Nonlinear response analysis and experimental verification for thin-walled plates to thermal-acoustic loads

Yundong SHA,Jian WANG

Liaoning Key Laboratory of Advanced Test Technology for Aeronautical Propulsion System,Shenyang Aerospace University,Shenyang 110136,China

Nonlinear response analysis and experimental verification for thin-walled plates to thermal-acoustic loads

Yundong SHA,Jian WANG*

Liaoning Key Laboratory of Advanced Test Technology for Aeronautical Propulsion System,Shenyang Aerospace University,Shenyang 110136,China

Buckling; Experimental verification; Nonlinear response; Power spectral density; Probability spectrum density; Snap-through; Thermal-acoustic load; Thin-walled structure

For large deflection strongly nonlinear response problem of thin-walled structure to thermal-acoustic load,thermal-acoustic excitation test and corresponding simulation analysis for clamped metallic thin-walled plate have been implemented.Comparing calculated values with experimental values shows the consistency and veri fies the effectiveness of calculation method and model for thin-walled plate subjected to thermal-acoustic load.Then this paper further completes dynamic response calculation for the cross reinforcement plate under different thermal acoustic load combinations.Based on the obtained time-domain displacement response,analyses about structure vibration forms are mainly focused on three typical motions of post-buckled plate,indicating that the relative strength between thermal load and acoustic load determines jump forms of plate.The Probability spectrum Density Functions(PDF)of displacement response were drawn and analyzed by employing statistical analysis method,and it clearly shows that the PDF of postbuckled plate exhibits bimodal phenomena.Then the Power Spectral Density(PSD)functions were used to analyze variations of response frequencies and corresponding peaks with the increase of temperatures,as well as how softening and hardening areas of the plate are determined.In the last section,this paper discusses the change laws of tensile stress and compressive stress in pre/post buckling areas,and gives the reasons for N glyph trend of the stress Root Mean Square(RMS).

1.Introduction

The skin of hypersonic vehicle,the jet nozzle of ramjet engine,the flame tube of aircraft engine,afterburner liner,etc.will encounter severe work environment due to the joint actions of aerodynamic load,thermal load,acoustic load and mechanical load,where the surface temperature of the aircraft Thermal Protection System(TPS)may reach to 1648°C,and the Sound Pressure Level(SPL)may reach to 180 dB.1Thin-walled structure used to improve the aerospace craft flight performance will exhibit a complex nonlinear response2–6,including linear random vibration around the pre-buckled initial equilibrium position,snap-through motion around two post-buckled equilibrium positions and nonlinear vibration around one post-buckled equilibrium position,which lead to the premature fatigue failure of thin-walled structure.Therefore,in order to meet the structure design requirements of aerospace thin walled structure,the preparatory analysis for thermalacoustic dynamic response becomes the key of the current works.

The simulations of thin-walled structure play an important role in improving the effectiveness and reliability of the tests.Currently,the numerical simulation methods adopted to solve nonlinear response problem mainly contain:perturbation method,Fokker Planck Kolmogorov(FPK)equation,Von Karman-Herrmann large deflection plate equation,Equivalent Linearization(EL)theory,Reduced Order Method(ROM),Galerkin theory and Finite Element Method(FEM).The EL method has already been used to calculate the stress and strain response of the thermal buckling of plate.7–9Combining Galerkin method with Monte Carlo method,Vaicaitis and Kavallieratos studied the nonlinear response of metal and composite structure subjected to random excitation.10,11Mei et al.used FEM to calculate nonlinear random response of the shell structure to thermal-acoustic excitation.12

Meanwhile,considering the thermal-acoustic nonlinear response problems of aerospace thin-walled structure,NASA Langley research center and the US Air Force Wright-Patterson Flight Dynamics Laboratory(AFFDL)focused on the response characteristics of the thin-walled structure to thermal-acoustic load,and further completed the thermalacoustic test of aluminum plate in progressive wave tube.13Then Rizzi expounded the test methods for dynamic strain and sonic fatigue of thin-walled structure in thermal-acoustic environment.14Based on the single mode equation,Ng conducted the nonlinear response analysis tests by using clamped metallic plate and composite plate and obtained the response characteristics.15Jacobson performed thermal-acoustic tests to assess the suitable composite panels for ASTOV.16Through using high-temperature random fatigue equipment and hightemperature progressive wave tube,Jacobs et al.of McDonnell Douglas studied fatigue performance of ceramic matrix composites.17In recent years,NASA carried out a series of structure thermal tests and thermal-acoustic projects,which are based on the flight environments of reusable flight vehicles X-33,X-37 and hypersonic validation machine X-43A.

Based on Ref.6,this paper selects a new-type material of aircraft engine,and conducts the acoustic vibration test of the clamped plate in thermal environment,obtaining the results of acceleration response and strain response.Then,based on the Von Karman large deflection theory,the FEM/ROM method is used to calculate dynamic response of the plate.Through comparing the response results between simulation and test,the results show the consistency,which veri fies the effectiveness of thermal-acoustic calculation method.Finally,the calculation model is used to calculate thermalacoustic response of the clamped cross reinforced metal plate,and the detailed analysis is performed.

2.Nonlinear response analysis

2.1.Large deformation nonlinear equation for thin-walled structure

Based on Von Karman large deflection plate theory18and Kirchhoff’s assumptions,the strain equation of a point which has the arbitrary distance from mid-plane is:

where u and v are x and y direction displacements in mid-plane respectively,and w is lateral deflection of z direction.

The strain compatibility equation which is obtained by performing differential operation between strain and stress is:

where Nx,Nyand Nxydenote the membrane force,and h is the thickness of plate.

Taking Eq.(3)into Eq.(2),the strain compatibility equation20presented by stress function is:

where F=Fh+Fpis composed of harmonic solution Fhand particular solution Fp;E is modulus of elasticity;α is coefficient of thermal expansionis the average temperature along the thickness direction,∇2is Laplace operator and∇4is biharmonic operator.By assuming that temperature is linearly distributed along the thickness,the temperature is:

where T(x，y，z)denotes the temperature function of plate,and θ is the temperature gradient along the thickness of plate.

Considering damping force,acoustic load,inertia force,the stress and corresponding shear force,membrane force,and bending moment comprehensively,the stress analysis of plate is carried out to obtain Von Karman large deflection motion equation with temperature.where ρ is density,ξ is damping coefficient,μ is Poisson’s ratio,p(x，y，t)is random stress of simulation acoustic load,D is bending stiffness.

2.2.Thermal buckling theory of thin-walled structure

When thin-walled structure is in temperature field,the compressive stress causes the buckling of plate,which is widely called thermal buckling.The temperature corresponding to thermal buckling is critical buckling temperature.21

Thermal buckling is the quasi-static process.Due to isotropic characteristic of metal,thermal buckling governing equation can be simpli fied to be:

Considering the in-plane displacement of the plate around the initial equilibrium position is zero,the in-plane strain is zero.According to the derivation of internal stress at the mid-plane,we can derive:

where a,b are length and width;m,n are order numbers.The above suggests that plate maintains initial equilibrium state,or maintains the balance in the form of displacement function below the critical buckling temperature.The first-order critical temperature of clamped plate(m=1,n=1)can be determined as

where β=b/a,TCis critical temperature and Trefis reference temperature.

2.3.Large deflection equation of plate simplified by single degree of freedom

The equation of single degree of freedom can be simplified as

where ω0is the first-order frequency of clamped plate,ξ0is the first-order damping coefficient andis thermal buckling coefficient,A,˙A,¨A,respectively denotes the modal displacement,the first derivative of the modal displacement,and the second derivative of the modal displacement.

Considering that the potential energy is the product of restoring force and displacement,it can be expressed as

As we can see from Fig.1,when the plate is in the prebuckling region,S<1,potential energy curves present concave wells.The plate has the minimum potential energy in the origin,which is the initial equilibrium position.When the plate is in the post-buckling region,S>1,there are two points with the minimum potential energy,which correspond to two post-buckling equilibrium positions.What’s more,the relative strength between thermal load and acoustic load determines the jump form of structure.(1)Given the relatively weak excitation,plate will make the small-amplitude vibration around the one of the post-buckling equilibrium positions.(2)Once the plate subjects to considerable excitation,it will generate the snap-through motion around two post-buckling equilibrium positions.(3)As excitation levels keep invariant,potential energy wells are deeper,resulting in the decrease of snapthrough motions,until they are restricted in the single potential energy well.22

3.Experimental verification for thermal-acoustic calculation method

3.1.Thermal-acoustic test of thin-walled structure

In this paper,the rectangular metallic plate was used to carry out dynamic response test14of thin-walled structure subjected to thermal-acoustic load,and as shown in Fig.2,thickness is 1.5 mm.The material properties of plate are listed in Table 1,K is heat transfer coefficient and OSPL means overall sound pressure level.In order to measure strain data of the xdirection on the mid-point of short side,strain gauges(#1,#3)were adopted.Meanwhile,strain gauges(#2,#4)were used to obtain strain data of the y-direction on the midpoint of long side.The fixture with a mouth shape was used to tightly pressure the sides of rectangular metallic plate to achieve clamped constraints.Both sides of the plate were subjected to asymmetric thermal load,as shown in Fig.3.The considered plate was exposed to evenly distributed,band limited Gaussian white noise with a bandwidth frequency from 100 to 1250 Hz,which enabled a wide-band multi-mode response.The combinations of thermal-acoustic load are listed in Table 2.In the process of test,gas paths should be activated firstly,and then thermal control system was opened;when the surface temperature accorded with the requirement,the acoustic load was applied to the front of plate.

Table 1 Material properties at different temperatures.

Table 2 Thermal-acoustic loading combinations.

The test was divided into two sections.The aim of the first section was to verify the effectiveness of test system about thermal-acoustic loading capacity.The results show that the test system can perform the thermal-acoustic combined loading,meanwhile,the acoustic load was up to 157 dB with a specific bandwidth frequency and the surface temperature can be up to 500°C.The second section was to test the dynamic response of plate to high temperature and intense noise.However,in the process of test,several strain gauges had not valid response data caused by intensified thermal load.Therefore,in the process of comparison between test values and simulation values,only relevant response data at temperature below 300°C were given,which had already included the processes of pre-buckling and post-buckling,guaranteeing the effectiveness of comparison and validation.

3.2.Simulation contrast verification

Firstly,in order to achieve accurate simulation,the properties,boundary conditions and thermal-acoustic load of simulation object were same as test.Then,the method of FEM/ROM was adopted to calculate thermal-acoustic dynamic response of plate,extracting strain response that conformed to the same location as test.In the end,it performed the contrast verification between simulation and test,which mainly involved the following works.

In the process of test,the acceleration response of the center position in different thermal-acoustic loading combinations had been obtained by using laser vibrometer,as shown in Fig.4.And the first-order thermal modal frequencies of plate under different temperatures can be concluded by analyzing acceleration response and strain response together.The comparison results of test values and simulation values are listed in Table 3,which show that the first-order thermal modal frequencies keep the consistency and display the trend3of first increasing and then decreasing with the increase of temperature,validating the natural variation characteristic that plate will be kept in softening region in pre-buckling and then convert into hardening region in post-buckling.Fig.5 presents,as SPL=151 dB,the representative acceleration response spectrums.Where FFT is the logogram of fast fourier transform,and g denotes acceleration.Therefore,it is obvious that as T=50°C in the pre-buckling region,the first-order frequency was 347 Hz,and as T=150°C in the post-buckling region,the first-order frequency was 306 Hz.

Table 3 First-order thermal modal frequencies of plate under different temperatures.

The locations of strain gauges and installation position of plate are shown in Fig.6.At 145 dB and 151 dB,the strain response values corresponding to fundamental frequencies in mid-points of long side and short side in test had been measured to compare with the simulation values,as listed in Table 4 and Table 5,where SM and LM represent the midpoints of short side and long side respectively.What’s more,the results indicate that the strain values of simulation have the pretty agreement with test for the same order of magnitude,verifying the effectiveness of calculation method and model for nonlinear dynamic response of thin-walled plate to thermal-acoustic load.Fig.7 gives,as SPL=151 dB,the representative strain response spectrums in the mid-point(#1)of short side,and obviously,it shows that as T=50°C in prebuckling region,the strain value is 10.747×10-6,and as T=150°C in post-buckling region,the strain value is 16.596×10-6.

From a case study of Table 5,we can further analyze the contrast results.At 50°C,the test and simulation strain values of short side point were 10.7×10-6and 9.5×10-6respectively,and the test and simulation values of long side were 13.9×10-6and 12.1×10-6respectively.Obviously,because the structure was in the linear response section of prebuckling region,the contrast results showed the high-level consistency.When the temperature was up to 100°C,the test and simulation strain values of short side point were 15.1×10-6and 20.5×10-6respectively,and the test and simulation values of long side were 22.5×10-6and 27.6×10-6respectively.Besides,when the temperature was up to 150°C,the large deflection nonlinear response of structure post-buckling region made structure be in the unstable vibration section,which visibly reduced the consistency of contrast results.Meanwhile,the test and simulation strain values of short side point were 16.6×10-6and 24.5×10-6respectively,and the test and simulation values of long side were 26.0×10-6and 30.8×10-6respectively.With the increase of temperature in postbuckling region,especially at 200 °C and 250 °C,the contrast results achieved consistency again.

Based on the relevant theories3–6of buckling and snapthrough,the test strain response results in Table 5 are analyzed in four parts as follows.First,in pre-buckling region as T=50°C,the plate subjected to acoustic excitation would exhibit the random vibration around the initial equilibrium position,and the strain responses at mid-points of long side and short side were weak,corresponding to unidirectional strain values for 13.9×10-6and 10.7×10-6respectively.Second,once the temperature was approximately close to the critical buckling temperature as T=100 °C or T=150 °C,and the SPL was up to 151 dB,the plate would start to make snapthrough motion around two post-buckled equilibrium positions and show the strong nonlinear response.As a result,strain response values increased sharply,where the strain response value of short side and long side mid-points increased to 16.6×10-6and 26.0×10-6respectively.Third,with the continuous increase of temperature to 200°C,compared with the acoustic load,the heat load was stronger,and the plate would do random vibration around a post-buckling equilibrium position,resulting in the decrease of amplitude response and the increase of average response,but the first mentioned of two played a leading role in plate response,where the strain response values of short side and long side mid-points reducedto 12.1×10-6and 14.4×10-6respectively.Fourth,when the temperature increased to 250°C,the average response persistently increased and became the important in fluence factor in plate response,where the strain response values of short side and long side mid-points increased to 14.7×10-6and 15.1×10-6respectively.

Table 4 Comparison of results of strain values between simulation and test(SPL=145 dB).

Table 5 Comparison of results of strain values between simulation and test(SPL=151 dB).

In conclusion,by conducting thermal-acoustic experiment and numerical simulation of thin-walled plate and making comparisons of modal frequencies and strain values between simulation and experiment,the effectiveness of calculation method and model for nonlinear dynamic response of thinwalled plate to thermal-acoustic load was verified adequately.

4.Analysis on thermal-acoustic nonlinear response

The clamped cross reinforced plate was selected as the simulation calculation object as shown in Fig.8,the properties were kept the same as the test plate,and Table 6 lists the specific geometry parameters.It is assumed that thermal load was uniformly distributed on the surface of plate,and acoustic load was band limited Gaussian white noise with a bandwidth frequency from 0 to 1500 Hz.The thermal-acoustic response results of node 9 were mainly extracted to analyze the vibration characteristics in this paper,as shown in Fig.9.And Table 7 shows the first eight-order modal frequencies of the plate at room temperature.

The first critical buckling temperature of clamped plate was 70.6°C calculated by using finite element method.For the convenience of expression,the coefficient of thermal buckling S was used to denote temperature,where S=T/Tcr,Tcrwas the first-order critical buckling temperature.Therefore,there were three kinds of typical state,S<1,S=1 and S>1 which represent pre-buckling,critical buckling and postbuckling respectively.In this paper,dynamic response of the plate under different S and SPL combinations had been obtained,where S was from 0 to 20 with an interval of 0.1 and SPL ranged from 124 to 172 dB with an interval of 3 dB;therefore,a total of 3417 thermal-acoustic combinations had been obtained.(1,172)would be used to represent S=1 and SPL=172 dB in the following section.

4.1.Lateral displacement

Fig.10 presents,as SPL=169 dB,the lateral displacement time history and corresponding Probability Density Functions(PDF)of plate.In pre-buckling,the plate exhibited the random vibration around the initial equilibrium position,resulting in intensified vibration response and increased displacement amplitude with the increase of temperature,as shown in Fig.10(a)and(c).Meanwhile,Fig.10(b)and(d)indicate that owing to the intensified acoustic load,the PDF of lateral displacement no longer obeyed normal distribution and the nonlinear response increased gradually.When the temperature was close to critical buckling,evidently enhancing nonlinear response,the plate began to make reciprocating vibration around two vibration equilibrium positions8,as shown in Fig.10(e)and(f).

In the post-buckling region,the plate precisely reached to snap-through motions,which was decided by the relative strength of thermal load and acoustic load.Namely,considering stronger acoustic load,the plate would make persistent snap-through around two post-buckled equilibrium positions,showing the peak of nonlinear response15,as shown in Fig.10(g).The PDF of lateral displacement visibly exhibited bimodal phenomenon,and by analyzing Fig.10(h),it is concluded that when compared with the probability for plate vibrating around the convex equilibrium position,the probability for plate vibrating around the concave equilibrium position was greater,which suggested that the intensified acoustic load would drive a deepened potential energy well in the concave equilibrium position;therefore,the plate was more likely to do random vibration around the concave equilibrium position.Further,when thermal load matched acoustic load,the plate would make intermittent snap-through,as shown in Fig.10(i).Meanwhile,it could be observed that due to the intensified thermal load,the potential energy in convex equilibrium position became deeper,resulting in greater probability of vibrating around convex equilibrium position,as shown in Fig.10(j).

Table 7 The first eight-order modal frequencies.

Table 6 Geometry parameters.

Then,with temperature increasing persistently,owing to the stronger thermal load,the potential energy well would be dramatically deeper,giving rise to the significantly increased mean and the reduced amplitude,and nonlinearly vibrate around one of the post buckling equilibrium positions,which is shown in Fig.10(k).What’s more,the PDF of lateral displacement gradually tended to be normal distribution with the subdued nonlinear response through the analysis of Fig.10(l).

The RMS of the lateral displacement in different thermalacoustic loading combinations is shown in Fig.11.Obviously,it is discerned that in pre-buckling region the RMS increased with the increase of thermal load,which is reasonably affected by the sound pressure level,and when it came to critical buckling,the nonlinear response RMS increased significantly.In post-buckling region,thermal load played the important role in the increased RMS of the lateral displacement,and on the contrary the RMS curves in different SPL were approximately coincident.

The change law of the variance of lateral displacement in different SPL is shown in Fig.12.Due to the intensified nonlinear response with the increase of thermal load,the fluctuation degree of lateral displacement amplitude increased with thermal load increasing in pre-buckling.Then the variance of lateral displacement reached to the maximum value with the severe dynamic response in critical buckling,and the fluctuation degree of lateral displacement amplitude increased significantly.In post-buckling,owing to the plate vibrating around one of the post-buckled equilibrium positions,the lateral displacement response verged on the steady state and the variance turned to reduce.Meanwhile,acoustic load became the main reason for the fluctuation degree of data.Besides,when the thermal load got to be fairly intensified,as S=16,acoustic excitation almost had not the in fluence on the amplitude of lateral displacement response.

Fig.13 gives the Power Spectral Density(PSD)of lateral displacement in pre-buckling and post-buckling.5Further analysis showed that in the pre-buckling region,the fundamental frequencies of the plate decreased with temperature increasing;in addition,the PSD showed the response peaks in the high frequencies and the peak values increased with the increase of thermal load,which came to the maximum value near the critical buckling,as shown in Fig.13(a).Obviously,in pre-buckling,considering the increase of thermal stress,the plate was into the softening region with the reduced stiffness,leading to severer vibration response.However,when compared with the pre-buckled response results,the significant difference was that the PSD experienced the response peaks at low frequencies,and with the increase of thermal load,the peak values decreased and the fundamental frequencies of the plate increased,as shown in Fig.13(b).It indicates that in post-buckling,owing to the intensified thermal stress,the plate was into the hardening region with the fortified stiffness,reducing vibration response.

4.2.Stress component

Fig.14 shows the time history of x stress component extracted in the center of the plate.The features could be obtained as follows:

(1)The vibration equilibrium position of the x stress component time history gradually deviated from the X axis.From Fig.14(a),we can see that,in critical buckling,the equilibrium position deviated to-100 MPa,and the higher the temperature was,the larger the offset was.Meanwhile,the tensile stress and compressive stress existed at the same time with the alternating effects on the plate,but the compressive stress was the key factor.

(2)When the snap-through motions of the plate occurred,on the one hand,the plate vibrated around the convex position,corresponding to the larger x stress component amplitude and the smaller mean;on the other hand,the plate vibrated around the concave position,and the conclusion was just opposite,as shown in Fig.14(b).What’s more,with the increase of temperature,the x stress component amplitude and mean corresponded to be smaller and larger respectively.For example,Fig.14(b)shows that as S=1.1 the maximum value of amplitude was around 140 MPa,and the mean value was around 10 MPa.Furthermore,as S=1.3 the maximum value of amplitude was around 100 MPa,and the mean value was around 90 MPa.

(3)Fig.14(d)indicates that when the thermal load was intensified enough,the plate vibrated around the concave equilibrium position,resulting in the significantly decreased amplitude and increased mean of the x stress component,and meanwhile,the plate mainly exhibited compressive stress.

Fig.15 shows that when the SPL kept constant,the RMS of x stress component presented the N glyph with the increase of temperature.And the curves can be divided into four parts:

(1)In the pre-buckling region,as S<1 owing to the simultaneous increase between amplitude and mean with the increase of temperature,the RMS of x stress component increased gradually.

(2)The RMS of x stress component persistently increased from critical buckling to end of snap-through,like the range as SPL=169 dB from S=1.0 to S=1.3 and as SPL=172 dB from S=1.0 to S=1.4.

(3)From the end of snap-through to completely tensile area,the plate would go through the compression area,resulting in the decreased RMS of x stress component.

(4)Once the plate got into the totally tensile area,the mean of x stress component would increase with the increase of thermal stress and played the main role in the plate,resulting in the increased RMS of x stress component.

5.Conclusion

(1)The clamped thin-walled metallic plate is selected to carry out dynamic response test in thermal-acoustic environment.Comparing the thermal modal frequencies between simulation and test,we can see that the results keep consistent and with the increase of temperature the first-order modal frequencies show the same trend of if rst increase in the pre-buckling region and then decrease in the post-buckling region.Then,this paper makes the comparison of strain response value,and the results indicate that the strain values of simulation have the pretty agreement with test for the same order of magnitude,verifying the effectiveness of calculation method and model for nonlinear dynamic response of thin-walled plate to thermal-acoustic load.

(2)This paper completes dynamic response calculation for the cross reinforcement plate under different thermalacoustic loading combinations.Through the analysis on lateral displacement,the following conclusions can be drawn.In the pre-buckling region,owing to intensified thermal stress,the fundamental frequencies gradually decrease in softening area,corresponding to the peaks with the opposite change law,and due to the Gaussian distribution of the response,the plate vibrates randomly around the initial equilibrium position.In the post-buckling region,the fundamental frequencies gradually increase in hardening area,and most importantly,the snap-through motions of the plate occur,which is decided by the relative strength of thermal load and acoustic load.In addition,considering the non-Gaussian distribution of the response,the plate with two potential energy wells exhibits the bimodal phenomenon.Furthermore,acoustic load and thermal load turn to be the main effect on the RMS of lateral displacement in pre-buckling and post-buckling regions respectively.

(3)When the plate vibrates two equilibrium positions,tensile stress and compressive stress alternately affect the plate,and the amplitude of x stress component becomes the primary impact on the plate.Once given the intensified thermal load,the plate will vibrate around one of two equilibrium positions,and the x stress component is into the completely tensile area with the increased mean and RMS.

(4)The thermal-acoustic tests for thin-walled structure need to be carried out to obtain enough experiment data,providing the reference to further research the post-buckled response characteristics of thin-walled structure.

Acknowledgements

This study was supported by Aviation Basic Science Fund Project of China(No.20151554002).

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6 September 2016;revised 20 December 2016;accepted 18 July 2017

Available online 16 October 2017

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*Corresponding author.

E-mail address:j_wang2001@sina.com(J.WANG).

Peer review under responsibility of Editorial Committee of CJA.