Vibration Control and Separation of a Device Scanning an Elastic Plate

Shueei-Muh Linand Min-Jun Teng

Nomenclature

1 Introduction

The problem of moving mass has many engineering applications,most notably in the design of railroad tracks for high-speed train,roadways and airport runways for aircraft,bridges and elevated roadways for moving vehicles[Beskou and Theodorakopoulos,2011],computer storage disk drives[Huang and Mote Jr.,1996]high speed precision machining[Esen,2013]and the motion of moving beams[Lin,(2009a,2009b,2011)].

This problem is generally simulated using two models:(1)the moving-load model,and(2)the moving-mass model.The major difference is that the inertial effects of the moving body are incorporated into the model formulations in the movingmass model,but not in the moving-load model Due to the complexity of the moving-mass model,the majority of studies available in the literature only consider the vertical translational component of the moving mass acceleration in the full term formulation of the problem,neglecting the other convective acceleration terms leading to error Akin and Mofid[1989]investigated a beam with moving mass and found that the moving-load model causes an error of 2～80%.In addition,Nikkhooet al.[2007]found for an Euler–Bernoulli beam,ignoring the convective terms in the formulation could lead to a remarkable error for mass velocities greater than a socalled critical velocity.Intuitively,for plate problems with more convective terms,this could be even more important.

In the moving-load model,Gbadeyan and Oni[1995]investigated the dynamic behavior of beams and rectangular plates under moving loads Huang and Thambiratnam[2001]investigated deflection response of plate on Winkler foundation in response to moving accelerated loads.Kim[2004]investigated the buckling and vibration of a plate on an elastic foundation subjected to in-plane compression and moving loads.Lee and Yhim[2004]investigated the dynamic response of composite plates subjected to multi-moving loads based on a third order theory.Moving velocities made greater contributions to the dynamic responses of the composite plates for higher speed.Moreover,the dynamic resistance for plates made of composite materials was excellent and stable.Wu[2005]proposed the approximated method predicting the dynamic responses of a two-dimensional rectangular plate undergoing a transverse moving line load by using the one-dimensional equivalent beam model.Au and Wang[2005]investigated sound radiation from forced vibration of rectangular orthotropic plates under moving loads.Based on the Rayleigh integral and the dynamic response of the plate,the acoustic pressure distributions around the plate were obtained in the time domain.Law et al.[2007]investigated the dynamic identification of moving loads from a vehicle traveling on top of a beam-slab type bridge deck using numerical and experimental studies.Elliottet al.[2007]described the location tracking of a moving load with an unknown,harmonically varying magnitude on a plate using a distributive sensing method.Using this method,the actual position of forces moving at various speeds can be determined to within 2%error at speeds less than 3.2 m/s.Malekzadehet al.[2009,2010]studied the dynamic response of thick laminated rectangular and annular sector plates subjected to moving load by using a three-dimensional hybrid numerical method.The hybrid method is composed of a series solution,layerwise theory and the differential quadrature method in conjunction with the finite difference method.Sheng and Wang[2011]investigated the response and control of functionally graded lami-nated piezoelectric shells under thermal shock and movement loadings.They found that the maximum value of the displacement increases with increase in velocity of moving loads until a critical speed,and then decrease after this critical speed Zhanget al.[2011]proposed an approximate solution for the dual-duct simply supported rectangular plate subjected to a moving load using a stepped plate approximation theory.Mart?’nez-Rodrigo and Museros[2011]studied the optimal design of passive viscous dampers for controlling the resonant response of orthotropic plates under high-speed moving loads.They found that for a particular set of auxiliary beams,there exist optimum parameters of passive viscous dampers that minimized the plate resonance.

In the moving-mass model,Gbadeyan and Dada[2006]took the finite difference method to solve the moving mass problem.Their study presented that the maximum shearing forces,bending and twisting moments occurred almost at the same time.Also,the values of the maximum deflections were higher for Mindlin plates than for non-Mindlin plates.Wu[2007]proposed the finite moving mass element method and studied the influence of moving-load-induced inertia force,Coriolis force and centrifugal force on the dynamic behavior of inclined plates subjected to moving loads.He concluded the effects of Coriolis force and centrifugal force were perceptible only in the case of higher moving-load speed.Rofooei and Nikkhoo[2009]derived the constitutive equation of motion for a thin rectangular plate with a number of piezo patches bonded on its surface under the excitation of a moving mass.Eigenfunction expansion was used to transform the equation of motion into a number of coupled ordinary differential equations.A classical closed-loop optimal control algorithm was employed to effectively suppress the resonant dynamic response of the system.Ghafoori and Asghari[2010]used the finite element method based on the first-order shear deformation theory and the Newmark direct integration method to study the dynamic behavior of composite plate Their research presented that the most sensitive lamination to the inertia of moving mass was[45/45/45/45]lamination so that the moving mass analysis gave poor results in the moving-load model Eftekhari and Jafari[2012]proposed the methodology composed of the Ritz method,differential quadrature method,integral quadrature method and Newmark time integration scheme to study the transient response of rectangular plates subjected to linearly varying inplane stresses and moving masses.Amiriet al.[2013]took into account the moving mass inertia effect all the convective terms of its out-of-plane acceleration components for the Mindlin plate.The eigenfunction expansion method transformed the governing equation into a set of ordinary differential equations and then solved by using the matrix-exponential based solution method.Their results showed,for moderately thick plates,there was the remarkable differences between the results in the Mindlin plate theory and the classical plate theory.Esen[2013]used the finite element method to study the transverse vibration of rectangular thin plates under a moving mass.The literature presented that the vibration effect of the change in velocity was more significant when compared to the change in mass.

So far,little research has been devoted to the investigation of separation and vibration control of plate in the moving-mass model.This study is investigates the control and separation of a concentrated mass moving along an arbitrary trajectory on a plate.The semi-analytical solutions for these systems are presented.Moreover,the effects of several parameters on the separation and vibration control are investigated also.

2 Moving mass model

2.1 Governing Equation and Boundary Conditions

Figure 1:Geometry and coordinate system of a simply-supported plate subjected to a moving mass.

A concentrated mass moves along an arbitrary trajectory on a rectangular isotropic elastic plate,as shown in Figure 1a.The governing equation is

In terms of the dimensionless parameters in the nomenclature,the corresponding dimensionless governing equation is

2.2 Semi-analytical solution

From Eq.(12c)the absolute accelerationd2wc/dτ2includes the Coriolis accelerationsand time-dependent coefficients{ξc(τ),ζc(τ)}.Because the forcefcof Eq.(12b)includes the product of the unknown variablewcand the time-dependent coefficients,this system composed of Eqs.(12)-(20)is implicit and very difficult to solve directly The semi-analytical methodology is presented.First,using the approximated acceleration method,the implicit system is transformed into an explicit system.Secondly,the analytical solution of the transformed system can be derived by using the mode superposition method.Finally,error due to transformation must be near zero.The details are demonstrated below

2.2.1Approximated acceleration method

It is assumed that the overall dynamic behavior of the general system is composed of the behaviors of several time subintervals.The overall dynamic performance of the system is derived step by step.If each time subinterval?τis very small,the absolute acceleration can be linearly approximated as

The approximated absolute acceleration is defined asWhen the unknown parameterλis correctly chosen the error of the acceleration atτ=τi+1approaches zero,i.e.,

It should be noted that ifλis correctly chosen the approximate acceleration(τi+1)/dτ2is calculated via Eq.(21).Substituting Eq.(21)into the governing equation(12),the transformed system is obtained:

2.2.2Mode superposition method

The dynamic solution and the forcing term of Eq.(23)are respectively assumed to be

In summary,if parameterλof Eq.(21)in the domainsatisfies the minimum error condition(22),accurate time functions??can be obtained via Eq.(26).Substituting these back into Eq.(24)will determine the displacement in the domain.Moreover,the overall dynamic behavior can be determined step by step.When the time subinterval?τapproaches to zero,the overall accurate solution is successfully determined[Lin,201].

Table 1:Convergence of the proposed methodkc=0].

Table 1:Convergence of the proposed methodkc=0].

?

Without the loss of generality,assume a concentrated mass moving fromx=0 tox=L1with constant speeddξc/dτ=v.Table 1 verifies the effect of the number of time subintervals and the number of terms(M×N)of Eq.24)on the numerical result of the response ratiow(0.5,0.5,τ(ξc=1))/ws(0.5,0.5)wherew(0.5,0.5,τ(ξc=1))is the dynamic displacement atξgζg 0.5 when the concentrated mass moves to the positionξc=1ws(0.5,0.5)is the static displacement at the center of plate,ξgζg 0.5,subjected to the same concentrated weight atξc=ζc=0.5 Szilard[1974]presented the displacement of a s-s-s-s plate subjected to a concentrated load,shown in Figure 1b,as

The numerical result determined by the proposed method converges very rapidly.Even when the number of subintervals is only one thousand,the difference between the present displacement and the converged displacement is less than 0.29%.However,when the mass ratio and the moving speed are increased,more modes are required for the accurate results.

Table 2:Comparison of dynamic amplification factors,wmax(0.5,0.5,τ)/ws(0.5,0.5),of a simply supported 2(m)×2(m)×17(cm)aluminum plate by the proposed method compared to the method of Nikkhoo and Rofooei[2012]when the concentrated mass moves at a constant speed along a trajectory parallel to the plate edge,i.e.,ζc(τ)=0.5,from ξc=0 to ξc=1.[E=731 × 1010pa,ρ =2700 kgm-3,μg=0.33c=cc=k=kc=0,r=1.0,and the moving speed V= βv′,v′=2L1/T1in which T1is the first period of vibration of the plate].

Table 2 demonstrates the comparison of dynamic amplification factors(DAF),wmax(0.5,0.5,τ)/ws(0.5,0.5),of a simply supported 2(m)×2(m)×17(cm)aluminum plate by the presented method versus the results from Nikkhoo and Rofooei[34]when the concentrated mass moves at a constant speed along a parallel trajectory to the plate edge,i.e.,ζc(τ)=0.5,fromξc=0 toξc=1.The dynamic amplification factor is the ratio of the plate’s absolute maximum dynamic deflection to its maximum static response at the plate’s center point.If the mass ratio mcand the moving speedvare small,the results determined by the presented method and Nikkhoo and Rofooei[2012]are very consistent.However,for larger mass ratio mcand the moving speed,the difference between the approaches is significant.By the presented method,the separation phenomenon is found.

2.3 Mechanism of Separation and Factors

2.3.1Critical condition of separation

The mechanism of a vehicle separating from a plate is described in the following.If there is no guide keeping the vehicle in connection with the plate as shown in Figure 1a,the vehicle may separate from the plate when the interacting contact forcefcchanges from compressive to tensional.When compressive or positive contact forcefcexists,the vehicle will move along the plate.However,when normal contact forcefcis decreased to be zero,the vehicle will separate from the plate.Therefore,the critical condition of separation is‘fc(τ)=0’.

2.3.2Effect of parallel trajectory

Consider a concentrated mass moves at a constant speed along a parallel trajectory to the plate edge,i.e.,ζc=0.5 andξc=vτfromξc=0 toξc=1.One investigates the influence of the moving speedvand the aspect ratio(L1/L2)ron two dynamic responses{w(0.5,0.5,τ),w(ξc,ζc,τ)}wherew(0.5,0.5,τ)is the dynamic displacement at the center of plate andw(ξc,ζc,τ)is the dynamic displacement of the vehicle position.From Figure 2a when the aspect ratior=0.5 and moving speedv=0.5 if the coordinate of vehicleξcis increased fromξc=0,the vibration responses are increased.When the vehicle moves toξc=0.5,the maximum responses{wmax(0.5,0.5,τ),wmax(ξc,ζc,τ)}occur.Forv=1.0 when the vehicle moves toξc=0.43,the maximum responses occur.In addition,forv=2.0 whenξc=0.7,the maximum responsewmax(ξc,ζc,τ)occurs.However,the maximum responsewmax(0.5,0.5,τ)happens atξc=0.75.Obviously,the higher the moving speedvis,the larger the maximum responses are.Moreover,for higher speedv=2.0,when the vehicle moves toξc=0.9235,the vehicle will separate from the plate.In other words,for the aspect ratio r=0.5 if the moving speed is increased to v=2.0 separation will occur.

Figure 2:Influence of moving speed v and aspect ratio r on response ratio w/ws(0.5,0.5)when a concentrated mass moves at a constant speed along a trajectory parallel to the plate edge,i.e.,ζc(τ)=0.5 from ξc=0 to ξc=1[mc=0.1,ˉg=0.1,ξc(τ)=vτ,c=cc=k=kc=0;(a):r=0.5,(b):r=1.0,(c):r=2.0].

Furthermore,in Figure 2b with aspect ratior=1 andv=2.0 when the concentrated mass moves fromξc=0 toξc=1,separation will not occur.This differs to the plate withr=0.5 as shown in Figure 2a.The reason is that the plate withr=0.5 is more flexible than the plate withr=1.Moreover,if moving speedvis increased to the value of 3,the vehicle will separate from the plate atξc=0.9526 Finally,in Figure 2c with the aspect ratior=2 though the moving speed is increased tov=5,separation will not occur.Finally,if the moving speed v is increased to the value of 10,the vehicle will separate from the plate atξc=0.6921 It is concluded from Figure 2 that the effect of moving speedvon the maximum responses is significant.When the moving speed is over the critical speed,the moving mass will separate from the plate.Also,the larger the aspect ratioris,the higher the critical speedvcriticalis.

2.3.3Effect of diagonal trajectory

Consider a concentrated mass moving at a constant speed along a diagonal trajectory from{ξc,ζc}={0,0}to{ξc,ζc}={1,1/r},i.e.,ξc(τ)=v1τ,ζc(τ)=(v1/r)τ.We investigate the influence of the moving speedv1on two dynamic responses{w(0.5,0.5,τ),w(ξc,ζc,τ)}and separation From Figure 2b where v=1.0 or 2.0,the moving mass does not separate from plate However,when the speed increased to 3.0,the moving mass will separate from plate at the 0.9526 position.Figure 3 shows when v=2 or 3 and the vehicle moves toξc=0.9575 or 0.8858,it will separate from the plate.Further,it is observed from Figures 2b and 3 with aspect ratior=1 and moving speedv=2,separation will occur along a diagonal trajectory instead of a parallel to the plate edge.This demonstrates the effects of aspect ratiorand moving trajectory on critical speed are significant.A detailed investigation follows.

2.3.4Effects of aspect ratio and mass ratio

Figure 4 demonstrates the relationship among the aspect ratior,the concentrated mass ratiomcand the critical speedvcritical.It shows the larger the aspect ratioris,the higher the critical speedvcritical.However,the larger the concentrated massmcis,the lower the critical speedvcritical.Moreover,if aspect ratioris small,the critical speed of a diagonal trajectory is lower than that of a trajectory parallel to the plate edge.On the other hand,if the aspect ratioris large,the critical speed of a diagonal trajectory is higher than that of a parallel one.At the critical aspect ratiorcritical,the critical speeds are same.Note the larger the massmcis,the smaller the critical aspect ratiorcritical.

2.3.5Effect of foundation

Figure 3:Influence of moving speed v on response ratio w/ws(0.5,0.5)when a concentrated mass moves at a constant speed along a diagonal trajectory from{ξc,ζc}={0,0}to{ξc,ζc}={1,1}[mc=0.1,ˉg=0.1ξc(τ)=v1τ;ζc(τ)=(v1/r)τ?r=1,c=cc=k=kc=0].

Figure 4:Influence of moving mass mcand aspect ratio ron critical speed v1,critical when a concentrated mass moves at a constant speed.[ˉg=0.01c=cc=k=kc=0;solid line:along the trajectory parallel to the plate edge, ξc(τ)=v1τ;ζc(τ)=0.5;dashed line:along the diagonal trajectory ξc(τ)=v1τ;ζc(τ)=v2τv1/v2=r].

Figure 5:Influence of the spring constant k and damping coefficientc on the critical speed when a concentrated mass moves at a constant speed along a trajectory parallel to the plate edge,i.e.,ζc(τ)=0.5 from ξc=0 to ξc=1[mc=0.1,ˉg=0.1,r=1].

Figure 5 demonstrates the influence of the spring and damping constants{k,c}of foundation on the critical speedvcriticalwhen the concentrated mass moves at a constant speed along a trajectory parallel to the plate edge,i.e.,ζc(τ)=0.5,fromξc=0toξc=1We find the spring constantkis smaller than the value of105and the critical speedvcriticalis almost constant.Further,the critical speedvcriticalincreases greatly with the spring constantkMoreover,the larger the damping coefficientsc,the higher the critical speedvcritical.

2.3.6Effect of nonconstant moving speed

We investigate the effects of three different movements on the dynamic response ratiow(ξc,ζc,τ)/ws(0.5,0.5)when the concentrated mass moves along a parallel trajectory to the plate edge and the required time fromξc=0 toξc=1,is ‘T’.The three movements are described as the following:

1.The first moving speed is constant,dξc/dτ=v0.The corresponding position of the vehicle isξc(τ)=v0τand the required time period to cross the plate fromξc=0 toξc=1,isT=1/v0.

2.The second moving speed isdξc/dτ=vs0[1-sin(πτ/T)].The corresponding position of vehicle isξc(τ)=vs0[τ+(T/π)(cos(πτ/T)-1)].The corresponding parametervs0=1/[T(1-2/π)]

3.The third moving speed isThe corresponding position of vehicle isξc(τ)=ˉvs0(T/π)[1-cos(πτ/T)].The corresponding parameter

From Figure 6a withT=1 the vibration response of the third movement is the smoothest and that of the second movement is the worst.For the second movement when the vehicle moves toξc=0.9593,the vehicle will separate from the plate.In Figure 6b withT=0.5 the vibration response of the second movement is the worst.When the vehicle moves toξc=0.9868,it will separate from the plate.This shows that if the moving speeddξc/dτ=ˉvs0sin(πτ/T),the vibration response can be significantly suppressed.

Figure 6:Influence of the different movements dξc/dτby the response ratio w(ξc,ζc,τ)/ws(0.5,0.5)when a concentrated mass moves at a constant speed along a trajectory parallel to the plate edge,i.e.,ζc(τ)=0.5 from ξc=0 to ξc=1.[mc=0.1,ˉg=0.1,c=cc=k=kc=0,r=1 vs0=1/[T(1-2/π)]ˉvs0=π/2T;(a)T=1/v=1,(b):T=1/v=0.5].

3 Active Control of moving mass

Based on the above facts,there exists the phenomenon of separation and significant vibration response when there is no active control.For suppressing vibration and preventing separation the following active control law is presented.

3.1 Governing Equation and Boundary Conditions

Consider a scanning device is supported by spring and damper and moving on a plate.This can be expressed as a mass-spring-damper model moving on a plate as shown in Figure 1b.In addition to the x-y movement in the horizontal plane,z-direction movement is considered.In other words,the supporting displacementηcis time-dependent.The dimensionless governing equation is the same as Eq.(12a)excepting the contact force:

The associated boundary conditions are the same as Eqs.(13-2).This model is applied to simulate the system of a device scanning a plate.

3.2 Solution method

In a similar approach,the time variabletis divided intonsections and the dynamic performance of the system is derived step by step.The contact force is approximated by

Except Eqs.(30,31),the solution method is the same as the moving mass system.

3.3 Control Law

For suppression of vibration the following control law is assumed:

The velocity of scanner foundationdηc(τ)/dτis proportional to the acceleration of the scanner device position.If the control parameterGis positive,velocity is increased with acceleration.Conversely,if parameterGis negative,velocity is decreased with acceleration.The scanner device moves at a constant speed along a trajectory parallel to the plate edge fromξc=0 toξc=1 The effect of control parameterGon the dynamic response is investigated and plotted in Figure 7.It shows when the control parameterG=-2000,the vibration of the scanner device approaches zero demonstrating this control law is effective.Further,Figure 8 demonstrates the influence of the spring constantkcand the damping coefficientccon the suppression of vibration.It is found that the effect of the spring constantkcon the suppression of vibration displacement is significant,but that of the damping coefficientccnegligible.

4 Applicability of the moving-load model

Figure 7:Influence of control gain G on the suppression of vibration when a concentrated mass moves at a constant speed along a trajectory parallel to the plate edge,i.e., ζc(τ)=0.5 from ξc=0 to ξc=1.[cc=5kc=5?c=k=0,mc=0.1ˉg=0.1?ξc(τ)=2τr=1].

Figure 8:Influence of the spring constant kcand the damping coefficient ccon the suppression of vibration when a concentrated mass moves at a constant speed along a trajectory parallel to the plate edge,i.e.,ζc(τ)=0.5,from ξc=0 to ξc=1.[G=-10,c=k=0mc=0.1?ˉg=0.1ξc(τ)=2τ?r=1].

This theory refers to the Appendix(moving load model).It is well known that separation cannot be studied using the moving load model.Figure 9 compares the vibration displacements at the moving position in the moving-load and movingmass models.From Figure 9a and 9b with aspect ratior=1,the displacementsw(ξc,ζc,τ)in the two models are almost consistent at the initial part of trajectory.But the difference gradually increases with the coordinates of the vehicle,ξc.Forr=1 or 2,when the vehicle moves with the speed ofv=3 or 10,it will separate from the plate atξc=0.9526 or 0.6921,respectively Note,the larger the aspect ratiorand the moving speedvare,the greater their difference between the displacementsw(ξc,ζc,τ)of the two models.This shows the moving mass problem may be accurately approximated by the moving load model only when the moving speedvis very slow and at the initial part of the trajectory.

Figure 9:Influence of moving speed v and aspect ratio r on the response ratio w/ws(0.5,0.5)when a concentrated mass moves at a constant speed v along a trajectory parallel to the plate edge,i.e., ζc(τ)=0.5,from ξc=0 to ξc=1[c=cc=k=kc=0,mc=0.1,ˉg=0.1,ξc(τ)=vτ;(a):r=1,(b):r=2;solid line:moving mass model;dashed line:moving load model].

5 Conclusion

The moving mass problem may be accurately approximated by the moving load model with constraints of low moving speedvand only at the initial part of the trajectory.Separation cannot be studied by using the moving-load model.An effective control methodology for the suppression of vibration of a device moving on a plate is proposed in this work and the effects of several parameters on the separation and the vibration control of system are discovered as follows:

1.For a diagonal or parallel trajectory the larger the aspect ratioris,the higher the critical speedvcritical.

2.For a diagonal or parallel trajectory the larger the concentrated massmcis,the lower the critical speedvcritical.

3.If the aspect ratioris small,the critical speed of a diagonal trajectory is lower than that of a parallel one.But if the aspect ratio r is large,the critical speed of a diagonal trajectory is higher compared to a parallel trajectory.

4.If the spring constantkof the Winkler foundation is smaller than the value of 105,critical speedvcriticalis almost constant.At the same time,the larger the damping coefficientscis,the higher the critical speedvcritical.

5.If varying moving speed such asis considered,vibration response can be significantly suppressed.

6.The effect of the spring constantkcon the suppression of vibration displacement is significant but the effect of the damping coefficientccis negligible.

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