonlinear models which are described as the energy dissipation throughout viscous, material or structural hysteretic and frictional damping mechanisms. In the presence of a mathematical model of the damping mechanisms, it is possible to estimate the damping ratio from the theoretical comparison of the damped and un-damped systems. On the other hand, solving the inverse problem of the input force estimation and its distribution to each SDOFs, from the measured structural responses plays an important role in structural identification process. In this paper model-based damping approximation method and a model-less structural input estimation are considered. The effectiveness of proposed methods has been carried out through analytical and numerical simulation of the lumped mass system and the results are compared with reference data. Consequently, high convergence of the comparison results illustrates the satisfactory of proposed approximation methods.

Keywords

structural modal parameters / damping identification method / input excitation force identification / Inverse problem

Introduction

The structural identification is classified in two scale like entire structure and material scales. In the whole structural level, assorted analysis methods have been extensively used to extract dynamic characteristic based on measurement vibration data as an inverse problem, as well as identifying damage on the structures. The structural dynamic characteristics are the most significant parameters that play a decisive role in structural damage assessment. The significance of these determinants came from the fact that their changes have the physical meaning, such as the variability of the modal frequency, mode shapes and modal damping are related to variability of the structural stiffness, mass and energy dissipation behavior respectively. On the other hand, changing in modal properties is the result of changing in geometry shape and material properties of the structure. In the material scale, damage identification and tracking of effects on mechanical properties are considered as multiscale analysis that contains nano, micro, meso and macro scales. Nanthakumar et al. [] proposed an iterative method to identify damage in piezoelectric structures. In each iteration for different defect configuration, the inverse problem of cracks and voids detection is treated by solving the forward problem with using extended finite element methods (EFEM). The cost function is minimized by performing multilevel coordinate search (MCS) method. Also, the numerical method based on combination of classical shape derivative and the level-set method is used to minimize the cost function in structural optimization [,]. Integrity of XFEM-MCS methodology and proposed numerical method is demonstrated throughout the test problems []. Rabczuk et al. [] considered to the fracture and fragmentation of concrete through simulation of the concrete fragmentation under explosive loading by a meshfree lagrangian and smooth particle hydrodynamics methods. Also, three dimensional discrete cracks modeling in meshfree are considered in Refs. [–]. Goangseup et al. [] proposed an extended meshfree method for cohesive cracks. Rabczuk et al. [] proposed a three-dimensional meshfree method for arbitrary crack initiation and propagation. Moreover, a three-dimensional meshfree methods for modeling random crack initiation and growth are presented in Ref. []. Various crack tracking techniques are considered which is applicable in three-dimensions in the context of partition of unity methods, especially meshfree methods []. The meshfree method based on the local partition of unity for cohesive cracks are presented in Ref. []. Analysis of prestressed concrete beams under quasi-static loading by choosing coupled element free Galerkin finite element approach are investigated in Ref. []. Two-dimensional approach to model fracture of reinforced concrete structures under ascending static loading conditions are described in Ref. []. Particle methods for modeling reinforced concrete [] which can easily handle large deformations and fracture and the dynamic failure of concrete structures under blast and impact loading are presented [].

There are wide ranges of approaches to the structural identification. Accurate identification of modal parameters and specially tracking the variability of modal frequencies and mode shapes are discussed in literatures [–]. The more sensitive parameter to the damage is the damping behavior of the structure. The damping properties of a system are described as the ratio of energy absorbing ability. The complexity of structural damping mechanisms has made this parameter to be one of the ongoing research topics. The uncertainty of which state variables of the motion that are affected by the damping forces is hardening the modeling of the damping. Damping modeling are considered as linear and nonlinear models which are described as the energy dissipation throughout viscous, material or structural hysteretic and frictional damping mechanisms [–]. Moreover, measuring the input force and its distribution on each SDOFs are not possible in many cases. Structures are exposed to the various forcing scenarios at the same time, solving the inverse problem of the input force estimation and its distribution to each SDOFs, from the measured structural responses plays an important role in structural identification process [–].

Rayleigh damping is the stiffness and mass proportional model which has been used for decades. It is assumed that the only damping related state of the motion is the instantaneous velocity variable []. Despite the all of damping modeling advances, the real structural energy dissipation mechanism is still complicated. Lee [] developed a direct method to identify damping from frequency response function (FRF). To gain more accurate results, the natural mode information is not used. The both viscous damping and internal structural damping mechanisms are identified in separate matrices. H. Yamaguchi, R. Adhikari [] identified modal damping of structural cables by using energy-based representation of modal damping. According to this damping ratio definition the modal damping for each mode is considered as the ratio of the modal dissipated energy per cycle to the modal potential energy. Finally, the damping ratio is semi-analytically evaluated in a different type of bridges to investigate the applicability of energy-based method. Xu et al. [] proposed a neural network-based algorithm for direct identification of structural stiffness and damping parameters by using time domain velocity and displacement responses. The proposed method theoretically based on the comparison of an object structure response with a reference structure which has the same topology and a degree of freedom with that. To investigate the performance of the neural network, the proposed root mean square (RMS) difference vector of velocity and displacement responses are evaluated for both reference and object structure. The results prove that it is an effective index for system identification with a parametric evaluation neural network. To identify multi degree of freedom system’s damping, Slavic et al. [] presented a continuous wavelet transform (CWT) method based on the Gabor wavelet function for possibility of adapting its time and frequency spread. Some uncertainties such as: a description of the instantaneous noise, the edge-effect of the CWT, the frequency-shift of the CWT and the bandwidth of the wavelet function and the selection of the parameters of the Gabor wavelet function of the CWT are considered throughout presenting of three damping identification methods: the cross-section method, the amplitude method and the phase method. The results demonstrate the advantages of using the amplitude and phase methods, which give information about the instantaneous noise and are appropriate for automating the identification process. Min et al. [] proposed a direct identification method by using experimental data. System matrices are identified for non-proportional damping structures by using modal parameters. Consequently, this method can accurately identify the stiffness, mass and damping matrices of the highly damped system, and is the perfect mathematical model for a lumped mass system. Arora [] proposed a new direct structural damping identification method by using complex FRFs and accurate mass and stiffness matrices. In this method damping is modeled as structural damping by using the complex stiffness model. First, the mass and stiffness matrices are updated by using FRF-based updating method, then the damping is identified by using updated mass and stiffness matrices as proposed in this study. The advantage of this method is its ability in the identification of damping in closely spaced mode cases. This method requires a complex FRF matrix of structure so it’s not practicable for using in complicated structures. Different numerical and experimental cases with various damping levels are considered in this study, consequently proposed method is able to identify the experimental FRFs with all levels of damping in the system. Pan and Wang [] presented a new potential damping model in order to identify the exponential damping model. Application of complex modes analysis and its damping identification procedure are investigated. As a result, it’s applicable to identifying both viscous and non-viscous damping in structures. Also, an iterative method is proposed for relaxation factor. Finally, the FE model updating method for the systems with exponential damping is presented based on FRF, in which it is suitable to predict the natural frequencies and FRFs of the systems.

Consequently, it is obvious that the complication of the various approaches due to the intrinsic complexity of the physical phenomenon. Mystery of the damping comes from the fact that unlike the elastic modulus which is accounted in stiffness computation, the damping properties of materials are not developed well, so it could not be possible to identify damping from properties like as the stiffness account. The structural damping matrix should be calculated from the modal damping ratios. On the other hand, because of various mechanisms of damping sources and uncertainty of damping relate state of the motion, it’s not feasible to identify or describe a mathematical model thus it should be idealized like as viscous or equivalent viscous damping models. In the presence of any mathematical model of the damping mechanisms, it is possible to estimate the damping ratio from the theoretical comparison of the damped and un-damped systems, also it is possible to measure structural responses and fix damping related state of the motion to be zero in order to find structural un-damped responses. The both of these facts are used in the process of the further proposed method. The idea behind the proposed damping identification method is the comparison of damped system models with un-damped forms.

Theoretical development of the proposed methods

Mathematical model-based damping identification

Despite all the difficulties in the modeling of damping, there are some approaches like as a viscous and structural (hysteretic) damping models that have been used for decades. If the exact damping related state of the equation of motion is known, it is possible to identify the exact damping ratio from the comparison of the damped and un-damped systems, but if the exact damping related state of the equation of motion is not known then it is possible to identify the equivalent damping ratio by mentioned comparison. The importance of this kind of approach is that clarification in a damping variation of the structure in the case of unknown damping model. The viscous and hysteretic damping models are used in this study to identify the damping ratio. First, the damped and un-damped systems with viscous damping models are introduced as the second order differential equation of motion for a single degree-of-freedom (SDOF), given by:

(1)m{\ddot{X}}_d(t)+c{\dot{X}}_d(t)+k{X}_d(t)=F_d(t),

(2)m{\ddot{X}}_u(t)+k{\dot{X}}_u(t)=F_u(t),

where {\ddot{X}}_d,{\dot{X}}_d,X_dand F_d are the acceleration, velocity, displacement responses and excitation force of the damped system, and {\ddot{X}}_u, X_u, F_u are the acceleration, displacement responses and excitation force of un-damped system, respectively. Also,m is the mass, c is the viscous damping coefficient, k is the stiffness, and t is the time variable. For harmonic excitation, Fourier transform solution is X(t)=X(\omega )e^{i\omega t} andF(t)=F(\omega )e^{i\omega t}. Equation (1) and Eq. (2) become:

(3)(k-{\omega }_d^{\mathrm{2}}m)X_d(\omega )+(i\omega c)X_d(\omega )=F_d(\omega ),

(4)(k-{\omega }_u^{\mathrm{2}}m)X_u(\omega )=F_u(\omega ).

Normal or real dynamic stiffness matrix for both systems are given by:

(5)S_d=k-{\omega }_d^{\mathrm{2}}m,

(6)S_u=k-{\omega }_u^{\mathrm{2}}m.

The excitation force is known or acquired from the lowest level response of the structure or could be estimated from the second part of proposed methods. As assumptions: first, the mass and stiffness of the un-damped system are updated with considered to the damped system responses by using the Finite element (FE) model updating in order to gain un-damped system responses. Secondly, excitation forces of the both systems are equal. Then from Eq. (3) and Eq. (4):

(7)F_d(\omega )=F_u(\omega ).

By substituting Eq. (5)-Eq. (6) and also using Eq. (3) and Eq. (4), Eq. (7) becomes:

(8)S_d{X}_d(\omega )+(i\omega c)X_d(\omega )=S_u{X}_u(\omega ).

Equation (8) can be rewritten as:

(9)(i\omega c)X_d(\omega )=S_u{X}_u(\omega )-S_d{X}_d(\omega ).

For each \omega value, Eq. (9) can be written as:

(10)i\omega c={\displaystyle \frac{S_u{X}_u(\omega )-S_d{X}_d(\omega )}{X_d(\omega )}}.

Magnitude of complex value of c is giving viscous damping coefficient directly.

Energy dissipation per vibration cycle which is mentioned in many researches, prove that it is proportional to the velocity of motion and also over wide frequency range it’s independent from frequency and proportional to the square of the vibration amplitude [,]. Dissipated energy per cycle for viscous damping is:

(11)E_{dissipation}={\displaystyle {\int }_{\mathrm{0}}^T(f)}\dot{X}dt=\pi c\omega X^{\mathrm{2}},

(12)\Delta E_{viscous}={\displaystyle {\int }_{\mathrm{0}}^T(c\dot{X})}\dot{X}dt=\pi c\omega X_{\mathrm{0}}^{\mathrm{2}},

where X_{\mathrm{0}} is the initial vibration amplitude, and integral is taken over period T. By using the identified damping coefficient, viscous damping ratio, \xi can be as:

(13)\xi ={\displaystyle \frac{\Delta E_{viscous}}{\mathrm{4}\pi E_{potential}}},

where E_{potential} is the maximum strain potential energy of the structure and given by:

(14)E_{potential}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}k{X}_{\mathrm{0}}^{\mathrm{2}}.

By substituting Eq. (12), Eq. (14) in to Eq. (13), and using identified damping coefficient then:

(15)\xi ={\displaystyle \frac{c\omega }{\mathrm{2}k}}.

Finally, structural (hysteretic) damping model is used here to identify the damping. The damped and un-damped systems with hysteretic damping model are introduced as the second order differential equation of motion for a single degree-of-freedom (SDOF) by using the concept of complex stiffness, given by:

(16)m{\ddot{X}}_d(t)+(k+iD)X_d(t)=F_d(t),

(17)m{\ddot{X}}_u(t)+k{X}_u(t)=F_u(t).

For harmonic excitation, Fourier transform solution is X(t)=X(\omega )e^{i\omega t} andF(t)=F(\omega )e^{i\omega t}. Equation (16) and Eq. (17) become:

(18)(k-{\omega }_d^{\mathrm{2}}m)X_d(\omega )+(iD)X_d(\omega )=F_d(\omega ),

(19)(k-{\omega }_u^{\mathrm{2}}m)X_u(\omega )=F_u(\omega ).

Normal or real dynamic stiffness matrix for both systems are given by:

(20)S_d=k-{\omega }_d^{\mathrm{2}}m,

(21)S_d=k-{\omega }_u^{\mathrm{2}}m.

According to the same assumptions and procedure as mentioned above for viscous damping model (Eq. (3)-Eq. (9)), for each\omega values, hysteretic damping ratio can be identified.

(22)F_d(\omega )=F_u(\omega ).

By substituting Eq. (20)-Eq. (21) and also using Eq. (18) and Eq. (19), Eq. (22) becomes:

(23)S_d{X}_d(\omega )+(iD)X_d(\omega )=S_u{X}_u(\omega ).

Equation (23) can be rewritten as:

(24)(iD)X_d(\omega )=S_u{X}_u(\omega )-S_d{X}_d(\omega ).

For each \omega value, Eq. (24) can be written as:

(25)iD={\displaystyle \frac{S_u{X}_u(\omega )-S_d{X}_d(\omega )}{X_d(\omega )}}.

On the other hand, whereT=\mathrm{2}\pi /\omega is the time period of vibration. According to many researches which determined that the damping due to internal friction (material hysteresis) is nearly independent of the forcing frequency but still proportional to the square of the response amplitude [–].

(26)E_{dissipation}\propto a{X}^{\mathrm{2}}.

Then from Eq. (11) and Eq. (26);

(27)\pi c_{equivalent}\omega X^{\mathrm{2}}=a{X}^{\mathrm{2}}.

Then equivalent damping coefficient is:

(28)c_{equivalent}={\displaystyle \frac{a}{\pi \omega }}.

By substituting equivalent viscous damping coefficient instead of viscous damping coefficient in Eq. (1) and using the\dot{X}=i\omega X for a harmonic vibration F{e}^{i\omega t}, it becomes:

(29)m\ddot{X}+\left({\displaystyle \frac{a}{\pi \omega }}\right)i\omega X+kX=F{e}^{i\omega t}.

Let \eta ={\displaystyle \frac{a}{\pi k}} and D={\displaystyle \frac{a}{\pi }}: m\ddot{X}+(k+iD)X=F{e}^{i\omega t}.

By factoring out the stiffness and using identified hysteretic damping ratio, then:

(31)m\ddot{X}+k(\mathrm{1}+i\eta )X=F{e}^{i\omega t},

(32)\eta ={\displaystyle \frac{D}{k}},

where k(1+i\eta) is the complex stiffness description and, \eta is the hysteretic damping ratio or structural damping factor.

Mathematical model-less input force estimation and damping ratio

While the structures are exposed to the different forcing like a random earthquake, wind and environmental forcing patterns, so measuring the input force and its distribution on each SDOFs are not possible in many cases. Solving the inverse problem of the input force estimation and its distribution to each SDOF, from the measured structural responses plays an important role in structural identification process [–]. One of the fundamental concepts in the field of vibration analysis is the Frequency Response Function (FRF). FRF is the function of frequency that transforms the system input force to the response Fig. (1). It can be expressed as the Fourier series expansion of the input and output of the system.

Multiplying both sides of Eq. (34) with H_d(\omega ):

(35)H_d{ }^{-\mathrm{1}}(\omega )H_d(\omega )X_d(\omega )=H_d(\omega )F_d(\omega ).

In the first part of Eq. (35), the state of H_d(\omega )X_d(\omega ) theoretically means that, the damped system is inversely excited by the external excitation forceX_d(t), and the second part should be equal to the X_d(\omega ). Then Eq. (35) comes to:

(36)H_d{ }^{-\mathrm{1}}(\omega ){\chi }^f(\omega )=X_d(\omega ).

In which, {\chi }^f(\omega ) is the response of damped system to the external excitation forceX_d(t). If it is not possible to excite the system by this kind of excitation force then, for each SDOF, the{\chi }^f(\omega ) could be produce by using a numerical method. The mass and stiffness of the system are updated and only by guessing the damping coefficient value process should be continuing. The approximated damping coefficient should be iteratively checked with damping coefficient that identified by using the first part of the proposed method, or by checking the similarity of the estimated response to the measured response. High convergence will give the exact damping ratio. Once, the{\chi }^f(\omega ) response is acquired, then the external excitation force (F_d{ }^i) and FRF (H_d{ }^i) of the damped system should be identified by:

(37)H_d{ }^i(\omega )={\chi }^f(\omega )X_d{ }^{-\mathrm{1}}(\omega ),

(38)F_d{ }^i(\omega )=X_d(\omega ){\chi }^{f-\mathrm{1}}(\omega )X_d(\omega ).

As it is seen from Eq. (37) and Eq. (38), for all identified FRF and excitation force, they must produce the response of the system (X_d(\omega )), but the target FRF and excitation force should be justified in two steps. First, checking the damping coefficient from the first part of the proposed method, by using identified excitation force, and second measuring the similarity (in the mean of measuring the similarity of the shape) of the produced response by using new methods and the acquired response from the real system.

(39)H_d{ }^i(\omega )F_d{ }^i(\omega )=X_d(\omega ),

(40)X^i(\omega )=X_d(\omega ).

However, the proposed method seems to be simplistic, but according to the result of numerical and analytical approaches in the next sections, it is very effective in decomposition of the structural responses to their prior product parts. In the case of complicate models, just by guessing the damping ratio and controlling by the similarity of the identified responses with primarily measured responses, it is possible to estimate the structure’s input force and subsequently damping ratio.

Numerical investigation

Mathematical model base damping identification

In this part, the effectiveness of proposed identification methods is investigated by using two various data simulation approaches such as: mathematical model by using the Mathworks Matlab-Simulink programming environment and numerical evaluation of dynamic response by using a numerical method based on interpolation of excitation force []. In all two approaches zero mean band limited Gaussian white noise (WN) process (bandwidth between 0.05 and 50 Hz) with noise root mean square (RMS) acceleration of 0.04 g, is used as an external excitation force. Finally, the identified damping ratios are compared to the results which are identified by using an advanced output-only structural identification method which is named Natura Excitation Technique combined with Eigen system Realization Algorithm (NeXT-ERA).

In the first section, simple lumped mass model of SDOF system is used to acquire simulated data. The mass, stiffness and damping coefficient are assumed to be known. The simple mass-spring-damper model is built with Matlab Simulink programming environment which is shown in Fig. 2 is described by lumped massm of 500 kg and spring stiffness k of 5× 10⁵ N/m, the viscous damping coefficient c is assumed in four different values of 3 × 10³, 5 × 10³, 1 × 10⁴ and 2× 10⁴ N/m. To gain displacement response of damped and un-damped systems, 10 s external WN excitation force is input to the model.

According to the assumed values for m, k and c, the estimated damping ratio as the reference damping ratios is {\xi }_r, and also the identified damping ratios by using the proposed method is {\xi }_i which are shown in Table 1.

As it shown in Table 1, proposed damping identification results perfectly match with reference assumed damping ratios.

In the second part of this section, in order to determine accuracy of the identified damping coefficient by using the proposed method, the displacement response of the damped system is inversely simulated from a numerical method. All above mentioned assumptions for the mass, stiffness and excitation force are the same assumptions as here. Once, the response is numerically simulated then the comparison of the reference mathematical model response and the response which produced numerically are considered in Fig. 3. For the different values of damping ratio, the similarity of them is given in Table 2. It should be accounted that there is around 0.3-1% bias from the numerical method accounting process error.

Mathematical models-less input excitation force and damping ratio estimation

In this section, accuracy of the proposed method for inverse identification of the input force and FRF of the damped SDOF system is considered through numerical implementation. The displacement response of the system is produced by using a numerical method based on interpolation of excitation force []. The zero mean band limited Gaussian white noise (WN) process (bandwidth between 0.05 and 50 Hz) with noise root mean square (RMS) acceleration of 0.04 g, is used as an external excitation force. The system physical parameters, mass, stiffness and damping coefficient are assumed to be known and described as mass (m) of 50 kg and stiffness (k) of 2×10⁶ N/m, the damping coefficient (c) is assumed in four different values of C1= 3 × 10³,C2= 5 × 10³, C3= 1 × 10⁴ and C4= 1.9 × 10⁴ N/m. Firstly, the excitation force and FRF are identified by using the proposed method. Identified and reference excitation forces are shown in Fig. 4.

In the second part, in order to determine the accuracy of estimated force, the same system is excited by the identified force. The displacement response of both, identified and reference response of the system must be the same as each other. The process results are shown in Fig. 5 and Table 3.

Analytical approaches by using the finite element modeling method

In this part, the effectiveness of the mathematical model-less input excitation force method is investigated by using an analytical approach. Analytical modeling of the simple structure is investigated by using Finite element modeling. In the case of complicate models, just by guessing the damping ratio and controlling by the similarity of the identified responses with primarily measured responses, it is possible to estimate the structure’s input force and subsequently damping ratio. To find the effectiveness of the proposed method, a simple lumped mass SDOF system is considered for identification. This steel structure is modeled in the SAP2000 finite element modeling program which is shown in Fig. (6). The damping ratio is considered as constant damping in this model. The mass and stiffness in the response direction arem= 400 kg and k = 1.1139×10⁵ N/m. FE model is excited in x direction by the zero mean band limited Gaussian white noise (WN) process (bandwidth between 0.05Hz and 50Hz) with noise root mean square (RMS) acceleration of 0.9 g.

Time history analysis is done and the response of the structure is acquired from the joint which is shown in Fig. (6). By using the proposed method, the forces on structure are identified and then compared to the reference excitation forces which are shown in Fig. (7).

In the second step, FE model is excited by the identified excitation forces and then the new response of the system is measured from the same joint. The key point here is a selection of the damping ratio. The Target damping ratio should produce the maximum similarity of the measured response. Comparison between the reference response which is measured from the FE model with identified response and their similarity are shown in Fig. (8).

Conclusions

In the first part of this study, a model-based structural damping identification method is proposed. The idea behind this proposed method is a theoretical comparison of the damped and un-damped system models. In the second part, estimating of the input excitation force is considered. This method is based on the fundamental concept of the FRF. The effectiveness of them is investigated by numerical and analytical approaches. The importance of these methods is their ability to identifying target parameters just by using measured responses and also for each frequency content of a response it can be possible to track each SDOFs behavior. In addition, capability of proposed damping identification method in different modeling of the damping or following the damping variation of the structure in the case of unknown damping model. The identified responses and input excitation force by using these methods are compared with reference ones that are used in the simulation process. High convergence results illustrate the satisfactory of the proposed approximation methods.

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