This paper presents a new method for identifying the nonlinear parameters of a connection used to couple two structures. The proposed Nonlinear Internal Force Identification Method (NIFIM) calculates the nonlinear internal forces and the corresponding relative displacements at the nonlinear connections by using the measured nonlinear responses of the system in the frequency domain. The method requires the measured nonlinear frequency response functions (FRFs) of the coupled structure, as well as the FRFs of the underlying linear system. In practical jointed systems, the linear FRFs correspond to the “stuck” mode of the nonlinear dry friction elements and can be obtained experimentally under low-amplitude excitation or computed numerically. The identified nonlinear internal forces are then used to estimate the relative displacements at joint degrees of freedom, and the parameters of the nonlinearity are obtained. The proposed method eliminates the need for measurements at the connection DOFs, a benefit of the technique as it is often difficult, if not impossible, to make such measurements in many engineering applications. The formulation developed to identify the joint forces at the connections requires measurements either at a single frequency with different excitation force amplitudes or at several frequencies in a specific range with a single excitation amplitude. The paper presents three case studies based on simulated experiments to demonstrate the performance of the proposed method and its sensitivity to measurement errors. These case studies include a lumped parameter system with dry friction nonlinearity and finite element models of two beams connected by a single bolted connection, and three bolted connections. The results show that the proposed method is robust to measurement noise and can accurately identify the nonlinear parameters of the connection.

This study presents a high-fidelity investigation into the coupled in-plane and out-of-plane nonlinear vibration characteristics and modal interactions of rotating pre-twisted blades with variable thickness and chord length. A geometrically nonlinear structural model is developed based on first-order shear deformation theory, with all nonlinear terms of the Green’s strain tensor retained to accurately capture large deformation effects. The formulation is constructed within a surface-based framework that incorporates pre-set, pre-twist, spanwise and chordwise cross-sectional variation, and chord tapering. Two centrifugal stiffening strategies, i.e., Direct Integration of Centrifugal Forces (DICFs) and Pre-Stressed Analysis (PA), are systematically compared to evaluate their influence on both free and forced vibration responses. The spatial domain is discretized using the Spectral Chebyshev Technique (SCT), allowing a high number of modes to be retained across complex geometries. An enhanced reduced-order modeling framework is employed to preserve key nonlinear restoring forces and multi-mode interactions. The resulting equations are solved using the harmonic balance method with arc-length continuation to compute steady-state solutions and nonlinear frequency response curves. Numerical results reveal significant differences in resonance behavior and internal modal couplings under different centrifugal stiffening assumptions. This comprehensive approach offers new insights into the nonlinear dynamics of rotating blades, highlighting the critical influence of modeling strategy and model order on accurately capturing the full spectrum of nonlinear dynamics.

Dry friction dampers are employed in many engineering applications to reduce vibrations and increase the fatigue life of mechanical parts. In supercritical helicopter tail drive shafts, dry friction dampers with gaps are used to minimize maintenance downtime by activating only when shaft vibrations exceed a predefined threshold. The presence of a gap mitigates wear and prolongs the damper’s lifespan, but introduces strong nonlinearities associated with contact and stick–slip behavior. However, many existing dry friction models either neglect this gap or simplify its effects by omitting contact engagement, stick–slip transitions, and contact loss. Therefore, in this paper, an analytical one-dimensional dry friction contact model incorporating a gap is proposed, and explicit analytical transition criteria for contact engagement, stick–slip and contact loss phenomena are obtained. For simple harmonic motion, closed-form transition angles and Fourier coefficients of the friction force are derived, enabling frequency-domain analysis using the Harmonic Balance Method and the analytical characterization of nonlinear stiffness and damping properties of the friction damper. Utilizing these transition criteria, hysteresis curves representing the equivalent stiffness and damping characteristics of the friction contact are constructed and the effect of key parameters of the friction contact are systematically investigated. As a case study, a helicopter tail drive shaft equipped with a dry friction damper with a gap is analyzed. The results demonstrate the influence of friction damper parameters on the vibration amplitude reduction of the tail drive shaft and provide design-relevant insights for optimizing damper performance in helicopter applications.

Rotating blades operating under high-temperature experience intense mechanical and thermal stresses induced by centrifugal forces and thermal shock. In this research, to address this critical yet understudied topic, coupled thermo-elastic dynamics of rotating blades incorporating precise rotational effects is investigated. A comprehensive modeling approach, characterizing blade geometry in terms of pre-twist and pre-set angles is employed based on the theory of surfaces. Thermal strain-displacement field is defined based on third-order shear deformation theory in the model integrated with a third-order expansion of temperature change distribution within the blade. Rotational factors related to Coriolis, centrifugal stiffening, and rotational softening effects are considered. In the case of stiffening effects, two different modeling approaches including direct integration of centrifugal forces (DICFs), and pre-stressed dynamics about steady-state equilibrium deformations (SSEDs) are employed. When incorporating large- amplitude SSEDs in the latter, nonlinear rotational effects are incorporated into the model. To overcome the complexity arising from coupled thermo-elasticity, and rotational motion, the spectral Chebyshev technique is applied to the derived integral boundary value problems and coupled energy equations. Finally, natural frequencies, and thermal and centrifugal deformations are investigated comprehensively by including DICFs and pre-stressed dynamics. Results show that thermo-elastic dynamics of the system is highly affected by the modeling approaches used for centrifugal stiffening effects.

Geometrically nonlinear effects commonly arise in thin-walled structures, such as beams, plates, and shells, when subjected to large-amplitude vibrations. Accurate modeling of these effects is crucial in various applications, as nonlinear coupling between in-plane and out-of-plane motions generates unique dynamic responses. Conventional methods for spatial discretization, including series-based and finite element techniques, often face challenges in efficiency, adaptability, and computational cost when applied to structures with intricate geometries or boundary conditions. To address these challenges, this study extends the spectral Chebyshev technique (SCT), renowned for its efficiency and high convergence in linear systems, to geometrically nonlinear systems. By further advancing SCT and utilizing its compact formulation, nonlinear restoring forces-including contributions from in-plane tension, shear, and moments-are precisely discretized, while an analytical Jacobian is derived to enhance computational efficiency in iterative solutions for nonlinear problems. Additionally, a novel reduced-order modeling framework, enhancing the classical modal truncation method, is developed to capture the full dynamics of both nonlinear responses and restoring forces, enabling efficient large-scale analysis. Nonlinear frequency responses are computed using the harmonic balance method across various configurations, including flat, curved, and twisted structures with arbitrary boundary shapes. The results are validated against commercial finite element analysis software and existing literature, demonstrating the computational efficiency and accuracy of the developed framework. This approach establishes a robust foundation for analyzing the complex, large-scale dynamics of thin-walled structures in engineering applications.

This study presents the spectral Chebyshev technique (SCT) for nonlinear vibrations of rotating beams based on a weak formulation. In addition to providing a fast-converging and precise solution for linear vibrations of structures with complex geometry, material, and physics, this method is further advanced to be able to analyze the nonlinear vibration behavior of continuous systems. Rotational motion and material gradation further complicate this nonlinear behavior. Accordingly, the beam is considered to be axially functionally graded (FG) and a model representing the forced nonlinear vibrations of the beam about steady-state equilibrium deformations (SSEDs) is developed. The model includes Coriolis, centrifugal softening, and nonlinear stiffening effects caused by coupling of the axial, chordwise, and flapwise motions, and large amplitude deformations. The integral boundary value problem for the rotating structure is discretized using the SCT and element-wise multiplication definition. As a result, mass, damping, and stiffness matrices, as well as internal nonlinear forcing functions and external forcing vectors, are obtained for a given rotating beam. This formulation provides a general representation of nonlinear strain relations in matrix form and circumvents the complexity rising from obtaining and solving the partial differential equations directly. In addition, nonlinear forcing functions are obtained in matrix form which facilitates the application of harmonic balance method easier to obtain the forced nonlinear response.

Flutter instability occurs when the modal damping of the system becomes negative while the natural frequency is nonzero, in which the energy of the system increases while the structure is undergoing vibrations. As an aeroelastic problem, flutter occurs when wing-like or panel structures undergo self-excited vibrations where the vibration amplitude increases greatly while the fluid flow speed is at or higher than a critical threshold. Structural system is generally modeled using finite element or the Rayleigh-Ritz method; one other alternative is the generalized differential quadrature method (GDQM), where the derivative in the domain is approximated by the function values in the domain. In this work, the aeroelastic flutter problem of simply supported and cantilever plates under supersonic flow is modeled using linear Kirchhoff-Love plate theory and first-order piston theory to model the fluid-structure interaction. Equation of motion is discretized using GDQM in the spatial domain to transform differential equations into algebraic equations. Resulting algebraic equations are solved as a state space eigenvalue problem to obtain complex mode shapes and complex eigenvalues. Results are compared with those found in the literature, and the ability of GDQM to be applied to aeroelastic flutter analysis is assessed.

During the structural analysis of micro- and nanostructures, classical elasticity theory fails to capture the small-scale size effects that are present in the experimental results. To incorporate the small-scale size effects into the equation of motion, either intermolecular forces over relatively long distances or higher-order derivatives of displacement field need to be considered. Two such theories employing the mentioned models are nonlocal elasticity theory and strain gradient theory, respectively. In this work, nonlocal strain gradient theory, which includes small-scale size effects, is used to model carbon nanotubes with the assumptions of Timoshenko beam theory. Natural frequencies are found with the differential quadrature method, and genetic algorithm is used to determine the nonlocal and length scale parameters present in the theory using molecular dynamic results for different aspect ratios to obtain the best overall fit. Instead of obtaining different small-scale parameters for each mode number, optimization of all modes at once is carried out, which is extended to also include the classical material properties. The results are inclined toward nonlocal elasticity theory for the problem at hand.

Periodic forced response analysis of nonlinear real systems is a computationally demanding task. In order to reduce the computational burden, different approaches are proposed in the literature. The reduced computational effort required for the response-dependent nonlinear normal modes (RDNMs) developed recently, making them suitable for the computation of the steady-state harmonic response of nonlinear systems by employing modal superposition method (MSM). RDNMs are obtained by representing the nonlinear internal forces as a nonlinearity matrix multiplied by the displacement vector using describing function method (DFM). The nonlinearity matrix is considered as a structural modification to the linear system, and RDNMs are calculated by solving the eigenvalue problem of this modified system. However, the solution of a large eigenvalue problem is computationally demanding. Therefore, a further reduction is made by applying the dual modal space method. A detailed study is conducted on the finite element model of a two-blade system having a shroud-to-shroud contact in order to investigate the performance of utilizing RDNMs in MSM. The finite element model of the system is obtained in commercial finite element software, and one-dimensional friction elements with normal load variation are used at the contact interface. Harmonic balance method (HBM) is used to obtain the nonlinear algebraic equations representing the steady-state response of the system which are solved by Newton's method. Several case studies are performed, and the effect of using different number of RDNMs is studied.

In this chapter, nonlinear vibration analysis of L-shaped beams is performed for different structural parameters, and the effects of these parameters are observed. The L-shaped beam is composed of two beams joined end to end and perpendicular to each other; therefore, the system is considered as two separate beams with the same boundary conditions at their mutual ends. In addition to this, concentrated masses are attached to each beam on the L-shaped beam. The dynamic model is obtained by using Euler-Bernoulli Beam Theory and Hamilton’s principle. These equations are further simplified by disregarding the axial motions of the beams and only the transverse motions are considered in calculations. Galerkin’s method is utilized to discretize the obtained nonlinear partial differential equations into a set of nonlinear ordinary differential equations. These nonlinear ordinary differential equations are converted into a set of nonlinear algebraic equations by using Harmonic Balance Method (HBM), which are then solved numerically by using Newton’s method with arc-length continuation. In order to observe the effect of the nonlinearity, a linear solution is also obtained and compared with the nonlinear solution. Several case studies are performed in order to observe the effect of system parameters on the nonlinear steady-state response of the L-shaped beam.

Dry friction damping is a commonly used approach in reducing resonant amplitudes in turbomachinery vibrations to prevent High Cycle Fatigue (HCF) failure of the blades in turbines or compressors. Intentional friction contact provides additional damping to the overall system by means of dry friction; however, predicting the forced response of frictionally constrained structures can be challenging due to the nonlinear nature of the problem. Although previous studies have worked on designing and mathematical modelling of various friction dampers, the effect of cumulative wear degradation on the structure over the operational lifetime is often underestimated. In this paper, the finite element model of a shrouded blade is used for this purpose as a case study. Harmonic Balance Method (HBM) in conjunction with Receptance Method (RM) is utilized to formulate the nonlinear algebraic equations of motion which are solved in frequency domain. In this study, one dimensional dry friction element with normal load variation is used in modeling friction contact. Alternating Frequency Time (AFT) method is used to obtain the harmonic coefficients, also known as Fourier coefficients, of the nonlinear forces introduced by frictional contact. Additionally, the well-known wear-energy method, which predicts the wear degradation at the contact interface is coupled to the nonlinear forced response analysis. This study aims to investigate how wear affects the dynamic response, performance, and reliability of the shrouded blade case study considered.

In this study, a single-degree-of-freedom structure with multiple nonlinear tuned mass dampers is investigated. A sinusoidal excitation is assumed to be applied to the structure. In order to obtain the results, a nonlinear mathematical model of the general system is obtained in the form of a set of nonlinear ordinary differential equations. Dry friction and cubic stiffness nonlinearities are considered for the nonlinear terms. To obtain steady-state frequency responses of the system, these nonlinear ordinary differential equations are transformed into a set of nonlinear algebraic equations by using Harmonic Balance Method. Fourier transformation is used in the calculation of the Fourier coefficients of the nonlinear forcing terms. Then, numerical solutions to the resulting nonlinear algebraic equations are obtained using Newton’s Method with Arc Length Continuation. The effect of the number of tuned mass dampers and their arrangement, i.e. in series or in parallel, on vibration reduction is observed by performing optimization on several cases and comparing them. In order to observe the separate and combined effects of the nonlinearities, and the number of tuned mass dampers used, several case studies are carried out using optimized case parameters with different combinations of the nonlinearities. The results of these case studies and the linear model are compared with each other to see the effect of the nonlinearities. Additionally, results of the system with optimum nonlinear tuned mass dampers are compared with that of the optimum linear tuned mass dampers and the effect of nonlinearities are discussed.

Due to ease of mounting, high strength feature and availability, bolted joints are one of the most common and widely used connection methods. Nonlinear modeling of bolted joints is a complex problem. One common approach of modeling bolted joints is the use of macroslip friction elements distributed around the bolted connection. The use of multiple macroslip friction elements increases the computational time of the analysis. Therefore, in this study, in order to decrease the computational time Modal Superposition Method (MSM) with Response Dependent Nonlinear Modes (RDNMs) is utilized to obtain the solution of the jointed structures. RDNMs, which are calculated during the solution, are a special set of normal modes which captures the dynamics of the nonlinear problem much more accurately than linear modes of the system. Therefore, use of RDNMs decreases the number of modes that should be used in the analysis drastically. Hence, the dimension of the nonlinear algebraic equation set to be solved and the resultant solution time decrease significantly. The obtained form of nonlinear algebraic equation set is solved by using Newton’s method with arc-length continuation. A half-lap bolted joint is considered as a case study. Performance of classical MSM that uses linear modes of the system is compared with the MSM with RDNMs.

Vibration analysis of shrouded bladed disk systems often becomes expensive due to friction nonlinearities and randomness stemming from mistuning phenomena. This implies a great demand for a highly efficient model order reduction technique to not only reduce the computational effort but, more importantly, provide reliable displacement predictions on certain degrees of freedom (shrouds). The latter becomes more critical in bladed disks with shroud contacts, since the promising results from contact models are limited by the accuracy of displacements predicted by reduced-order models for shroud degrees of freedom. In this study, some notable reduction of order methodologies based on substructuring, namely, fixed interface (Craig-Bampton), free interface (Rubin), and dual Craig-Bampton, and the mixed interface method, which is a combination of free and fixed interface methods, are investigated. The center of attention in this work is the modal contributions of components to the final result and the influence of modal characteristics of substructures on the efficiency of a particular reduction technique. To this end, the methods are examined by a different number of retained modes. The effect of adding up more vibration modes to the reduction basis on the accuracy and computational cost for each reduction technique is compared for predefined error tolerance. It is concluded that the physical characteristics of the blade and disk components significantly affect the forced response of the bladed disk system. Consequently, it can be capitalized on to find a more effective reduction technique for the specific geometry of shrouded blisks to address high computational cost and accurate forced response required in specific areas.

Frictional interfaces, which induce energy dissipation in the system, are intentionally included in bladed disk system of gas turbine engines. By this way, resonant amplitude of the blade is decreased while high cycle fatigue failure of these structures is postponed. Mathematical modeling of the problem becomes complicated due to the nonlinear nature of the frictional contact which also adds difficulty to the solution of the problem. Time domain solution methods are expensive compared to frequency domain method for the determination of steady state response of nonlinear systems. Therefore, Harmonic Balance Method (HBM) is employed to constitute the set of algebraic equations to be solved numerically. Alternating Frequency Time (AFT) Method is utilized to find the Fourier coefficients of the friction force in an iterative way. In addition, analytical Jacobian formulation is implemented to the friction model to enhance the solution performance. Although various friction dampers have been designed, mathematically modeled, and discussed before, fretting wear which is the inevitable outcome of the dry friction is underemphasized up until today. In this work, a blade platform system which represents a simplified underplatform damper is studied to investigate the effect of number of macroslip friction elements on wear estimation and the effect of wear on the dynamic response of the system.

Flutter is a phenomenon that occurs in wings or platelike structures as a result of aerodynamical forces when a certain flow speed, i.e., flutter speed, is reached. Flutter results in severe vibrations which eventually leads to fatigue failure of the wing. Many solutions are suggested against flutter phenomena. Wings or platelike structures under the effect of flowing air may contain nonlinearities due to connections or materials used. In this paper, effect of different structural nonlinear elements on the flutter speed is studied by using a 2D wing model. Aerodynamic lift and moment acting on the airfoil is obtained by utilizing Theodorsen’s unsteady aerodynamics which is only applicable to subsonic flow. In this paper, modified Theodorsen model for a 2D wing is used. To solve the flutter equation, several methods are suggested in the literature. Methods like k method and p-k method assume harmonic vibration in the generalized coordinates resulting in an eigenvalue problem. These methods are applied to linear systems. When nonlinearities are present in the system, numerical time marching solutions to differential equations are used; however, they are very costly in terms of computational time. In this study, state-space approach is utilized to obtain the flutter speed in frequency domain by using describing function method (DFM). The nonlinear system of differential equations is converted into a nonlinear eigenvalue problem utilizing state-space approach from which the flutter speed resulting in unstable solutions is obtained. Nonlinear eigenvalue problem obtained can be solved iteratively without time marching methods. This method of finding flutter speed is computationally much faster than solving nonlinear flutter problems in time domain. Free play nonlinearity is a frequently observed nonlinearity where there exists a gap at both sides of the wing after which it is restricted by stiffnesses. Piecewise linear stiffness is a symmetric nonlinearity similar to free play where finite stiffness exists instead of a zero stiffness in between the gap. Softening cubic stiffness is a nonlinearity where stiffness of the structure decreases as the amplitude of the vibration increases. In this study, free play (gap nonlinearity), piecewise linear stiffness, and cubic stiffness nonlinearities acting on the rotational degree of freedom are considered in the case studies. Results obtained for these nonlinearities are presented and compared with each other.

In recent years, the concept of nonlinear normal modes (NNMs) has gained interest for interpreting a broad range of nonlinear dynamic phenomena. Present study introduces a simple and efficient computational framework to compute NNMs of large order nonlinear structures overcoming difficulties associated with commonly used time-integration and discretization methods. In this paper, a Receptance Based Nonlinear Normal Mode Calculation Method (RBNM) is developed to determine the NNMs of large ordered realistic finite element models in frequency domain. Describing Function Method (DFM), which makes it possible to write the nonlinear internal forcing vector as a nonlinear displacement dependent stiffness matrix multiplied by displacement vector, is used to model the internal nonlinear forcing vector in frequency domain. A unique matrix manipulation based on dynamic stiffness and receptance concepts is used, as a result of which the number of governing nonlinear equations becomes independent from the total number of degrees of freedom of the model and it only depends on the number of degrees of freedom associated with the nonlinear elements. This brings a significant computational advantage over the classical reduced order modeling techniques. A simple 2-degree of freedom (DOF) system, the results of which are available in the literature, is used to verify the proposed method in determining NNMs. A 20-DOF lumped parameter system model and later a complex finite element model of a large structure are used to demonstrate the applicability of RBNM in determining NNMs of larger systems with several different types of nonlinear elements distributed among the degrees of freedom.

In this study, axisymmetric nonlinear natural frequencies of simply supported functionally graded (FG) annular microplate are investigated. In the formulation, it is considered that the mechanical properties such as modulus of elasticity, shear modulus of elasticity, and material properties such as density of the FG microplate change in the thickness direction. The size effect is taken into account by using the modified couple stress theory (MCST). Formulation is based on the Mindlin plate theory, and the nonlinearity in the formulation is due to von Karman’s large deformation assumption. The governing nonlinear partial differential equations of motion and boundary conditions for the FG microplate are derived by using the variational approach and Hamilton’s principle. The harmonic balance method is utilized in order to obtain a nonlinear eigenvalue problem. An iterative eigenvalue solver is used to determine the nonlinear natural frequencies of the FG microplate as a function of vibration amplitude.