Nonlinear Differential Equation in Anti-aging Test of Polymer Nanomaterials

Rutile nano-TiO2 is a non-toxic and odorless inorganic UV shielding agent with excellent performance in polymer materials. This material absorbs part of the UV light energy through electronic transitions and releases it as heat or fluorescence. In this way, the molecular chain of the polymer material is prevented from being broken after absorbing high-energy ultraviolet light. So it has a certain antiaging effect. This article takes this material as the research object. We set up a molecular antiaging experiment for parametrically excited nonlinear differential equations. In this paper, the amplitude-frequency response of the system, the stability, and the bifurcation characteristics of the system are analyzed using the multiscale method. At the same time, we use numerical simulation to verify the theoretical analysis results. The research shows that the surface-treated nano-TiO2 has good dispersibility in polymer materials.


Introduction
Polymers, fiber composite materials, biomaterials, etc., are widely used in aerospace, vehicle engineering, civil engineering, bioengineering, and materials engineering.Composite materials often exhibit significant nonlinear viscoelastic properties under load.The nonlinear vibration of viscoelastic structures has attracted more and more attention from scholars at home and abroad.Some scholars have studied the periodic and chaotic motion of viscoelastic rods under the action of simple harmonic excitation [1].Some scholars have studied viscoelastic Timoshenko beams.They derive the equations of motion of the system from assumed constitutive relations.At the same time, the system's nonlinear behavior is analyzed by numerical calculation.Some scholars have studied viscoelastic transmission belt lateral vibration's characteristics and chaotic dynamic behavior.Some scholars have studied the nonlinear vibration of carbon nano-reinforced composite sandwich cantilever beams.According to the Kelvin-Voigt model, they established the dynamic governing equations of the viscoelastic cantilever beam.At the same time, the stability of the system is studied by multiscale analysis and experiments.Some scholars have studied the lateral vibration of piles under nonlinear elastic and linear viscous conditions.Some scholars have studied the nonlinear vibration of simply supported viscoelastic beams with lumped masses at the ends [2].Some scholars have studied the nonlinear dynamics, bifurcation, and chaotic dynamic responses of simply supported piezoelectric composite laminated beams under axial and lateral loads combined.Some scholars have analyzed the lateral motion of viscoelastic beams under the excitation of axial load [3].They obtained the governing equations through viscoelastic constitutive relations.Some scholars have studied the nonlinear vibration of the isotropic viscoelastic layer and cantilever beam.Some scholars have studied the nonlinear vibration of variable-speed rotating viscoelastic beams.Some scholars have analyzed the bifurcation and chaotic characteristics of parametrically excited viscoelastic transmission belts.Nonlinear constitutive relations often lead to complex dynamics governing equations.Our further analysis of the system's stability, bifurcation, and chaotic motion adds to the difficulty [4].The experimental modeling method provides a way for the viscoelastic material to establish a mathematical model which is convenient for nonlinear dynamic analysis.We use ABS resin as the base material.At the same time, we made a series of composite samples with rutile nano-titanium dioxide filled with 1%-10%.This paper establishes an experimental system for parametric excitation of nonlinear vibrating beams [5].The experimental setup is the lateral vibration of a viscoelastic simply supported beam subjected to the axial excitation force and applied with controllable dry friction damping.The nonlinear dynamics governing equations of viscoelastic parametric beams are obtained through experimental data and incremental harmonic balance nonlinear identification.
Stability and bifurcation properties of our equation (1) solution at 1/2 subharmonic resonance [6].We analyze the system's amplitude-frequency response, the solution's stability, and the bifurcation properties using a multiscale approach.At the same time, we use numerical simulation to verify the theoretical analysis results.

Experimental Modeling
We establish the dynamic governing equations of viscoelastic composite beams employing experimental modeling [7].The article starts with the preparation of materials.At the same time, we prepared six different composite beam series samples according to the composition ratio of ABS and rutile nano-titanium dioxide (Table 1).The experiment uses ABS material as the base material.Filled with other nanoscale components to modify ABS materials [8].In this way, composite materials with corresponding functions can be obtained.Rutile nano titanium dioxide is widely used in plastic, rubber, and functional fiber products.It can improve the product's antiaging ability, anti-pulverization ability, weather resistance, and product strength.At the same time, it can maintain the product's color luster and prolong the product's service life.In this paper, the experimental data of No. 3 material are used for parameter identification.Then use the experimental data of other materials for model verification.In this way, a dynamic governing equation suitable for nonlinear parametric vibration of a class of nanocomposite materials can be obtained.The article first establishes an experimental system corresponding to the model.The dynamic properties of viscoelastic composite beams under parametric excitation are studied [9].Therefore, the experimental device is fixed at one end and slid at the other.In the experiment, we need to eliminate the influence of gravity.We will arrange the buckling beam longitudinally (Figure 1).The frequency modulation range of the system excitation frequency is between 0Hz-200Hz.The excitation amplitude f is controlled by adjusting the voltage.The nonlinear damping of the system is produced by the dry friction structure arranged at the sliding end.Force devices and force sensors control the nonlinear damping of the system.The lateral vibration response of the beam is measured by an accelerometer attached to the middle of the beam.Material No. 3 was chosen as the basic material for experimental modeling.Its size is 200mm×20mm×1.5mm.The mass is 7.3g.The first-order natural frequency is 6.5Hz.
Therefore, when considering the dynamic model of the ABS-TiO2 nano-viscoelastic material beam, we need to assume that the nonlinear term of the system is a 3rd-degree polynomial.
Then the dynamic model of the system is set to the following form: is a parameter that needs to be identified through experimental data and nonlinear parameter identification theory.We measure the vibration response data of the beam from the experimental system and apply the incremental harmonic balance nonlinear identification method to identify the parameter ( 0, ,9)  i i    in equation ( 3).Substitute the above parameters into equation (3) to solve the response ( ) x t .The results are then compared with the experimentally measured response results.Comparing the influence of various nonlinear terms in equation ( 3) on the identification results shows that the simulation results are in good agreement with the experimental results when xx  is not included in equation (3) (Fig. 2).Otherwise, the solution of the equation will show decay to zero or infinity.However, whether the 2 x term exists in the equation does not affect the recognition result.In any case, the equation can get good results [10].But other models of material keeping item 2 x will get better results.so the optimal dynamic control equation of viscoelastic beam under parametric excitation is: Where 0  is related to the natural frequency and initial conditions. 1  is linear damping.

Solution stability
We take the No. 3 nanocomposite beam as an example to study the nonlinear dynamic characteristics of the beam under parametric excitation [11].Therefore, we can write the governing equation (4) in the following form: Where  is the linear damping coefficient.0  is the natural frequency of the system.0 f and f are related to the parametric excitation, and ( 0, ,5 is the nonlinear parameter of the system.In the case of 1/2 subharmonic resonance, the system will have response and bifurcation characteristics.Suppose 2 Here 0  0 is the tuning parameter.We substitute (5).At the same time, we use the multiscale method (5) to write: Where 0 0 f     .Suppose the first-order asymptotic solution of equation ( 6) is of the form: , We substitute equation ( 7) into equation (6).We equalize the coefficients of the same power on both sides of the equation to obtain the following system of differential equations: The solution to equation ( 8) is: ( ) A t is the complex conjugate of 2 ( ) A t .For long term terms to not appear in the solution, then there must be:

dA t i A t A t A t i A t A t A t dt if A t i A t A t
We substitute equation ( 12) into equation (11) and separate the real and imaginary parts.Simplify to get the average equation in the polar coordinate form under the first approximation as At this point, we discuss the steady-state solution of equation ( 13) and the variation of the system response with the parameter 0 ,   .Let Equation ( 14) can be abbreviated as: From this, the following different solutions can be obtained.There is 1 0 A  , here is assumed: (1) When 0 C  , there are: (2) When (3) Other circumstances include: We judge the stability of the solution at 1/2 subharmonic resonance.At the same time, we transform the average equation ( 13) from polar form to Cartesian form: Here x and y are real functions of 2 t .We substitute equation (21) into equation ( 11) and separate the real and imaginary parts from obtaining the average equation in Cartesian coordinates: From the Jacobi matrix of equation ( 22), the Eigen equation corresponding to the zero solution can be obtained: The Eigen equation corresponding to the non-zero solution is: According to the different solutions of the bifurcation response equation ( 14) and the Eigen equation ( 23), the stable region where the solution of (24) can be obtained is shown in Fig. 4. The singularity of the average equation ( 13) and the periodic solution of the governing equation ( 6) have the same stable region on the parametric plane 0 ( , )   .

Zero solution stability analysis
We analyze the zero-solution stability in different regions according to the zero-solution Eigen equation ( 23).At this point, we get the following result: (1) When 0   is shown in Figure 4.The zero solution is stable in the region I .In the region II is unstable.
(2) When 0   , the eigenvalues of the zero solution have at least one eigenvalue whose real part is greater than zero.so the zero solution is always unstable.

Non-zero solution stability analysis
According to the non-zero solution characteristic root structure of Eq. ( 24), it can be known that the stability condition of the non-zero solution is: The stable and unstable regions of non-zero solutions can be obtained by stability analysis as follows: (1) When 0   is shown in Figure 4.The non-zero solution (16) is stable in region II, and the non-zero solution (18) is stable in region VIII.The non-zero solution of equation ( 19) is not stable in region VIII.
(2) When 0   , the non-zero solution of equation ( 16) is unstable in region IV and stable in region V.The non-zero solution (18) is stable in region VI but unstable in region VII.The non-zero solution of equation ( 19) is not stable at VI and VII.

Bifurcation Analysis
System stability varies with system parameters.At this time, the system exhibits different motion forms.Next, numerical simulation is used to analyze the influence of the parameter excitation amplitude f on the bifurcation behavior of the system [12].The main parameter of the system (5) is 0 0 1, 2, 0.1, 0.05, 0.1( 1, 2,3, 4,5) . Fig. 5 is a graph of the maximum Lyapunov exponent concerning the parameter excitation amplitude f , and Fig. 6 is a bifurcation graph obtained by the Runge-Kutta numerical integration method concerning the parameter excitation amplitude f .The results show that the system exhibits rich, dynamic phenomena with parameter variation f .1) When the excitation amplitude (0, 0.286) f  , the system begins to move in a decaying motion due to the damping action.The system is finally at rest.2) When the incentive effect is further strengthened, (0.286,12) f  , the system appears in periodic motion [13].The system goes through period 1→period 2→period 3. 3) From Figure 5 and Figure 6, it can be seen that the system will change from period 3 to period 4 with the increase of excitation amplitude f. 4) Then the system undergoes period-doubling bifurcation and enters chaos state.When f  (13.0445, 15.3655), the maximum Lyapunov exponent in Figure 5 is positive.This indicates that the system is in a chaotic state at this time.

Conclusion
We analyze the dynamics of the system.The influence of the linear damping coefficient on the system's stability under 1/2 subharmonic resonance is discussed.At this time, the amplitude-frequency response of the system and the stable region in the 0 ( , )   plane are obtained.Numerical simulation is used to analyze the influence of parameter excitation on bifurcation behavior.At the same time, we discussed the path of the system to chaos.It is found that chaos is entered through period-doubling bifurcation.At the same time, we found that the system has paroxysmal chaos.This provides a theoretical basis for better utilization of nanocomposites.

Figure 1 Figure 2
Figure 1 Schematic diagram of the experimental setup

2  and 5 
are nonlinear stiffnesses.46 7    , ,    and 8 are related to nonlinear factors such as dry friction, internal damping, etc., 9  is the parameter excitation amplitude.At this point, we verify the applicability of model (4).We use experimental data measured on beams of materials 1, 2, 4, and 5.The article compares the numerical simulation results with the experimental results (Figure3).It can be seen from the figure that the simulation results of the model are in good qualitative and quantitative agreement with the measured results.Model (4) is well validated.

Figure 3
Figure 3 Comparison and verification of experimental results and simulation results of materials with different proportions

Figure 4
Figure 4 Steady-state solution stability region

Figure 5 Figure 6
Figure 5 Maximum Lyapunov exponent plot of the system