CN107870074A

CN107870074A - A kind of non-linear internal resonance characterization test method of fiber composite plate and test system

Info

Publication number: CN107870074A
Application number: CN201711075567.5A
Authority: CN
Inventors: 李晖; 梁晓龙; 韩清凯; 翟敬宇; 孙伟
Original assignee: Northeastern University China
Current assignee: Northeastern University China
Priority date: 2017-11-06
Filing date: 2017-11-06
Publication date: 2018-04-03
Anticipated expiration: 2037-11-06
Also published as: CN107870074B

Abstract

A kind of non-linear internal resonance characterization test method of fiber composite plate and test system, method of the present invention includes：The internal resonance intrinsic frequency of certain two rank of composite plate is chosen, the dimensional parameters and material parameter of composite plate formation internal resonance are predefined based on integral multiple compatibility method；Processing prepares composite plate；The internal resonance intrinsic frequency of certain two rank of test compound plate；Whether the internal resonance intrinsic frequency for judging certain two rank is integral multiple relation, if it is not, then the clip position of fine setting composite plate and again certain two rank internal resonance intrinsic frequency of test compound plate, make the internal resonance intrinsic frequency of certain two rank that integral multiple relation be presented；Carry out the internal resonance characterization test of composite plate under different excitation amplitude conditions and the vibration characteristic of its non-linear internal resonance is analyzed, conclude the changing rule of nonlinear vibration response.The method of testing of the present invention proposes the intrinsic frequency of integral multiple compatibility method theoretical calculation composite plate, and determines the design parameter of composite plate, and measuring accuracy and testing efficiency are higher.

Description

A kind of non-linear internal resonance characterization test method of fiber composite plate and test system

Technical field

The invention belongs to vibration test technology field, and in particular to a kind of non-linear internal resonance characterization test of fiber composite plate Method and test system.

Background technology

The damping vibration attenuation ability that fibre reinforced composites specific strength is high, specific modulus is high, heat endurance is good, also certain, Therefore it is widely used in engineering in practice, such as solar energy sailboard, aeroengine fan blades and large scale wind power machine blade, At present, engineering exists in practice largely passes through exemplary complex thin-slab structure part made of the type material.However, because fiber increases Strong composite is different from conventional material, and its performance has obvious anisotropy, and its macroscopical thin-slab structure part shows a system The unexistent dynamic phenomena of linear system, such as hysteresis is with jumping, sub-harmonic resona and ultraharmonic resonance, period doubling bifurcation are with mixing Ignorant motion, internal resonance, damping are particularly non-linear interior with the Non-Linear Vibration feature such as dynamic excitation frequency and the dependence of amplitude Covibration, this brings very big difficulty and challenge to traditional vibration experiment and analytical technology based on linear equivalence.

It is even non-linear to its in order to effectively suppress to the non-linear internal resonance phenomenon of composite sheet structural member Internal resonance is utilized, it is necessary to be studied and form a set of scientific and reasonable characterization test method effectively to obtain fiber composite thin plate Non-linear internal resonance phenomenon, and then grasp its vibration behavioral characteristic under different excitation amplitudes.

At present, people have carried out certain research in non-linear internal resonance field, and devise some and utilize internal resonance The invention device of phenomenon, but the condition of internal resonance is produced to it and mechanism is illustrating and inadequate clear and definite.Patent CN201410007932.9, patent CN201510008831.8, patent CN201210146634.9 are using non-linear interior respectively Energy collecting device device and the sound absorption structure invention of covibration, but they do not clearly state the dimensional parameters of these structures And material parameter, these parameters can influence the ratio of structural natural frequencies, and then it can determine that internal resonance phenomenon whether there is.Patent CN201210345735.9 is a kind of based on 2:The flexible mechanical arm dissipative damping device of 1 internal resonance, the device are adjusted using inertia The position of section ball makes mechanical arm configuration reach 1:2 internal resonances, but the structure does not consider whether adjustable range can make the structure Intrinsic frequency reach 1:2, and then internal resonance phenomenon is produced to realize the passive energy dissipation of mechanical arm.Patent The method for solving of dynamic response when CN201410330339.8 proposes roach life internal resonance shallow for elastic boundary, but it is closed The solution of internal resonance theoretical response has been noted, has not been verified in actual test；Patent CN 201110027107.1 proposes deep-sea Platform mooring system parametric excitation-internal resonance coupled vibration analysis program and vibration control technology, but it is according to platform internal resonance Coupled motions amplitude changes with time, and judges whether platform occurs ginseng and swash-internal resonance, does not clearly state platform heaving and indulges Mode is shaken into the formation mechenism of 2: 1 internal resonance relations.

From the point of view of the data that can currently grasp, people non-linear internal resonance using and analysis carried out certain grind Study carefully, but most of inventions do not illustrate the necessary condition of internal resonance formation, while the analysis and research of some internal resonances only focus on The solution of vibratory response, a set of scientific and reasonable characterization test method is not formed effectively to obtain non-linear internal resonance phenomenon. Therefore, it is necessary to continue to study the problem, particularly internal resonance measuring accuracy and testing efficiency etc. input how are being improved Bigger energy.

The content of the invention

The embodiment of the present invention provides a kind of non-linear internal resonance characterization test method of fiber composite plate and test system, proposes The intrinsic frequency of integral multiple compatibility method theoretical calculation composite plate, and determine the design parameter of composite plate, measuring accuracy and test It is more efficient.

The present invention provides a kind of non-linear internal resonance characterization test method of fiber composite plate, comprises the following steps：

Step 1：The internal resonance intrinsic frequency of certain two rank of fiber composite plate is chosen, is predefined based on integral multiple compatibility method Fiber composite plate forms the dimensional parameters and material parameter of internal resonance；

Step 2：Fiber composite plate is prepared according to the predetermined dimensional parameters of step 1 and material parameter processing；

Step 3：Build sweep check system and test the internal resonance intrinsic frequency of certain two rank of the fiber composite plate；

Step 4：Whether the internal resonance intrinsic frequency for judging certain two rank of the fiber composite plate is integral multiple relation, if It is then to perform step 6, if otherwise performing step 5；

Step 5：Certain the two rank internal resonance finely tuned the clip position of fiber composite plate and retest fiber composite plate are intrinsic Frequency, so that integral multiple relation is presented in the internal resonance intrinsic frequency of certain two rank；

Step 6：Carry out the internal resonance characterization test of fiber composite plate under different excitation amplitude conditions and fiber is answered The vibration velocity response signal of plywood is analyzed and processed, and then the vibration characteristic of its non-linear internal resonance is analyzed, and is returned Receive the changing rule of nonlinear vibration response.

The present invention also provides a kind of fiber composite plate non-linear internal resonance characterization test system, including：Control machine, power are put Big device, electromagnetic exciter, laser doppler vibrometer, data acquisition equipment and computer；

Control machine, for exporting sinusoidal signal；

Power amplifier, for the sinusoidal signal amplification for exporting control machine；

Electromagnetic exciter, for receiving the sinusoidal signal after amplifying, to produce sinusoidal excitation, and in a manner of basic excitation Act in composite plate；

Laser doppler vibrometer and data acquisition equipment, for gathering pumping signal and vibration velocity response signal；

Computer, for setting the basic parameter needed for internal resonance characterization test, the vibration velocity sound to fiber composite plate Induction signal is analyzed and processed.

A kind of non-linear internal resonance characterization test method of fiber composite plate and test system of the present invention at least has following Beneficial effect：

(1) intrinsic frequency of integral multiple compatibility method theoretical calculation fiber composite plate proposed by the present invention, and determine that fiber is answered The design parameter of plywood, make the ratio between its internal resonance intrinsic frequency close to 1:2 or 1:3, the necessary condition that internal resonance is formed can be met.

(2) the fine tuning structure parameter in test process, it can make the intrinsic frequency of fiber composite plate that integer accurately be presented Times relation, and then realize the formation of internal resonance phenomenon.

(3) it is non-can to effectively obtain under different excitation amplitudes its for the fiber composite plate internal resonance characterization test step used Linear internal resonance phenomenon.

(4) test system used is easy to build, and method of testing step is succinctly clear and definite, and non-linear internal resonance phenomenon is obvious, Favorable repeatability.

Brief description of the drawings

Fig. 1 is the flow chart of the non-linear internal resonance characterization test method of fiber composite plate of the present invention；

The theoretical model of the fiber composite plate of Fig. 2 present invention；

Fig. 3 is the three-dimensional relationship scattergram of composite sheet preceding two ranks intrinsic frequency ratio under different length and width sizes；

Fig. 4 is the structured flowchart of the non-linear internal resonance characterization test system of fiber composite plate of the present invention；

Fig. 5 a are the time domain responses that test obtains when encouraging amplitude 0.3g and driving frequency 50.5Hz；

Fig. 5 b are the frequency domain responses that test obtains when encouraging amplitude 0.3g and driving frequency 50.5Hz；

Fig. 5 c are the phasor tracks that test obtains when encouraging amplitude 0.3g and driving frequency 50.5Hz；

Fig. 6 a are the time domain responses that test obtains when encouraging amplitude 0.6g and driving frequency 50.5Hz；

Fig. 6 b are the frequency domain responses that test obtains when encouraging amplitude 0.6g and driving frequency 50.5Hz；

Fig. 6 c are the phasor tracks that test obtains when encouraging amplitude 0.6g and driving frequency 50.5Hz；

Fig. 7 a are the time domain responses that test obtains when encouraging amplitude 1.2g and driving frequency 50.5Hz；

Fig. 7 b are the frequency domain responses that test obtains when encouraging amplitude 1.2g and driving frequency 50.5Hz；

Fig. 7 c are the phasor tracks that test obtains when encouraging amplitude 1.2g and driving frequency 50.5Hz；

Fig. 8 a are the time domain responses that test obtains when encouraging amplitude 0.3g and driving frequency 103.0Hz；

Fig. 8 b are the frequency domain responses that test obtains when encouraging amplitude 0.3g and driving frequency 103.0Hz；

Fig. 8 c are the phasor tracks that test obtains when encouraging amplitude 0.3g and driving frequency 103.0Hz；

Fig. 9 a are the time domain responses that test obtains when encouraging amplitude 0.6g and driving frequency 103.0Hz；

Fig. 9 b are the frequency domain responses that test obtains when encouraging amplitude 0.6g and driving frequency 103.0Hz；

Fig. 9 c are the phasor tracks that test obtains when encouraging amplitude 0.6g and driving frequency 103.0Hz；

Figure 10 a are the time domain responses that test obtains when encouraging amplitude 2g and driving frequency 103.0Hz；

Figure 10 b are the frequency domain responses that test obtains when encouraging amplitude 2g and driving frequency 103.0Hz；

Figure 10 c are the phasor tracks that test obtains when encouraging amplitude 2g and driving frequency 103.0Hz；

Figure 11 a are the time domain responses that test obtains when encouraging amplitude 6g and driving frequency 103.0Hz；

Figure 11 b are the frequency domain responses that test obtains when encouraging amplitude 6g and driving frequency 103.0Hz；

Figure 11 c are the phasor tracks that test obtains when encouraging amplitude 6g and driving frequency 103.0Hz；

Figure 12 a are the time domain responses that test obtains when encouraging amplitude 10g and driving frequency 103.0Hz；

Figure 12 b are the frequency domain responses that test obtains when encouraging amplitude 10g and driving frequency 103.0Hz；

Figure 12 c are the phasor tracks that test obtains when encouraging amplitude 10g and driving frequency 103.0Hz.

Embodiment

The non-linear internal resonance characterization test method of fiber composite plate of the present invention as shown in Figure 1, comprises the following steps：

Step 1：The internal resonance intrinsic frequency of certain two rank of fiber composite plate is chosen, is predefined based on integral multiple compatibility method Fiber composite plate forms the dimensional parameters and material parameter of internal resonance；Specifically, step 1 includes：

Step 1.1：The internal resonance intrinsic frequency of fiber composite plate is calculated based on integral multiple compatibility method, comprising The explicit expression of the dimensional parameters of fiber composite plate and the internal resonance intrinsic frequency of material parameter；Wherein, dimensional parameters include Long, wide, the macro-size such as thickness, material parameter includes the parameters such as modulus of elasticity, modulus of shearing, Poisson's ratio and density.

Step 1.2：According to the size range that can be clamped under Machinability Evaluation and experiment condition, dimensional parameters and material are determined Parameter, make certain two rank intrinsic frequency f of internal resonance₁And f₂It is presented 1:2、1:The integral multiple relations such as 3, and then meet fiber composite plate Form the necessary condition of internal resonance.

It is final to determine fiber composite thin plate with reference to manufacturer production condition, and on the premise of reliable technological requirement is ensured Each layer laying parameter, and prepare the macroscopical plate for meeting internal resonance test request.

Step 3：Test system building and the internal resonance intrinsic frequency for testing certain two rank of the fiber composite plate；Specifically Ground, step 3 include：

Step 3.1：Build connecting test system and determine to test required restrained boundary condition；

The boundary condition for determining the non-linear internal resonance characterization test of fiber composite plate is cantilever, and uses torque Spanner tightens two M12 bolts on fixture with moment values 50Nm.The arrangement of point position will avoid each rank of cantilever composite plate At node-line.

Step 3.2：Basic parameter needed for sweep check is set, including：The sensitivity of laser doppler vialog, sampling frequency Rate, frequency resolution, the signal type of signal generator；

When it is implemented, the sensitivity for setting laser doppler vialog is 8000mv/ (m/s)；It is of interest according to testing Bandwidth is analyzed, it is 3200Hz to select sample frequency；Frequency resolution is 0.25Hz；Signal type is sine sweep signal.

Step 3.3：The internal resonance intrinsic frequency of certain two rank of composite plate is chosen, in a manner of basic excitation, respectively to quilt Survey composite plate and carry out tentering swept frequency excitation, frequency sweep section can select the 75%~125% of certain rank intrinsic frequency, in this frequency range Sweep check (typically smaller than 1Hz/s) is carried out can effectively eliminate the sweep velocity of transient oscillation, is surveyed using laser doppler Vibration Meter carry out response signal test, and to time domain initial data carry out at times FFT handle, by recognize peak value of response come The accurate each rank internal resonance intrinsic frequency for obtaining composite plate.

The intrinsic frequency surveyed out may have minute differences with the calculated results, at this time, it may be necessary to which discriminating test obtains Composite sheet internal resonance intrinsic frequency f₁And f₂Whether integral multiple relation is met, if f₁And f₂Integral multiple relation is unsatisfactory for, then Need to carry out the fine setting of structural parameters, composite sheet is clamped again using fixture, and repeat step 3 tests its intrinsic frequency, Eventually through the method constantly coordinated, ensure f₁And f₂Numerical value be in 1:2 or 1:The integral multiple relations such as 3.

The step 1.1 obtains the aobvious of the internal resonance intrinsic frequency of the dimensional parameters comprising fiber composite plate and material parameter Formula expression formula specifically includes：

Step 1.1.1：Be illustrated in figure 2 the theoretical model of fiber composite plate, fiber composite plate be had by n-layer it is orthogonal each What the fiber and matrix material combinations of anisotropy feature formed, its dimensional parameters length a, width b and thickness h, each layer of thickness All same is spent, sets the intermediate layer of fiber composite plate as a reference plane, using fiber composite plate length direction as x-axis direction, Width establishes xoy coordinate systems as y-axis direction, and 1 in figure represents fiber longitudinal direction, and 2 represent fiber transverse direction, 3 represent it is vertical In the direction of 1-2 planes；If the machine direction of fiber composite plate and the angle in x-axis direction are θ, E₁Represent along fiber longitudinal elasticity Modulus, E₂Represent along fiber transverse modulus of elasticity, G₁₂Represent modulus of shearing, ν₁₂Represent that fiber caused by the stress of fiber longitudinal direction is indulged To the Poisson's ratio with transverse strain, ν₂₁The Poisson's ratio of fiber vertical and horizontal strain caused by the horizontal stress of fiber is represented, ρ is Density；

Step 1.1.2：Calculate the strain-stress relation of material；

Step 1.1.3：It is horizontal according to face in the kinetic energy of thin plate transverse free vibration, strain energy, thin plate institute's bending moment and moment of torsion, thin plate To vibration displacement and the Hamilton's principle of elastodynamics, thin plate vibration shape variation equation is calculated；

Step 1.1.4：Set the model function of vibration of cantilever composite sheet；

Step 1.1.5：According to the model function of vibration of thin plate vibration shape variation equation and cantilever composite sheet, can obtain on intrinsic The algebraic equation of circular frequency, directly solves inherent circular frequency, obtains the dimensional parameters comprising fiber composite plate and material parameter The explicit expression of internal resonance intrinsic frequency.

In present embodiment, the fiber composite thin plate tested is prepared from TC300 carbon fibers/resin-based materials, Producer provide prepare thickness can in the range of 2~3mm unrestricted choice.On the premise of stable processing technology is ensured, selection Orthogonal Symmetric laying form, shares 11 layers (thickness 2.36mm), and specific parameter of laying is [(90 °/0 °)₂/90°/0°/90° (0°/90°)₂], its fiber longitudinal modulus of elasticity E₁=119GPa, fiber transverse modulus of elasticity E₂=8.7GPa, shear modulus G₁₂ =4GPa, Poisson's ratio ν₁₂=0.23, density p=1618kg/m3, each laying has identical thickness and fiber volume fraction. Then, it is solid using before the composite sheet under the different length and width sizes of integral multiple compatibility method calculating two under cantilever border of interest There is frequency f₁And f₂Three-dimensional relationship scattergram, as shown in figure 3, red solid dot is meets 1 in figure:2 internal resonance test requests Length and width result.Finally, the mould and the size of laboratory holder provided with reference to producer, what is finally prepared meets The length and width of the composite sheet of internal resonance test request, thick size are 310 × 160 × 2.36mm, in order to ensure cantilever marginal testing Effect, 30mm is reserved to be clamped by fixture in length direction, and fixture is tightened with moment values 50Nm using torque spanner On two M12 bolts.Table 1 gives the type composite sheet internal resonance intrinsic frequency f for calculating and obtaining₁And f₂Corresponding knot Fruit.

Table 1 is the composite sheet internal resonance intrinsic frequency (Hz) that theoretical calculation obtains.

Internal resonant frequency	f₁	f₂	f₁:f₂
				Numerical value (Hz)	51.5	100.2	1:1.95

Specifically, the strain-stress relation of the step 1.1.2 calculating material is：

It is theoretical according to Thin plate under small deflection, displacement field is write into following form：

W (x, y, z, t)=w₀(x,y,t)(1c)

Wherein, u, v, w represent the displacement at any point in plate；w₀Represent face displacement in plate；T represents the time；

It can be seen from plate theory, normal strain ε_zWith shearing strain γ_yz、γ_xzAll be 0, i.e. ε_z=γ_yz=γ_xz=0, by answering Become and the relation of displacement, in plate the strain at any point can be expressed as：

For orthotropic material, the strain-stress relation of material major axes orientation is：

σ₁Represent the direct stress along fiber longitudinal direction, σ₂Represent along the horizontal direct stress of fiber, σ₁₂Expression is put down along fleece-laying The shear stress in face；ε₁Represent the normal strain along fiber longitudinal direction, ε₂Represent along the horizontal normal strain of fiber, γ₁₂Represent along fiber lay down If the shearing strain of plane；

Wherein, off-axis stiffness coefficient Q₁₁、Q₁₂、Q₂₂、Q₆₆It is as follows：

Q₆₆=G₁₂ (4d)

When having certain angle theta between material major axes orientation and global coordinate system, calculated with stress-strain rotation axis formula It is as follows to strain-stress relation of the kth laminate under global coordinate system：

Wherein, off-axis stiffness coefficientIt is as follows：

In formula, k represents the kth layer of composite sheet, θ_kRepresent the machine direction of kth laminate and the folder of global coordinate system x-axis Angle.

Specifically, the step 1.1.3 calculating thin plate vibration shape variation equation is：

The kinetic energy of thin plate transverse free vibration is expressed as：

Strain energy is expressed as：

Wherein,

Moment M suffered by thin plate_x,M_yWith moment of torsion M_xyFor：

Wherein,

Formula (9) is substituted into formula (8), abbreviation obtains：

According to the Hamilton's principle of elastodynamics, Free Vibration of Thin Plate variation equation is expressed as：

Face transverse vibrational displacement can be set in thin plate：

w₀(x, y, t)=W (x, y) sin (ω t) (13)

Wherein, W (x, y) is model function of vibration；

The thin plate vibration shape variation equation formula of above-mentioned formula (14) can be obtained by formula (7) (11) (12) (13).

Specifically, the model function of vibration of the step 1.1.4 solutions cantilever composite sheet is：

, can be by cantilever THIN COMPOSITE below based on two-dimension beam function combined method due to being concerned with cantilever herein The model function of vibration of plate is set to：

W_mn(x, y)=A_mnX_m(x)Y_n(y) (15)

Wherein, A_mnFor undetermined coefficient, X_m(x) it is fixation-free beam m first order mode functions, is represented by：

In formula：λ_x1=1.875, λ_x2=4.694, λ_x3=7.854, σ_x1=0.7341, σ_x2=1.0185, σ_x3= 0.9992；

Y_n(y) it is freedom-free beam n-th order model function of vibration, is represented by：

Y₁(y)=1 (19a)

Y₂(y)=1-2y/b (19b)

In formula：λ_y3=4.730, σ_y3=0.9825；

M, n represent that the vibration shape along x, half wave number in y directions, can use different positive integers, represent the different orders of the vibration shape respectively.

Specifically, the step 1.1.5 obtains the dimensional parameters comprising fiber composite plate and the internal resonance of material parameter is consolidated The explicit expression for having frequency is：

Model function of vibration (15) is substituted into vibration shape variation equation (14), can be obtained on inherent circular frequency ω_mnAlgebraically side Journey, directly solve inherent circular frequency：

Wherein：

Specifically, step 6 concludes the changing rule of nonlinear vibration response, including：

Step 6.1：On the basis of edge-restraint condition is not changed, set needed for fiber composite plate internal resonance characterization test Basic parameter, including：Electromagnetic exciter determines frequency frequency values, the sensitivity of laser doppler vibrometer, sample frequency, frequency point Resolution, response signal add hanning windows to handle；

The sensitivity for setting laser doppler vialog is 8000mv/ (m/s)；According to analysis bandwidth of interest is tested, select It is 3200Hz to select sample frequency；Frequency resolution is 0.25Hz；Signal type is sinusoidal frequency-fixed signal.

Step 6.2：Changing in controller encourages the size of amplitude to increase successively from 0.3g~10g, in different excitation width Under degree, the internal resonance intrinsic frequency f of certain two rank is utilized respectively₁And f₂, in a manner of determining frequency excitation, excite composite sheet to produce altogether Shake；

Embodiment is：First under tri- excitation amplitudes of 0.3g, 0.6g, 1.2g (the first rank resonance amplitude is larger, Controller maximum can only be controlled in 1.2g or so excitation amplitude) utilize shake table with first natural frequency 50.5Hz to fiber Composite sheet enters row energization, excites composite sheet to produce resonance；Wherein, Fig. 5 a are when encouraging amplitude 0.3g and driving frequency 50.5Hz Test the time domain response obtained；Fig. 5 b are the frequency domain responses that test obtains when encouraging amplitude 0.3g and driving frequency 50.5Hz；Fig. 5 c It is the phasor track that test obtains when encouraging amplitude 0.3g and driving frequency 50.5Hz；Fig. 6 a are to encourage amplitude 0.6g and driving frequency The time domain response that test obtains during 50.5Hz；Fig. 6 b are that the frequency domain that test obtains when encouraging amplitude 0.6g and driving frequency 50.5Hz rings Should；Fig. 6 c are the phasor tracks that test obtains when encouraging amplitude 0.6g and driving frequency 50.5Hz；Fig. 7 a are to encourage amplitude 1.2g and swash The time domain response that test obtains when encouraging frequency 50.5Hz；Test obtains when Fig. 7 b encourage amplitude 1.2g and driving frequency 50.5Hz Frequency domain response；Fig. 7 c are the phasor tracks that test obtains when encouraging amplitude 1.2g and driving frequency 50.5Hz.

Then, when selection excitation amplitude is respectively 0.3g, 0.6g, 2g, 6g, 10g, using shake table with the intrinsic frequency of second-order Rate 103Hz enters row energization to composite sheet.Wherein, test obtains when Fig. 8 a encourage amplitude 0.3g and driving frequency 103.0Hz Time domain response；Fig. 8 b are the frequency domain responses that test obtains when encouraging amplitude 0.3g and driving frequency 103.0Hz；Fig. 8 c are the amplitudes of encouraging 0.3g and the phasor track that test obtains during driving frequency 103.0Hz；Fig. 9 a are when encouraging amplitude 0.6g and driving frequency 103.0Hz Test the time domain response obtained；Fig. 9 b are the frequency domain responses that test obtains when encouraging amplitude 0.6g and driving frequency 103.0Hz；Fig. 9 c It is the phasor track that test obtains when encouraging amplitude 0.6g and driving frequency 103.0Hz；Figure 10 a are to encourage amplitude 2g and driving frequency The time domain response that test obtains during 103.0Hz；Figure 10 b are the frequency domains that test obtains when encouraging amplitude 2g and driving frequency 103.0Hz Response；Figure 10 c are the phasor tracks that test obtains when encouraging amplitude 2g and driving frequency 103.0Hz；Figure 11 a are to encourage amplitude 6g and swash The time domain response that test obtains when encouraging frequency 103.0Hz；Figure 11 b are to test to obtain when encouraging amplitude 6g and driving frequency 103.0Hz Frequency domain response；Figure 11 c are the phasor tracks that test obtains when encouraging amplitude 6g and driving frequency 103.0Hz；Figure 12 a are the amplitudes of encouraging 10g and the time domain response that test obtains during driving frequency 103.0Hz；Figure 12 b are when encouraging amplitude 10g and driving frequency 103.0Hz Test the frequency domain response obtained；Figure 12 c are the phasor tracks that test obtains when encouraging amplitude 10g and driving frequency 103.0Hz.

Step 6.3：After structural vibration response is stable, pass through data acquisition equipment record pumping signal now and vibration Speed responsive signal, and in Real Time Observation spectrogram certain two rank internal resonance intrinsic frequency f₁And f₂Can mutually it eject, and In the presence of two basic frequencies into multiple proportion；

Step 6.4：Extraction step 6.3 tests recorded vibration velocity response signal, by Matlab softwares by speed Shifted version is integrated into, and is drawn using vibration displacement as abscissa, vibration velocity is the phasor track of ordinate；

Step 6.5：Phasor track of the binding fiber composite plate under different excitation amplitudes, when spectrogram and spectrogram sentence Its disconnected non-linear internal resonance phenomenon, and conclude the changing rule of its nonlinear vibration response.

As shown in figure 4, a kind of non-linear internal resonance characterization test system of fiber composite plate of the present invention, including：Control machine 1st, power amplifier 2, electromagnetic exciter 3, fiber composite plate 4, laser doppler vibrometer 5, data acquisition equipment 6 and calculating Machine 7.

Control machine 1 is used to export sinusoidal signal；Power amplifier 2 is used to amplify the sinusoidal signal of control machine output；Electricity Magnet exciter 3 is used to receive the sinusoidal signal after amplification, to produce sinusoidal excitation, and fiber is acted in a manner of basic excitation In composite plate 4；Laser doppler vibrometer 5 and data acquisition equipment 6 are used to gather pumping signal and vibration velocity response signal； Computer 7 is used to set basic parameter needed for internal resonance characterization test, the vibration velocity response signal to fiber composite plate to enter Row analyzing and processing.

The input of the output end connection power amplifier 2 of control machine 1, the output end connection electromagnetism of power amplifier 2 swash Shake device 3, is acted on by realizing sinusoidal excitation signal in the form of basic excitation on fiber composite plate 4；Laser doppler vibrometer 5 Output end connection data acquisition equipment 6 input, data acquisition equipment 6 output end connection computer 7 input, use To realize the collection of the vibratory response of fiber composite plate 4 and analyzing and processing.

Presently preferred embodiments of the present invention is the foregoing is only, the thought being not intended to limit the invention is all the present invention's Within spirit and principle, any modification, equivalent substitution and improvements made etc., it should be included in the scope of the protection.

Claims

A kind of 1. non-linear internal resonance characterization test method of fiber composite plate, it is characterised in that comprise the following steps：

Step 1：The internal resonance intrinsic frequency of certain two rank of fiber composite plate is chosen, fiber is predefined based on integral multiple compatibility method Composite plate forms the dimensional parameters and material parameter of internal resonance；

Step 2：Fiber composite plate is prepared according to the predetermined dimensional parameters of step 1 and material parameter processing；

Step 3：Build sweep check system and test the internal resonance intrinsic frequency of certain two rank of the fiber composite plate；

Step 4：Whether the internal resonance intrinsic frequency for judging certain two rank of the fiber composite plate is integral multiple relation, if it is Step 6 is performed, if otherwise performing step 5；

Step 5：Finely tune the clip position of fiber composite plate and retest certain two rank internal resonance intrinsic frequency of fiber composite plate, So that integral multiple relation is presented in the internal resonance intrinsic frequency of certain two rank；

Step 6：Carry out the internal resonance characterization test of fiber composite plate under different excitation amplitude conditions and to fiber composite plate Vibration velocity response signal analyzed and processed, and then the vibration characteristic of its non-linear internal resonance is analyzed, concluded non- The changing rule of linear oscillator response.
2. the non-linear internal resonance characterization test method of fiber composite plate as claimed in claim 1, it is characterised in that the step 1 includes：

Step 1.1：The internal resonance intrinsic frequency of fiber composite plate is calculated based on integral multiple compatibility method, acquisition includes fiber The explicit expression of the dimensional parameters of composite plate and the internal resonance intrinsic frequency of material parameter；

Step 1.2：According to the size range that can be clamped under Machinability Evaluation and experiment condition, dimensional parameters and material ginseng are determined Number, make certain two rank intrinsic frequency of internal resonance that integral multiple relation be presented, and then meet that fiber composite plate forms necessity of internal resonance Condition.
3. the non-linear internal resonance characterization test method of fiber composite plate as claimed in claim 2, it is characterised in that the step 1.1 including：

Step 1.1.1：Fiber composite plate is that the fiber that has orthotropy feature by n-layer and matrix material combinations form , its dimensional parameters length a, width b and thickness h, each layer of thickness all same, set the intermediate layer conduct of fiber composite plate Reference planes, using fiber composite plate length direction as x-axis direction, width establishes xoy coordinate systems as y-axis direction, if The machine direction of fiber composite plate and the angle in x-axis direction are θ, E₁Represent along fiber longitudinal modulus of elasticity, E₂Represent horizontal along fiber To modulus of elasticity, G₁₂Represent modulus of shearing, ν₁₂Represent the Poisson of fiber vertical and horizontal strain caused by the stress of fiber longitudinal direction Than ν₂₁The Poisson's ratio of fiber vertical and horizontal strain caused by the horizontal stress of fiber is represented, ρ is density；

Step 1.1.2：Calculate the strain-stress relation of material；

Step 1.1.3：According to face transverse-vibration in the kinetic energy of thin plate transverse free vibration, strain energy, thin plate institute's bending moment and moment of torsion, thin plate Dynamic displacement and the Hamilton's principle of elastodynamics, calculate thin plate vibration shape variation equation；

Step 1.1.4：Set the model function of vibration of cantilever composite sheet；

Step 1.1.5：According to the model function of vibration of thin plate vibration shape variation equation and cantilever composite sheet, can obtain on intrinsic circle frequency The algebraic equation of rate, directly solves inherent circular frequency, obtains the interior common of the dimensional parameters comprising fiber composite plate and material parameter The explicit expression for intrinsic frequency of shaking.
4. the non-linear internal resonance characterization test method of fiber composite plate as claimed in claim 3, it is characterised in that the step 1.1.2 it is specially：

It is theoretical according to Thin plate under small deflection, displacement field is write into following form：

<mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mi>z</mi> <mfrac> <mrow> <mo>&part;</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&part;</mo> <mi>x</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mi>a</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mi>z</mi> <mfrac> <mrow> <mo>&part;</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&part;</mo> <mi>y</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mi>b</mi> <mo>)</mo> </mrow> </mrow>

W (x, y, z, t)=w₀(x, y, t) (1c) wherein, u, v, w represent the displacement at any point in plate；w₀Represent face position in plate Move；T represents the time；

It can be seen from plate theory, normal strain ε_zWith shearing strain γ_yz、γ_xzAll be 0, i.e. ε_z=γ_yz=γ_xz=0, by strain and The relation of displacement, in plate the strain at any point can be expressed as：

<mrow> <msub> <mi>&epsiv;</mi> <mi>x</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&part;</mo> <mi>u</mi> </mrow> <mrow> <mo>&part;</mo> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <mi>z</mi> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>a</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>&epsiv;</mi> <mi>y</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&part;</mo> <mi>v</mi> </mrow> <mrow> <mo>&part;</mo> <mi>y</mi> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <mi>z</mi> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>b</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>&gamma;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>&part;</mo> <mi>u</mi> </mrow> <mrow> <mo>&part;</mo> <mi>y</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>&part;</mo> <mi>v</mi> </mrow> <mrow> <mo>&part;</mo> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <mn>2</mn> <mi>z</mi> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>&part;</mo> <mi>x</mi> <mo>&part;</mo> <mi>y</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>c</mi> <mo>)</mo> </mrow> </mrow>

For orthotropic material, the strain-stress relation of material major axes orientation is：

<mrow> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>&sigma;</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&sigma;</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&sigma;</mi> <mn>12</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>Q</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>Q</mi> <mn>12</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Q</mi> <mn>21</mn> </msub> </mtd> <mtd> <msub> <mi>Q</mi> <mn>22</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>Q</mi> <mn>66</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>&epsiv;</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&epsiv;</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&gamma;</mi> <mn>12</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

σ₁Represent the direct stress along fiber longitudinal direction, σ₂Represent along the horizontal direct stress of fiber, σ₁₂Represent cutting along fleece-laying plane Stress；ε₁Represent the normal strain along fiber longitudinal direction, ε₂Represent along the horizontal normal strain of fiber, γ₁₂Represent along fleece-laying plane Shearing strain；

Wherein, off-axis stiffness coefficient Q₁₁、Q₁₂、Q₂₂、Q₆₆It is as follows：

<mrow> <msub> <mi>Q</mi> <mn>11</mn> </msub> <mo>=</mo> <mfrac> <msub> <mi>E</mi> <mn>1</mn> </msub> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>v</mi> <mn>12</mn> </msub> <msub> <mi>v</mi> <mn>21</mn> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mi>a</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>Q</mi> <mn>12</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>v</mi> <mn>12</mn> </msub> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>v</mi> <mn>12</mn> </msub> <msub> <mi>v</mi> <mn>21</mn> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mi>b</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>Q</mi> <mn>22</mn> </msub> <mo>=</mo> <mfrac> <msub> <mi>E</mi> <mn>2</mn> </msub> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>v</mi> <mn>12</mn> </msub> <msub> <mi>v</mi> <mn>21</mn> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mi>c</mi> <mo>)</mo> </mrow> </mrow>

Q₆₆=G₁₂ (4d)

<mrow> <msub> <mi>v</mi> <mn>21</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>12</mn> </msub> <mfrac> <msub> <mi>E</mi> <mn>2</mn> </msub> <msub> <mi>E</mi> <mn>1</mn> </msub> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mi>e</mi> <mo>)</mo> </mrow> </mrow>

When having certain angle theta between material major axes orientation and global coordinate system, is calculated with stress-strain rotation axis formula Strain-stress relation of the k laminates under global coordinate system is as follows：

<mrow> <msup> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>&sigma;</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&sigma;</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&sigma;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>12</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>16</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>12</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>22</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>26</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>16</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>26</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>66</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>&epsiv;</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&epsiv;</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&gamma;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

Wherein, off-axis stiffness coefficientIt is as follows：

<mrow> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>11</mn> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mn>11</mn> </msub> <msup> <mi>cos</mi> <mn>4</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>12</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>Q</mi> <mn>66</mn> </msub> <mo>)</mo> </mrow> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mn>22</mn> </msub> <msup> <mi>sin</mi> <mn>4</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mi>a</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>12</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>11</mn> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mn>22</mn> </msub> <mo>-</mo> <mn>4</mn> <msub> <mi>Q</mi> <mn>66</mn> </msub> <mo>)</mo> </mrow> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>sin</mi> <mn>4</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>+</mo> <msup> <mi>cos</mi> <mn>4</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mi>b</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>22</mn> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mn>11</mn> </msub> <msup> <mi>sin</mi> <mn>4</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>12</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>Q</mi> <mn>66</mn> </msub> <mo>)</mo> </mrow> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mn>22</mn> </msub> <msup> <mi>cos</mi> <mn>4</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mi>c</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>16</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>11</mn> </msub> <mo>-</mo> <msub> <mi>Q</mi> <mn>12</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>Q</mi> <mn>66</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>sin&theta;</mi> <mi>k</mi> </msub> <msup> <mi>cos</mi> <mn>3</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>12</mn> </msub> <mo>-</mo> <msub> <mi>Q</mi> <mn>22</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>Q</mi> <mn>66</mn> </msub> <mo>)</mo> </mrow> <msup> <mi>sin</mi> <mn>3</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <msub> <mi>cos&theta;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mi>d</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>26</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>11</mn> </msub> <mo>-</mo> <msub> <mi>Q</mi> <mn>12</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>Q</mi> <mn>66</mn> </msub> <mo>)</mo> </mrow> <msup> <mi>sin</mi> <mn>3</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <msub> <mi>cos&theta;</mi> <mi>k</mi> </msub> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>12</mn> </msub> <mo>-</mo> <msub> <mi>Q</mi> <mn>22</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>Q</mi> <mn>66</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>sin&theta;</mi> <mi>k</mi> </msub> <msup> <mi>cos</mi> <mn>3</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mi>e</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>66</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>11</mn> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mn>22</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>Q</mi> <mn>12</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>Q</mi> <mn>66</mn> </msub> <mo>)</mo> </mrow> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mn>66</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>sin</mi> <mn>4</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>+</mo> <msup> <mi>cos</mi> <mn>4</mn> </msup> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mi>f</mi> <mo>)</mo> </mrow> </mrow>

In formula, k represents the kth layer of composite sheet, θ_kRepresent the machine direction of kth laminate and the angle of global coordinate system x-axis.
5. the non-linear internal resonance characterization test method of fiber composite plate as claimed in claim 3, it is characterised in that the step 1.1.3 it is specially：

The kinetic energy of thin plate transverse free vibration is expressed as：

<mrow> <mi>T</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>&rho;</mi> <mi>h</mi> <msub> <mo>&Integral;</mo> <mi>A</mi> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&part;</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>&part;</mo> <mi>t</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>A</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

Strain energy is expressed as：

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>U</mi> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mo>&Integral;</mo> <mi>A</mi> </msub> <msubsup> <mo>&Integral;</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mi>k</mi> </msub> </msubsup> <mrow> <mo>(</mo> <msup> <msub> <mi>&sigma;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <msub> <mi>&epsiv;</mi> <mi>x</mi> </msub> <mo>+</mo> <msup> <msub> <mi>&sigma;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <msub> <mi>&epsiv;</mi> <mi>y</mi> </msub> <mo>+</mo> <msup> <msub> <mi>&sigma;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <msub> <mi>&gamma;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>d</mi> <mi>A</mi> <mi>d</mi> <mi>z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mo>&Integral;</mo> <mi>A</mi> </msub> <msubsup> <mo>&Integral;</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mi>k</mi> </msub> </msubsup> <mrow> <mo>(</mo> <msup> <msub> <mi>&sigma;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <msub> <mi>z&kappa;</mi> <mi>x</mi> </msub> <mo>+</mo> <msup> <msub> <mi>&sigma;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <msub> <mi>z&kappa;</mi> <mi>y</mi> </msub> <mo>+</mo> <msup> <msub> <mi>&sigma;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <msub> <mi>z&kappa;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>d</mi> <mi>A</mi> <mi>d</mi> <mi>z</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

Wherein,

Moment M suffered by thin plate_x,M_yWith moment of torsion M_xyFor：

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>M</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>M</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mo>&Integral;</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mi>k</mi> </msub> </msubsup> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>&sigma;</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&sigma;</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&sigma;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <mi>z</mi> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mo>&Integral;</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mi>k</mi> </msub> </msubsup> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>12</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>16</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>12</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>22</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>26</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>16</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>26</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>Q</mi> <mo>&OverBar;</mo> </mover> <mn>66</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>z&kappa;</mi> <mi>x</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z&kappa;</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z&kappa;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>z</mi> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>D</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>D</mi> <mn>12</mn> </msub> </mtd> <mtd> <msub> <mi>D</mi> <mn>16</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>D</mi> <mn>12</mn> </msub> </mtd> <mtd> <msub> <mi>D</mi> <mn>22</mn> </msub> </mtd> <mtd> <msub> <mi>D</mi> <mn>26</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>D</mi> <mn>16</mn> </msub> </mtd> <mtd> <msub> <mi>D</mi> <mn>26</mn> </msub> </mtd> <mtd> <msub> <mi>D</mi> <mn>66</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>&kappa;</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&kappa;</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&kappa;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>

Wherein,

Formula (9) is substituted into formula (8), abbreviation obtains：

<mrow> <mi>U</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mo>&Integral;</mo> <mi>A</mi> </msub> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mo>(</mo> <msub> <mi>D</mi> <mn>11</mn> </msub> <msub> <mi>&kappa;</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>12</mn> </msub> <msub> <mi>&kappa;</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>16</mn> </msub> <msub> <mi>&kappa;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>)</mo> <msub> <mi>&kappa;</mi> <mi>x</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>D</mi> <mn>12</mn> </msub> <msub> <mi>&kappa;</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>22</mn> </msub> <msub> <mi>&kappa;</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>26</mn> </msub> <msub> <mi>&kappa;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>&kappa;</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>D</mi> <mn>16</mn> </msub> <msub> <mi>&kappa;</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>26</mn> </msub> <msub> <mi>&kappa;</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>66</mn> </msub> <msub> <mi>&kappa;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>&kappa;</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>d</mi> <mi>A</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>

According to the Hamilton's principle of elastodynamics, Free Vibration of Thin Plate variation equation is expressed as：

<mrow> <mi>&delta;</mi> <msubsup> <mo>&Integral;</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <msub> <mi>t</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <mi>U</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>

Face transverse vibrational displacement can be set in thin plate：

w₀(x, y, t)=W (x, y) sin (ω t) (13)

Wherein, W (x, y) is model function of vibration；

<mrow> <mi>&delta;</mi> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>a</mi> </msubsup> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <msup> <mi>&rho;h&omega;</mi> <mn>2</mn> </msup> <msup> <mi>W</mi> <mn>2</mn> </msup> <mo>-</mo> <msub> <mi>D</mi> <mn>11</mn> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <mi>W</mi> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <msub> <mi>D</mi> <mn>12</mn> </msub> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <mi>W</mi> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <mi>W</mi> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>4</mn> <msub> <mi>D</mi> <mn>16</mn> </msub> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <mi>W</mi> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <mi>W</mi> </mrow> <mrow> <mo>&part;</mo> <mi>x</mi> <mo>&part;</mo> <mi>y</mi> </mrow> </mfrac> <mo>-</mo> <msub> <mi>D</mi> <mn>22</mn> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <mi>W</mi> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>4</mn> <msub> <mi>D</mi> <mn>26</mn> </msub> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <mi>W</mi> </mrow> <mrow> <mo>&part;</mo> <mi>x</mi> <mo>&part;</mo> <mi>y</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <mi>W</mi> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>-</mo> <mn>4</mn> <msub> <mi>D</mi> <mn>66</mn> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <mo>&part;</mo> <mn>2</mn> </msup> <mi>W</mi> </mrow> <mrow> <mo>&part;</mo> <mi>x</mi> <mo>&part;</mo> <mi>y</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>

The thin plate vibration shape variation equation formula of above-mentioned formula (14) can be obtained by formula (7) (11) (12) (13).
6. the non-linear internal resonance characterization test method of fiber composite plate as claimed in claim 3, it is characterised in that the step 1.1.4 it is specially：

, can be by cantilever composite sheet below based on two-dimension beam function combined method due to being concerned with cantilever herein Model function of vibration is set to：

W_mn(x, y)=A_mnX_m(x)Y_n(y) (15)

Wherein, A_mnFor undetermined coefficient, X_m(x) it is fixation-free beam m first order mode functions, is represented by：

<mrow> <msub> <mi>X</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cosh</mi> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mrow> <mi>x</mi> <mi>m</mi> </mrow> </msub> <mi>x</mi> </mrow> <mi>a</mi> </mfrac> <mo>)</mo> <mo>-</mo> <mi>cos</mi> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mrow> <mi>x</mi> <mi>m</mi> </mrow> </msub> <mi>x</mi> </mrow> <mi>a</mi> </mfrac> <mo>)</mo> <mo>-</mo> <msub> <mi>&sigma;</mi> <mi>m</mi> </msub> <mo>(</mo> <mrow> <mi>sinh</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mrow> <mi>x</mi> <mi>m</mi> </mrow> </msub> <mi>x</mi> </mrow> <mi>a</mi> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mrow> <mi>x</mi> <mi>m</mi> </mrow> </msub> <mi>x</mi> </mrow> <mi>a</mi> </mfrac> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>

In formula：λ_x1=1.875, λ_x2=4.694, λ_x3=7.854, σ_x1=0.7341, σ_x2=1.0185, σ_x3=0.9992；

<mrow> <msub> <mi>&lambda;</mi> <mrow> <mi>x</mi> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> <mi>&pi;</mi> <mo>,</mo> <mrow> <mo>(</mo> <mi>m</mi> <mo>&GreaterEqual;</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>&sigma;</mi> <mrow> <mi>x</mi> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>cosh</mi> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mrow> <mi>x</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mrow> <mi>x</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sinh</mi> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mrow> <mi>x</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mrow> <mi>x</mi> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <mrow> <mo>(</mo> <mi>m</mi> <mo>&GreaterEqual;</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>

Y_n(y) it is freedom-free beam n-th order model function of vibration, is represented by：

Y₁(y)=1 (19a)

Y₂(y)=1-2y/b (19b)

<mrow> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>cosh</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mrow> <mi>y</mi> <mi>n</mi> </mrow> </msub> <mi>y</mi> </mrow> <mi>b</mi> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mrow> <mi>y</mi> <mi>n</mi> </mrow> </msub> <mi>y</mi> </mrow> <mi>b</mi> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&sigma;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>sinh</mi> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mrow> <mi>y</mi> <mi>n</mi> </mrow> </msub> <mi>y</mi> </mrow> <mi>b</mi> </mfrac> <mo>)</mo> <mo>+</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mrow> <mi>y</mi> <mi>n</mi> </mrow> </msub> <mi>y</mi> </mrow> <mi>b</mi> </mfrac> <mo>)</mo> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <mi>n</mi> <mo>></mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mi>c</mi> <mo>)</mo> </mrow> </mrow>

In formula：λ_y3=4.730, σ_y3=0.9825；

<mrow> <msub> <mi>&lambda;</mi> <mrow> <mi>y</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>3</mn> </mrow> <mn>2</mn> </mfrac> <mi>&pi;</mi> <mo>,</mo> <mrow> <mo>(</mo> <mi>n</mi> <mo>&GreaterEqual;</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>&sigma;</mi> <mrow> <mi>y</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>cosh</mi> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mrow> <mi>y</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mrow> <mi>y</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sinh</mi> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mrow> <mi>y</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mrow> <mi>y</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <mrow> <mo>(</mo> <mi>n</mi> <mo>&GreaterEqual;</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>

M, n represent that the vibration shape along x, half wave number in y directions, can use different positive integers, represent the different orders of the vibration shape respectively.
7. the non-linear internal resonance characterization test method of fiber composite plate as claimed in claim 3, it is characterised in that the step 1.1.5 it is specially：

Model function of vibration (15) is substituted into vibration shape variation equation (14), can be obtained on inherent circular frequency ω_mnAlgebraic equation, directly Connect and solve inherent circular frequency：

<mrow> <msub> <mi>&omega;</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msqrt> <mfrac> <mrow> <mo>(</mo> <msub> <mi>D</mi> <mn>11</mn> </msub> <msub> <mi>T</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>D</mi> <mn>12</mn> </msub> <msub> <mi>T</mi> <mn>3</mn> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>D</mi> <mn>16</mn> </msub> <msub> <mi>T</mi> <mn>4</mn> </msub> <mo>+</mo> <msub> <mi>D</mi> <mn>22</mn> </msub> <msub> <mi>T</mi> <mn>5</mn> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>D</mi> <mn>26</mn> </msub> <msub> <mi>T</mi> <mn>6</mn> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>D</mi> <mn>66</mn> </msub> <msub> <mi>T</mi> <mn>7</mn> </msub> <mo>)</mo> </mrow> <mrow> <msub> <mi>&rho;hT</mi> <mn>1</mn> </msub> </mrow> </mfrac> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>

Wherein：

<mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msup> <msub> <mi>X</mi> <mi>m</mi> </msub> <mn>2</mn> </msup> <msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mn>2</mn> </msup> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mi>a</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>T</mi> <mn>2</mn> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <msup> <mi>dx</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mn>2</mn> </msup> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mi>b</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>T</mi> <mn>3</mn> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <msup> <mi>dx</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <msup> <mi>dy</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mi>c</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>T</mi> <mn>4</mn> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>b</mi> </msubsup> <mfrac> <mrow> <msub> <mi>dX</mi> <mi>m</mi> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> <msub> <mi>X</mi> <mi>m</mi> </msub> </mrow> <mrow> <msup> <mi>dx</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mfrac> <mrow> <msub> <mi>dY</mi> <mi>n</mi> </msub> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mi>d</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>T</mi> <mn>5</mn> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msup> <msub> <mi>X</mi> <mi>m</mi> </msub> <mn>2</mn> </msup> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <msup> <mi>dy</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mi>e</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>T</mi> <mn>6</mn> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msub> <mi>X</mi> <mi>m</mi> </msub> <mfrac> <mrow> <msub> <mi>dX</mi> <mi>m</mi> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mfrac> <mrow> <msub> <mi>dY</mi> <mi>n</mi> </msub> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> <mfrac> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow> <mrow> <msup> <mi>dy</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mi>f</mi> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>T</mi> <mn>7</mn> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>b</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>dX</mi> <mi>m</mi> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>dY</mi> <mi>n</mi> </msub> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mi>g</mi> <mo>)</mo> </mrow> </mrow>
8. the non-linear internal resonance characterization test method of fiber composite plate as claimed in claim 1, it is characterised in that the step 3 include：

Step 3.1：Build connecting test system and determine that the boundary condition of the non-linear internal resonance characterization test of fiber composite plate is Cantilever；

Step 3.2：Basic parameter needed for sweep check is set, including：The sensitivity of laser doppler vialog, sample frequency, The signal type of frequency resolution, signal generator；

Step 3.3：The internal resonance intrinsic frequency of certain two rank of composite plate is chosen, in a manner of basic excitation, respectively to tested multiple Plywood carries out tentering swept frequency excitation, carries out response signal test using laser doppler vialog, and time domain initial data is entered FFT is handled row at times, by recognizing peak value of response come the accurate each rank internal resonance intrinsic frequency for obtaining composite plate.
9. the non-linear internal resonance characterization test method of fiber composite plate as claimed in claim 1, it is characterised in that the step 6 include：

Step 6.1：Basic parameter needed for fiber composite plate internal resonance characterization test is set, including：Electromagnetic exciter is surely frequent Rate value, the sensitivity of laser doppler vibrometer, sample frequency, frequency resolution, response signal add hanning windows to handle；

Step 6.2：Under different excitation amplitudes, the internal resonance intrinsic frequency of certain two rank is utilized respectively, to determine the side of frequency excitation Formula, composite sheet is excited to produce resonance；

Step 6.3：After structural vibration response is stable, pass through data acquisition equipment record pumping signal now and vibration velocity Response signal, and can the internal resonance intrinsic frequency of certain two rank mutually eject in Real Time Observation spectrogram, and have two Into the basic frequency of multiple proportion；

Step 6.4：Extraction step 6.3 tests recorded vibration velocity response signal, by Matlab softwares by rate integrating Into shifted version, and draw using vibration displacement as abscissa, vibration velocity is the phasor track of ordinate；

Step 6.5：Phasor track, time-domain diagram and frequency domain figure of the binding fiber composite plate under different excitation amplitudes judge it Non-linear internal resonance phenomenon, and conclude the changing rule of its nonlinear vibration response.
A kind of 10. non-linear internal resonance characterization test system of fiber composite plate, it is characterised in that including：Control machine, power amplification Device, electromagnetic exciter, laser doppler vibrometer, data acquisition equipment and computer；

Control machine, for exporting sinusoidal signal；

Power amplifier, for the sinusoidal signal amplification for exporting control machine；

Electromagnetic exciter, for receiving the sinusoidal signal after amplifying, to produce sinusoidal excitation, and acted in a manner of basic excitation In composite plate；

Laser doppler vibrometer and data acquisition equipment, for gathering pumping signal and vibration velocity response signal；

Computer, for setting the basic parameter needed for internal resonance characterization test, the vibration velocity response letter to fiber composite plate Number analyzed and processed.