Large amplitude free vibration analysis of Timoshenko beams using a relatively simple finite element formulation

Free vibration analysis of uniform isotropic Timoshenko beams with geometric nonlinearity is investigated through a relatively simple finite element formulation, applicable to homogenous cubic nonlinear temporal equation (homogenous Duffing equation). Geometric nonlinearity is considered using von-Karman strain displacement relations. The finite element formulation begins with the assumption of the simple harmonic motion and is subsequently corrected using the harmonic balance method. Empirical formulas for the non-linear to linear radian frequency ratios, for the boundary conditions considered, are presented using the least square fit from the solutions of the same obtained for various central amplitude ratios. Numerical results using the empirical formulas compare very well with the results available from the literature for the classical boundary conditions such as the hinged–hinged, clamped–clamped and clamped–hinged beams. Numerical results for the beams with non-classical boundary conditions such as the hinged-guided and clamped-guided, hitherto not studied, are also presented.     

The influence of corner stress singularities on the vibration characteristics of rhombic plates with combinations of simply supported and free edges

This paper offers accurate flexural vibration solutions for rhombic plates with simply supported and free edge conditions. A cornerstone here is that the analysis explicitly considers the bending stress singularities that occur in the two opposite, hinged–hinged and/or hinged–free corners having obtuse angles of the rhombic plates. These singularities become significant to the vibration solution as the rhombic plate becomes highly skewed (i.e. the obtuse angles increase). The classical Ritz method is employed with the assumed normal displacement field constructed from a hybrid set of (1) admissible and mathematically complete algebraic polynomials, and (2) comparison functions (termed here as “corner functions”) which account for the bending stress singularities at the obtuse hinged–hinged and/or hinged–free corners. It is shown that the corner functions accelerate the convergence of solutions, and that these functions are required if accurate solutions are to be obtained for highly skewed plates. Accurate nondimensional frequencies and normalized contours of the vibratory transverse displacement are presented for rhombic plates having a large enough skew angle of 75° (i.e. obtuse angles of 165°), so that the influence of the stress singularities is large. Frequencies and mode shapes of isosceles triangular, hinged–free plates are also available from the data presented.     

Free vibration of Timoshenko beam with finite mass rigid tip load and flexural–torsional coupling

In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural frequencies on various parameters of the tip mass is discussed. Explicit expressions for mode shapes and orthogonality condition are also obtained. Finally, the results obtained by the application of the Timoshenko beam model are compared with those of three other beam models, i.e. Euler–Bernoulli, shear and Rayleigh beam models. In this way, the effects of shear deformation and rotary inertia in the response of the beam are evaluated.     

Analysis of dynamic properties of boring bars concerning different clamping conditions

Boring bars are frequently used in the manufacturing industry to turn deep cavities in workpieces and are usually associated with vibration problems. This paper focuses on the clamping properties’ influence on the dynamic properties of clamped boring bars. A standard clamping housing of the variety commonly used in industry today has been used. Both a standard boring bar and a modified boring bar have been considered. Two methods have been used: Euler–Bernoulli beam modeling and experimental modal analysis. It is demonstrated that the number of clamping screws, the clamping screw diameter sizes, the screw tightening torques, the order the screws are tightened has a significant influence on a clamped boring bars eigenfrequencies and its mode shapes orientation in the cutting speed—cutting depth plane. Also, the damping of the modes is influenced. The results indicate that multi-span Euler–Bernoulli beam models with pinned boundary condition or elastic boundary condition modeling the clamping are preferable as compared to a fixed-free Euler–Bernoulli beam for modeling dynamic properties of a clamped boring bar. It is also demonstrated that a standard clamping housing clamping a boring bar with clamping screws imposes non-linear dynamic boring bar behavior.     

Dynamic finite element method for generally laminated composite beams

A dynamic finite element method for free vibration analysis of generally laminated composite beams is introduced on the basis of first-order shear deformation theory. The influences of Poisson effect, couplings among extensional, bending and torsional deformations, shear deformation and rotary inertia are incorporated in the formulation. The dynamic stiffness matrix is formulated based on the exact solutions of the differential equations of motion governing the free vibration of generally laminated composite beam. The effects of Poisson effect, material anisotropy, slender ratio, shear deformation and boundary condition on the natural frequencies of the composite beams are studied in detail by particular carefully selected examples. The numerical results of natural frequencies and mode shapes are presented and, whenever possible, compared to those previously published solutions in order to demonstrate the correctness and accuracy of the present method.     

FITTING MEASURED FREQUENCY RESPONSE USING NON-LINEAR MODES

The article is devoted to curve fitting of the measured frequency response functions (FRFs) of an actual beam with a non-linear component. A frequency response model, based on the non-linear normal modes (NNMs) approximated by the Ritz–Galerkin method and on a superposition assumption is built up. Identification of NNM is performed by minimising a cost function involving synthetised FRF and measured data and leads to natural frequencies, damping factors, mode shapes as functions of modal amplitudes.     

Non-linear flap-lag-extensional vibrations of rotating, pretwisted, preconed beams including Coriolis effects

The effects of pretwist, precone, setting angle, Coriolis forces and second degree geometric non-linearities on the steady state deflections, coupled frequencies and mode shapes of rotating, torsionally rigid, cantilevered beams are studied in this investigation. The equations governing flap-lag-extensional motion are derived including the effects of large precone (a component of sweep) and retaining geometric non-linearities up to second degree. The Galerkin method, with non-rotating normal modes, is used for the solution of both steady state non-linear equations and linear perturbation equations. The results indicate that the second degree geometric non-linear terms, which vanish for zero precone, can produce frequency changes of engineering significance (of the order of 20% on the fundamental mode, and about ±4% on the second mode). The results further indicate that the linear and non-linear Coriolis effects must be included in analyzing thick blades while these effects can be neglected in analyzing thin blades, typical of advanced turboprop blade configurations. For those cases where the effect is significant, the linear and non-linear Coriolis effects oppose one another, the non-linear effects generally being stronger.     

Effect of axial force on free vibration of Timoshenko multi-span beam carrying multiple spring-mass systems

The situation of structural elements supporting motors or engines attached to them is usual in technological applications. The operation of machine may introduce severe dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli–Euler single-span beams carrying a number of spring-mass system and Bernoulli–Euler multi-span beams carrying multiple spring-mass systems are plenty, but that of Timoshenko multi-span beams carrying multiple spring-mass systems with axial force effect is fewer. This paper aims at determining the exact solutions for the first five natural frequencies and mode shapes of a Timoshenko multi-span beam subjected to the axial force. The model allows analyzing the influence of the shear and axial force effects and spring-mass systems on the dynamic behavior of the beams by using Timoshenko Beam Theory (TBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The calculated natural frequencies of Timoshenko multi-span beam by using secant method for non-trivial solution for the different values of axial force are given in tables. The mode shapes are presented in graphs.     

Free periodic vibrations of beams with large displacements and initial plastic strains

This paper intends to analyse free vibrations of beams in the geometrically non-linear regime and with plastic strains. The specific goal is to find how plastic strains combined with large displacements influence the non-linear modes of vibration, by analysing the influence of the former two factors in mode shapes and natural frequencies of vibration. The geometrical non-linearity is represented by the Von Kármán type strain-displacement relations. A stress-strain relation of the bilinear type, with isotropic strain hardening, is assumed, the Von Mises yield criterion is employed and the flow theory of plasticity applied. To obtain the time domain ordinary differential equations of motion the principle of virtual work is used and a Timoshenko p-version finite element model with hierarchical basis functions is adopted. The equations of motion are naturally different from the usual large displacement equations, due to the appearance of matrices and vectors related with plastic terms. In the cases studied, plastic strains are imposed on the beam by equally distributed static forces; the forces are then removed and a study on the free vibrations is carried out. It is assumed that, once defined, the plastic strain field does not change. The time domain equations are transformed to the frequency domain by the harmonic balance method and these frequency domain equations are solved by an arc-length continuation method. The variations of mode shapes of vibration and of natural frequencies with vibration amplitude are investigated. It is found that the plastic strain distribution defines if and how much softening spring effect occurs. Hardening spring effect always appears, but with some plastic strain fields hardening spring takes place only at large vibration amplitudes. Plastic deformations also have an important effect in the vibration shapes.     

Effects of joint on dynamics of space deployable structure

Joints are necessary components in large space deployable truss structures which have significant effects on dynamic behavior of these joint dominated structures.Previous researches usually analyzed effects of one or fewer joint characters on dynamics of jointed structures.Effects of joint stiffness,damping,location,number,clearance and contact stiffness on dynamics of jointed structures are systematically analyzed.Cantilever beam model containing linear joints is developed based on finite element method,influence of joint on natural frequencies and mode shapes of the jointed system are analyzed.Analytical results show that frequencies of jointed system decrease dramatically when peak mode shapes occur at joint locations,and there are cusp shapes present in mode shapes.System frequencies increase with joint damping increasing,there are different joint damping to achieve maximum system damping for different joint stiffness.Joint nonlinear force-displacement is described by describing function method,one-DOF model containing nonlinear joints is established to analyze joints freeplay and hysteresis nonlinearities.Analysis results show that nonlinear effects of freeplay and hysteresis make dynamic responses switch from one resonance frequency to another frequency when amplitude exceed demarcation values.Joint contact stiffness determine degree of system nonlinearity,while exciting force level,clearance and slipping force affect amplitude of dynamic response.Dynamic responses of joint dominated deployable truss structure under different sinusoidal exciting force levels are tested.The test results show obvious nonlinear behaviors contributed by joints,dynamic response shifts to lower frequency and higher amplitude as exciting force increasing.The test results are further compared with analytical results,and joint nonlinearity tested is coincident with hysteresis nonlinearity.Analysis method of joint effects on dynamic characteristics of jointed system is proposed,which can be used in optimal design of joint parameters     

Natural frequencies,modes and critical speeds of axially moving Timoshenko beams with different boundary conditions

In this paper, natural frequencies, modes and critical speeds of axially moving beams on different supports are analyzed based on Timoshenko model. The governing differential equation of motion is derived from Newton's second law. The expressions for various boundary conditions are established based on the balance of forces. The complex mode approach is performed. The transverse vibration modes and the natural frequencies are investigated for the beams on different supports. The effects of some parameters, such as axially moving speed, the moment of inertia, and the shear deformation, are examined, respectively, as other parameters are fixed. Some numerical examples are presented to demonstrate the comparisons of natural frequencies for four beam models, namely, Timoshenko model, Rayleigh model, Shear model and Euler–Bernoulli model. Finally, the critical speeds for different boundary conditions are determined and numerically investigated.     

Refined theory for vibrations and buckling of laminated isotropic annular plates

Hamilton's variational principle is used for the derivation of equations of transversally isotropic laminated annular plates motion. Nonlinear strain—displacements relations are considered. Linearized vibration and buckling equations are obtained for the annular plates uniformly compressed in the radial direction. The effects of transverse shear and rotational inertia are included. A closed form solution is given for the mode shapes in terms of Bessel, power and trigonometric functions. The eigenvalue equations are derived for natural frequencies and buckling loads of annular and circular plates elastically restrained against rotation along edges. Classical-type plate theory results are obtained then by letting the transverse shear stiffness go to infinity and rotational inertia go to zero. Numerical examples are presented by tables and figures for 2- and 3-layered plates with various geometrical and physical parameters. The transverse shear, rotational inertia and boundary conditions effects are discussed.     

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.     

Four-beam model for vibration analysis of a cantilever beam with an embedded horizontal crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton's principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.     

Static, dynamic, and buckling analysis of partial interaction composite members using Timoshenko's beam theory

The static, dynamic, and buckling behavior of partial interaction composite members is investigated in this paper by taking into account for the influences of rotary inertia and shear deformations. The governing differential equations obtained are very comprehensive, covering and extending the current models for the problems that are based on Euler–Bernoulli beam theory. The analytical solutions of the deflection are then found for the beam with uniformly distributing load under common boundary conditions. The free vibration and buckling behavior are also studied and the analytical expressions of the frequencies of the simply supported beam are obtained explicitly, as are the buckling loads. For other boundary conditions, the eigen-equations are transcendental and thus some numerical examples are presented to demonstrate the effects of the shear deformation and rotary inertia on the resonant frequencies and buckling loads.     

考虑翘曲影响的Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀薄壁梁单元弯扭耦合振动的运动偏微分方程，推导其自由振动时的精确动态传递矩阵。采用考虑翘曲影响的Bernoulli-Euler梁理论，且假定薄壁梁单元的横截面是单对称的。动态传递矩阵可以用于计算薄壁梁集合体的精确固有频率和模态形状。针对两个薄壁梁算例，采用自动Muller法和结合频率扫描法的二分法求解频率特征方程，并讨论翘曲刚度对弯扭耦合：Bernoulli-Euler薄壁梁固有频率的影响。数值结果验证了本文方法的精确性和有效性，并指出翘曲刚度可以显著改变薄壁开口截面梁的固有频率。     

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bernoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method (DTM) has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.     

The dynamic analysis of nonuniformly pretwisted Timoshenko beams with elastic boundary conditions

The coupled governing differential equations and the general elastic boundary conditions for the coupled bending–bending forced vibration of a nonuniform pretwisted Timoshenko beam are derived by Hamilton's principle. The closed-form static solution for the general system is obtained. The relation between the static solution and the field transfer matrix is derived. Further, a simple and accurate modified transfer matrix method for studying the dynamic behavior of a Timoshenko beam with arbitrary pretwist is presented. The relation between the steady solution and the frequency equation is revealed. The systems of Rayleigh and Bernoulli–Euler beams can be easily examined by taking the corresponding limiting procedures. The results are compared with those in the literature. Finally, the effects of the shear deformation, the rotary inertia, the ratio of bending rigidities, and the pretwist angle on the natural frequencies are investigated.     

Rotary inertia and temperature effects on non-linear vibration, steady-state response and stability of an axially moving beam with time-dependent velocity

Free non-linear transverse vibration of an axially moving beam in which rotary inertia and temperature variation effects have been considered, is investigated. The beam is moving with a harmonic velocity about a constant mean velocity. The governing partial-differential equations are derived from the Hamilton's principle and geometrical relations. Under special assumptions, the two partial-differential equations can be mixed to form one integro-partial-differential equation. The multiple scales method is applied to obtain steady-state response. Elimination of secular terms will give us the amplitude of vibration. Additionally, the stability and bifurcation of trivial and non-trivial steady-state responses are analyzed using Routh-Hurwitz criterion. Eventually, numerical examples are presented to show rotary inertia, non-linear term, temperature gradient and mean velocity variation effects on natural frequencies, critical speeds, bifurcation points and stability of trivial and non-trivial solutions.     

Non-linear free vibration analysis of laminated orthotropic cylindrical shells

A method to predict the influence of geometric non-linearities on the natural frequencies of an empty laminated orthotropic cylindrical shell is presented in this paper. It is a hybrid of finite element and classical thin shell theories. Sanders—Koiter non-linear and strain-displacement relations are used. Displacement functions are evaluated using linearized equations of motion. Modal coefficients are then obtained for these displacement functions. Expressions for the mass, linear and non-linear stiffness matrices are derived through the finite element method (in terms of the elements of the elasticity matrix). The uncoupled equations are solved with the help of elliptic functions. The frequency variations are first determined as a function of shell amplitudes and then compared with the results in the literature.