Supercritical Nonlinear Vibration of a Fluid-Conveying Pipe Subjected to a Strong External Excitation
Nonlinear vibration of a fluid-conveying pipe subjected to a transverse external harmonic excitation is investigated in the case with two-to-one internal resonance. The excitation amplitude is in the same magnitude of the transverse displacement. The fluid in the pipes flows in the speed larger than the critical speed so that the straight configuration becomes an unstable equilibrium and two curved configurations bifurcate as stable equilibriums. The motion measured from each of curved equilibrium configurations is governed by a nonlinear integro-partial-differential equation with variable coefficients. The Galerkin method is employed to discretize the governing equation into a gyroscopic system consisting of a set of coupled nonlinear ordinary differential equations. The method of multiple scales is applied to analyze approximately the gyroscopic system. A set of first-order ordinary differential equations governing the modulations of the amplitude and the phase are derived via the method. In the supercritical regime, the subharmonic, superharmonic, and combination resonances are examined in the presence of the 2 : 1 internal resonance. The steady-state responses and their stabilities are determined. The various jump phenomena in the amplitude-frequency response curves are demonstrated. The effects of the viscosity, the excitation amplitude, the nonlinearity, and the flow speed are observed. The analytical results are supported by the numerical integration.
Pipes conveying fluid have been found in many engineering systems such as automobile, aerospace structures, nuclear reactors, boilers, heat exchangers, and steam generators. Due to the widespread applications in many industry fields, the vibration and the stability of fluid-conveying pipes have been extensively investigated, as summarized by Païdoussis [1, 2] and Ibrahim [3, 4].
The fluid flowing speed plays a crucial role in the dynamics of pipes. Under the critical speed, the straight configuration is the stable equilibrium of the pipe. In such a subcritical regime, Thurman and Mote  firstly treated nonlinear vibration and highlighted the significance of the nonlinearity in the case with large speeds. Since then, more and more efforts have been devoted to the study of the nonlinear vibration, including the benchmark paper of Holmes  on the subject. It should be motioned that these analyses were extended further for the nonplanar motion of pipes by Ghayesh et al. [7, 8]. If the fluid speed is larger than the critical speed, the straight pipe configuration becomes unstable and two curved configurations occur as stable equilibriums. Nikolić and Rajković  employed the Lyapunov-Schmidt reduction and the singularity theory to analyze stationary bifurcations in fluid-conveying pipes. Plaut  applied a shooting method to examine the equilibriums and the vibrations of fluid-conveying pipes. Modarres-Sadeghi and Païdoussis  used the finite difference method to investigate supercritical behaviors of extensible pipes. For the case of analysis, Zhang and Chen  revealed 2 : 1 internal resonance of the fluid-conveying pipes in the supercritical regime. Sinir  showed that many internal resonances might be activated among the vibration modes around the same or different buckled configurations. The supercritical problem of pipes with pulsating fluid flow has been studied by Zhang and Chen , while few studies have been devoted to the force dynamics of pipes conveying fluid because the external instruction of the governing equation is complicated. All of the above-mentioned works have not accounted external excitations.
The pipes are subjected to different environmental actions such as repeated operational startup and shutdown produces. If external excitations cannot be ignored, the pipe motion should be regarded as forced vibration. Chen  calculated the response of a cantilevered linear pipe conveying fluids to time-dependent external forces and arbitrary initial conditions. Gulyayev and Tolbatov  employed the transfer matrix method to simulate the forced behavior of a pipe containing inner nonhomogeneous flows of a boiling fluid. Seo et al.  applied the finite element method to compute the stability and the forced response of a pipe conveying harmonically pulsating fluid. Liang and Wen  used the multidimensional Lindstedt-Poincaré method to determine the frequency-amplitude response curves of forced nonlinear vibration of fluid-conveying pipes. It should be remarked that these literatures on pipes with external excitations concern the flow speed in subcritical ranges. To the authors’ best knowledge, there is no published literatures on nonlinear forced vibration of the fluid-conveying pipes in the supercritical regime.
To address the lack of researches in the aspect, the present work focuses on nonlinear forced vibration of fluid-conveying pipes in the supercritical regime. The amplitude of the external excitation is assumed at the same order of the transverse displacement. In addition to the external resonance, internal resonance is taken into account. Internal resonance with resulting modal interaction among different linear modes is a typical nonlinear phenomenon (Nayfeh and Balachandran  Nayfeh ). It has been observed in nonlinear vibration of pipes conveying fluids. McDonald and Sri Namachchivaya [21, 22] investigated the local and global dynamics of pipes conveying fluid near 0 : 1 internal resonance. Xu and Yang  employed the method of multiple scales to treat the nonlinear modal interaction of the first two modes under external sinusoidal excitation at certain flow velocity. Panda and Kar [24, 25] applied the method of multiple scales to investigate combination and principal parametric resonances in the presence of 3 : 1 internal resonances in pipes conveying pulsating fluid. Ghayesh  and Ghayesh et al.  highlighted the effects of internal resonance on nonlinear forced dynamics of an axially moving beam, a system similar to a fluid-conveying pipe. All these works on pipes conveying fluids are in the subcritical regime. The published work on internal resonance of fluid-conveying pipes in the supercritical regime is [12–14] while they did not consider external excitations.
The paper is organized as follows. Section 2 presents the equation governing motion measured form a specified curved equilibrium. In Section 3, the frequency and amplitude relationships of subharmonic, superharmonic, and combination and internal resonances occurring simultaneously are derived from the Galerkin truncation and the multiscale analysis. In Section 4, phenomenon of various jumps is explored in the frequency-response curves. In Section 5, the analytical results are compared with the numerical integration results. Section 6 ends the paper with concluding remarks.
2. Mathematical Model
A fluid-conveying pipe at both ends hinged to a transversely moving base is illustrated schematically in Figure 1. The dynamics and stability of tubes conveying fluid are reexamined by means of Euler-Bernoulli beam theory for the tube and a cylindrical shell fluid-mechanical model for the fluid flow. When compared with those equations derived by the cylindrical shell theory (Païdoussis [28–30]), some differences were reported, which were associated with the assumptions. It is shown that this refined theory is necessary for describing adequately the dynamical behaviour of the cross-sectional dimensions of the pipe with respect to its length, although the cylindrical shell theory is quite satisfactory; for long tubes, Euler-Bernoulli beam theory is perfectly adequate.
For the work of the assumptions of pipes conveying fluid the reader is strongly recommended to consult the review articles (Païdoussis ). The fluid is assumed to be incompressible, inviscid, and irrotational. The profile of the velocity inside the pipe is constant throughout the pipe. The pipe is modeled as an Euler-Bernoulli beam. Its motion is confined in a plane. Thus the motion of pipe can be described by transverse displacement at neutral axis coordinate and time . The geometric nonlinearity due to the stretching effect of the midplane of the pipe is accounted. The effect of external damping is neglected here. The gravity effect is also neglected and the pipe is nominally horizontal. If the gravity effect is taken into account, the pipe is with a curved statics equilibrium configuration. However, the analytical procedure is similar if the transverse displacement is measured from the curved equilibrium. With the introduction of the external harmonic excitation, the equation of transverse motion of the pipe is given by (Païdoussis )with the boundary conditions of simply supportswhere is the fluid velocity, is the externally imposed axial tension, and are, respectively, the mass per unit length of pipe and fluid materials, is the cross-sectional area of the pipe, is the length, is the flexural stiffness of the pipe material, and is the viscosity coefficient of internal dissipation of the pipe material which is assumed to be viscoelastic and of the Kelvin-Voigt type. and represent external excitation amplitude and frequency, and the comma notation preceding or denotes partial derivatives with respect to or .
Incorporating the following dimensionless quantities:one can nondimensionalize (1) asand boundary conditions (2) aswhere the comma-subscript notation now denotes the partial differentiation with respect to the dimensionless coordinate and time.
The pipe conveying fluid is considered in the supercritical regime. Its pair of curved equilibriums can be derived in a similar way to the case of axially moving beams (Wickert ) asfor , where is the critical speed. Over the supercritical range, , there exist the three solutions (unstable), (stable), and (stable). After the transformation of , (4) becomeswhere right superscript ′ denotes the differentiation with coordinate. Equation (7) governs the motion measured form the specified curved equilibrium configuration. To express the smallness of the amplitude of pipe motion and the viscosity in the pipe material, they are rescaled as and , where the small parameter is a book-keeping device in the subsequent multiscale analysis. The strong external excitations studied here are with amplitude that can be rescaled as . Substitution of (6) into (7) for leads to a nonlinear integro-partial-differential equation with variable coefficientswhere
3. Approach of Analysis
3.1. Truncation via the Galerkin Method
Equation (8) can be cast into a set of ordinary differential equations via the method of Galerkin. Suppose can be approximated by the finite order truncationwhere are eigenfunctions for the free undamped vibrations of a beam satisfying the pinned-pinned boundary conditions (5); namely , and is the th modal response. In linear vibrations, the low frequency modal responses predominate in the modal expansion because the amplitude of a modal response is proportional to the reciprocal of the square of the frequency. In weakly nonlinear vibrations, it seems reasonable to assume the predomination of the low frequency modal responses. Therefore, truncation order will be chosen as low as possible here to simplify the problem. To account the internal resonance, set . The truncation may result in discretization errors and cannot account for all possible internal or external responses.
For greater truncation order, it is necessary to include all of the modes that participate significantly in the response and the larger the number of modes is, the more complicated the dynamics of the system can be. Although the investigation reported in this paper was restricted to a structural system having only two modes, the phenomena described also can occur between any two modes in a multimode system having the appropriate internal resonance due to nonlinear (modal) interactions.
As the flow-rate was increased past a critical value, the position of equilibrium was found to get unstable and bifurcate into the symmetry of steady-state curved equilibrium configuration of a fluid-conveying pipe. The equations governing of positive motion have a form similar to those derived for negative. Then, in the supercritical regime, the strong forced vibration about a curved equilibrium is investigated in the following. Thus, in the case of , substituting (10) (with ) into (8), multiplying the resulting equation by weighted function and integrating the product from 0 to 1 yield a linear gyroscopic system with time-depending forcing terms and small nonlinear termswhereThe dot represents the differentiation with respect to dimensionless time .
3.2. Perturbation via the Method of Multiple Scales
The method of multiple scales (Nayfeh and Mook ) can be employed to seek for an approximate solution to (11). The first-order asymptotic expansion of the solutions to (11) can be assumed in the formwhere and are, respectively, the fast and slow time scales. Substituting (13) into (11) and equating coefficients of like powers of and , one obtains the following.
Order is as follows:Order is as follows:where and .
Equation (14) defines a 2-degree-of-freedom linear constant-coefficient nonhomogeneous gyroscopic system with a time-depending forcing term. The solution of (14) can be expressed as the general solution to the corresponding homogeneous equation plus a particular solution to the nonhomogeneous equations; namely,where stands for complex conjugate of the proceeding terms andand the first two natural frequencies of system can be solved from (14) without the forcing term asSubstituting (16) into (15) yieldswhere stands for terms that do not produce secular or small-divisor terms and the overbar indicates the complex conjugate. Although the method of multiple scales is an established approach, to authors’ best knowledge, there has no treatment on the perturbation of time-dependent gyroscopic systems. In what follows, the case with a two-to-one internal resonance and four fundamental resonances occurring simultaneously is considered.
3.3. Subharmonic Resonance of First Mode: Modulation Equation and Steady-State Responses
Examine the subharmonic resonance of the first mode in the presence of 2 : 1 internal resonance. To describe the nearness of to and to , introduce the detuning parameters and such that
To establish the solvability conditions of gyroscopic system, one assumes a particular solution to (19) in the form
Substituting (21) into (19), using (20), and equating the coefficients of and on both sides, one obtainswhereThe natural frequencies, given by (18), make the determinant of the coefficient matrix (22) vanish. The existence of solutions to (22) demandsThus, the solvability condition of subharmonic resonance of the first mode in presence of 2 : 1 internal resonance is derived from (24) asSubstituting (23) into (25) and rearranging the resulting equation lead towhere
To express the amplitude of motion conveniently, the polar transformation for the complex amplitude can be introduced aswhere and are the amplitude and the phase that are real valued functions. Substituting (28) into (26) and separating the resulting equation into real and imaginary parts yield the modulation equation:where and . The right superscripts and denote, respectively, the real and the imaginary parts of each coefficient.
For the steady-state response, and are constants. Therefore the left-hand side of (29) should be zero. The resulting equations have two possible solutions. In the case termed as the single-mode solution (, ), one has trivial linear solutions:In the second case, there are coupled two-mode solutions (, ) that can be numerically solved.
To determine the stability of the nontrivial coupled two-mode solutions, one derives the disturbance equation of (29) aswhere right superscript denotes transpose and is the Jacobian matrix calculated at the coupled two-mode solution. The eigenvalues of determine the stability of steady-state responses corresponding to the two-mode solutions. The solution is stable if and only if the real parts of eigenvalues are less than or equal to zero.
3.4. Superharmonic Resonance of Second Mode: Modulation Equation and Steady-State Responses
The superharmonic resonance of second mode and modal interactions may occur simultaneously. In this case, to describe quantitatively the nearness of these resonances, one introduces the detuning parameters and such that In order to avoid secular terms, the solvability conditions of second superharmonic resonance in presence of internal resonance can be derived in the similar way in Section 3.3. Substituting particular solution (21) into (19), considering (32), and equating the coefficients of and on both sides, one obtainswhereThus, (33) has solutions only when the solvability conditionholds. Substituting (34) into (35) and rearranging terms in the resulting equation lead to the modulation equations:where the , , , and terms are defined by (27) in Section 3.3 andSubstituting (28) into (36) and separating the resulting equation into real and imaginary parts, one obtains the polar equations:where and .
The steady-state response is with constants and defined by (38) with zero left-hand side. There are two types of steady-state responses. The single-mode solution (, ) isThe two-mode solutions (, ) can be numerically solved. Their stabilities can be determined by the eigenvalues of the Jacobian matrix of first-order differential equations (38). If the real parts of the eigenvalues are all negative then the steady-state solution is stable.
3.5. Summation Resonance: Modulation Equation and Steady-State Responses
In this case, the frequency relations for the 2 : 1 internal resonance and the combination resonance of the sum type arewhere and are the detuning parameters.
Following the similar arguments as in Section 3.3, the solvability conditions can be determined for the summation resonance in presence of internal resonance. Substituting particular solution (21) into (19), using conditions (40), and equating the coefficients of and on both sides lead towhereThen the solvability condition of the simultaneous summation and internal resonances areThus, substituting (42) into (43) and rearranging the resulting equation yield the modulation equations:where the , , , and terms are defined by (27) andSubstituting the polar expression (28) into (44) and separating the resulting equation into real and imaginary parts yieldwhere and .
For the steady-state response, the constants and are defined by (46) with zero left-hand side. There are two possible types of steady-state response. The first is the trivial zero solution , corresponding to the equilibrium. The second is the nontrivial coupled solutions , that can be numerically solved. The stability of the nontrivial steady-state response can be analyzed by examining the eigenvalues of the Jacobian matrix of (46).
3.6. Difference Resonance: Modulation Equation and Steady-State Responses
In this case, the combination resonance of the differential type with 2 : 1 internal resonance is considered. Use the detuning parameters and to describe the nearness of to and to , respectively. Thus, the frequency relations are given bywhere and are the detuning parameters.
Thus, substituting particular solutions (21) into (19), using (47), and equating the coefficients of and on both sides yieldwhereAccording to similar lines as in previous case, the solvability condition of the differences resonance in the presence of internal resonance case isSubstituting (49) into (50) and rearranging the resulting equation givewhere the , , , and term are defined by (27) andSubstituting (28) into (51) and separating the resulting equation into real and imaginary parts yield