Axially moving beams can represent many engineering devices, such as band saws, power transmission belts, aerial cable tramways, crane hoist cables, flexible robotic manipulators, and spacecraft deploying appendages. Despite usefulness and advantages of these devices, vibrations associated with the devices have limited their applications. Therefore, understanding transverse vibrations of axially moving beams is important for the design of the devices. The investigations on vibrations of axially moving beams have theoretical importance as well, because an axially moving beam is a typical representative of distributed gyroscopic systems. The term “gyroscopic” arose in recognition of an early problem in gyrodynamics. Actually, the Coriolis acceleration component experienced by axially moving materials imparts a skew-symmetric or gyroscopic term to their governing equations. Due to particular characteristics of the gyroscopic term, the approaches developed in analyzing vibrations of an axially moving string can be applied to other more complicated distributed gyroscopic systems. Because of the practical and theoretical significance, the investigation on nonlinear vibrations of axially moving beams is a challenging subject which has been studied for many years and is still of interest today.
The relevant researches on transverse vibrations of axially moving strings can be dated back to (Aiken, 1878). There are several excellent and comprehensive survey papers, notably (Mote, 1972), (Ulsoy and Mote, 1978), (Mote et al., 1982), (D’Angelo et al., 1985), (Wickert and Mote, 1988), (Wang and Liu, 1991), (Abrate, 1992), (Zhu, 2000), reviewing the state-of-the-art in different time phases of investigations related to vibrations of axially moving beams. The present chapter emphasizes on the recent achievements, although some early results are mentioned for the sake of completeness. Besides, the chapter focuses the nonlinear problem only. If the vibration amplitude is large, the nonlinearity should be taken into account. Hence the chapter, unlike (Chen, 2005a) for axially moving strings, is not a comprehensive survey with a complete and detailed representation of current researches. Instead, the chapter is a counterpart of (Chen et al., 2008) for axially moving beams. The author tries to put the some available results into a general framework, as well as to highlight the work of the author and his collaborators. It is hoped that the chapter serves as a collection of ideas, approaches, and main results in investigations on nonlinear vibration of axially moving beams.
The chapter is organized as follows: Section 2 focuses on the mathematical models of nonlinear transverse vibration. The special attentions are paid to the comparison of two different nonlinear models and the introduction of the material time derivative into the viscoelastic constitutive relations. Section 3 covers the developments and the applications of approximately analytical methods, including the asymptotic method, the Lindstedt-Poincaré method, the method of normal forms, the method of nonlinear, complex modes, the method of multiple scales, and the incremental harmonic balance method. Section 4 is devoted to the numerical approaches, including the Galerkin discretization, the finite difference, and the differential quadrature. Section 5 reveals the nonlinear behaviors such as bifurcation and chaos based on the numerical solutions. Section 6 discusses energetics, conserved quantity and the applications. The final section recommends future research directions.
2. Governing equations
2.1. Coupled vibration
The governing equation is the base of all analytical or numerical investigations. Generally, an axially moving beam undergoes both the longitudinal vibration and the transverse vibration, and they are coupled. (Thurman & Mote, 1969) obtained the governing equation of coupled longitudinal and transverse vibrations of an axially moving beam. (Koivurova & Salonen 1999) revisited the same modeling problem and clarified its kinematic aspects. Their nonlinear formulation for the moving beam problem has two limitations: the material of the beam is linear elastic constituted by Hooke’s law, and the axial speed of the beam is a constant. As (Wickert & Mote 1988) pointed out, modeling of dissipative mechanisms is an important vibrations analysis topic of axially moving materials. An effective approach is to model the beam as a viscoelastic material. Therefore, it is necessary to deal with constitutive laws other than Hooke’s law. Axial transport acceleration frequently appears in engineering systems. For example, if an axially moving beam models a belt on a pair of rotating pulleys, the rotation vibration of the pulleys will result in a small fluctuation in the axial speed of the belt. The nonlinear model in (Thurman & Mote, 1969) for coupled vibration can be generalized to an axially accelerating viscoelastic beam as follows.
Consider a uniform axially moving beam of density
Study the motion of the beam in a reference frame moving in the axial direction and at speed
where a comma preceding
For a slender beam (for example, with
In the moving reference frame, the beam itself is without any axial transportation, while the boundaries are moving at speed
where . To avoid the moving boundary conditions (5), which are difficult to tackle, the transformation of coordinates is introduced as follows
Then, expressed in the new coordinates, the boundary conditions have a simpler form
Under the new coordinates, the partial derivatives with respect to
If other viscoelastic constitutive relations are used to describe the beam materials, they can be incorporated into the governing equation in the similar way. However, a controversial issue arises concerning the application of differential-type constitutive laws including the Kelvin relation in axially moving materials. Some investigators used the partial time derivative in the Kelvin model for axially moving strings (Zhang & Zu, 1998) (Zhang & Song, 2007), (Chen et al., 2007) and (Ghayesh, 2008), or beams (Chen & Yang, 2005, 2006a,b), (Ghayesh & Balar, 2008), (Ghayesh & Khadem, 2008), (Yang et al., 2009), and (Özhan & Pakdemirli, 2009). However, (Mochensturm & Guo, 2005) convincingly argued that the Kelvin model generalized to axially moving materials should contain the material time derivative to account for the added “steady state” dissipation of an axially moving viscoelastic string. Actually the material time derivative was also used in other works on axially moving viscoelastic beams (Marynowski, 2002, 2004, 2006), (Marynowski & Kapitaniak, 2002, 2007), (Yang & Chen, 2005), (Ding & Chen, 2008), (Chen & Ding, 2008, 2009), (Chen & Wang, 2009) and (Chen, et al., 2008, 2009, 2010). Here a coordinate transform will be proposed to develop the governing equations, which can introduce naturally the material time derivative in the viscoelastic constitutive relations.
In small but finite stretching problems in literatures of nonlinear oscillations, only the lowest order nonlinear terms need to be retained so that the governing equation of small-amplitude motion will be obtained. Such simplified coupled governing equations were used in analytical investigations on axially moving elastic beams (Thurman & Mote, 1969), (Riedel & Tan, 2002), and (Sze et al., 2005).
It should be remarked that there are different types of governing equations for axially moving beams (Tabarrok et al., 1974), (Wang & Mote, 1986, 1987), (Wang, 1991), (Hwang & Perkins, 1992a,b, 1994), (Vu-Quoc & Li 1995), (Behdinan, et al, 1997), (Hochlenert et al., 2007), (Pratiher & Dwivedy 2008), (Spelsbrg-Korspeter et al., 2008), and (Humer & Irschik, 2009). Actually, there are various beam theories such as Euler-Bernoulli theory, shear-deformable theories, and three-dimensional theories, and geometric nonlinearities may take different forms. Correspondingly, there are various governing equations of axially moving beams. Even if an axially stationary slender structure is prescribed by more sophisticated governing equations, the coordinate transform (7) is a still convenient approach to derive the governing equations of the slender structure undergoing an axial motion.
2.2. Transverse vibration
Although the transverse vibration is generally coupled with the longitudinal vibration, many researchers considered only the transverse vibration in order to derive a tractable equation. Inserting
The quasi-static stretch assumption means that one can use the averaged value of the disturbed tension to replace the exact value
Both the nonlinear partial-differential equation and the nonlinear integro-partial-differential equation have been applied to some special cases such as free vibration without external excitation (
The nonlinear integro-partial-differential equation can also be obtained through uncoupling the governing equation for coupled longitudinal and transverse vibration under the quasi-static stretch assumption in small but finite stretching problems, and a special case of free vibration of axially moving elastic beam was treated in (Wickert, 1992). Under quasi-static stretch assumption, the dynamic tension to be a function of time alone. In traditional derivation in (Wickert, 1992), the nonlinear integro-partial-differential equation seems more exact than the nonlinear partial-differential equation because it is the transverse equation of motion in which the longitudinal displacement field is taken into account. However, the derivation here indicates that the nonlinear partial-differential equation can be reduced to the nonlinear integro-partial-differential equation based on the quasi-static stretch assumption. Numerical investigations on free vibration of axially moving elastic beams (Ding & Chen, 2008) and forced vibration of axially moving viscoelastic beams (Chen & Ding, 2009) indicated that the nonlinear integro-partial-differential equation is superior to the partial-differential equation, in the sense that approximates the coupled governing equation of planar motion better (some details in Subsection 4.2). However, since there has no decisive evidence to favor any models of transverse nonlinear vibration of axially moving beams, it is still an open problem.
3. Approximate analytical methods
3.1. Direct-perturbation approaches
As exact solutions are usually unavailable, approximate analytical methods are widely applied to investigate nonlinear vibration of axially moving beams. The approximate analytical methods can be applied to the nonlinear (integro-)partial-differential equations without discretization. Such a treatment is regarded as a direct-perturbation. The practice can be dated back to (Thurman & Mote, 1969) in which a modified Lindstedt method was used to calculate the fundamental frequency.
The method of multiple scales can be employed to analyze nonlinear vibration of axially moving beams. Actually, a general framework of the multi-scale analysis has been proposed for a linear gyroscopic continuous system under small nonlinear time-dependent disturbances (Chen & Zu, 2008). Consider a gyroscopic continuous system with a weak disturbance
for complex functions
for all functions
and the boundary conditions, and
It should be noticed that the solvability condition (25) holds providing the boundary conditions are appropriate. That is,
The method of multiple scales has been applied in various transverse nonlinear vibration problems of axially moving beams. These problems include free (Öz et al, 2001) and (Chen & Yang, 2007), forced (Özhan & Pakdemirli, 2009), and parametric(Öz et al, 2001) and (Özhan & Pakdemirli, 2009) vibration of axially moving elastic beams, as well as forced (Yang & Chen, 2006) and parametric (Chakraborty & Mallik, 1999), (Chen & Yang, 2005) and (Chen & Ding, 2008) vibrations of axially moving viscoelastic beams. In addition to these works on the base of the Euler-Bernoulli beam theory, the method of multiple scales has also be applied to study free vibration of an axially moving beam with rotary inertia and temperature variation effects (Ghayesh & Khadem, 2008), parametric vibration of axially moving viscoelastic Rayleigh beams (Ghayesh & Balar, 2008), and forced (Tang et al. 2009) and parametric (Tang et al., 2010) vibrations of axially moving elastic Timoshenko beams, while the multi-scale analysis on axially moving viscoelastic Timoshenko beams has been only limited to linear parametric vibration (Chen et al., 2010).
Addition to the method of multiple scales, the method of asymptotic analysis is also an effective approach to treat parametric or nonlinear vibration. Based on the idea of Krylov, Bogoliubov, and Mitropolsky, (Wickert, 1992) developed a asymptotic method for general gyroscopic continuous systems with weak nonlinearities, and the method was specialized to free nonlinear vibration of an axially moving elastic beam with supercritical transport speed. (Maccari, 1999) proposed another asymptotic approach for analyzing transverse vibration of axially stationary beams, which are disturbed conservative continuous systems, and determined external force-response and frequency-response curves in the cases of primary resonance and subharmonic resonance for a weakly periodically forced beam with quadratic and cubic nonlinearities. The approach was extended to the gyroscopic continuous system with a weak nonlinear and time-dependent disturbance in order to analyze transverse vibration of an axially accelerating viscoelastic string constituted by the Kelvin model (Chen et al., 2008) and the standard linear solid model (Chen & Chen, 2009). The method of asymptotic analysis has been also presented for nonlinear parametric vibration of axially accelerating viscoelastic beams constituted by the Kelvin model (Chen et al., 2009) as well as linear parametric vibration of axially accelerating viscoelastic beams constituted by the Kelvin model (Chen & Wang, 2009) and the standard linear solid model (Wang & Chen, 2010).
Nonlinear normal modes whose shapes depend on the amplitude provide a possible direct treatment on nonlinear vibration of axially moving beams. (Chakraborty et al., 1999) used a temporal harmonic balance and a spatial perturbation technique to determine the nonlinear complex normal modes for free and forced vibrations of axially moving elastic beams. The approach was adopted to study the response of a parametrically excited axially moving beam both without and with an external harmonic force (Chakraborty & Mallik, 1998). The results were justified by the wave propagation analysis (Chakraborty & Mallik, 2000a,b).
3.2. Discretization-perturbation approaches
Discretization of governing equations is a commonly used approach to obtain approximate solutions of vibration problems of continuous systems. For the governing equations (11) of coupled vibration of axially moving beams, one assumes an approximate solution in the form
After the discretization, various perturbation techniques such as the method of multiple scales can be employed to analyze the resulting nonlinear ordinary-differential equations approximately. Such a treatment is regarded as a discretization-perturbation
In practical problems,
Discretization-perturbation approaches have also been used in analyzing transverse nonlinear vibration of axially moving beams. In this case, equation (27) will be substituted into equation (13) or (14) and then the Galerkin procedure can be used to discretize equation (13) or (14) into
4. Numerical approaches
4.1. Galerkin procedure
Numerical calculation is an effective approach to studying nonlinear vibration of axially moving beams. Based on the numerical solutions of the governing equations, some changing tendencies of vibration characteristics, such as frequencies or amplitudes, with related parameters can be predicted, the approximate analytical results can be verified, and the nonlinear dynamical behaviours can be revealed.
Among numerical approaches, the Galerkin procedure can be applied to discretize the governing equation of nonlinear vibration of axially moving beams. The Galerkin discretization is not only the priority of discretization-perturbations reviewed in Subsection 3.2, but also feasible approach to numerical solutions. Using the 3 order Galerkin discretization of governing equation (in the type of equation (18)) for transverse motion of axially moving viscoelastic beams excited by the changing tension, (Marynowski, 2002) and (Marynowski & Kapitaniak, 2002) numerically investigated the effects of different viscoelastic models, such as the Kelvin model, the Maxwell model, and the standard linear solid model, on the dynamic response and found that different viscoelastic models yield very close numerical results for small damping. The Galerkin procedure has been mainly use to calculate long time nonlinear dynamical behaviors, which will be addressed in Subsection 5.1.
In the application of the Galerkin discretization, the main problem in the actual computations is the complexity of the resulting discretized equations. If the number of terms retained in the Galerkin discretization is rather large, the explicit expression of nonlinear terms is very difficult to obtain. (Chen & Yang, 2006b) proposed a technique to simplify the nonlinear terms in the equations derived from the Galerkin discretization. All nonlinear terms are regrouped to combine the repeated terms and cancel the zero terms. Therefore, the resulting equations can be easily coded for computers and then be effectively calculated. For example, the Galerkin discretization of the governing equation (18) for transverse motion (in the dimensionless form) is
If both the base and weight functions are chosen as sine functions, the stationary mode shapes for the simply supported beams, equation (29) can be cast into a from convenient to compute. Evaluating the corresponding inner products, regrouping the nonlinear terms to combine the same terms and canceling all null terms in the resulting equation, one obtains
where the sum is defined to be zero if its lower limit is larger than its upper limit. Although equation (30) seems rather complicated, it is very efficient when used for computer implementing, because almost all repeated nonlinear terms are put together, and terms with zero coefficients are eliminated. In fact, equation (29) contains 2
It should be remarked that, based on stationary mode shapes, the even order Galerkin discretization can take the linear gyroscopic terms into full account, while the odd order discretization will miss some effects of the gyroscopic terms.
4.2. Finite difference
The finite difference method is a numerical procedure to solve partial differential equations. The method can be used to discretize both spatial coordinates and time or to discretize spatial coordinates only. In the former case, the procedure consists of four steps: 1 Discretize the continuous spatial domain and temporal interval, on which a partial differential equation is defined, into a discrete finite difference grid; 2 Approximate the individual exact partial derivatives in the partial differential equation by algebraic finite difference approximations; 3 Substitute the finite difference approximations into the partial differential equation in order to derive a set of algebraic finite difference equations; 4 Solve the resulting algebraic finite difference equations.
The finite difference method can be applied to calculated nonlinear vibration of axially moving beams. For example, the method will be employed to solve numerically equation (11) (Chen & Ding, 2010). Introduce the
Substitution of equations (31) and (32) into equation (11) leads to a set of algebraic equations with respect to
The finite difference method was applied to examine the validity of the two nonlinear transverse models (equations (13) and (14)) and to determine the superiority in the sense of approximating the coupled governing equation (11) of planar vibration. For forced vibration of axially moving viscoelastic beams (Chen & Ding, 2010), the steady-state transverse responses of the beam center calculated from the two transverse models are contrasted with the results based on the coupled equations of planar vibration. Qualitatively, the three models predict the same tendencies with the changing parameters. Quantitatively, there are certain differences. In the view of both the center amplitude and the beam shape, the nonlinear integro-partial-differential equation yields the results closer to those from the governing equation of coupled vibration. The similar result was obtained by the finite difference method for response in free vibrations of axially moving elastic beams (Ding & Chen, 2009a).
The finite difference method was used to confirm the analytical results of nonlinear transverse vibration of axially moving beams. For free vibration of axially moving elastic beams, (Pellicano & Zirilli, 1997) compared the beam center displacement changing with time via the Lindstedt- Poincaré method and the normal form method with the numerical solutions via the finite difference method, and found that they are in good agreement. For parametric vibration of axially accelerating viscoelastic beams, (Chen & Ding, 2008) compared the stable steady-state response of the beam center via the method of multiple scales with the numerical solutions via the finite difference method, and demonstrated that they have the same qualitative tendencies changing with the related parameters and are quantitatively with rather high precision.
4.3. Differential quadrature
The differential quadrature method, initiated from the idea of integral quadrature, is an efficient discretization technique to seek accurate numerical solutions using a considerably small number of grid points. The method can be used to discretize both spatial coordinates and time or to discretize spatial coordinates only. In later case, the differential quadrature discretization of a partial-differential equation yields a set of differential-algebraic equations via the following four steps. 1 Discretize the continuous spatial domain, on which a partial differential equation is defined, by grid points; 2 Approximate the individual exact partial derivatives in the partial differential equation by a linear weighted sum of all the functional values at all grid points; 3 Substitute the differential quadrature approximations into the partial differential equation to obtain a set of ordinary-differential-algebraic equations; 4 Solve the resulting ordinary-differential-algebraic equations. The two extensively decisive issues in the applications of the differential quadrature method are to choose grid points and to determine the weighting coefficients for the discretization of a derivative of necessary order.
The differential quadrature method can be applied to calculated nonlinear vibration of axially moving beams. Equation (11) is treated as an example to show the application of the differential quadrature method. Introduce
The quadrature rules for the derivatives of a function at the grid points yield
where the weighting coefficients are the expression
and the recurrence relationship
Consider the beam with simply supports at both ends (
which can be numerically solved via the convenient integration routines.
The differential quadrature method was applied to check the validity and the superiority of the two nonlinear transverse models. For free vibration of axially moving elastic beams (Ding & Chen, 2009a), the transverse responses of the beam center calculated from the two transverse models are contrasted with the results based on the coupled equations of planar vibration. The computational investigation leads to the following conclusions: 1 The differences between the two models are both relatively small for not very large vibration; 2 The model differences increase with the vibration amplitude and the axial speed; 3 The integro-partial-differential equation yields better results.
The differential quadrature method was used to validate the analytical results of nonlinear transverse vibration of axially moving beams. (Chen et al., 2009) developed a differential quadrature scheme to verify the approximate analytical results of stable steady-state response in parametric vibration of axially accelerating viscoelastic beams. Figure 3 shows the comparison, in which the solid and dot lines represent the results of the asymptotic analysis method and the differential quadrature method respectively. The amplitudes from both methods are almost coincided, especially near the exact-resonance (
5. Nonlinear dynamical behaviours
5.1. Galerkin discretization
Axially moving beams undergo periodic vibrations in the aforementioned researches. Nonlinear system may exhibit chaos, steady-state response sensitive to initial conditions thus unpredictable after a certain time and recurrent but either periodic or quasiperiodic hence like a random single with the continuous frequency spectrum. Besides, the dynamical behaviors of nonlinear system may change qualitatively at the critical value of the parametric variation, and the qualitative change is termed as bifurcation.
Many investigations on bifurcation and chaos are based on the Galerkin discretization of various transverse models of axially moving beams. For transverse free vibration of accelerating elastic beams in the supercritical regime, based on 1 order Galerkin discretization, (Ravindra & Zhu, 1998) applied Melnikov’s criterion to find out the parameter condition of occurring chaos and performed numerical simulations to show both period-doubling and intermittent routes to chaos. For transverse harmonically forced vibration of axially moving elastic beams in the supercritical regime, based on 8 order Galerkin discretization, (Pellicano & Vestroni, 2002) observed intricate scenario of chaos, including cascade of bifurcations, blue-sky catastrophes and coexisting chaotic and periodic orbits. Actually, they also considered 12 order Galerkin discretization and found that a few number of degree-of-freedom is sufficient to furnish a good spatial representation and to follow the actual dynamical behaviors. For transverse parametric vibration of axially moving viscoelastic beams excited by the time-dependent tension, based on 4 order Galerkin discretization, (Marynowski, 2004) and (Marynowski & Kapitaniak, 2007) observed the inverse period doubling and inverse Hope bifurcation and occurrences of regular and chaotic motions for beams constituted by the Kelvin model and the standard linear solid model respectively. For transverse parametric vibration of axially accelerating viscoelastic beams, based on 4 order Galerkin discretization, (Chen & Yang, 2006b) constructed numerically the bifurcation diagrams in the case that the axial speed perturbation amplitude, the mean axial speed, or the viscosity coefficient is respectively varied while other parameters are fixed. They also calculated the largest Lyapunov exponent from the discretized governing equation. Numerical results show that, with the increasing speed perturbation amplitude, the increasing mean speed, and the decreasing viscosity coefficient, the equilibrium loses its stability and bifurcates into a periodic motion, and the periodic motion becomes chaotic motion via period doubling bifurcation. In addition, the chaotic motion and the periodic motion exchange alternately for the sufficiently large speed
perturbation amplitude and mean speed, and for the sufficiently small viscosity coefficient. Figures 4 and 5 show respectively the bifurcation diagrams versus the dimensionless speed fluctuation amplitude and the dimensionless viscosity coefficient.
5.2. Differential quadrature and time series
The differential quadrature method is an effective numerical technique for initial and boundary problems, and it has much higher precision than the few term Galerkin discretization. However, it has not been applied to calculate nonlinear behaviors of axially moving materials until (Ding & Chen, 2009b). They used the differential quadrature method to investigate bifurcation and chaos of an axially accelerating viscoelastic beam constituted by the Kelvin model. Based on the numerical solutions, analysis of the time series yielded the Lyapunov exponent to identify periodic and chaotic motions. Numerical results show that, with the increasing mean axial speed, the equilibrium loses its stability and bifurcates into a periodic motion, and the periodic motion becomes chaotic motion. The chaotic motion and the periodic motion exchange alternately for the sufficiently large mean axial speed and speed perturbation amplitude. Figures 6 and 7 show the Poincaré map and the largest Lyapunov exponent of periodic and chaotic motions respectively.
6. Energetics and conserved quantity
It is well known that the total mechanical energy in free vibration of an undamped axially stationary elastic beam with pinned or fixed ends is constant. However, many investigations found that the total mechanical energy associated with free vibration of an axially moving elastic beam is not constant even if the beam travels between two motionless supports. (Barakat, 1968) considered the energetics of an axially moving beam and found that energy flux through the supports can invalidate the linear theories of axially moving beams at sufficiently high transporting speed. (Tabarrok, 1974) showed that the total energy of a traveling beam without tension is periodic in time. (Wickert & Mote, 1989) presented the temporal variation of the total energy related to the local rate of change and calculated the temporal variation of energy associated with modes of moving beams. Considering the case that there were nonconservative forces acting on two boundaries, (Lee & Mote 1997) presented a generalized treatment of energetics of translating beams. (Renshaw et al., 1998) examined the energy of axially moving beams from both Lagrangian and Eulerian views and found that Lagrangian and Eulerian energy functionals are not conserved for axially moving beams. (Zhu & Ni, 2000) investigated energetics of axially moving strings and beams with arbitrarily varying lengths. (Chen & Zu, 2004) proposed energetics of axially moving beams with geometric nonlinearity due to small but finite stretching of the beams. Hence the variation of the total mechanical energy is a fundamental feature of free transverse vibration of axially moving beams. However, all aforementioned investigations on energetics and conserved quantities of axially moving beams have only been limited to transverse vibration, in which longitudinal motion is assumed to be uncoupled and thus neglectable. Actually, the energetics of coupled vibration of axially moving elastic strings (Chen, 2006) can be extended to beams.
Assume that the axially moving beam described at the beginning of Subsection 2.1 is elastic (
Then the time-rate of energy change is
Interchanging the order of differentiation and integration and inserting
are respectively the longitudinal and transverse components of the tension in the beam,
is the total mechanical energy per unit length. Hence the second term in equation (40) stands for the energy change due to the axial motion of the beam. Physically, equation (40) means the change rate of the energy consisting of two parts: the power of the beam tension, moment and shear applying at two ends and the energy variation resulted from the axial motion.
For an axially stationary beam,
If the axially stationary beam is with pinned or fixed ends, equation (45) leads to the conservation of the mechanical energy, which is a well known fact.
6.2. Conserved quantity
Although the total mechanical energy of axially moving beams is generally not constant, there does exist an alternative conserved quantity. (Renshaw et al. 1998) presented both Eulerian and Lagrangian conserved functionals for axially moving beams. (Chen & Zu, 2004) generalized their results to nonlinear free vibration of axially moving beams. They adopted the partial-differential equation (a special case of equation (13)) for axially moving beams undergoing nonlinear transverse vibration. (Chen & Zhao, 2005) also present a conserved functional for a beam modeled by an integro-partial-differential equation derived from the quasi-static assumption (a special case of equation (14)). They applied the conserved functional to verify that the straight equilibrium configuration is stable for beams at low axial speed.
Define the functional
Evaluation of the temporal differentiation by parts yield
At a pinned or fixed end,
The conserved quantity in a mechanical system is not only mathematically the first integral leading to a reduction in the order of the system, but also reflects the physical essence of the system closely related to the symmetries of the system. Therefore, it is theoretically significant to investigate the conserved quantities. The conserved quantity in a mechanical system can be used to check and develop numerical simulation algorithms. It is also useful for stability analysis and controller design.
7. Concluding remarks
Because an axially moving beam is an effective mechanical model that can be used in diverse engineering fields, many research activities in the area have been witnessed. The chapter summarizes some resent works on modeling, analysis and simulations of nonlinear vibrations of axially moving beams. It will remain to be an active research field. There are many promising topics for future researches, including but surely not limited to the follows: (1) modeling slender structures via sophisticated beam theories such as three-dimensional beams or composite beams, (2) incorporating functionally graded, theromviscoelastic or other advanced materials, (3) accounting for aerodynamic forces and heating and other actions coupled with the vibration, (4) considering complex constraints and coupling such as belts in drive systems, (5) developing analytical approaches especially for coupled vibrations and strongly nonlinear vibrations, (6) investigating convergence, consistency, and stability of numerical procedures, (7) exploring energetics of nonlinear and time-dependent beams under general constraint conditions, (8) understanding complicated dynamical behaviors such as global bifurcations, chaos, patterns, and spatio-temporal chaos.
This work was supported by the National Outstanding Young Scientists Foundation of China (No. 10725209), the National Natural Science Foundation of China (No. 90816001), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20093108110005), Shanghai Subject Chief Scientist Project (No. 09XD1401700), Shanghai Leading Talent Program, Shanghai Leading Academic Discipline Project (No. S30106), and the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT0844).
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