Study of the Response Prediction of a Parametric Vibration Based on Modulation Feedback

Abstract

The problem of response prediction is investigated for parametric vibration in terms of a new concept. The response solution is presented in the special form of Fourier series for signal degree freedom of parametric vibration based on modulation feedback. By applying harmonic balance and limitation operation, all coefficients of a harmonic component are fully determined with a set of series. Meanwhile, some important dynamic behaviors are exposed through mathematical deduction, and an instability phenomenon can be discussed through given frequency factors. The investigation result shows that the new approach has an advantage in the complete response expression, and it is very significant for the theoretical research and engineering application concerning parametric vibration.

1. Introduction

The problem of parametric vibration arises in many branches of physics and engineering, and investigation of stability and response prediction are the two most significant dynamic problems in the parametric vibration system. In the past, several methods have been used to study the stability of systems with periodic coefficients. These include Hill’s Method [1], the perturbation method [2], the averaging approach, Floquet theory with numerical integration [3], and Sinha’s numerical scheme with the shifted Chebyshev polynomial [4, 5].

Hill’s infinite determinant was used to compute the transition curve or the instability boundary. However, it does not yield the response vector at an arbitrary point in the parametric space. The perturbation method supplies a method for obtaining the solution vector and instability boundary. The solution vector is expanded as a power series in terms of the small parametric and substituted into the original equation. The method is applicable only if the parametric multiplying the period term is small [6]. Authors [7–9] have tried to use the approximate approach to determine the stability and response from an approximate system of equations, which is usually obtained by replacing the elements of the periodic coefficients matrix with piecewise constants or linear functions. Sinha and Wu presented a numerical scheme for computing the Floquet transition matrix associated with a class of periodic systems. In the authors’ approach, the solution vector is expanded in terms of the shifted Chebyshev polynomials. This technique reduces the original differential system to a system of linear algebraic equation from which the solutions in the interval of one period can be easily obtained.

In science and engineering practice, it will be very useful to express the response solution in the form of a Fourier series, which is convenient in an online vibration monitoring system. Especially in a mechanical fault diagnosis, such expression is important and not replaceable. One typical example is to diagnose the mechanical fault in a rotational rotor with a cross crack by analyzing a characteristic of the parametric system [10]. However, so far, no related response solution has directly been considered in all of the above approaches.

In this paper, the concept of modulation feedback is introduced to set a response form that is similar to a Fourier series. Using harmonic balance in the equation and limitation operation, a full response expression is conducted. Though this concept comes from the content of a modulation system, it reflects the intrinsic physical character of a parametric system. Moreover, the presented solution conducts the response prediction that could be seen as an expansion of the Perturbation solution.

2. Modulation Feedback Conception

Consider a classical parametric vibration equation:𝑑²𝑥𝑑𝑡²+𝜔²_𝑛1+𝛽cos𝜔𝑡𝑥=0, (1) and rewrite it as 𝑑²𝑥𝑑𝑡²+𝜔²_𝑛𝑥=−𝑥𝜔²_𝑛𝛽cos𝜔𝑡. (2) Based on (2), the problem of parametric vibration can be schematically described as a system drawn in Figure 1, and it is a special feedback system that contains a second-order linear unit and a frequency modulation. The output of the system is the general solution of parametric vibration.

Figure 1 

Modulation feedback system.

In this system, in addition to enhancing the input and feedback signal, the second-order linear unit will inherently generate the harmonic component 𝐶₀₁cos𝜔_𝑛𝑡 after the moment 𝑡=0. However, the generated component 𝐶₀₁cos𝜔_𝑛𝑡 will be multiplied by 𝜔²_𝑛𝛽cos𝜔𝑡, and this operation is called the frequency modulation, which conducts frequency fission in the system.

The phenomenon of frequency fission can be stated as the following: the component 𝜔_𝑛 is fissile into the combination of harmonic components 𝜔_𝑛−𝜔 and 𝜔_𝑛+𝜔 by a modulation operation within the first Δ𝑡. Then, the modulation result directly feeds back to the second-order linear unit as an input of the new harmonic components. Thus, within the second Δ𝑡, three components 𝜔_𝑛,𝜔_𝑛−𝜔, and 𝜔_𝑛+𝜔 will be the output of the system and be involved in another fission operation, a sequential combination of five harmonic components 𝜔_𝑛,𝜔_𝑛−𝜔,𝜔_𝑛−2𝜔, 𝜔_𝑛+𝜔, and 𝜔_𝑛+2𝜔 will become the modulation results and will feed back to the second-order linear unit, and the iterate fission operation continues. The whole physical procedure of the frequency fission is schematically described in Figure 2.

Figure 2 

Frequency fission procedure (𝑡≥0,Δ𝑡→0).

Assuming the frequency fission procedure continues, except for the case of the instable state, the frequency fission will attain a state of balance. Therefore, there are many linear combinations of harmonic components related to 𝜔_𝑛 and 𝜔 in the output of the system as the frequency fission balance, and they can be mathematically expressed as follows: 𝑥=^∞_𝑘=−∞𝐶_𝑘cos𝜔_𝑛+𝑘𝜔𝑡. (3) Because the energy of the harmonic components is concentrated on the narrow band range in the frequency domain, the coefficient 𝐶_𝑘→0 while 𝑘→∞.

Therefore, the response solution problem of the parametric vibration equation is simplified to the determination of coefficient values of 𝐶_𝑘 in (3).

3. Fission Coordinate Equation

After substituting the solution (3) into (1), we obtain the following infinite set of linear, algebraic, homogeneous equations for the 𝐶_𝑘 by equating the coefficients of each harmonic on both sides of (1):𝜔²_𝑛𝛽2𝐶_𝑘−1+𝜔²_𝑛−𝜔_𝑛+𝑘𝜔²𝐶_𝑘+𝜔²_𝑛𝛽2𝐶_𝑘+1(()).=0𝑘=⋯−𝑚,−𝑚−1,…,−3,−2,−1,0,1,2,3⋯𝑚−1,𝑚⋯ (4) Here, we introduce notations:𝜛_𝑘=𝜔²_𝑛−𝜔_𝑛+𝑘𝜔²(()),𝜔𝑘=⋯−𝑚,−𝑚−1,…,−3,−2,−1,0,1,2,3⋯𝑚−1,𝑚⋯𝛾=^2𝑛𝛽2. (5) First, we consider 2𝑚+1 finite equations, and put all coefficient relations together and form a linear algebra equation called a fission coordinate equation (which is similar to Hill’s form).𝜛_−𝑚𝛾𝛾𝜛_−𝑚+1𝛾⋅⋅⋅𝛾𝜛₋₃𝛾𝛾𝜛₋₂𝛾𝛾𝜛₋₁𝛾𝛾0𝛾𝛾𝜛₁𝛾𝛾𝜛₂𝛾𝛾𝜛₃𝛾⋅⋅⋅𝛾𝜛_𝑚−1𝛾𝛾𝜛_𝑚𝐶_−𝑚𝐶_−𝑚+1⋅𝐶₋₃𝐶₋₂𝐶₋₁𝐶₀𝐶₁𝐶₂𝐶₃⋅𝐶_𝑚−1𝐶_𝑚=−𝛾𝐶_−(𝑚+1)0⋅0000000⋅0−𝛾𝐶_𝑚+1. (6)

4. Determination of Coefficient 𝐶_𝑘

To get a full response expression, the determination of coefficient 𝐶_𝑘 is carried out by the following operation.

(a) Consider the lower half of (6):𝜛₁−𝜛₋₁𝛾𝛾𝜛₂𝛾𝛾𝜛₃𝛾⋅⋅⋅𝛾𝜛_𝑚𝐶₁𝐶₂𝐶₃⋅𝐶_𝑚=−𝛾2𝐶₀+𝐶₋₂00⋅−𝛾𝐶_𝑚+1. (7) Let frequency factors be 𝛼₁=𝜛₁−𝜛₋₁,𝛼₂=𝜛₂−𝛾²𝛼₁,⋮𝛼_𝑚=𝜛_𝑚−𝛾²𝛼_(𝑚−1). (8) Then 𝛼₁𝛾𝛼₂𝛾𝛼₃𝛾𝛼⋅⋅⋅_𝑚𝐶₁𝐶₂𝐶₃⋅𝐶_𝑚=−𝛾2𝐶₀+𝐶₋₂𝛾²2𝐶₀+𝐶₋₂𝛼₁−𝛾³2𝐶₀+𝐶₋₂𝛼₁𝛼₂⋅(−1)^𝑚𝛾^𝑚2𝐶₀+𝐶₋₂∏^𝑚−1_𝑖=1𝛼_𝑖−𝛾𝐶_𝑚+1. (9) One of the solutions to (9): 𝐶₁=2𝐶₀+𝐶₋₂×−𝛾𝛼₁−𝛾³𝛼²₁𝛼₂−𝛾⁵𝛼²₁𝛼²₂𝛼₃−𝛾−⋯_2𝑘−1∏^𝑘−1_𝑖=1𝛼²_𝑖𝛼_𝑘−⋯+(−1)^𝑚+2𝛾^𝑚∏_𝑖^𝑚−1𝛼_𝑖𝐶_𝑚+1. (10) As mentioned in the above section, 𝐶_𝑚+1→0 while 𝑚→∞. Let 𝑘=𝑛; thus the coefficient 𝐶₁ can be expanded as 𝐶₁=2𝐶₀+𝐶₋₂𝑆₁, (11) where 𝑆₁𝛾=−𝛼₁−𝛾³𝛼²₁𝛼₂−𝛾⁵𝛼²₁𝛼²₂𝛼₃𝛾−⋯−^2𝑛−1∏^𝑛−1_𝑖=1𝛼²_𝑖𝛼_𝑛−⋯. (12) For the same reason, 𝐶₂,𝐶₃,…𝐶_𝑘 can be determined. They are 𝐶₂=2𝐶₀+𝐶₋₂𝑆₂𝐶₃,=2𝐶₀+𝐶₋₂𝑆₃,⋮𝐶_𝑘=2𝐶₀+𝐶₋₂𝑆_𝑘, (13)(14)(15) where 𝑆₂=𝛾²𝛼₁𝛼₂+𝛾⁴𝛼₁𝛼²₂𝛼₃+𝛾⁶𝛼₁𝛼²₂𝛼²₃𝛼₄𝛾+⋯+^2𝑛∏^𝑛_𝑖=2𝛼²_𝑖𝛼₁𝛼_𝑛+1𝑆−⋯,₃𝛾=−³𝛼₁𝛼₂𝛼₃−𝛾⁵𝛼₁𝛼₂𝛼²₃𝛼₄−𝛾⁷𝛼₁𝛼₂𝛼²₃𝛼²₄𝛼₅𝛾−⋯−^2𝑛+1∏^𝑛+1_𝑖=3𝛼²_𝑖𝛼₁𝛼₂𝛼_𝑛+2⋮𝑆−⋯,_𝑘=(−1)^𝑘∞_𝑛=1𝛾^{2𝑛+𝑘−2}∏^𝑘−1_𝑗=1𝛼_𝑗∏^{𝑛+𝑘−2}_𝑖=𝑘𝛼²_𝑖𝛼_{𝑛+𝑘−1}. (16)

(b) Consider the upper half of (6):𝛼₋₁𝛾𝛼₋₂𝛾𝛼₋₃𝛾𝛼⋅⋅⋅_−𝑚𝐶₋₁𝐶₋₂𝐶₋₃⋅𝐶_−𝑚=−𝛾2𝐶₀+𝐶₂𝛾²2𝐶₀+𝐶₂𝛼₁−𝛾³2𝐶₀+𝐶₂𝛼₁𝛼₂⋅(−1)^𝑚𝛾^𝑚2𝐶₀+𝐶₂∏^𝑚−1_𝑖=1𝛼_𝑖−𝛾𝐶_−(𝑚+1), (17) where the frequency factors 𝛼₋₁=𝜛₋₁−𝜛₁,𝛼₋₂=𝜛₋₂−𝛾²𝛼₋₁,⋮𝛼_−𝑚=𝜛_−𝑚−𝛾²𝛼_{−(𝑚−1)}. (18) The solution of (17) is 𝐶₋₁=2𝐶₀+𝐶₂𝑅₁,𝐶₋₂=2𝐶₀+𝐶₂𝑅₂𝐶₋₃,=2𝐶₀+𝐶₂𝑅₃,⋮𝐶_−𝑘=2𝐶₀+𝐶₂𝑅_𝑘, (19)(20)(21)(22) where 𝑅₁𝛾=−𝛼₋₁−𝛾³𝛼²₋₁𝛼₋₂−𝛾⁵𝛼²₋₁𝛼²₋₂𝛼₋₃𝛾−⋯−^2𝑛−1∏^𝑛−1_𝑖=1𝛼²_−𝑖𝛼_−𝑛𝑅−⋯,₂=𝛾²𝛼₋₁𝛼₋₂+𝛾⁴𝛼₋₁𝛼²₋₂𝛼₋₃+𝛾⁶𝛼₋₁𝛼²₋₂𝛼²₋₃𝛼₋₄𝛾+⋯+^2𝑛∏^𝑛_𝑖=2𝛼²_−𝑖𝛼₋₁𝛼_−(𝑛+1)𝑅−⋯,₃𝛾=−³𝛼₋₁𝛼₋₂𝛼₋₃−𝛾⁵𝛼₋₁𝛼₋₂𝛼²₋₃𝛼₋₄−𝛾⁷𝛼₋₁𝛼₋₂𝛼²₋₃𝛼²₋₄𝛼₋₅𝛾−⋯−^2𝑛+1∏^𝑛+1_𝑖=3𝛼²_−𝑖𝛼₋₁𝛼₋₂𝛼_−(𝑛+2)⋮𝑅−⋯,_𝑘=(−1)^𝑘∞_𝑛=1𝛾^{2𝑛+𝑘−2}∏^𝑘−1_𝑗=1𝛼_−𝑗∏^{𝑛+𝑘−2}_𝑖=𝑘𝛼²_−𝑖𝛼_{−(𝑛+𝑘−1)}. (23)(24)(25)(26) From (11), (13), (19), and (20), we obtain𝐶₋₂=−2𝐶₀𝑅1+¹𝑆₂𝑅₁+𝑆₁,𝐶₂=−2𝐶₀𝑆1+¹𝑅₂𝑆₁+𝑅₁. (27) The final solution form of (4) is 𝐶_−𝑘=−2𝐶₀𝑆₁𝑅₂𝑆₁+𝑅₁𝑅_𝑘,(),𝐶𝑘=1,2,3,…_𝑘=−2𝐶₀𝑅₁𝑆₂𝑅₁+𝑆₁𝑆_𝑘,(𝑘=1,2,3,…). (28)

5. Special Case

While 𝜔=2𝜔_𝑛/𝑞 (𝑞 is an integral number) and 𝛽>0, the response solution (3) is changed to 𝑥=^∞_𝑘=−∞𝐶_𝑘𝜔cos𝑞+2𝑘^𝑛𝑞𝑡. (29) If 𝑞 is an even number, one gets 𝑥=^{−(1+𝑞/2)}_𝑘=−∞𝐶_𝑘𝜔cos𝑞+2𝑘^𝑛𝑡𝑞+^∞_{𝑘=−𝑞/2}𝐶_𝑘𝜔cos𝑞+2𝑘^𝑛𝑡𝑞=𝐶_{−(𝑞/2+1)}cos2𝜔_𝑛𝑡𝑞+𝐶_{−(𝑞/2+2)}cos4𝜔_𝑛𝑡𝑞+𝐶_{−(𝑞/2+3)}cos6𝜔_𝑛𝑡𝑞+⋯+𝐶_−𝑞/2+𝐶_−𝑞/2+1cos2𝜔_𝑛𝑡𝑞+𝐶_−𝑞/2+2cos4𝜔_𝑛𝑡𝑞+𝐶_−𝑞/2+3cos6𝜔_𝑛𝑡𝑞+⋯=𝐶_−𝑞/2+^∞_𝑚=1𝐶_{−𝑞/2−𝑚}+𝐶_{−𝑞/2+𝑚}cos2𝑚𝜔_𝑛𝑡𝑞. (30) If 𝑞 is an odd number, one gets 𝑥=^{−(𝑞+1)/2}_𝑘=−∞𝐶_𝑘𝜔cos𝑞+2𝑘^𝑛𝑡𝑞+^∞_{𝑘=−(𝑞−1)/2}𝐶_𝑘𝜔cos𝑞+2𝑘^𝑛𝑡𝑞=𝐶_{−(𝑞+1)/2}𝜔cos^𝑛𝑡𝑞+𝐶_{−(𝑞+1)/2−1}cos3𝜔_𝑛𝑡𝑞+𝐶_{−(𝑞+1)/2−2}cos5𝜔_𝑛𝑡𝑞+⋯+𝐶_{−(𝑞−1)/2}𝜔cos^𝑛𝑡𝑞+𝐶_{−(𝑞−1)/2+1}cos3𝜔_𝑛𝑡𝑞+𝐶_{−(𝑞−1)/2+2}cos5𝜔_𝑛𝑡𝑞+⋯=^∞_𝑚=0𝐶_{−(𝑞+1)/2−𝑚}+𝐶_{−(𝑞−1)/2+𝑚}()𝜔cos2𝑚+1_𝑛𝑡𝑞. (31)(a)There is a direct component 𝐶_−𝑞/2 in the response 𝑥 while 𝑞 is an even number in (29). This is a kind of particular characteristic exposed by the given approach in the response study of parametric vibration.(b)Based on (30) and (31), the theoretical spectrum of the response of the parametric vibration can be easily predicted.(c)While 𝑚=𝑞/2 in (30) or 𝑚=(𝑞−1)/2 in (31), if the coefficients 𝐶₀+𝐶_−𝑞≫𝐶₀, (32) the response will be the case of the instability state.

6. Complete Form of the Solution

6.1. Complete Form of the Solution

According to the theory of second-order differential equation, the strict form of a general solution for the given parametric vibration equation should be 𝑥=^∞_𝑘=−∞𝐶_𝑘cos𝜔_𝑛+𝑘𝜔𝑡+𝐷_𝑘sin𝜔_𝑛+𝑘𝜔𝑡. (33) Meanwhile, it needs to meet the sufficient and necessary condition that the Wronskian determinant is equal to the constant.

In fact, 𝑑𝑤(𝑡)=𝑑𝑑𝑡𝑢𝑑𝑡¹(𝑡)𝑢₂(𝑡)𝑢̇₁(𝑡)𝑢̇₂(𝑡)=0. (34)𝑤(𝑡) has no relation with time 𝑡, where 𝑢₁(𝑡)=^∞_𝑘=−∞𝐶_𝑘cos𝜔_𝑛𝑢+𝑘𝜔𝑡,₂(𝑡)=^∞_𝑘=−∞𝐷_𝑘sin𝜔_𝑛+𝑘𝜔𝑡. (35) Because 𝑤(𝑡) is a constant, (33) is the complete form of the general solution of the given parametric vibration equation, where 𝐶₀ and 𝐷₀ are the arbitrary constants in the solution in (33) and depend on the initial condition.

6.2. Determination of Coefficient 𝐷_𝑘

The coefficient 𝐷_𝑘 is determined by using the harmonic balance as the determination of the coefficient value of 𝐶_𝑘: 𝐷_−𝑘=−2𝐷₀𝑆₁𝑅₂𝑆₁+𝑅₁𝑅_𝑘,𝐷_𝑘=−2𝐷₀𝑅₁𝑆₂𝑅₁+𝑆₁𝑆_𝑘. (36) Thus, in the special case of 𝜔=2𝜔_𝑛/𝑞, if 𝑞 is an even number, the complete form of the solution will be 𝑥=𝐶_−𝑞/2+^∞_𝑚=1𝐶_{−𝑞/2−𝑚}+𝐶_{−𝑞/2+𝑚}cos2𝑚𝜔_𝑛𝑡𝑞+𝐷_−𝑞/2+^∞_𝑚=1−𝐷_{−𝑞/2−𝑚}+𝐷_{−𝑞/2+𝑚}sin2𝑚𝜔_𝑛𝑡𝑞. (37) If 𝑞 is an odd number, the complete form of the solution will be 𝑥=^∞_𝑚=0𝐶_{−(𝑞+1)/2−𝑚}+𝐶_{−(𝑞−1)/2+𝑚}()𝜔cos2𝑚+1_𝑛𝑡𝑞+^∞_𝑚=0−𝐷_{−(𝑞+1)/2−𝑚}+𝐷_{−(𝑞−1)/2+𝑚}()𝜔sin2𝑚+1_𝑛𝑡𝑞. (38)

6.3. Initial Condition

Equation (33) is the parametric vibration solution after the system attains a state of balance through frequency fission. However, when 𝑡=0, equation (1) degenerates to a linear equation. Therefore, the coefficients 𝐶₀ and 𝐷₀ in (33) can be determined by 𝐶₀=𝑥|_𝑡=0,𝐷₀=1𝜔_𝑛𝑑𝑥𝑑𝑡_𝑡=0. (39)

6.4. Comparison with Another Method

If 𝑘=−1,0,1 in (33), we obtain 𝑥=¹_𝑘=−1𝐶_𝑘cos𝜔_𝑛+𝑘𝜔𝑡+𝐷_𝑘sin𝜔_𝑛+𝑘𝜔𝑡. (40) This is a form of the parametric vibration solution conducted by singular Perturbation called the two variables expansion. Thus, the presented method conducts the response solution that is the expansion of the solution obtained by Perturbation.

In fact, equation (6) is similar to Hill’s form, which was used in the study of the instability boundary of parametric vibration during that time [11]. However, this paper determines the coefficients of the harmonic series by using a special property of 𝜔₀=0 in (6).

7. Instability

There are infinite terms of frequency factor 𝛼_𝑘(𝑘=…,−3,−2,−1,1,2,3,…) in the expression of the obtained solution, which causes instability in the system while 𝛼_𝑘=0. The frequency factors 𝛼_𝑘 are related to 𝜔 and 𝛽, and the roots of 𝛼_𝑘=0 distribute around the instability areas, which are summarized as follows.

7.1. Principal Instability Area

While 𝜔=2𝜔_𝑛 and 𝑘=−1 are selected, the frequency factor 𝛼₋₁=𝜔²_𝑛−(𝜔_𝑛−𝜔)²∣_{𝜔=2𝜔𝑛}=0. As a result, the values of 𝛾/𝛼₋₁ and 𝛾³/𝛼²₋₁𝛼₋₂ in (23) will tend to be infinite while 𝛽≠0. This means the harmonic response corresponding to 𝐶₋₁ tends to be infinite in the principal instability area.

7.2. High-Order Instability Boundary Curve

Based on the roots of 𝛼_𝑘=𝜔²_𝑛−(𝜔_𝑛+𝑘𝜔)²−𝛾²/𝛼_𝑘+1=0 (𝑘<−1), the group of harmonic resonance (higher-order instability) boundary curves are drawn in the parameter plan of 𝜔 and 𝛽, as shown in Figures 3(a)–3(d) (generated by commercial software Maple 9.5), which have the following features.

(a) Parameter plan of 𝜔 and 𝛽 while 𝛼−2=0

(b) Parameter plan of 𝜔 and 𝛽 while 𝛼−3=0

(c) Parameter plan of 𝜔 and 𝛽 while 𝛼−4=0

(d) Parameter plan of 𝜔 and𝛽 while 𝛼−5=0

Figure 3 

High-order instability boundary curve.

Left Bias
Due to the term 𝛾²/𝛼_𝑘+1 having influence on frequency factor 𝛼_𝑘, each group of multiple harmonic resonance curves is under bias toward 2𝜔_𝑛/𝑚 (𝑚=2,3,4,…, and 𝑘=−2,−3,−4,…). In other words, each harmonic resonance boundary curve is located at the left side of the point 2𝜔_𝑛/𝑚 that is the high-order instability point while 𝛽 is close to 0.

Instability Region Overlap
There are different order harmonic resonance boundary curves in each instability area around 2𝜔_𝑛/𝑚 (𝑚=1,2,3,…). For example, as shown in Figure 4, the multiple harmonic resonance regions exit near the third-order instability area, and they are based on 𝛼₋₃=0, 𝛼₋₄=0⋯; but for 𝛼₋₄=0, 𝛼₋₅=0⋯ almost overlap in the same region while 0<𝛽<1. Wherein the curve 𝛼₋₃=0 is the left side of the instability region, and the curve 𝛼_−𝑘=0 (𝑘→∞, in theory) is the right side of the instability region. Therefore, the harmonic resonance boundary curves (𝛼_−𝑘=0) are composed of the boundaries of the instability region.

Figure 4 

Third instability region and boundary curve.

Cross Direction
As known from (26), the harmonic resonance strength is different between 𝛼_𝑘→0 and 𝛼_𝑘+1→0. Thus, the harmonic resonance strength is sensitive to the 𝜔 cross direction (from left to right or from right to left) to the high-order instability area.

8. Example

Let 𝜔_𝑛=1, 𝜔=0.5, and 𝛽=0.3 in (1), the initial conditions are 𝑥∣_𝑡=0=1 and 𝑑𝑥/𝑑𝑡∣_𝑡=0=0.

From (37), the response solution can be expressed as𝑥(𝑡)=𝐶₋₂+𝐶₋₃+𝐶₋₁cos0.5𝑡+𝐶₋₄+𝐶₀+𝐶cos𝑡₋₅+𝐶₁cos1.5𝑡+𝐶₋₆+𝐶₂cos2𝑡+⋯. (41) On the initial condition, obtain 𝐶₀=1.

The calculation result of coefficient values 𝐶_𝑘 and the corresponding frequency component 𝜔_𝑛+𝑘𝜔 are summarized in Table 1 in which the theoretical spectrum of the response solution of parametric vibration can be predicted. The spectrum is schematically shown in Figure 5.

Table 1 

Calculation result of the value 𝐶_𝑘 and corresponding frequency component 𝜔_𝑛+𝑘𝜔 (𝜔_𝑛=1,𝛽=0.3, and 𝜔=0.5).

Figure 5 

Spectrum of the parametric vibration response solution (𝜔𝑛=1, 𝛽=0.3, 𝜔=0.5).

The response solution is very sensitive to 𝜔 while 𝜔 is closed to 0.5. If 𝜔 is a little less than 0.5, the high-order instability will be reflected in the solution (33). For example, in the case of 𝜔=0.49845877 in the above example, we get 𝐶₋₄=2825.8. The result can be seen as a case of infinity, and it is governed by the left bias of higher-order instability. Simultaneously, the direct component 𝐶₋₂ cannot be neglected. The detailed results are listed in Table 2, and the theoretical response spectrum is drawn in Figure 6.

Table 2 

Calculation result of the value 𝐶_𝑘 and corresponding frequency component 𝜔_𝑛+𝑘𝜔 (𝜔_𝑛=1, 𝛽=0.3, and 𝜔=0.49845877).

Figure 6 

Spectrum of the parametric vibration response solution in the case of higher-order harmonic resonance (𝜔𝑛=1, 𝛽=0.3, and 𝜔=0.49845877).

9. Conclusion

The response prediction of the single freedom of parametric vibration can be determined with a set of combinations of harmonic components related to 𝜔_𝑛 and 𝜔. Once the coefficients 𝐶₀ and 𝐷₀ are given, whether the general case or the special case of 𝜔=2𝜔_𝑛/𝑞, the harmonic components coefficients 𝐶_𝑘 and 𝐷_𝑘 can be fully determined by a set of series. On the coefficients 𝐶_𝑘 and 𝐷_𝑘, the corresponding theoretical spectrum can be drawn; it will be very useful in identifying and predicting the characteristics in the mechanical system on a model of parametric vibration, especially in the case of 𝜔=2𝜔_𝑛/𝑞.

The instability region can be determined when the frequency factor 𝛼_𝑘 is equal to zero. The investigation result matches the case of Hill’s instability boundary curve. The harmonic resonance in the high-order instability region has three features, which are left bias, region overlap, and cross direction.

The presented concept of modulation feedback reflects the physical inherency in the parametric vibration system, though it comes from the scope of a modulation system. This can become an available tool for us to analyze the response in the parametric vibration. Therefore, it is expected to obtain a response prediction of parametric vibration with an external force excitation and of a multiple freedom of parametric vibration, and so on.

Acknowledgment

The authors wish to acknowledge the support of the 211 Leading Academic Discipline Project of Shanghai University.