Mode Shape Functions Research Articles

A new and simple analytical approach to determining the natural frequencies of framed tube structures

This paper presents a new and simple solution for determining the natural frequencies of framed tube combined with shear-walls and tube-in-tube systems. The novelty of the presented approach is based on the bending moment function approximation instead of the mode shape function approximation. This novelty makes the presented solution very simpler and very shorter in the mathematical calculations process. The shear stiffness, flexural stiffness and mass per unit length of the structure are variable along the height. The effect of the structure weight on its natural frequencies is considered using a variable axial force. The effects of shear lag phenomena has been investigated on the natural frequencies of the structure. The whole structure is modeled by an equivalent non-prismatic shear-flexural cantilever beam under variable axial forces. The governing differential equation of motion is converted into a system of linear algebraic equations and the natural frequencies are calculated by determining a non-trivial solution for the system of equations. The accuracy of the proposed method is verified through several numerical examples and the results are compared with the literature.

Free Vibration Analysis of a Uniform Continuous Beam with an Arbitrary Number of Cracks and Spring-Mass Systems

This paper presents a method on free vibration analysis of a uniform continuous beam with an arbitrary number of cracks and spring-mass systems and a damage identification algorithm. Firstly, based on an equivalent model of a single-span beam obtained by using fictitious cracks or fictitious spring-mass systems, mode shape functions of a single-span uniform beam are derived according to fundamental solutions and recurrence formulas. Next, the motion equations and compatibility conditions of spring-mass system at intermediate supports are used to establish the characteristic equation of continuous beam with an arbitrary number of cracks and spring-mass systems. The order of determinant for the characteristic equation is lower, and that of m-spans continuous beam with an arbitrary number of cracks and T spring-mass systems is only $$3m+T-1$$ . Then, according to the formed free vibration analysis method, a damage identification algorithm for the uniform continuous beam with an arbitrary number of cracks and spring-mass systems is presented. This algorithm only utilizes the first-order mode to identify crack position and depth. Finally, the results of this paper are compared with those of the finite element method and other existing references to validate the proposed method. The effects of parameters for cracks and spring-mass systems on natural frequencies of continuous beams are studied. Inverse problem examples of a three-span cracked continuous beam carrying one spring-mass system are also discussed.

Distributed microscopic actuation analysis of paraboloidal membrane shells of different geometric parameters

Abstract Paraboloidal membrane shells of revolution are commonly used as key components for advanced aerospace structures and aviation mechanical systems. Due to their high flexibility and low damping property, active vibration control is of significant importance for these in-orbit membrane structures. To explore the dynamic control behavior of space flexible paraboloidal membrane shells, precision distributed actuation and control effectiveness of free-floating paraboloidal membrane shells with piezoelectric actuators are investigated. Governing equations of the shell structronic system are presented first. Then, distributed control forces and control actions are formulated. A transverse mode shape function of the paraboloidal shell based on the membrane approximation theory and specified boundary condition is assumed in the modal control force analysis. The actuator induced modal control forces on the paraboloidal shell are derived. The expressions of microscopic local modal control forces are obtained by shrinking the actuator area into infinitesimal and the four control components are investigated respectively to predict the spatial microscopic actuation behavior. Geometric parameter (height-radius ratio and shell thickness) effects on the modal actuation behavior are explored when evaluating the micro-control efficiency. Four different cases are discussed and the results reveal the fact that shallow (e.g., antennas/reflectors) and deep (e.g., rocket/missile fairing) paraboloidal shells exhibit totally different modal actuation behaviors due to their curvature differences. Analytical results in this paper can serve as guidelines for optimal actuator placement for vibration control of different paraboloidal structures.

On the formulation of a finite element method for the stiffened multi-layered airfoil/hydrofoil structure: Post buckling analysis for the wings of underwater gliders

A finite element method is developed for the stiffened multi-layered airfoil/hydrofoil structure for the large deformation and finite strain problem. The kinematics of the airfoil/hydrofoil is set up. The Consistent Orthogonal Basis Function Space is applied for the airfoil/hydrofoil structure. Given the airfoil/hydrofoil configuration and boundary conditions, the basis function space can be uniquely determined, such that the diagonal mass matrix is obtained accurately and the basis functions are very identical with the mode shape functions of the structure. In order to satisfy the displacement compatibility condition between adjacent layers of the airfoil/hydrofoil, the traction degree-of-freedom is also induced.The post buckling analysis is presented for the wing (hydrofoil) structure of the underwater glider. The water pressure is applied on the outer surface and the critical buckling pressure is calculated. The post buckling equilibrium path is also given. The results are verified with ANSYS. The present study of the buckling analysis of the airfoil/hydrofoil under water pressure is helpful to the design of underwater glider.

Transmission loss characteristics of an annular cavity with arbitrary port locations using Green's function method.

The prediction of transmission loss characteristics of a short chamber requires the study of both longitudinal as well as transverse modes. This prompts the development of a three-dimensional (3D) transfer matrix, which can address the wave propagation in all three directions. Accordingly, an analytical methodology for transmission loss characteristics of an annular cavity using Green's function method is presented in this paper. The 3D wave equation is used to derive the mode shape functions for the cavity. The average pressure on the inlet and outlet ports is predicted using the Green's function derived from the mode shapes. Three different port location configurations of the annular cavity are considered for establishing the analytical methodology, viz., face-inlet-face-outlet (FIFO), side-inlet-side-outlet (SISO), and face-inlet-side-outlet (FISO). Numerical model results are used to corroborate the analytical results. The effects of geometrical parameters on the transmission loss are studied.

Buckling analysis of composite plates using moving least squares differential quadrature method

ABSTRACTThis work concerns with buckling and vibration analysis of composite plates based on a transverse shear theory. A numerical scheme is introduced to determine the angular frequencies and critical buckling loads of such plates. Moving least square differential quadrature method is employed to reduce the problem to that of eigen value problem. The accuracy and efficiency of the proposed scheme is examined with different computational characteristics, (radius of support domain, basis completeness order, and scaling factors). The obtained results agreed, at less execution time, with the previous ones. Further, a parametric study is introduced to investigate the influence of elastic and geometric characteristics, (Young's modulus gradation ratio, shear modulus gradation ratio, Poisson's ratio, loading parameter, and aspect ratio), of the composite on the values of critical buckling load, natural frequencies, and behavior of mode shape functions.

Principal-internal resonance of an axially moving current-carrying beam in magnetic field

The magneto-elastic 1:3 principal-internal resonance of axially moving current-carrying beams in a magnetic field is investigated. With the elastic theory of Euler–Bernoulli beam and the basic theory of electromagnetic field taken into account, the magneto-elastic vibration equation of an axially moving beam in a magnetic field is therefore developed by the Hamiltonian principle. On the basis of the hypothetical two-order modal shape function satisfying the fixed-hinged boundary condition, amplitude–frequency response equations of the 1:3 principal-internal resonance are obtained by Galerkin and multi-scale method. Calculation results based on relevant MATLAB programming, show that: When the system first-order primary resonance occurs, the second-order modal amplitude that should be disappeared excited by internal resonance, but its value is smaller than the first-order modal amplitude. It has only two changing stages and lacks a small single-value area in the curve of amplitude and external excitation force amplitude for the first-order primary resonance of the system; the curves of amplitude versus magnetic field strength, influenced by applied current, exhibit an asymmetric distribution property along the longitudinal line where the value of the magnetic field strength is zero; the response can be dominated by the first mode in the second-order primary-internal resonance of the system, in which the amplitude of the external excitation must exceed a critical value before the first mode can be excited. In addition, such as the axial velocity, the magnetic field strength, the excitation amplitude and the tuning parameters have a significant effect on the amplitude of the system, especially the magnetic field has an obvious inhibitory effect on the vibration.

Magneto-elastic dynamics and bifurcation of rotating annular plate**Project supported by the National Natural Science Foundation of China (Grant No. 11472239), the Hebei Provincial Natural Science Foundation of China (Grant No. A2015203023), and the Key Project of Science and Technology Research of Higher Education of Hebei Province of China (Grant No. ZD20131055)

In this paper, magneto-elastic dynamic behavior, bifurcation, and chaos of a rotating annular thin plate with various boundary conditions are investigated. Based on the thin plate theory and the Maxwell equations, the magneto-elastic dynamic equations of rotating annular plate are derived by means of Hamilton’s principle. Bessel function as a mode shape function and the Galerkin method are used to achieve the transverse vibration differential equation of the rotating annular plate with different boundary conditions. By numerical analysis, the bifurcation diagrams with magnetic induction, amplitude and frequency of transverse excitation force as the control parameters are respectively plotted under different boundary conditions such as clamped supported sides, simply supported sides, and clamped-one-side combined with simply-anotherside. Poincaré maps, time history charts, power spectrum charts, and phase diagrams are obtained under certain conditions, and the influence of the bifurcation parameters on the bifurcation and chaos of the system is discussed. The results show that the motion of the system is a complicated and repeated process from multi-periodic motion to quasi-period motion to chaotic motion, which is accompanied by intermittent chaos, when the bifurcation parameters change. If the amplitude of transverse excitation force is bigger or magnetic induction intensity is smaller or boundary constraints level is lower, the system can be more prone to chaos.

Nonlinear dynamical behaviors of deploying wings in subsonic air flow

In recent years, the morphing wing, which is able to improve the lift–drag characteristics of the aircraft by changing wingspans and aspect ratios, has become a popular research issue. However, aeroelasticity is still a bottleneck for the development of morphing wings. The development of the aerodynamic calculations of deploying wings in subsonic air flow will be further promoted by this research. In this paper, we consider the aerodynamics of a deploying wing and investigate the nonlinear dynamic behaviors of the deploying wing. Under the flow condition of ideal incompressible fluid, the flow field is a potential field and the potential function must satisfy the Laplace linear equation and the superposition principle. When the thickness, curvature and angle of attack of the airfoil are small, the thin airfoil theory can be used to calculate the effects of the mean camber line to obtain the circulation distribution of the deploying wings in subsonic air flow. The steady aerodynamic lift on the deploying wing is derived by using the Kutta–Joukowski lift theory. Then, the aerodynamic lift is applied on a deploying wing, which is modeled as a cantilevered thin shell deploying in the axial direction. The nonlinear partial differential governing equations of motion for the deploying cantilevered thin shell subjected to the aerodynamic force in subsonic air flow are established based on Hamilton’s principle. The time-varying dependent vibration mode-shape functions are chosen using the boundary conditions, and then the Galerkin method is employed to transform the partial differential equation into two time-varying nonlinear ordinary differential equations. Numerical simulations are performed for the nonlinear dynamic responses of the deploying wing subjected to the aerodynamic force, and then the influence of different parameters, including the extending velocity and disturbance velocity, on the stability of the wing are analyzed. The effects of deploying velocities on the nonlinear vibrations of the first-order and second-order modes for the deploying wing are studied.

Nonlinear Dynamics of Z-Shaped Folding Wings with 1:1 Inner Resonance

Predicting the nonlinear vibration responses of a Z-shaped folding wing during the morphing process is a prerequisite for structural design analysis. Therefore, the present study focuses on the nonlinear dynamical characteristics of a Z-shaped folding wing. The folding wing is divided into three carbon fiber composite plates connected by rigid hinges. The nonlinear dynamic equations of the system are derived using Hamilton’s principle based on the von Kármán equations and classical laminate plate theory. The mode shape functions of the system are then obtained using finite element analysis. Galerkin’s approach is employed to discretize the partial differential governing equations into a two-degree-of-freedom nonlinear system. The case of 1:1 inner resonance is considered. The method of multiple scales is employed to obtain the averaged equations of the system. Finally, numerical simulation is performed to investigate the nonlinear dynamical characteristics of the system. Bifurcation diagrams and wave-form diagrams illustrate the different motions of the Z-shaped folding wing, including periodic and chaotic motions under given conditions. The influence of transverse excitations on the bifurcations and chaotic motion of the Z-shaped folding wing is investigated numerically.

On the formulation of a 3-D smooth curved pipe finite element with arbitrary variable cross-section

A smooth curved pipe finite element is developed. The cross-section can be defined by any shape geometrically variable along the axial direction of the curved pipeline. The curved axial line of the pipe can be defined by any curve and the exact smooth configuration is obtained. the kinematics of the local cross-section with arbitrary shapes is set up. Then, the global 3-D curved pipe is obtained by extruding the local cross-section along the curved pipe axial line. The Consistent Orthogonal Basis Function Space is developed, such that the basis functions are very similar to mode shape functions and the diagonal mass matrix is accurately obtained. The nonlinear finite element implementation details are presented. Numerical tests are presented for four kinds of curved pipes and the post buckling analysis is presented. The results are verified with a commercial FEM package.

Solution of free harmonic vibration equation of simply supported Kirchhoff plate by Galerkin-Vlasov method

This work studies the dynamic characteristics of simply supported rectangular thin plates undergoing natural transverse vibrations in harmonic motion. The governing partial differential equation for the free transverse vibration of the plate was solved by the Galerkin-Vlasov variational technique. The assumption of free harmonic motions reduced the governing equation to an algebraic eigen value eigen­vector problem, which was solved in the space domain to obtain the eigen frequencies and modal shape functions of the vibrating Kirchhoff plate. The eigen frequencies and modal shape functions obtained were found to be identical with the results obtained by the classical methods of Navier and Levy for the same problem. http://dx.doi.org/10.4314/njt.v36i2.6

Roll System and Stock’s Multi-parameter Coupling Dynamic Modeling Based on the Shape Control of Steel Strip

The existence of rolling deformation area in the rolling mill system is the main characteristic which distinguishes the other machinery. In order to analyze the dynamic property of roll system’s flexural deformation, it is necessary to consider the transverse periodic movement of stock in the rolling deformation area which is caused by the flexural deformation movement of roll system simultaneously. Therefore, the displacement field of roll system and flow of metal in the deformation area is described by kinematic analysis in the dynamic system. Through introducing the lateral displacement function of metal in the deformation area, the dynamic variation of per unit width rolling force can be determined at the same time. Then the coupling law caused by the co-effect of rigid movement and flexural deformation of the system structural elements is determined. Furthermore, a multi-parameter coupling dynamic model of the roll system and stock is established by the principle of virtual work. More explicitly, the coupled motion modal analysis was made for the roll system. Meanwhile, the analytical solutions for the flexural deformation movement’s mode shape functions of rolls are discussed. In addition, the dynamic characteristic of the lateral flow of metal in the rolling deformation area has been analyzed at the same time. The establishment of dynamic lateral displacement function of metal in the deformation area makes the foundation for analyzing the coupling law between roll system and rolling deformation area, and provides a theoretical basis for the realization of the dynamic shape control of steel strip.

Theoretical analysis based on fundamental functions of thin plate and experimental measurement for vibration characteristics of a plate coupled with liquid

This study combined theoretical, experimental, and numerical analysis to investigate the vibration characteristics of a thin rectangular plate positioned horizontally at the bottom of a rectangular container filled with liquid. Flow field pressure was derived using an equation governing the behavior of incompressible fluids. Analytic solutions to vibrations in a thin plate in air served as the fundamental function of the thin plate coupled with liquid. We then used liquid pressure, and the out-of-plane deflection of the thin plate for the construction of frequency response functions for the analysis of vibration characteristics in the liquid-plate coupling system. Two experimental methods were employed to measure the vibration characteristics of the thin plate immersed in water. The first involved using sensors of polyvinylidene difluoride (PVDF) to measure transient signals of fluid-plate system subjected an impact at the thin plate. These were then converted to the frequency domain in order to obtain the resonant frequencies of the fluid-plate coupling system. The second method was amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), which was used to measure the dynamic characteristics of the thin plate in the flow field. This method was paired with the image processing techniques, temporal speckle pattern interferometry (TSPI) and temporal standard deviation (TSTD), to obtain clear mode shapes of the thin plate and resonant frequencies. Comparison of the results from theoretical analysis, finite element method, and experimental measurements confirmed the accuracy of our theoretical analysis, which was superior to the conventional approach based on beam mode shape functions. The experimental methods proposed in this study can be used to measure the resonant frequencies of underwater thin plates, and clear mode shapes can be obtained using AF-ESPI. Our results indicate that the resonant frequencies of thin plates underwater are lower than those in air. To a limited extent, increasing the depth of water enhanced the reduction of resonant frequencies.

Gallium nitride MEMS resonators: how residual stress impacts design and performances

Starting from Gallium Nitride epitaxially grown on silicon, pre-stressed micro-resonators with integrated piezoelectric transducers have been designed, fabricated, and characterized. In clamped–clamped beams, it is well known that tensile stress can be used to increase the resonant frequency. Here we calculate the mode shape functions of out-of-plane flexural modes in pre-stressed beams and we derive a model to predict both the resonant frequency and the piezoelectric actuation factor. We show that a good agreement between theory and experimental results can be obtained and we derive the optimal design for the electromechanical transduction. Finally, our model predicts an increase of the quality factor due to the tensile stress, which is confirmed by experimental measurements under vacuum. This study demonstrates how to take advantage from the material quality and initial stress resulting of the epitaxial process.

Automotive brake squeal analysis with rotating finite elements of asymmetric disc in time

The new finite element brake squeal model is proposed where the finite elements of a real brake disc rotate in time. Contact nodal forces between the rotating disc and stationary pads are allocated to the moving contact area at every time step. When the proposed model is applied to an asymmetric automotive brake disc, it becomes the periodic time-varying brake system. The stability boundary of the discrete time-varying system is numerically calculated by the Floquet theory. Also, the quasi-static linearized eigenvalue analysis is conducted to show that the unstable modes repeatedly appear at the short interval of the disc rotation angle. The results are consistent with the angle-dependent local phenomenon of squeal termed squeal periodicity in the squeal experiment. In the nonlinear time-domain analysis, the squeal vibration increases and then decays in time for the rotating mode shape functions. It demonstrates that the rotation of an asymmetric disc can change the nonlinear squeal behavior as well as the linear stability character drastically.

Comparison of 2D boundary curving methods with modal shape functions and a piecewise linear target mesh

It is well known that high-order simulation techniques demand an accurate geometric representation and a coarse mesh. To fulfill both requirements, curved meshes are generated. In most cases, curving methods assume that the exact geometry is known. But it can be useful to develop curving methods with only a limited knowledge of the target geometry. In this paper, three curving methods are described that take a piecewise fine linear mesh as input: a least squares approach, a direct optimisation in the H1-seminorm, and a H1-seminorm optimisation in a reference space. Hierarchic, modal shape functions are used as basis for the geometric approximation. The methods are compared on two test geometries, a unit circle and a distorted ellipse. Considering both test cases, the direct optimisation approach shows the most promising results. Finally, the main steps for the extension to 3D are outlined.

Nonlinear Dynamic Analysis of Macrofiber Composites Laminated Shells

This work presents the nonlinear dynamical analysis of a multilayer d31 piezoelectric macrofiber composite (MFC) laminated shell. The effects of transverse excitations and piezoelectric properties on the dynamic stability of the structure are studied. Firstly, the nonlinear dynamic models of the MFC laminated shell are established. Based on known selected geometrical and material properties of its constituents, the electric field of MFC is presented. The vibration mode-shape functions are obtained according to the boundary conditions, and then the Galerkin method is employed to transform partial differential equations into two nonlinear ordinary differential equations. Next, the effects of the transverse excitations on the nonlinear vibration of MFC laminated shells are analyzed in numerical simulation and moderating effects of piezoelectric coefficients on the stability of the system are also presented here. Bifurcation diagram, two-dimensional and three-dimensional phase portraits, waveforms phases, and Poincare diagrams are shown to find different kinds of periodic and chaotic motions of MFC shells. The results indicate that piezoelectric parameters have strong effects on the vibration control of the MFC laminated shell.

Optimal Design of a High-Agility Satellite with Composite Solar Panels

This paper defines mode shape function of a composite solar panel assumed as Kirchhoff-Love plate for considering a torsional mode of composite solar panel. It then goes on to define dynamic model of a high-agility satellite considering the flexibility of composite solar panel as well as stiffness of a solar panel’s hinge using Lagrange’s theorem, Ritz method and the mode shape function. Furthermore, this paper verifies the validity of dynamic model by comparing numerical results from the finite element analysis. In addition, this paper performs a dynamic response analysis of a rigid satellite which includes only natural modes for solar panel’s hinges and a flexible satellite which includes not only natural modes of solar panel’s hinges, but also structural modes of composite solar panels. According to the results, we confirm that the torsional mode of solar panel should be considered for the structural design of high-agility satellite. Finally, we performed optimization of high-agility satellite for minimizing mass with solar panel’s area limit using the defined dynamic model. Consequently, we observed that the defined dynamic model for a high-agility satellite and result of the optimal design are very useful not only because of their optimal structural design but also because of the dynamic analysis of the satellite.

Size-dependent nonlinear vibration of nonhomogeneous shell embedded with a piezoelectric layer based on surface/interface theory

Based on the Donnell’s nonlinear shallow-shell theory and electro-elastic surface/interface theory, the size-dependent nonlinear vibration of nonhomogeneous nano-shell covered with a piezoelectric nano-layer is addressed, and the surface energy effect on the vibration is analyzed. The Donnell’s nonlinear shallow-shell theory is introduced to derive the governing differential equations. The nonlinear mode shape functions for the longitudinal circumferential and radial variables are proposed to characterize the displacement fields and electric potential. In numerical examples, the amplitude frequency curves of the nano-shell are analyzed. The effects of the internal residual stress and the surface material properties on the nonlinear vibration of nano-shell are discussed. Comparison with the results without surface energy effect is presented to validate the results in this paper.