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School of Mechanics and Engineering Science，Shanghai University，Shanghai 200444，P.R.China

Abstract:The ideally straight hydraulic pipe is inexistent in reality.The initial curve caused by the manufacturing or the creep deformation during the service life will change the dynamic character of the system.The current work discusses the effect of the initial curve on the hydraulic pipe fixed at two ends for the first time.Based on the governing equation obtained via the generalized Hamilton’s principle，the potential energy changing with the height of the initial curve is discussed.The initial curve makes the potential energy curve asymmetric，but the system is always monostable.The initial curve also has very important influence on natural frequencies.It hardens the stiffness of the first natural mode at first and then has no effect on this mode after a critical value.On the contrast，the second natural frequency is constant before the critical value but increases while the height of the initial curve exceeds the critical value.On account of the initial value，the quadratic nonlinearity appears in the system.Forced resonance is very different from that of the ideally straight pipe under the same condition.Although the 2∶1 internal resonance is established by adjusting the height of the initial curve and the fluid speed，the typical double-jumping phenomenon does not occur under the initial curve given in the current work.This is very different from the straight pipe in the supercritical region.The work here claims that the initial curve of the hydraulic pipe should be taken into consideration.Besides，more arduous work is needed to reveal the dynamic characters of it.

Key words：forced resonance；hydraulic pipe；slightly?curved pipe；harmonic balance method

0 Introduction

The hydraulic system is an important part for machine，engine，airplane and many engineering machineries.It usually contains lots of pipes，which convey the hydraulic oil with high pressure and high speed.Under excitations and pulsating pressure or speed，the piping system may yield drastic vibra?tion，which will damage the system.Hence，many scholars paid their attention to this field，including pipes in other fields［1-2］.

For those curved pipes designed on purpose，the dynamic character is discussed based on the ini?tial curve intrinsically.The early model of the semicircular arc pipe conveying fluid was proposed by Chen［3］.He found that the curved pipe showed insta?bility similarly to a straight pipe when the fluid ve?locity exceeded a certain critical value based on an assumption that the centerline of the curved pipe was inextensible.However，it is controversial.The stability was not to be lost while one treated the cen?terline extensible.Consequently，Chung and Jung proposed a more accurate model for the semi-circu?lar fluid-conveying pipe［4］.They discussed the natu?ral character of this kind of pipes.Ni and his cowork?ers developed this model to the microscale region［5］.They also discussed the dynamic response of the semi-circular arc pipes conveying fluid via the simu?lation［6］.In 2017，Zhao et al.discussed the dynamic response of this kind of pipe by the analytical meth?od［7］.These investigations indicate that the dynamic character of the curved pipe is complex.For straight pipes，the model is based on that the static equilibri?um configuration is always straight.However，the manufacturing error and the creep deformation dur?ing the service life will make this assumption out of operation.The straight static equilibrium configura?tion becomes slightly curved.Consequently，the slightly curved hydraulic pipe must be discussed.

Sinir investigated a slightly curved pipe with the SIN function shape in 2010［8］.He claimed that the initial curve would increase the critical flow ve?locities and the natural frequencies of the system，and decrease the amplitude of the resultant motions.Latterly，Wang et al.also found that the critical flow velocity at which buckling instability occurred was higher than that for a pipe without imperfec?tions［9］.In 2017，Li et al.discussed the forced reso?nance of a slightly curved pipe［10］.With the ampli?tude of initial imperfection increasing，the hardening behavior of the system was transformed into the soft behavior.Recently，the model of slightly curved pipe has been developed to the non-planar type［11］and the supercritical region［12］.All these investiga?tions assumed that the pipe was simply supported at two ends.But for a real hydraulic pipe，the supports at two ends are more close to the fixed restriction.

The softening phenomenon also can be found for slightly curved beams.Oz et al.produced con?tributory work on this field［13］.They discussed the influence of different initial curve on the nonlinear re?sponse［14］and found that the 2∶1 internal resonance was possible for the case of parabolic curvature but not for that of sinusoidal curvature for a simply sup?ported beam［15］.Huang and Chen discussed the bi?furcation of the simply?supported slightly?curved beam via the incremental harmonic balance meth?od［16-17］.The response was more complex than the beam without the initial curve as the system con?tained the quadratic nonlinearity and the cubic non?linearity at the same time.Consequently，the vibra?tion of the slightly curved beam must be con?trolled［17］.The boundary also has very important in?fluence to the slightly curved beam［18］.These inves?tigations also focused on the simply supported struc?ture.

The current work tries to discuss the natural character and the resonance of the hydraulic pipe fixed at two ends.The 2∶1 internal resonance condi?tion is given to test that whether the 2∶1 internal res?onance would happen under the given initial curve proposed in the current work.The harmonic balance method together with the pseudo-arc-length method is used to solve the complex steady state response.

1 Mechanical Modeling and Natu?ral Frequencies

Fig.1 shows a section of hydraulic pipe of a fight?er.The boundary of the pipe is simplified as the fixed constraint.The distance between the two sup?ports isLpwhile the external and the inner diameters of the pipe areDandd，respectively.The density of the pipe isρpand the Young’s modulus isE.In the pipe，there is full of aviation hydraulic oil.To simplify the theoretical model，one can consider the hydraulic oil as incompressible Newtonian fluid without viscosity.With this assumption，the oil pressure is the same everywhere in the pipe.There?fore，the effect of the pressure of the hydraulic oil can be neglected.The hydraulic oil has the density ofρfand moves with the velocityΓ.Values of these parameters are given in Table 1.

Fig.1 Schematic diagram of hydraulic pipe fixed at two ends

Table 1 Physical parameters

In ideal conditions，the pipe will be straight during the service life.But considering the manufac?turing defect and the creep deformation，the pipe may have an initial curve.According to Ref.［10］，the initial curve affects the dynamic characters of the system a lot.This will also be very important for the hydraulic pipe as it plays a very important role in the equipment.The initial curve is marked asY（x）.Based on it，the governing equation of the hydraulic pipe will be established by the generalized Hamilton’s principle.

The bending deformation of the pipe isu（x，t）.Thus，the kinetic energy of the pipe is

whereuis the lateral displacement along the pipe andApthe cross section of the pipe.The kinetic en?ergy of the hydraulic oil along the pipe is

whereAfdenotes the cross section of the fluid.Ac?cording to the Hamilton’s principle，the potential energy needs to be taken into consideration.Thus，the potential energy of the pipe is

and it is the strain of a micro-body along thexcoor?dinate.Considering the virtual work of the uniform?ly distributed harmonic force along the pipe，the governing equation of the hydraulic pipe will be ob?tained as

with the boundary condition

whereF0is the amplitude of the excitation force andΩthe excitation frequency.

The Kelvin’s material derivative is used in the current work to describe the viscoelasticity of the pipe，which means the following equation needs to be substituted into Eq.（5）.

whereμis the visco-elastic coefficient of the pipe.

At last，the governing equation is

Based on Eq.（8），the natural characters and dy?namic characters can be discussed.Besides，one can find from it that the linear restoring force is affected by the initial curve.But the initial curve does not af?fect the linear damping of the system.Both of the nonlinear restoring force and the nonlinear damping are changed by the initial curve.Different with the bi?stable supercritical pipes conveying fluid［19］，the slight?ly curved pipe here is a monostable system，although the governing equation here seems like that of the pipes conveying fluid.To validate this conclusion，the potential energy of the slightly curved pipe will be discussed.Before that，the initial curve should be defined.Considering the fixed boundary，one can use the following function to describe the initial curve

The shape of it is given in Fig.2.One can ob?serve from it that the displacement and the rotation angle at two ends are zero，which satisfies the boundary condition Eq.（6）.Consequently，the poten?tial energy can be demonstrated changing with the initial curve in Fig.3.WhileA0is zero，the potential energy curve is symmetric about the 0 position.The symmetry disappears asA0is increasing.But the po?tential energy always has just one well.

Fig.2 Initial curve of the slightly curved pipe

Fig.3 Potential energy changing with initial curve

By eliminating the excitation，the nonlinearity and the viscoelasticity in the governing equation，the natural frequencies can be obtained from the de?rived linear system.As a gyroscopic system，natu?ral frequencies and resonance will be discussed based on natural modes of a static beam with fixed boundaries［20-21］.This means natural modes are dis?cussed via

with the boundary value given below

where?is the natural mode andβthe modal eigen?value.Things different from Refs.［20-21］are that the hydraulic pipe in the current work is slightly curved.But this is not a problem for solving the gov?erning equation here as Refs.［20-21］have success?fully used the series expansion to get the natural fre?quency and the resonance.Hence，the solution of the slightly-curved hydraulic pipe is still based on the modal expansion.The key for the accuracy of the so?lution is the convergence of the expansion.By com?paring natural frequencies based on different expan?sions，an expansion with four modes can obtain con?vergent results for the first two natural frequencies.

As can be observed from Figs.4（a，b），natural frequenciesω1，ω2are decreasing with the increase of the fluid speed.This is a recognized conclusion of the pipe conveying fluid.Meanwhile，the first natu?ral frequencyω1will increase with the height of ini?tial curve at first.But this tendency disappears while the height exceeds a critical value.After that，the first natural frequencyω2keeps the same.On the contrast，the second natural frequency is constant at first.After the critical height，it will increase with the height of the initial condition.Figs.4（c，d）ex?plain this phenomenon more clearly in the 2D view.These two subpictures demonstrate the first two nat?ural frequencies under the fluid speeds 0 m/s and 50 m/s，respectively.They all indicate the effect of the initial curve on the first two natural frequencies as described earlier.Besides，the first two natural frequencies have a point of intersection while the flu?id speed is 0 m/s.But this intersection point disap?pears once the fluid flows.

Figs.4（e，f）describe the first two natural fre?quencies changing with the initial curve and without the initial curve.With the given initial curve，the first natural frequency increases dramatically.But the second one changes little while the speed is not too high.The initial curve also changes the critical speed，which makes the first natural frequency be zero，to a high value.This indicates that the initial curve must be considered as the influence of it on the natural characters of the hydraulic system is con?siderable.

Fig.4 The first two natural frequencies changing with dif?ferent parameters

As mentioned in Ref.［15］，the 2∶1 internal resonance occurs under a special initial curve.In the current work，the first two natural frequencies are set to satisfy this internal resonance condition to vali?date that whether the 2∶1 internal resonance could happen with the shape of Eq.（9）.The height of the initial curve in the current is set to 6 mm，which is just 0.6% of the length of the pipe.While the hy?draulic oil flows at the speed 121.5 m/s，the 2∶1 in?ternal resonance condition is established.Under this condition，the first natural frequency is 214.57 rad/s and the second one is 429.14 rad/s.

2 Harmonic Balance Method

The section above introduces the governing equation of the hydraulic pipe based on the initial curve，including the influence of it on the natural fre?quencies.In this section，the influence of the initial curve on the dynamic response will be discussed.Considering the nonlinearity may be strong，the har?monic balance methodφis used in the current work.The solution of the system is expressed as

where?is the mode of the static beam with the fixed restraint.The partial differential governing equation Eq.（8）will be projected to the modal space via the Galerkin method.In this way，four coupled ordinary differential equations about variatesqi（i=1，2，3，4）will be produced.According to HBM，they are expanded as

Substituting Eq.（10）into the governing equa?tion，making use of Eq.（11），there comes corre?sponding functions of harmonics.Collecting coeffi?cients of different harmonics，one can obtain the steady state response functions.The response changing with the excitation frequency can be solved out from them.

As the governing equation contains the cubic nonlinearity，min Eq.（11）will be set as 3 during the calculation.The strong nonlinearity may trigger complex resonance，which introduces jumping phe?nomenon to the response curve.The multiplicity of solution makes the coupled function difficult to be solved out.To overcome this trap，the Newton’s method together with the pseudo-arc-length method is proposed.The detailed introduction of it can be found in Ref.［22］.

All the natural modes of the static beam under the same restriction，the natural frequencies of the hydraulic pipe，the Galerkin truncation and the solu?tion of the steady state response are carried out by the software MAPLE.The computational accuracy is set as 10-16.The judging criteria of the conver?gence of the solution is，where Erriis the scrap value of each coefficient function.At this step，the resonance of the hydraulic pipe with an initial curve can be obtained by substituting all parameters.

3 Numeric Examples and Discussion

The viscoelastic coefficientμis set to 5×107N·s/m2while the excitation force is 0.5 N/m.Substituting these parameters into the steady state response functions，coefficients of harmonics are solved out.Reponses of each harmonic on the first two modes are demonstrated in Figs.5 and 6，respec?tively.On account of the quadratic nonlinearity，the zero-drift coefficient is nonzero.It is a component of the harmonic solution with zero-frequency.Besides，it arouses the response on the second harmonic.

For the harmonics on the first mode，the 2nd order harmonic has the same magnitude with the 1st order one.The vibrating energy is transmitted to the quadratic nonlinearity.It is strange that，the re?sponse of the 3rd harmonic has a resonance peak at the second order of natural frequency.On the con?trast，the responses of the zero-drift，the 1st har?monic and the 2nd harmonic are tiny in the second natural frequency region.

Fig.5 Harmonic responses of the first mode

Fig.6 shows the response of corresponding har?monics on the second mode.Although both of the structure and the excitation are symmetric，reso?nance also occurs on the asymmetric mode.This is a classical character of the pipe conveying fluid.However，this does not mean the secondary reso?nance will happen while the excitation frequency nears the second natural frequency.

Fig.6 Harmonic responses of the second mode

The total response of the hydraulic pipe can be superposed by these harmonics together，including the spatial modes.Fig.7 shows the maximal and minimal responses of the middle point，the quarter point and the three-quarter point on the slightly curved hydraulic pipe.Obviously，the response curve is bended to the left，which means the initial curve softens the nonlinearity of the response.The response curve is asymmetric about the initial posi?tion.This is caused by the zero-drift part of the har?monic response.Although the forced resonance hap?pens under the 2∶1 internal resonance condition，the typical double-jumping phenomenon does not occur.It means the 2∶1 internal resonance condition is not triggered under the given initial curve in the current work.This means the 2∶1 internal resonance is not just dominated by the internal resonance condition and the quadratic nonlinearity.For continuous，the shape of initial curve［15］and the boundary condi?tion［23］also influences the 2∶1 internal resonance.

Fig.7 Total responses along the pipe

4 Conclusions

The work here investigated the force resonance under the 2∶1 internal resonance condition.The steady state response was solved out by the Harmonic bal?ance method.Based on the discussions above，some conclusions were obtained.

（1）The initial curve hardens the stiffness of the first natural mode，which enlarges the first natu?ral frequency and the critical fluid speed.

（2）The initial curve softens the nonlinear re?sponse.More harmonics are produced by the qua?dratic nonlinearity and the cubic nonlinearity.

（3）The 2∶1 internal resonance will not happen under the given shape in the current work，although the internal resonance condition and the quadratic nonlinearity occurs together.