阅读说明：本技术 一种柔性机械臂的主共振控制方法 (Main resonance control method of flexible mechanical arm ) 是由 边宇枢 耿瑞海 刘鹏 靳博涵 彭佳乐 于 2021-07-30 设计创作，主要内容包括：本发明公开了一种柔性机械臂的主共振控制方法,该方法主要包括：将柔性机械臂的横向弯曲变形离散化,得到柔性机械臂的变形方程；引入主共振吸振器的刚度反馈系数、阻尼反馈系数以及非线性耦合反馈系数,构造吸振器控制信号模型；利用凯恩方法建立安装主共振吸振器的柔性机械臂振动控制模型；采用有限元模态仿真方法,研究吸振器的安装对柔性机械臂振动特性的影响；应用多尺度法求解方程的近似解析解,得到振动系统稳态响应的单模态解与双模态解；最后在柔性机械臂的主共振条件下,基于饱和原理,利用主共振吸振器对柔性机械臂的主共振进行控制,从而抑制柔性机械臂被控模态的模态幅值。(The invention discloses a main resonance control method of a flexible mechanical arm, which mainly comprises the following steps: discretizing the transverse bending deformation of the flexible mechanical arm to obtain a deformation equation of the flexible mechanical arm; introducing a rigidity feedback coefficient, a damping feedback coefficient and a nonlinear coupling feedback coefficient of the main resonance vibration absorber to construct a vibration absorber control signal model; establishing a flexible mechanical arm vibration control model for installing a main resonance vibration absorber by using a Kenner method; the method comprises the following steps of (1) researching the influence of the installation of a vibration absorber on the vibration characteristic of the flexible mechanical arm by adopting a finite element modal simulation method; solving an approximate analytic solution of an equation by using a multi-scale method to obtain a monomodal solution and a bimodal solution of the steady-state response of the vibration system; and finally, under the condition of main resonance of the flexible mechanical arm, based on the saturation principle, controlling the main resonance of the flexible mechanical arm by using the main resonance vibration absorber, so as to inhibit the modal amplitude of the controlled mode of the flexible mechanical arm.)

1. A main resonance control method of a flexible mechanical arm comprises the following specific steps:

the method comprises the following steps: making reasonable assumption, discretizing the transverse bending deformation of the flexible mechanical arm, and obtaining the deformation of the flexible mechanical arm as follows:

in the formula (1), w (x, t) is the transverse deformation of the flexible mechanical arm,representing the ith order mode function, q_i(t) represents the modal coordinates.

Step two: constructing a vibration absorber control model, and introducing a rigidity feedback coefficient and a damping feedback coefficient of the vibration absorber to form linear feedback; on the basis of linear feedback, introducing a vibration response signal of the flexible mechanical arm, and constructing a nonlinear coupling term for adjusting the motion coupling relation of the vibration absorber and the flexible mechanical arm; and finally, forming a control model by coupling the linear terms and the nonlinear terms.

Step three: establishing a flexible mechanical arm vibration control model with a main resonance vibration absorber; adjusting the stiffness feedback coefficient to form internal resonance between the flexible mechanical arm and the vibration absorber; and analyzing the influence of the damping feedback coefficient and the coupling feedback coefficient on the internal resonance by adopting a control variable method, thereby verifying the effectiveness of the main resonance vibration absorber.

Step four: introducing simple harmonic external excitation into a system of the flexible mechanical arm, establishing a weak forced vibration dynamic model, and carrying out dimensionless processing.

Step five: and extracting a vibration equation of the controlled mode of the flexible mechanical arm and the mode of the vibration absorber, and solving an approximate analytic solution of the equation by adopting a multi-scale method to obtain a monomodal solution and a bimodal solution of the steady-state response of the vibration system.

Step six: analyzing the existence forms of the single-mode solution and the double-mode solution, solving an external excitation saturation threshold value, revealing a saturation control mechanism, and realizing the main resonance control of the flexible mechanical arm.

2. The method as claimed in claim 1, wherein in the step one, the controlled mode is discretized. In the modes of the flexible mechanical arm in the bending deformation stages, the vibration energy of the first-stage mode (n is 1) is larger and plays a main role in transverse vibration response, so that the first-stage mode is selected as a controlled mode for vibration control of the vibration absorber. Recording the first-order modal response of the flexible mechanical arm as q₁Then the deformation of the flexible mechanical arm is approximately:

w(x,t)＝φ₁(x)q₁(t) (2)。

3. the method for controlling the main resonance of the flexible mechanical arm as claimed in claim 1, wherein in the second step, the control signal is constructed by the stiffness feedback coefficient, the damping feedback coefficient and the introduced nonlinear coupling feedback coefficient:

in the formula (3), τ₃For constructed control signals, k_p、k_dRepresenting the stiffness feedback coefficient and the damping feedback coefficient of the motor, representing the desired rigid strut angular displacement and velocity, respectively, and χ representing the non-linearly coupled feedback term of the formation.

Wherein the desired angular displacement of the rigid strutAnd angular velocityAre all zero, the nonlinear coupling term of the structure isWherein e is₁Representing the coupling feedback coefficient of the motor, the control signal τ in equation (3)₃Can be adjusted as follows:

4. the method of claim 1, wherein the method comprises the step three, whereinBy adjusting the feedback coefficient k of the motor rigidity_pThe vibration frequencies of the flexible mechanical arm and the vibration absorber meet the degree of common relation of 2:1, and the two vibration frequencies are shown as a formula (7); and (3) establishing a vibration control equation of the system by using a Kane method, and performing Taylor expansion on the nonlinear term, wherein the final simplification result is shown as a formula (5) and a formula (6).

In formulae (5) and (6):

5. the method of claim 1, wherein in step four, an external excitation is applied to the end of the free end of the flexible manipulatorThe obtained vibration control equation is shown as the formula (8) and the formula (9).

In formulae (8) and (9):

6. the method for controlling the main resonance of the flexible mechanical arm as claimed in claim 1, wherein in the fifth step, a single-mode solution and a double-mode solution of the steady-state response are obtained by a multi-scale method, as shown in formulas (10) to (17).

Single-mode solution:

a₂＝0 (11)

bimodal solution:

single-mode solution:

a₂＝0 (11)

bimodal solution:

7. the method as claimed in claim 1, wherein in step six, the saturation phenomenon reveals an energy transfer relationship between the external excitation and the internal resonance mode when the flexible mechanical arm is subjected to the external excitation of the 1:1 primary resonance type based on the existence of the 2:1 internal resonance of the flexible mechanical arm and the vibration absorber. When the external excitation amplitude is larger than the critical pointNamely, when the external excitation threshold value is reached, the flexible mechanical arm enters a nonlinear response region, and the first-order modal amplitude of the flexible mechanical arm is suppressed through a channel for internal resonance energy exchange between the flexible mechanical arm and the vibration absorber; when the external excitation amplitude is continuously increased, the first-order modal amplitude of the flexible mechanical arm is saturated and redundant energy is transmitted to the vibration absorber modal. Therefore, the control of the main resonance of the flexible mechanical arm can be realized based on the saturation principle of the main resonance vibration absorber.

Technical Field

The invention relates to a main resonance control method of a flexible mechanical arm, which takes a main resonance vibration absorber as a carrier, researches the problem of nonlinear dynamic characteristics of the flexible mechanical arm containing 1:1 main resonance, provides a main resonance vibration reduction method based on a nonlinear saturation control principle, and can effectively inhibit the main resonance of the flexible mechanical arm.

Background

With the progress of modern science and technology, the high-speed, precise and light flexible mechanical arm has the remarkable advantages of low power consumption, high efficiency, good environmental adaptability and the like in practical application, and is widely applied to the fields of industrial manufacturing, aerospace and the like. However, the influence of the flexibility factor on the structure design and control is more prominent, which is mainly reflected in that after the large-scale and large-flexibility structure generates vibration, the self-damping is small, so that the structure is difficult to quickly attenuate in a short time. Moreover, if the flexible mechanical arm generates long-time continuous vibration, the structural performance, the working stability and the service life of the flexible mechanical arm are directly influenced. In addition, the vibration magnitude level of the flexible structure can be increased along with the increase of the working speed; meanwhile, the increase of the working speed can also cause the change of the excitation frequency, and when the excitation frequency is close to or even falls within the range of the modal frequency of the mechanical arm, the resonance problem can be caused. Therefore, it is of great importance to analyze the kinematics and dynamics problems caused by flexibility and to explore methods and means for improving the dynamic performance of flexible mechanical arms.

The main resonance is forced vibration, and when the frequency of external excitation is close to the natural frequency of the vibration mode of the flexible mechanical arm, the flexible mechanical arm generates the main resonance. Under the condition of main resonance, even small-amplitude excitation can excite violent vibration response very quickly, thus destroying system stability and deteriorating dynamic performance. The large-scale and large-flexible mechanical arm always receives periodic acting force, the flexible system has dense mode, and the mode parameters can change along with the adjustment of the working pose. In this case, the main resonance relationship is easily formed between the modal frequency of the flexible mechanical arm and the frequency of the exciting force. When the small excitation acts on the main resonance flexible mechanical arm, the flexible mechanical arm can generate large-amplitude vibration due to main resonance in a short time, and at the moment, the vibration of the flexible mechanical arm can not be inhibited in a short time by depending on the structure of the flexible arm, so that the stability of the system is damaged, and the complete instability of the whole flexible mechanical arm system is seriously caused. The main resonance phenomenon cannot be avoided in the flexible mechanical arm system, so the research on the main resonance control method of the flexible mechanical arm is particularly critical.

At present, the main resonance control research of the flexible mechanical arm mainly focuses on a passive control method, and the research on an active control method needs to be enhanced. Therefore, the invention provides a main resonance control method for a flexible mechanical arm with large scale, large flexibility and small damping. An energy transfer path is established between the flexible mechanical arm mode and the main resonance vibration absorber, when the amplitude of external excitation exceeds a saturation threshold, vibration energy is transferred into the vibration absorber and is dissipated through damping of the vibration absorber, and the amplitude of the controlled mode of the flexible mechanical arm which generates main resonance with the external excitation cannot be increased along with the increase of the amplitude of the external excitation, so that the main resonance of the flexible mechanical arm is well inhibited.

Disclosure of Invention

The invention aims to solve the technical problem of providing a main resonance control method of a flexible mechanical arm, which can effectively inhibit the main resonance of the flexible mechanical arm based on an internal resonance principle and a nonlinear saturation control principle.

The technical scheme for solving the technical problems is as follows: a main resonance control method of a flexible mechanical arm comprises the following specific steps:

the method comprises the following steps: making reasonable assumption, discretizing the transverse bending deformation of the flexible mechanical arm, and obtaining the deformation of the flexible mechanical arm as follows:

in the formula (1), w (x, t) is the transverse deformation of the flexible mechanical arm,representing the ith order mode function, q_i(t) represents the modal coordinates.

Step two: constructing a vibration absorber control model, and introducing a rigidity feedback coefficient and a damping feedback coefficient of the vibration absorber to form linear feedback; on the basis of linear feedback, introducing a vibration response signal of the flexible mechanical arm, and constructing a nonlinear coupling term for adjusting the motion coupling relation of the vibration absorber and the flexible mechanical arm; and finally, forming a control model by coupling the linear terms and the nonlinear terms.

Step three: establishing a flexible mechanical arm vibration control model with a main resonance vibration absorber; adjusting the stiffness feedback coefficient to form internal resonance between the flexible mechanical arm and the vibration absorber; and analyzing the influence of the damping feedback coefficient and the coupling feedback coefficient on the internal resonance by adopting a control variable method, thereby verifying the effectiveness of the main resonance vibration absorber.

Step four: introducing simple harmonic external excitation into a system of the flexible mechanical arm, establishing a weak forced vibration dynamic model, and carrying out dimensionless processing.

Step five: and extracting a vibration equation of the controlled mode of the flexible mechanical arm and the mode of the vibration absorber, and solving an approximate analytic solution of the equation by adopting a multi-scale method to obtain a monomodal solution and a bimodal solution of the steady-state response of the vibration system.

Step six: analyzing the existence forms of the single-mode solution and the double-mode solution, solving an external excitation saturation threshold value, revealing a saturation control mechanism, and realizing the main resonance control of the flexible mechanical arm.

Preferably, the controlled mode is discretized. In the modes of the flexible mechanical arm in the bending deformation stages, the vibration energy of the first-stage mode (n is 1) is larger and plays a main role in transverse vibration response, so that the first-stage mode is selected as a controlled mode for vibration control of the vibration absorber. Recording the first-order modal response of the flexible mechanical arm as q₁Then the deformation of the flexible mechanical arm is approximately:

w(x,t)＝φ₁(x)q₁(t) (2)

preferably, the control signal is constructed by the stiffness feedback coefficient, the damping feedback coefficient and the introduced nonlinear coupling feedback coefficient:

in the formula (3), τ₃For constructed control signals, k_p、k_dRepresenting the stiffness feedback coefficient and the damping feedback coefficient of the motor, representing the desired rigid strut angular displacement and velocity, respectively, and χ representing the non-linearly coupled feedback term of the formation.

Wherein the desired angular displacement of the rigid strutAnd angular velocityAre all zero, the nonlinear coupling term of the structure isWherein e is₁Representing the coupling feedback coefficient of the motor, the control signal τ in equation (3)₃Can be adjusted as follows:

preferably, wherein the feedback coefficient k of the motor rigidity is adjusted_pThe vibration frequencies of the flexible mechanical arm and the vibration absorber meet the degree of common relation of 2:1, and the two vibration frequencies are shown as a formula (7); and (3) establishing a vibration control equation of the system by using a Kane method, and performing Taylor expansion on the nonlinear term, wherein the final simplification result is shown as a formula (5) and a formula (6).

In formulae (5) and (6):

preferably, the external excitation is applied to the end of the free end of the flexible mechanical armThe obtained vibration control equation is shown as the formula (8) and the formula (9).

In formulae (8) and (9):

preferably, the single-mode solution and the double-mode solution of the steady-state response are obtained by a multi-scale method, as shown in formulas (10) to (17).

Single-mode solution:

a₂＝0 (11)

bimodal solution:

preferably, on the basis of the existence of 2:1 internal resonance of the flexible mechanical arm and the vibration absorber, the saturation phenomenon reveals an energy transfer relationship between the external excitation and the internal resonance mode when the flexible mechanical arm is subjected to the external excitation of the 1:1 main resonance type. When the external excitation amplitude is larger than the critical pointNamely, when the external excitation threshold value is reached, the flexible mechanical arm enters a nonlinear response area and passes through a channel for internal resonance energy exchange between the flexible mechanical arm and the vibration absorber to enable a first-order mode of the flexible mechanical armThe state amplitude is restrained; when the external excitation amplitude is continuously increased, the first-order modal amplitude of the flexible mechanical arm is saturated and redundant energy is transmitted to the vibration absorber modal. Therefore, the control of the main resonance of the flexible mechanical arm can be realized based on the saturation principle of the main resonance vibration absorber.

Compared with the prior art, the invention has the advantages that:

aiming at the defects of the related research of the main resonance control of the flexible mechanical arm, a vibration suppression method is provided, and the problem of the main resonance control of the flexible mechanical arm with large scale, large flexibility and small damping at present is solved; the method comprises the steps that a cross-mode energy exchange channel between a flexible mechanical arm and a main resonance vibration absorber is constructed by utilizing a nonlinear saturation principle, when the amplitude of external excitation exceeds a saturation threshold, energy is transferred into the vibration absorber and is dissipated through damping of the main resonance vibration absorber; therefore, the modal amplitude of the flexible mechanical arm cannot be increased along with the increase of the external excitation amplitude, the control of the main resonance of the flexible mechanical arm is achieved, and the dynamic performance of the flexible mechanical arm is improved.

Drawings

FIG. 1 is a schematic view of the overall structure of a flexible manipulator according to the present invention;

fig. 2 is a control block diagram of a vibration absorber motor in the present invention;

FIG. 3 is a simulation diagram of transverse vibration mode of the flexible mechanical arm without a vibration absorber;

FIG. 4 is a simulation diagram of the transverse vibration mode of the flexible mechanical arm when the vibration absorber is installed in the invention;

FIG. 5 is a modal corresponding curve of the vibration absorber to the first-order mode and the vibration absorber mode of the flexible mechanical arm under the undamped condition in the present invention;

FIG. 6 shows the small external resonance detuning (ε 0.05, σ) of the present invention₂0.1), internal resonance full tuning (σ)₁0) and Γ₁Response curve < 0;

FIG. 7 shows the complete tuning of internal and external resonances (e ═ 0.05, σ) in accordance with the present invention₁＝σ₂0) and Γ₁Response curve < 0;

FIG. 8 is a comparison graph of the response of the end of the main resonance of the flexible manipulator without the vibration absorber and under the optimal state of the vibration absorber in the numerical simulation process of the present invention.

The symbols in fig. 1 are illustrated as follows:

1 denotes a hinge joint, 2 denotes a rigid robot arm, 3 denotes a flexible robot arm, 4 denotes a swing link of a vibration absorber motor mover, 5 denotes a vibration absorber motor, and 6 denotes a main resonance vibration absorber system.

The symbols in fig. 2 are illustrated as follows:

w_Dindicating the response of the end of the flexible robot arm,representing a first order modal assumption function, q₁Representing a first order modal response of the flexible manipulator, e₁Representing the coupling feedback coefficient, k, of the motor_pRepresenting the stiffness feedback coefficient, k, of the motor_dRepresenting the damping feedback coefficient of the motor, s representing the proportionality coefficient, beta_dRepresenting the desired angular displacement of the rigid strut, beta representing the actual angular displacement of the shock absorber pendulum, tau₃Representing the motor control torque.

The specific implementation mode is as follows:

the present invention will be described in further detail with reference to the following drawings and specific examples, but the present invention is not limited thereto.

The invention provides a main resonance control method of a flexible mechanical arm, which combines an internal resonance mechanism, a main resonance mechanism and a saturation mechanism to realize vibration control of the mechanical arm.

The method comprises nonlinear dynamics modeling, main resonance analysis and saturation mechanism analysis of the flexible mechanical arm with the vibration absorber. In order to clearly and concisely illustrate the main resonance suppression method proposed by the present invention, a simplified model is used in fig. 1 instead of a specific flexible mechanical arm model.

The method comprises the following specific steps:

the method comprises the following steps: making reasonable assumptions on the flexible mechanical arm and the joint; the material of the flexible mechanical arm is assumed to be uniform and isotropic, and the structural characteristics of the flexible mechanical arm meet the characteristics of an Euler-Bernoulli beam; the deformation of the flexible mechanical arm only occurs in the transverse direction, and the longitudinal deformation of the flexible mechanical arm is not counted; the influence of the flexibility of the joint on the flexible mechanical arm is not considered, and the rotational inertia of the joint is ignored relative to the overall rotational inertia of the flexible mechanical arm.

Discretizing the transverse bending deformation of the flexible mechanical arm by using a modal hypothesis method to obtain the deformation of the flexible mechanical arm as follows:

in the formula (1), w (x, t) is the transverse deformation of the flexible mechanical arm,representing the ith order mode function, q_i(t) represents the modal coordinates.

As shown in the modal simulations of fig. 3 and 4, of the flexural deformation of the flexible mechanical arm, the vibration energy of the first-order mode (n ═ 1) is large and plays a major role in the lateral vibration response, so that the first-order mode is selected as the controlled mode for vibration control of the vibration absorber. Recording the first-order modal response of the flexible mechanical arm as q₁Then the deformation of the flexible mechanical arm is approximately:

w(x,t)＝φ₁(x)q₁(t) (2)

step two: constructing a vibration absorber control model, wherein the control object of the vibration absorber is to design a motor control moment tau₃So that the vibration absorber can dynamically track the vibration response change of the flexible mechanical arm, and the energy of the controlled flexible mechanical arm is transferred to the vibration absorber and dissipated based on the internal resonance principle, and the control block diagram of the motor of the vibration absorber is shown in fig. 2.

The vibration absorber adopts a feedback control principle, and forms linear feedback by introducing a rigidity feedback coefficient and a damping feedback coefficient of the vibration absorber; on the basis of linear feedback, introducing a vibration response signal of the flexible mechanical arm, and constructing a nonlinear coupling term for adjusting the motion coupling relation of the vibration absorber and the flexible mechanical arm; and finally, coupling the linear term and the nonlinear term to form a control signal model:

in the formula (3), τ₃For constructed control signals, k_p、k_dRepresenting the stiffness feedback coefficient and the damping feedback coefficient of the motor, representing the desired rigid strut angular displacement and velocity, respectively, and χ representing the non-linearly coupled feedback term of the formation.

In which desired angular displacement of the rigid strutAnd angular velocityAre all zero, the nonlinear coupling term of the structure isWherein e₁Representing the coupling feedback coefficient of the motor, the control signal τ in equation (3)₃Can be adjusted as follows:

step three: assuming that the vibration absorber is a lumped parameter system and the mass of the rigid support rod is equivalent to the end mass; and (3) establishing a flexible mechanical arm vibration control model containing the main resonance vibration absorber by using a Kennel method, and establishing a flexible mechanical arm dynamic model and a coordinate system of each component, which are included in the main resonance vibration absorber, according to the model assumption in the step one, as shown in the figure 1.

Because the flexible mechanical arm has light weight, the corner deformation caused by transverse bending vibration is very small, and the calculation of the generalized force is slightly influenced by the corner deformation, the flexible mechanical arm has the advantages that the flexible mechanical arm is light in weight, the calculation of the generalized force is slightly influenced by the corner deformationThe corner change caused by transverse bending can be ignored in the mechanical arm dynamics modeling process; the generalized rate is taken as:

first-order modal response q of transverse vibration of flexible mechanical arm₁And the response beta of the rigid strut of the vibration absorber is a generalized coordinate; substituting the generalized main force, the generalized inertia force and the control moment of the vibration absorber into the Kane equationRecording the first-order modal damping of the flexible mechanical arm as f₁The vibration control equation of the system is established as follows:

in formulae (5) and (6):

the vibration frequency of the flexible mechanical arm and the main resonance vibration absorber can be obtained by a vibration control equation as follows:

wherein, the motor rigidity feedback coefficient k is adjusted_pEnabling the vibration frequency of the flexible mechanical arm and the vibration absorber to meet the 2:1 axiality relation, so as to form internal resonance, wherein the vibration energy of the rigid-flexible coupling mechanical arm can be transferred into the vibration absorber in a cross-mode manner, and as shown in fig. 5, an internal resonance energy exchange curve is formed; damping feedback coefficient k of regulating motor_dThe damping of the vibration absorber can be adjusted, so that the energy is dissipated, and the optimal damping feedback coefficient k can be obtained by controlling the variable method_d(ii) a Coupling feedback coefficient e₁The energy exchange frequency and amplitude between the vibration absorber and the flexible mechanical arm system can be adjusted, specific numerical values need to be selected according to the actual vibration condition of the flexible mechanical arm system, and the vibration absorber is guaranteed to be efficient and quickly achieve energy dissipation and vibration reduction.

Step four: applying external excitation to the end of the free end of a flexible manipulatorEstablishing a weak forced vibration dynamics model, wherein a vibration control equation is as follows:

in the formula:

the following dimensionless parameters are introduced into the equations (8) and (9) for the subsequent solution, and the parameters in the equations are subjected to dimensionless processing.

The dimensionless equations are obtained as shown in formula (18) and formula (19).

Step five: solving the steady state solution of equations (18) and (19) by using a multi-scale method, firstly introducing a small quantity parameter 0 & ltepsilon & lt 1, and substituting variables in a dimensionless equation to specify:

the object studied by the invention is the main resonance of the flexible mechanical arm under the action of small-amplitude external excitation in a certain frequency band range, so that the external excitation is weak, and the external excitation is assumed as follows:

substituting the above substituted variables into equations (18) and (19), omitting ε²And higher terms, to give ε⁰、ε¹Corresponding two equations, let ε⁰The solution to the equation of order is:

note the bookWill epsilon⁰Substituting the solution of the equation of order into ε¹The equation for order yields:

where cc denotes the conjugate of the preceding terms, NST denotes all terms that do not contribute long term terms to the solution, and (') denotes T for slow varying times₁The first derivative of (a).Representing a correction factor that accounts for the coupling feedback factor.

In the following step 2: based on the existence of 1 internal resonance, two detuning parameters sigma are introduced₁And σ₂：

ω_S2＝0.5ω_S1+εσ₁ (26)

ω_S3＝ω_S1+εσ₂ (27)

According to the formulas (26) and (27), the solvability of the formulas (24) and (25) is searched, and the solvability is extracted from the formulas (24) and (25) respectivelyEliminating the long term, resulting in solvability conditions:

for the first order approximate solution of the forced vibration equation, only A is considered₁、A₂Is a slowly varying time T₁As a function of (c). Let complex function A in equation (28) and equation (29)₁、A₂Expressed in polar form:

in the formula, a₁、α₁、a₂、α₂The following equations (28) and (29) are used to transform the equations (28) and (29) into an autonomous system, and are introduced:

γ₁＝2σ₁T₁+2α₂-α₁ (31)

a is to be₁、A₂The polar coordinate expression (30) of (2) is substituted for expressions (28) and (29), and a 'can be obtained by using the imaginary part and the real part as 0'₁、a₁α′₁、a′₂、a₂α′₂The expression of (1); when analyzing the steady-state response of the primary resonance, the amplitude of the response no longer changes with time, so that a of the primary resonance occurs_iAnd phase angle gamma_iAre all constant, bringing this condition into a'₁、a₁α′₁、a′₂、a₂α′₂The expression (c) is obtained by mathematical transformation:

the steady state response solution is performed on equations (33) - (36), the first result being a monomodal solution a₁≠0、a₂0, expressed as linear vibration; the second result is a bimodal solution a₁≠0、a₂Not equal to 0, expressed as nonlinear vibration, the conditions are substituted into equations (33) to (36), and the result is obtained:

single-mode solution:

a₂＝0 (11)

bimodal solution:

step six: analyzing the existence forms of the monomodal solution and the bimodal solution to obtain an external excitation saturation threshold value, so that a saturation mechanism can be revealed; according to the formula (10), the unimodal solution is independent of the nonlinear coefficient, the modal amplitude of the flexible mechanical arm and the external excitation amplitude are in a linear relation, and the vibration absorber does not work; as shown in the formula (12), the bimodal solution is a nonlinear solution in which a is directly excited₁Modal amplitude to external excitationIndependent of the vibration absorber damping coefficient and the physical parameters of the flexible mechanical arm.

Thus, for a in the non-linear solution₂The condition which needs to be satisfied when a real number solution exists is analyzed, and the dimensionless external excitation amplitude parameter can be determinedTwo critical values of (a).

The real number solution of the comprehensive linear solution and the nonlinear solution exists in a form, and the steady state response of the system is as follows:

(1) external excitation amplitude

Only one linear solution is present and,a₂＝0。

(2) external excitation amplitude

If gamma is₁＜0，a₂There are three solutions, one linear solution, two non-linear solutions:

if gamma is₁If the value is more than 0, only one linear solution is obtained,a₂＝0。

(3) when the external excitation amplitude is equal toWhen a is₂There are two solutions, a linear solution, a non-linear solution:

amplitude of external excitation modeFor controlling parameters, numerical simulation is carried out to obtain the controlled mode of the flexible mechanical arm and the motion mode curve of the rigid supporting rod, as shown in comparison between fig. 6 and 7, when the amplitude of the external excitation mode is largeIncrease from 0First-order modal excitation and modal amplitude a of long and flexible mechanical arm₁Gradually increasing, at the moment, the vibration mode of the vibration absorber is not excited temporarily, and the mode amplitude a₂Keeping to zero, the vibration absorber does not start to work, and the flexible mechanical arm moves linearly under the linear solution state.

When the modal amplitude a of the flexible mechanical arm₁Reaches a critical valueAt this time, the linear solution is no longer stable, and at this time, the nonlinear solution can become a stable solution, the modal amplitude a of the vibration absorber₂No longer remains zero. The amplitude of external excitation increases, the first-order mode of the flexible mechanical arm tends to be saturated, and the mode amplitude is stabilized at a fixed value(namely saturation amplitude), the vibration absorber starts to work, the vibration amplitude of the flexible mechanical arm reaches an energy saturation state, all additional energy caused by external excitation permeates into the mode of the vibration absorber from the first-order mode of the flexible mechanical arm, and therefore the control on the main resonance of the flexible mechanical arm can be realized, and fig. 8 is a terminal response comparison diagram of the main resonance of the flexible mechanical arm in the optimal state without the vibration absorber and the vibration absorber in the numerical simulation process.