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In this article, a precise integral method (PIM) for the load identification of continuous structures based on the polynomial nonlinear hypothesis is presented. Essentially, load identification can be regarded as solving a dynamic equation in an inverse process. PIM is famous for its high precision in positive engineering problems. It has been already used in load reconstruction, but most cases are with discrete structures. In each time step, the dynamic responses of the measured points are used to identify the load without establishing a recursive chain, so PIM is not sensitive to the initial value and has no accumulated error. Legendrepolynomials (LPIM) is a computational scheme. The dynamic load is assumed to be a polynomial fitting nonlinear function in each integral time step with LPIM. Just with the first modal, LPIM can reliably identify the concentrated load. In addition, as uncertainty is getting more and more attention in engineer, it is also needed to be considered in load identification. In linear systems, interval vertex is a reliable method to obtain the load bounds in the virtue of multisource uncertainties. The results of identified load are too lumpy because of the polynomial fitting. To improve the precision, Savitsky–Golay (S–G) filters which fit the curves by polynomials in a frame length is introduced to smooth down the load in high fidelity. Eight numerical examples are investigated to demonstrated the efficiency and precision of the developed method. [ABSTRACT FROM AUTHOR]

A new UBC-constrained (ultimate-bearing-capacity-constrained) non-probabilistic reliability-based optimization method for structures with uncertain-but-bounded parameters is proposed. Different from the traditional stress-constrained optimization, the ultimate bearing capacity (UBC) is taken as the constraints in the non-probabilistic reliability-based optimization, which reflect the system safety of structures. Based on the interval mathematics, the UBC constraint is transformed into the format of non-probabilistic reliability, in which a novel measuring index, namely, the UBC, is defined by an interval interference model. Thus, structural optimization is converted by the lightweight design problem with UBC reliability constraints. Moreover, the gradient-based Taylor expansion method is employed to obtain the lower and upper bounds of the UBC, which transforms a double-layer optimization into an efficient single-layer one. Finally, 2D and 3D structural examples are given to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

• A novel adaptive interval uncertainty analysis method is proposed. • The strategy of alternate iteration based on the Lipschitz upper and lower bounds avoids twice search to obtain response bounds. • The method does not need adjust parameter specifically when facing different problems. • The performance of convergence, efficiency and accuracy are proved theoretically. Multisource uncertainties, including property dispersibility of materials and fluctuating service environments, complicate structural design and reliability assessment. In this paper, a novel method named the adaptive alternating Lipschitz search method for structural analysis with unknown-but-bounded uncertainties (or interval uncertainties) is proposed. In contrast to traditional optimization methods that search twice to obtain response bounds, an adaptive alternate iteration strategy is proposed. By sampling step by step, two acquisition functions—named the Lipschitz upper bound and the Lipschitz lower bound—are defined. Structural response bounds can be simultaneously obtained by alternately optimizing the two acquisition functions. The parameter settings do not require adjustments for different types of problems. Additionally, the Bayesian Adaptive Direct Search method is adopted to improve the performance of the strategy. Numerical and experimental cases are presented to demonstrate the validity, accuracy, and efficiency of the proposed methodology. Detailed comparisons indicate that the proposed method is competitive when addressing complicated structural systems with different ranges of uncertainty. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes a bidirectional sequential Kriging (BSK) method for nonlinear interval uncertainty quantification of dynamic systems. Different from existing surrogate based methods, which build surrogate model at preselected samples, the proposed method constructs surrogate models sequentially. Small amount of initial samples are generated by Latin hypercube sampling (LHS) and a crude Kriging surrogate model (KSM) is constructed. Then, two index functions, named lower confidence bound (LCB) and upper confidence bound (UCB) respectively, are defined for sample collocation. The index functions guide the search for the optima of structural response. After iterations, a sophisticated surrogate model will be obtained and the interval bounds of dynamic response can be calculated quickly with some auxiliary algorithms such as genetic algorithm (GA). Numerical examples and engineering application are studied to verify the effectiveness of BSK. The results show that compared with existing methods, the proposed method can achieve higher accuracy with good efficiency in nonlinear interval analysis. Highlights • A BSK method is proposed for interval analysis of dynamic systems. • Two index functions, LCB and UCB, are introduced for sample collocation. • Numerical examples demonstrate its merits in nonlinear interval analysis. [ABSTRACT FROM AUTHOR]

Model validation is the principal strategy to evaluate the accuracy and reliability of computational simulations. A systematic model validation procedure including uncertainty quantification, model update and prediction is described based on a non-probabilistic interval model. The crucial technical challenge in model validation is limited data, thus the non-probabilistic interval model is adopted to describe uncertain parameters. To establish the model update formula, the concepts of the interval escape rate and interval coverage rate are first described. Then, not only can the possibility of failure be estimated but also the credibility of the possibility of failure based on the proposed model validation method. The data in the validation experiment are used to update the credibility of each interval model, while the data from the accreditation experiment are used to conduct a final check of the validated models. To demonstrate that the proposed method can be applied to model validation problems successfully, a validation benchmark, the static frame challenge problem, is implemented. In addition, a practical aviation structure engineering validation problem is described. The results of these two validation problems show the feasibility and effectiveness of the proposed model validation method. The theoretical framework proposed in this paper is also suitable for model validation of computational simulations in other research fields. [ABSTRACT FROM AUTHOR]

Sensitivity analysis is a vital part in the optimization design of coupled vibro-acoustic systems. A new interval sensitivity-analysis method for vibro-acoustic systems is proposed in this paper. This method relies on only interval perturbation analysis instead of partial derivatives and difference operations. For strongly nonlinear systems, in particular, this methodology requires parameter variation over narrower ranges in comparison with other methods. To implement sensitivity analysis based on this method, the interval ranges of the responses of the vibro-acoustic system with interval parameters should first be obtained. Therefore, an interval perturbation-analysis method is presented for obtaining the interval bounds of the sound-pressure responses of a coupled vibro-acoustic system with interval parameters. The interval perturbation method is then compared with the Monte Carlo method, which can be taken as the benchmark for comparative accuracy. Two numerical examples involving sensitivity analysis of vibro-acoustic systems illustrate the feasibility and effectiveness of the proposed interval-based method. [ABSTRACT FROM AUTHOR]

Considering that numerous sample data points are required in the probabilistic method, a non-probabilistic interval analysis method can be an alternative when the information is insufficient. In the paper, new strategies, which are iterative algorithm based interval uncertainty analysis methods (IA-IUAMs), are developed to acquire the bounds of the responses in multidisciplinary system. Two iterative processes, Jacobi iteration and Seidel iteration, are applied in the new methods respectively. The Jacobi iteration based interval uncertainty analysis method (JI-IUAM) utilizes the strategy of concurrent subsystem analysis to improve computational efficiency while the Seidel iteration based interval uncertainty analysis method (SI-IUAM) can accelerate convergence by utilizing the newest information. Both IA-IUAMs are able to evaluate the bounds of responses accurately and quickly. The presented methods are compared with general sensitivity analysis based interval uncertainty analysis method (SIUAM) and conventional Monte Carlo simulation approach (MCS). The validity and efficiency of the new methods are demonstrated by two numerical examples and two engineering examples. Results show that, on the one hand, IA-IUAMs are more efficient than MCS by avoiding hundreds of system analyses, on the other hand, IA-IUAMs are more accurate and have a wider range of application than SIUAM by avoiding linear approximation and global sensitivity calculation. [ABSTRACT FROM AUTHOR]

Based on the availability of measured acceleration signals of structures, the interval analysis technique and set-membership identification concept are combined to identify the structural damage in this paper. Because of the insufficiency and uncertainty of information obtained from measurements, the noises of measurements are enveloped by interval numbers. Via the first-order Taylor series expansion, the interval bounds of the element stiffness parameters (ESPs) of both undamaged and damaged structures are derived by updating the reference finite element model. Through the intersection operations of intervals of the ESP obtained from dynamic responses in different time periods, the estimate intervals of the ESP are refined. Three damage indexes as stiffness reduce factor, possibility of damage existence, and damage measure index are introduced to identify the damage in the structure. Even though the dynamic responses are with low signal-to-noise ratio, the injury of structure can be detected by the proposed method. Two numerical examples and an experimental example are performed to demonstrate the feasibility and effectiveness of the proposed technique. The results show that the proposed method can improve the accuracy of damage diagnosis compared with the deterministic damage identification method. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a novel credible structural interval analysis method is proposed. Traditionally, nonprobabilistic uncertainties are modeled as interval or convex models, and no description of their credibility is given. Based on credible nonprobability uncertainty theory, the credible interval and ellipsoid are quantified to model irrelevant and relevant uncertain structural parameters. An efficient adaptive kriging (EAK) method is developed to solve the interval dynamic responses of structures with credible intervals and ellipsoids. In contrast to existing propagation methods, the proposed EAK method shares the samples of different time steps to improve the computational efficiency. A numerical example and two engineering applications are used to show the validity, accuracy and efficiency of the proposed method. Comparisons show that the proposed method can effectively calculate interval dynamic responses with credibility and is competitive when addressing nonlinear or coupled structural systems. • Nonprobabilistic credibility is introduced into structural interval analysis. • An EAK method is developed to solve interval response with ellipsoid uncertainties. • A sample sharing technique for higher computational efficiency for dynamic analysis. [ABSTRACT FROM AUTHOR]

The analysis of the time-dependent kinematic reliability aims at measuring the possibility of satisfying the motion requirements for mechanisms over a specific time interval. Considering that the exact statistic characteristics of uncertain parameters such as the clearances and dimensions may not be available by virtue of the limited sample information in practical engineering, this paper investigates a new time-dependent reliability assessment method with insufficient uncertainty information. Firstly, to quantify the uncertainty effects of clearances and dimensions on kinematic performance, the clearances are assumed to be convex variables which confined within the clearance circle and the dimensions are regarded as interval variables. Furthermore, an integral procedure of non-probabilistic time-dependent reliability assessment by combination of the interval mathematics and the first-passage theory is proposed, and with the help of the regularization treatment, its solution strategy is mathematically conducted as well. By comparison with Monte-Carlo simulation method, two engineering examples are eventually presented to demonstrate the validity and applicability of the developed method. [ABSTRACT FROM AUTHOR]

Model reduction is a significant issue in dynamic system simulation and control, as a consequence of the unmanageable levels of storage and computational requirements for large-scale systems. In this paper, the concept of a balanced truncation approximation method is extended to large-scale systems with interval uncertainties to get the reduced-order model with uncertainties. In order to get the balanced system, the balancing transformation matrix is introduced by using the nominal system, and the reduced-order model with uncertainties is obtained by using balanced truncation. A major characteristic of this model reduction method is that the reduced-order model obtained in this way is also as uncertain as the original model. The closeness of the reduced-order model to the original model relies on the upper bounds of the ignored Hankel singular values. To compare the original model and the reduced-order model, a perturbation method is proposed to give the interval bounds of the responses of the original model and the reduced-order model. As applications of the proposed method, three numerical examples are given. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to focus on the influences of the uncertain dynamic responses on the reconstruction of loads. Design/methodology/approach – Based on the assumption of unknown-but-bounded (UBB) noise, a time-domain approach to estimate the uncertain time-dependent external loads is presented by combining the inverse system method in modern control theory and interval analysis in interval mathematics. Inspired by the concept of set membership identification in control theory, an interval analysis model of external loads time history, which is indeed a region or feasible set containing all possible loads being consistent with the bounded structural acceleration responses is established and further solved by two interval algorithms. Findings – Unlike traditional loads identification methods which only give a point estimation, an interval estimation of external loads time history, which is a region containing all the possible loads being consistent with the uncertain structural responses, is determined. The correlation characteristics among the responses of acceleration, velocity, and displacement are also discussed in consideration of the UBB uncertainty. Originality/value – For one hand, the solution of the inverse problem in original system is transformed to the solution of the direct problem in inverse system; for another, the authors deal with the uncertainty by use of interval analysis method, and the identified interval process, which contains any possible external loads time history being consistent with the bounded structural responses can be approximately obtained. [ABSTRACT FROM AUTHOR]

In this paper, a time-domain inverse method that combines the set membership identification theory with interval analysis is proposed to stably identify structural dynamic loads, in which circumstances the unknown-but-bounded (UBB) uncertainty in measured responses is considered. Instead of knowing precise statistical information, the noisy responses are treated as intervals, and hence only their bounds are needed. The procedure of dynamic loads reconstruction from noisy displacement measurements is mainly studied, and the correlation between the uncertain responses of displacement and acceleration is also investigated in order to deal with the practical problems effectively, where only uncertainty information of acceleration can be known. Moreover, for purpose of comparison, the traditional Green’s function method is introduced and further applied into engineering simulations (a ten-bar plane truss and a twenty-five-bar space truss). A test example of a four-story frame structure is eventually presented with limited acceleration sample data. The identified results of the three examples above illustrate the feasibility and accuracy of the developed method. [ABSTRACT FROM AUTHOR]

Actuator configuration and optimization are of great significance in active control of vibration and noise. This paper proposes an actuator position optimization method for active control system with uncertainties. The uncertain parameters were modeled by interval numbers. Based on interval analysis, the boundaries of eigenvalues of the controllability grammian are obtained and uncertainty propagation analysis method for actuator configuration is presented. Mathematical operation on the eigenvalues of controllability grammian is considered to be the optimization criterion in order to maximize the norm of controllability grammian. Both nominal value and radius of the performance index are considered in the optimization model proposed in this paper. The uncertain optimization model for actuator configuration is transformed into a deterministic optimization model based on weighted processing. Node index number on the finite element mesh is selected as design variable. Since the objective function is non-convexity, genetic algorithm is adopted to get the optimal solution. The feasibility of the robust optimization method is demonstrated by two examples. [ABSTRACT FROM AUTHOR]

Abstract: An uncertain quantification and propagation procedure via interval analysis is proposed to deal with the uncertain structural problems in the case of the small sample measurement data in this study. By virtue of the construction of a membership function, a finite number of sample data on uncertain structural parameters are processed, and the effective interval estimation on uncertain parameters can be obtained. Moreover, uncertainty propagation based on interval analysis is performed to obtain the structural responses interval according to the quantified results of the uncertain structural parameters. The proposed method can decrease the demanding on the sample number of measurement data in comparison with the classical probabilistic method. For instance, the former only need several to tens of sample data, whereas the latter usually need several tens to several hundreds of them. The numerical examples illustrate the feasibility and validity of the proposed method for non-probabilistic quantification of limited uncertain information as well as propagation analysis. [Copyright &y& Elsevier]

Based on measured natural frequencies and static displacements, an improved interval analysis technique is proposed for structural damage detection by adopting membership-set identification and two-step model updating procedures. Due to the scarcity of uncertain information, the uncertainties are considered as interval numbers in this article. Via the first-order Taylor series expansion, the interval bounds of the elemental stiffness parameters of undamaged and damaged structures are obtained. The structural damage is detected by the quantitative measure of the possibility of damage existence in elements, which is more reasonable than the probability of damage existence in the condition of less measurement data. In this study, the conversation of the interval analysis method is remarkably reduced by the membership-set identification technique. The present method is applied to a truss structure and a steel cantilever plate for damage identification, and the damage identification results obtained by the interval analysis method and probabilistic method are compared. This article also discusses the effects of damage level and uncertainty level on detection results. The numerical examples show that the wide intervals resulting from the interval operation can be narrowed by the proposed non-probabilistic approach, and the feasibility and applicability of the present method are validated. [ABSTRACT FROM PUBLISHER]

Abstract: Considering that excessive sample data points are needed in the probabilistic method, in this paper, two non-probabilistic methods are proposed for uncertainty quantification and propagation analysis based on the Gray mathematical theory and the information entropy theory. These two methods can give the interval estimation of true value from the framework of non-probabilistic theory under the condition of few sample points for the uncertain parameters. The uncertainty propagation analysis for the structural responses is implemented based on the quantification results of the uncertain structural parameters. Research on the comparisons of these two methods is performed by a plane truss structure with ten bars, and the numerical results show the feasibility and validity of the proposed methods. [Copyright &y& Elsevier]

This study is devoted to two objectives to illustrate that the probability and convexity concepts are not antagonistic and to introduce a new non-probabilistic convex model for structural reliability analysis. It is shown that the new measure of safety is easier to evaluate than the corresponding measure utilizing the interval analysis. Moreover, interrelation between the classical probabilistic method and convex modeling method is demonstrated. The purpose of this study is not to replace the probabilistic approach by the convex modeling method, but to illustrate that the probability and convexity concepts are compatible. Some numerical examples are presented to illustrate the feasibility of the proposed methodology. [ABSTRACT FROM AUTHOR]

A hybrid experimental-theoretical method is proposed to investigate the influence of unavoidable scatter in elastic moduli on the natural frequency and axial buckling load of composite plate using ellipsoidal and interval analyses. The elastic moduli for material T300-QY8911 are quantified by use of the smallest ellipsoid or smallest hyper-rectangle based on a set of real experimental data. Then the bounds of the natural frequency of axial buckling load of composite plate in virtue of the obtained ellipsoid and hyper-rectangle are evaluated. Numerical examples are provided to illustrate the feasibility and validity of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: The post-buckling behavior of the thin stiffened plates with uncertain initial deflection under uni-direction compression load is investigated based on Von-Karman large deflection theory. The uncertain initial deflections are considered to be unknown except that they belong to a given set in the interval range. Interval analysis model for computing the bounds of curves of the post-buckling deflection versus load of the plate is presented. Interval analysis model is compared with the stochastic model, which is taken as the benchmarks of accuracy for judgment. The results indicate that the non-probabilistic and stochastic methods will produce similar results for a large deviation of uncertainty. If the probabilistic information is unavailable, one should not propose the probabilistic method based on an arbitrary assumption on the distribution of the deflection coefficients. Rather, one should use the non-probabilistic method to uncertainty. [Copyright &y& Elsevier]

Abstract: Hybrid theoretical, experimental and numerical method is proposed for free vibration and buckling of composite shell with unavoidable scatter in elastic moduli. Based on the Goggin’s measurement techniques, the elastic moduli for material T300-QY8911 are measured, and a set of experimental points are obtained. The measurements of elastic moduli are quantified by either (1) the smallest ellipsoid and (2) the smallest four-dimensional uncertainty hyper-rectangle. Then uncertainty propagation in vibration and buckling problems of composite shell by ellipsoidal analysis and interval analysis are, respectively, studied from the theoretical standpoint. Comparison between these analyses is performed numerically. [Copyright &y& Elsevier]

Abstract: The numerical estimation of the static displacement bounds of structures with uncertain-but-bounded parameters is considered in this paper. By representing each uncertain-but-bounded parameter as an interval number or vector, a static response analysis problem for the structure can be expressed in the form of a system of linear interval equations, in which the coefficient matrix and the right-hand side term are, respectively, the interval matrix and the interval vector. In this study, we present two new simple mathematical proofs of the vertex solution theorem using Cramer’s rule for solving linear interval equations, different from the other proof methods, to find the upper and lower bounds on the set of solutions. The first takes advantage of optimization theory, while the second is based on interval extension. By means of a typical example considered first by Hansen, it can be seen that the result calculated by the vertex solution theorem is the same as one predicted by the Oettli–Prager criterion. Examples of a three-stepped beam and a 10-bar truss are presented to illustrate the computational aspects of the vertex solution theorem in comparison with the interval perturbation method. [Copyright &y& Elsevier]

Abstract: The study was intended to evaluate the range of dynamic responses of structures with uncertain-but-bounded parameters by using the parameter perturbation method. The uncertain parameters were modeled as an interval vector. The first-order perturbation quantities of responses of the perturbed system were obtained through the parameter perturbation method, and then taking advantage of interval mathematics a new algorithm to estimate the response interval was presented. Comparisons between the parameter perturbation method and the probabilistic approach from mathematical proofs and numerical simulations were performed. The numerical results are in agreement with the mathematical proofs. The response range given by the parameter perturbation method encloses that obtained by the probabilistic approach. The results also show good robustness of the proposed method. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to evaluate the effects of uncertain-but-bounded parameters on the complex eigenvalues of the non-proportional damping structures. By combining the interval mathematics and the finite element analysis, the mass matrix, the damping matrix and the stiffness matrix were represented as the interval matrices. Firstly, with the help of the optimization theory, we presented an exact solution—the vertex solution theorem, for determining the exact upper bounds or maximum values and exact lower bounds or minimum values of complex eigenvalues of structures, where the extreme values are reached on the boundary of the interval mass, damping and stiffness matrices. Then, an interval perturbation method was proposed, which needs less computational efforts. A numerical example of a seven degree-of-freedom spring-damping-mass system was used to illustrate the computational aspects of the presented vertex solution theorem and the interval perturbation method in comparison with Deif’s method. [Copyright &y& Elsevier]

• An improved Bayesian collocation method (IBCM) is developed for structural steady-state response analysis with large interval uncertainties. • Sequential Gaussian process surrogate model is applied for interval analysis. • A bi-directional sampling strategy is proposed to guide to search the extrema. • A decayed weight function is presented to balance exploration and exploitation in highly nonlinear cases. This paper presents an improved Bayesian collocation method (IBCM) for steady-state response analysis of structural dynamic systems with large interval uncertainties. The main task of interval analysis is to search the extrema of steady-state response within the parametric intervals, so that the response bounds can be obtained. However, interval analysis problems with large parametric uncertainties are usually highly nonlinear. Thus, to improve efficiency and accuracy for nonlinear interval analysis, the IBCM executes a bi-directional global optimization process by using a sequential Gaussian process surrogate model. In this method, IBCM constructs crude surrogate models based on Gaussian process. Then a bi-directional sampling strategy is proposed to guide to search the extrema within the parametric interval. Meanwhile, the surrogate model will also be refined. A decayed weight function is presented to balance exploration and exploitation in highly nonlinear cases. The above process repeats until it converges. The interval of steady-state response can be calculated with low computational cost according to the refined Gaussian process surrogate model. The feasibility and validity of the IBCM are demonstrated by numerical examples and engineering applications. [ABSTRACT FROM AUTHOR]

A modified interval parameter perturbation finite element method (MIPPM) and a reliability-based optimization model are proposed for the coupled structural-acoustic field prediction and structural design with uncertainties in both the physical parameters and boundary conditions. Interval variables are used to quantitatively describe all the uncertain parameters with limited information. The interval matrix and vector are expanded by the modified Taylor series. Compared with the traditional perturbation method, the proposed MIPPM can yield more accurate ranges of the uncertain structural-acoustic field, in which the higher order terms of Neumann series are employed to approximate the interval matrix inverse. The reliability idea is introduced to establish an interval optimization model relying on the satisfaction degree of interval. The uncertain constraints can be transformed into deterministic ones if given the confidence level. The proposed MIPPM is used to predict the intervals of the constraints, and whereby eliminate the optimization nesting. Numerical results about a 3D car are given to demonstrate the feasibility and efficiency of the proposed model and algorithm. [ABSTRACT FROM AUTHOR]

Abstract: Considering limited available information on uncertainties in structural - acoustic coupled systems, two methods namely the vertex method and the Legendre orthogonal polynomial based method for predicting their dynamic behavior are developed based on the Statistical Energy Analysis (SEA) approach. For the vertex method, an efficient program for determining coordinates of all vertices of the rectangular spanned by entries of the involved interval input vector is coded, which is well suited for an interval input vector in arbitrary dimension. Instead of calculating the extremum of the response of interest, a method for determining its minimal and maximal point vectors dimension by dimension with respect to uncertain parameters is proposed based on the Legendre orthogonal polynomial approximation. Following the theoretical analysis of the accuracy and efficiency of the proposed methods, their validation is performed by one numerical example and two applications. [Copyright &y& Elsevier]

Effects of uncertainties on the dynamic response of the nonlinear vibration systems with general form are investigated. Based on interval mathematics, modeling the uncertain parameters as interval numbers, a non-probabilistic interval analysis method, which estimates the range of the nonlinear dynamic response with the help of Taylor series expansion, is presented, where the partial derivatives of the dynamic response with respect to uncertain parameters are considered to be interval numbers. The sensitivity matrices of dynamic response with the uncertain parameters are derived. For the presented method, only the bounds on uncertain parameters are needed, instead of probabilistic density distribution or statistical quantities. Numerical examples are used to illustrate the validity and feasibility of the presented method. [Copyright &y& Elsevier]

In this paper, the modal interval analysis method to estimate modal parameters, frequency response function (FRF), and mode shapes of structures with uncertain-but-bounded is presented. Although the system parameters or properties are uncertain in many engineering problems, but their probable range of values i.e. upper and lower bounds, can be provided from practical experience and engineering knowledge. Moreover, to avoid the resonance of a structure and to consider the dynamic response of an uncertain one, and also for reliability and stability analysis, we often need the bounds of the ranges of structural characteristic parameters such as natural frequency and normal mode. By using modal analysis and interval calculus, we investigate the method of computing upper and lower bounds of paramenters such as, natural frequencies, modal shapes, and FRFs. Theoretically, it is possible to analyze the uncertain-but-bounded of a structure by using modal analysis and interval calculus. They can be estimated by modal interval analysis. On the basis of the estimated intervals, the engineering structure parameters can be applied into engineering design. A numerical example is presented for a tower structure, and the results illustrate that the proposed method is effective. A comparison of the modal interval method with the results of Monte Carlo simulation serves to validate the solutions and to identify the bounded ranges of parameters. [Copyright &y& Elsevier]