Boundary condition modelling and identif ication for cantilever-like structures using natural frequencies

Wei LIU , Zhichun YANG , Le WANG ,*, Ning GUO

a School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China

b Xi'an Aerospace Propulsion Institute, Xi'an 710100, China

KEYWORDS Boundary condition identif ication;Boundary condition modelling;Iterative method;Natural frequency;Sensitivity analysis

Abstract The actual boundary conditions of cantilever-like structures might be non-ideally clamped in engineering practice, and they can also vary with time due to damage or aging. Precise modelling of boundary conditions, in which both the boundary stiffness and the boundary mass should be modelled correctly, might be one of the most signif icant aspects in dynamic analysis and testing for such structures. However, only the boundary stiffness was considered in the most existing methods. In this paper, a boundary condition modelling and identif ication method for cantilever-like structures is proposed to precisely model both the boundary stiffness and the boundary mass using sensitivity analysis of natural frequencies. The boundary conditions of a cantilever-like structure can be parameterized by constant mass, constant rotational inertia,constant translational stiffness, and constant rotational stiffness. The relationship between natural frequencies and boundary parameters is deduced according to the vibration equation for the lateral vibration of a non-uniform beam. Then, an iterative identif ication formulation is established using the sensitivity analysis of natural frequencies with respect to the boundary parameters.The regularization technique is also used to solve the potential ill-posed problem in the identif ication procedure.Numerical simulations and experiments are performed to validate the feasibility and accuracy of the proposed method. Results show that the proposed method can be utilized to precisely model the boundary parameters of a cantilever-like structure.

1. Introduction

In the f ield of aeronautical engineering, a local structure such as a wing is always separated from a full aircraft for structural analysis or testing in the development of the aircraft, and the boundary conditions of the wing are always simplif ied as ideally clamped and supported, in which the stiffness and inertia of the local structure of the fuselage connected with the wing are neglected1,2.However, the stiffness and inertia of the local structure of the fuselage connected with the wing will affect the dynamic characteristics of the wing in some situations.Consequently,results of the structural analysis or testing of the wing with the ideally clamped boundary conditions will be imprecise. Furthermore, the connection between the wing and the fuselage will gradually change due to the aging of material and the accumulation of damage, so it may seriously affect the performance and service life of the structures3. Therefore,boundary conditions modelling and identif ication might be one of the most signif icant aspects for such structures.

As we know,precise modelling and identif ication of boundary conditions include two steps, the f irst step is reasonably modelling the boundary conditions,and the second step is correctly identifying the boundary conditions. The modelling of the boundary conditions is always based on engineering experience,and the boundary conditions of structures are modelled by elastic supports, i.e., several translational and rotational springs. The identif ication of the boundary conditions is achieved by using dynamic responses (such as natural frequency, modal shape and dynamics response) or static responses(such as static stiffness and static f lexibility).Ahmadian et al.4proposed a boundary condition identif ication method according to solving reduced-order characteristic equations, and verif ied the method using a plate with elastic supports. Utilizing the optimization toolbox in MATLAB,Ahmadian et al.5proposed another boundary condition identif ication method based on the residuals between measured and theoretical modal parameters (i.e., natural frequencies and damping ratios), and verif ied the method by boundary identif ications of a simulative square plate supported by springs and an experimental steel plate supported by rubber seal. Two identif ication methods of boundary conditions were investigated by Pabst and Hagedorn6, i.e., a direct identif ication method based on the characteristic equation using natural frequencies,and an iterative method based on sensitivity analysis of natural frequencies and modal shapes, and then the two methods were validated by a cantilever beam and a rectangular plate with torsional stiffness, respectively. Furthermore, Akhtyamov and Utyashev7utilized two natural frequencies to identify the boundary condition of a string, and verif ied the correctness of the method by several simulative cases. Except for modal parameters, the Frequency Response Function(FRF) also contains plenty information of boundary conditions, so Frikha et al.8demonstrated an approach that could be adopted to identify the boundary conditions of curvilinear structures based on a transfer matrix,and the feasibility of this approach was shown by tests. Operational deformation can also be used to identify boundary conditions when structures are in the operational status. Pai and Huang9used the sliding-window least-squares curve-f itting technique to decompose an Operational Def lection Shape (ODS) into central and boundary solutions, and then the central solutions could be used to identify the boundary conditions. They used a cantilever structure to validate the accuracy of the proposed method.

Based on static analysis,Waters et al.10proposed a boundary condition identif ication method based on quasi-static stiffness for a structure with partially embedded foundation support.In a similar way,Wang and Yang11proposed a static method to identify the boundary condition according to measuring the static f lexibility for non-uniform beam-like structures which were constrained at one end by a translational spring and a rotational spring, and they verif ied their method by experiments.Yi and Guo12proposed a boundary condition identif ication method for beam-like structures based on a neural network, and the static def lection and boundary condition parameters were used as the input and output data of the neural network.The accuracy of the method was verif ied by comparing identif ication results with analytical solutions. In addition,some researchers have proposed boundary identif ication methods based on uncertainty analysis. Mignolet et al.13proposed a probabilistic model with considerations of uncertainties on the boundary conditions of linear structures and the coupling between linear substructures. Ritto et al.14investigated an identif ication method based on the Bayesian approach for identifying the torsional stiffness parameter of a cantilever beam.

Boundary conditions were modelled by linear elastic supports in the aforementioned methods which only considered the boundary stiffness. Similar to the boundary stiffness, the boundary mass/rotational inertia will also affect the behaviors of structures. However, the effect of the boundary mass/rotational inertia cannot be considered by using a static response,and unrealistic identif ication results might be obtained in some particular cases sometimes11. Natural frequencies can ref lect the characteristics of boundary conditions; in addition, the measurement procedure for natural frequencies is more convenient compared to that for the FRF and time domain response. Therefore, the present study proposes a method using sensitivity analysis of natural frequencies to model and identify the boundary stiffness as well as the boundary mass/rotational inertia.The layout of this article is as follows:in Section 2, a theoretical formulation for calculating sensitivity of natural frequencies with respect to boundary parameters is deduced, and an algebraic equation for boundary parameters identif ication is established; simulative examples and experimental tests are performed in Sections 3 and 4 to verify the eff iciency and accuracy of the proposed method, respectively; f inally, some conclusions are summarized in Section 5.

2. Formulation of modelling and identif ication methods

2.1. Dimensionless eigenvalues of non-uniform beam

It is more suitable for these cantilever-like engineering structures,such as aircraft wings,to be simulated by a tapered Euler-Bernoulli beam when only the dynamic characteristics of the structures are concerned. As shown in Fig. 1, the tapered beam carries a mass block at one end, which can be simplif ied as the boundary mass m and the rotational inertia J. In addition,the tapered beam is constrained by a translational spring with stiffness ktand a rotational spring with stiffness kr,which can be utilized to simulate the boundary stiffness. The equation of motion for the lateral vibration of the non-uniform beam is given by

where y(x,t)is the lateral def lection,E is the elastic modulus,ρ is the mass density of the beam,l is the length of beam,A(x)is the rectangular cross-sectional area,and I(x)is the moment of inertia at position x.

The lateral def lection of the beam can be expressed using the method of separation of variables, and the time-

Fig. 1 Simplif ication of wing to tapered Euler-Bernoulli beam.

dependent component is assumed as a harmonic function.Consequently, the def lection can be written as

where Y(x)is the modal def lection,ω is the circular frequency of the beam, and then

Substituting Eqs. (2) and (3) into Eq. (4) gives

Assume that the width b(x) and the height h(x) of the rectangular cross-section are changed linearly along length x by

where βb=1-b1/b0and βh=1-h1/h0,which symbolize the degrees of taper in width and height,respectively.Accordingly,the cross-sectional area and moment of inertia at position x become

where A0=b0h0and I0=, which are the cross-sectional area and moment of inertia at the constraint end,respectively.The boundary conditions of the beam can be given by15

where J is the rotational inertia of the mass block m at the constrained end.

The dimensionless variable X=x/l is introduced to obtain generalized results,and the Adomian Modif ied Decomposition Method (AMDM) is adopted, which has been shown to be able to calculate the dimensionless eigenvalues of a beam by Hsu et al.16. Then, the solution of Eq. (4) becomes

The coeff icients of this polynomial can be given by

where superscript ‘‘′” indicates the derivation of the lateral def lection with respect to position x , and for n ≥4, the coeff icients can be expressed as

where λ=Ω2=ρA0ω2l4/(EI0)is the dimensionless eigenvalue of the non-uniform beam, and Ω is the dimensionless circular frequency16, which are inherent characteristics of structures.

However, Eq. (13) cannot be expressed by inf inite terms in practical calculation, and thus the solution may be generally approximated by a truncated series with N+1 terms,

To facilitate the analysis, the following dimensionless coeff icients are introduced:

where M indicates the total mass of the beam,and the expression becomes

By using the dimensionless coeff icients, the boundary conditions can be simplif ied as

Substituting Eq. (13) into Eqs. (18) and (19) leads to

Substituting Eqs. (22) and (23) into Eq. (15), polynomial coeff icients can be expressed as functions of C0,C1,βr,βt,βm,βIand the dimensionless eigenvalue λ as

where c1,k,c2,k,c3,k,c4,k,c5,k,c6,kare only related to βh,βband n,and function fix(·)denotes rounding an element to the nearest integer. By substituting the solution of Eq. (4) into Eqs. (20)and (21), it leads to

The relationship between polynomial coeff icients can be achieved based on the above deduction. Consequently,the following equations can be obtained by integrating Eqs. (22)-(26):

where

where c11,n,c12,n,c13,n,c14,n,c15,n,c16,nand c21,n,c22,n,c23,n,c24,n,c25,n,c26,nin Eqs. (28)-(31) can be determined by Eqs. (22)-(26). It is noted that the coeff icients are merely functions of the degree of taper and the truncated terms. For non-singular solutions, the characteristic function of the non-uniform beam can be written as Therefore, we can obtain the i th dimensionless eigenvalueof the tapered beam by solving Eq. (32), and the dimensionless circular frequency can be expressed asIn this work, the truncated terms are decided by the criteriawhen solving Eq.(32),and ε is a prescribed positive small value.The truncated term N is taken as 41 in the present study. The model of which tapered degrees are βb=0.5, βh=0 is used to illustrate the effects of dimensionless boundary parameters on dimensionless eigenvalues, as shown in Fig. 2. It can be seen from Fig. 2 that the f irstorder dimensionless eigenvalues will be apparently changed due to the changes of the dimensionless boundary parameters.

2.2. Sensitivity analysis of dimensionless eigenvalues of nonuniform beam

Therefore, we can obtain the sensitivity of the dimensionless eigenvaluewith respect to dimensionless boundary parameters βt, βr, βm, and βIby solving the partial derivatives ofThe following equations can be obtained by differentiating Eq. (33) with respect to boundary parameters, respectively:

Fig. 2 Effects of dimensionless boundary parameters on dimensionless eigenvalues.

The partial derivatives of f10,f11,f20,f21with respect to dimensionless parameters are given in Appendix A. The purpose of the current study is to estimate the changes of boundary parameters including the boundary stiffness and the boundary mass/rotational inertia. Moreover, it is convenient to omit the superscript ofin the following sections.Considering that an observational error is inevitable in practice, we should ensure that a change of the natural frequency caused by changes of boundary parameters is greater than that caused by a measuring error. Suppose that the measured value of the i th circular frequencycan be expressed as

where Δβt,minis the smallest identif iable change of the dimensionless translational stiffness, Δβr,minis the smallest identif iable change of the dimensionless rotational stiffness, Δβm,minis the smallest identif iable change of the dimensionless mass of the mass block, and ΔβI,minis the smallest identif iable change of the dimensionless rotational inertia of the mass block. Then, Eq. (40) becomes

If the sensitivity of the dimensionless eigenvalues meets the corresponding discriminants of Eq. (41), the dimensionless eigenvalues can be used to identify the corresponding parameters.It can also be seen from Eq.(41)that we should use a natural frequency with a higher sensitivity when the measurement error level is higher or the identif iable change of the dimensionless parameters is smaller.

2.3. Identif ication of boundary parameters based on sensitivity analysis

The sensitivity of the dimensionless eigenvalue with respect to the dimensionless boundary parameters has been deduced in Section 2.2.On this basis,assuming that the f lexural rigidity EI0of the constrained end is constant,then the sensitivity of the natural circular frequency with respect to the structural physical parameters can be derived as shown in the following equations:

Δω is the residual vector between the measured and calculated natural circular frequencies, which can be expressed as the product of the sensitivity matrix and the change of boundary parameters, i.e.,

where S ∈Cn×4is the sensitivity matrix, which is consisted of the sensitivity of each natural circular frequency with respect to boundary parameters, and the sensitivity matrix can be expressed as the following equation:

It is known that the identif ication of boundary parameters is an inverse problem of structural dynamics,and it is unstable to solve the above equation numerically, especially in the case of additional data having noises. Consequently, the sensitivity matrix may be ill-conditioned. This problem can be solved with the help of Tikhonov regularization technique17,18in this paper. The weighted least-square solutions of Eq. (46) can be replaced by the solution of the following optimization problem:

where W denotes the weighted matrix of each natural circular frequency, and λ*is the regularization parameter.

The role of the weighted matrix is to reduce the condition number of the coeff icient matrix (i.e., sensitivity matrix) in Eq. (46), so as to improve and alleviate the ill-condition of the coeff icient matrix. In this study, the elements of the weighted matrix are determined by normalizing the maximum element of each row of the sensitivity matrix,and the diagonal

elements of the weighted matrix are wii=1/[max(｜Si1｜,｜Si2｜,｜Si3｜,｜Si4｜)]2. An optimal parameter should yield a fair balance between the perturbation error and the regularization error in Eq.(46).There are several parameter-choice strategies,such as the generalized cross-validation,the quasi-optimality criterion,the L-curve criterion, etc. The regularization parameter is selected by the L-curve criterion in this study. The solution of Eq. (48) can be obtained by solving ?L/?(δp)=0 and can be expressed as

where I is the identity matrix,and the iterative formula for calculating boundary parameters can be expressed as

where ωmis the measured value of the natural circular frequency, and ωkis the calculated value of the natural circular frequency in the k th iterative step.

Fig. 3 shows the f low chart of the proposed identif ication method. In this method, boundary parameter identif ication is an iterative procedure based on sensitivity analysis.Like a general iterative procedure,the proposed method also requires the initial values of boundary parameters and the convergence criteria. Thus, the convergence criteria used in the iterations is def ined as

where ‖.‖2denotes the 2-norm, and κ is a pre-specif ied threshold.

3. Numerical examples and results

In this section,a tapered aluminum beam model is adopted to validate the feasibility and effectiveness of the proposed identif ication method. The tapered degrees of the non-uniform beam are βb=0.5, βh=0. Material properties and geometric properties are below: the elastic modulus is 71 GPa, the Poisson ratio is 0.3, and the density is 2700 kg/m3. The length of the model is 0.5 m, the width is 0.03 m, and the thickness is 0.003 m. The convergence criteria κ is given as 1×10-10in this study except otherwise specif ied. The relative error of the identif ied value of parameters is adopted to evaluate the accuracy of the method as

Fig. 3 Flow chart of proposed identif ication method.

where ε denotes the relative error of different boundary parameters,and p symbolizes the boundary parameters.Superscript i means the identif ied value of boundary parameters, while superscript t means the theoretical value of boundary parameters.

Three different types of Boundary Conditions(BC)are considered for simulation, i.e., BC I: βr=1,βt=1,βm=1,βI=0.1; BC II: βr=1,βt=1× 104,βm=1,βI=0.1; BC III:βr=1×104,βt=1,βm=1,βI=0.1.Their physical meanings can be interpreted as the mass and inertia of foundation being large in these three types of boundary conditions. Meanwhile,the translational stiffness and the rotational stiffness are very small in BC I, the translational stiffness is very large but the rotational stiffness is very small in BC II,and the translational stiffness is very small but the rotational stiffness is very large in BC III. With the consideration of the inf luence of measured noise, we introduce two different noise levels, 1% and 2%, in the simulations,respectively.The natural frequencies that take into account the effect of noise can be obtained by Eq. (38),and these frequencies will be regarded as measured values in simulations. The initial values of boundary parameters kt,kr,m,J should be deviated from the true values in the iterative algorithm, and the initial values of the above parameters are supposed as 1.45 times,1.4 times,1.45 times,and 1.4 times of their true values, respectively.

Before the simulations,we calculate the condition numbers of the sensitivity matrix for the three kinds of boundary conditions,and the condition numbers are 1.108×105,7.883×109,and 6.985×109,respectively.It can be seen that the sensitivity matrices for the three kinds of boundary conditions are severely ill-conditioned.Thus,we adopt the Tikhonov regularization method which is mentioned in Section 2.4 to solve the problem of ill-condition.

3.1. Selection of natural frequencies

The f irst four natural frequencies are selected for the identif ication of boundary parameters in this paper. Eq. (46) will be a statically-determinate equation when the f irst four natural frequencies are selected,and the solution of Eq.(46)is unique.If fewer or more natural frequencies are selected,Eq.(46)will be a statically-indeterminate or hyperstatic equation, and the solution of Eq. (46) will be a least-squares solution, so the accuracy of the identif ied boundary parameters might be poor when measurement errors are inevitable.A numerical example to verify the selection order of natural frequency is added as follows: BC I is considered for simulation, and the measurement noise level is assumed to be 1%.Then,the f irst modal frequency, f irst two modal frequencies, f irst three modal frequencies, f irst four modal frequencies, f irst f ive modal frequencies, and f irst six modal frequencies are used to identify the boundary parameters.

Fig. 4 shows the identif ied results of boundary parameters obtained by using different frequencies. It can be seen from Fig. 4 that the error of the identif ied boundary parameters is the smallest when the f irst four natural frequencies are selected.Therefore,the accuracy of identif ication is higher by using the f irst four natural frequencies.

3.2. Identif ication of boundary parameters for different types of boundary conditions

We obtain 1000 samples with different measured values for the identif ication of boundary parameters, and the mean value of the identif ied results of the 1000 samples will be regarded as the identif ied results. Finally, the relative error of the identif ied parameters will be calculated by Eq.(52).Meanwhile, we suppose that the smallest identif iable change of dimensionless boundary parameters is 10% in the present study. Then, we may determine which parameters can be identif ied for the three kinds of boundary conditions in advance by using Eq. (41).Table 1 lists the identif iable dimensionless parameters by using the f irst four dimensionless eigenvalues individually under the two measurement error levels, respectively.Figs.3-5 show the relative errors of the identif ied physical parameters under the two measurement error levels, respectively.

Fig. 4 Identif ied results of boundary parameters obtained by using different frequencies.

Table 1 Identif iable dimensionless parameters of different boundary conditions with different noise levels ηfre.

As can be seen from Table 1,the number of the identif iable dimensionless boundary parameters decreases gradually with an increase of the mode order under different types of boundary conditions, and this is mainly because high-order frequencies are insensitive to the changes of boundary parameters in this study.It can also be seen that the number of dimensionless parameters, which may be identif ied by using the f irst four natural frequencies, decreases as the noise level increases.Figs. 5-7 show the Probability Density Functions (PDFs) of the boundary parameters at a conf idence level of 99.7% (i.e.,3σ criterion) for the three types of boundary conditions,respectively. In Fig. 5, four types of boundary parameters can be identif ied, and in Figs. 6 and 7, only two types of boundary parameters can be identif ied. In addition, the conf idence interval of the boundary parameters is narrower under the low noise level for the same conf idence level and the same sample quantity.It indicates that the noise level will affect the discretization of identif ication results, i.e.,the higher the noise level is,the greater the discretization is.The results in Figs.5-7 show that the identif iable dimensionless parameters listed in Table 1 can be identif ied by using the f irst four natural frequencies.Moreover,the noise level will affect the identif ication results, i.e., the higher the noise level is, the greater the identif ied error of the boundary parameters is.

In this section, we adopt the tapered beam to validate the proposed method under three different types of boundary conditions, and the following qualitative discussions can be conducted:

(1) The measurement noise level of the natural frequency can affect the accuracy of the identif ied parameters,i.e., the larger the measurement error is, the larger the identif ication error is.

(2) It can be shown from Table 1 that the number of parameters that can be identif ied decreases with an increase of the mode order, because the sensitivities of high-order frequencies with respect to the boundary parameters become smaller.

(3) The accuracy of the discriminants in Eq. (41) has been validated by identifying the boundary parameters of three different types for the tapered beam,i.e.,the structural physical parameters can be identif ied as long as we can select the initial values of the parameters reasonably.

Fig. 5 PDFs of dimensionless boundary parameters under a conf idence level of 99.7% in BC I for different noise levels.

Fig. 6 PDFs of dimensionless boundary parameters under a conf idence level of 99.7% in BC II for different noise levels.

Fig. 7 PDFs of dimensionless boundary parameters under a conf idence level of 99.7% in BC III for different noise levels.

(4) It can be seen that the conclusion obtained by Eq.(41)is appropriate only when the identif iable change of dimensionless boundary parameters is suff iciently small. The reason for choosing a smaller identif iable variable is:(A)natural frequencies can be used to identify the boundary condition when its sensitivity is large enough; (B) the method proposed in this paper is applicable when the initial model is reasonably close to the f inal identif ied model.

(5) The proposed method can be used to identify specif ic boundary conditions. For different boundary conditions, all the boundary parameters can be identif ied for BC I; only some boundary parameters can be identif ied for BC II and BC III.

4. Experimental validation

Fig. 8 Experimental model.

A tapered aluminum beam is utilized as the experimental model to illustrate the effectiveness of the proposed method,as shown in Fig. 8. The degrees of taper, geometrical parameters, and material parameters of the beam are the same as the corresponding properties in Section 3.The boundary condition is simulated by using a steel mass block and a short steel beam clamped together at the end of the tapered beam, which are shown in Fig. 8(a). Three sets of steel beams and one set of mass block are combined to simulate three cases, i.e., Case A,Case B,and Case C,respectively,and the three cases belong to BC II in the simulations. The geometrical properties of the mass block and the steel beam are listed in Table 2. In addition, the mass block is designed as two halves to ensure the tapered beam located in the central plane of the mass block.A short uniform steel beam with f ixed-f ixed boundary conditions is utilized to simulate the foundation stiffness. Furthermore, the foundation mass and rotational inertia are provided by the mass block together with the steel short beam.The material properties of the mass block and the steel short beam are: Es=206 GPa, υs=0.3, ρs=7800 kg/m3.

As mentioned before, the foundation stiffness, foundation mass, and rotational inertia are provided by the uniform steel f ixed-f ixed beam and the mass block.Thus,the designed properties11can be approximated by Eq.(53),and the three types of boundary parameters in the tests are listed in Table 3. The boundary mass/inertia can be estimated by the length, width,height,and the location of center of gravity.The translational boundary stiffness can be determined by the deformation generated by the unit force applied on the intermediate position of the f ixed-f ixed beam,and the rotational boundary stiffness can be determined by the rotation angle generated by the unit moment applied on the intermediate position of the f ixedf ixed beam, respectively, i.e.,

where Is, Js, and leare the moment of inertia of the cross section, the torsion constant, and the effective length of the uni-form steel beam, respectively. The second moment of the area of the cross section Isand the torsion constant Jscan be calculated by the width bsand depth hsof the cross section of the steel beam,and lecan be approximately indicated by the length lsof the steel beam minus the width b0at the constrained end of the tapered beam.Meand Jeare the total mass and rotation inertia around the centroid axis of the mass block,respectively.H1and H2signify the distances between the centers of gravity of the two halves of the mass block and the central plane of the tapered beam, respectively.

Table 2 Geometric parameters for different cases.

Table 3 Designed values of boundary parameters (estimated values).

Fig. 9 Test setup for natural frequencies measurement.

Table 4 Measured frequencies for different cases

where H denotes the distance between the center of gravity to the short side in the x axis direction of each half of the mass block.

The model is installed on a test rig as shown in Fig. 9. A laser displacement sensor and an impact hammer are used as the response signal acquisition device and the excitation device, respectively, to avoid additional mass and additional stiffness to the structure. The Polymax algorithm of LMS is used to identify the modal parameters of the experimental model, and the natural frequencies of the structure are obtained from the measured FRFs.

The designed boundary parameters listed in Table 3 can be used as the initial values of the boundary parameters.Like the simulations in Section 3, we also adopt the f irst four modal frequencies to identify the boundary parameters of the tapered beam. Meanwhile, the f ifth- and sixth-order frequencies are utilized to validate the proposed method. In addition, the regularization technique is used to solve the ill-conditioned problem for each case. Table 4 lists the f irst six measured modal frequencies of the tapered beam for the three cases. Table 5 lists the identif ied values of the boundary parameters by using the f irst four natural frequencies for the three cases. It can be seen from Tables 3 and 5 that the identif ied values are close to the designed values of the boundary parameters.

Natural frequencies that are calculated by using the estimated and identif ied values of the boundary parameters are denoted as finand fid, respectively, and are listed in Table 6.The corresponding relative errors are denoted as Einand Eid,respectively, which are listed in Table 7 and plotted in Fig.10.The numbers in bold in Table 7 are the maximum relative errors of natural frequencies before and after the identif ication.It is found from Tables 6 and 7 that the relative errors of the f irst sixth natural frequencies are improved. Therefore,the theoretical model which is constructed by the identif iedparameters using the proposed method in this paper is very close to the experimental model.

Table 5 Identif ied boundary parameters for three cases.

Table 6 Natural frequencies before and after identif ication for different cases.

Table 7 Relative error natural frequencies before and after identif ication for different cases.

Fig. 10 Relative errors of natural frequencies before and after identif ication for different cases.

5. Conclusions

An iterative method for modelling and identifying the boundary parameters of cantilever-like structures is proposed in this paper by using the sensitivity analysis of natural frequencies. The boundary condition of a cantilever-like structure can be modelled by the mass,rotational inertia,and constant translational and rotational stiffness. Then, from the relationship between natural frequencies and boundary parameters,an iterative formulation of boundary parameter identif ication is established.A tapered beam as a numerical and experimental model is utilized to validate the feasibility and accuracy of this method.

The proposed method can be used to identify the boundary stiffness as well as the boundary mass/rotational inertia simultaneously, which means that the dynamics boundary of a structure might be precisely simulated. The current study can be treated as an improvement of existing methods,which only focus on the identif ication of the boundary stiffness. Meanwhile, it can be seen from identif ication results that the proposed method is robust to measurement noise.

In this study,whether low-order natural frequencies can be used to update the information of boundary conditions depends on the order of magnitude of boundary parameters. A precise result can be obtained when the order of boundary parameters is small. It is shown that low-order natural frequencies can be utilized for identifying the boundary conditions of cantileverlike structures in some particular conditions.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11402205), the Aeronautical Science Foundation of China(No.20171553014),and the Natural Science Basic Reasearch Plan in Shaanxi Province of China (No. 2018JM 5178).

Appendix A.

One needs to know the partial derivatives of f10,f11,f20,f21with respect to dimensionless parameters that are outlined in Section 2.2 when implementing the identif ication method presented inthisstudy,and theexpressionscanbewrittenasfollows: