CN107469702B - Nonlinear chaotic vibrating spear - Google Patents

Nonlinear chaotic vibrating spear Download PDF

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CN107469702B
CN107469702B CN201710657775.XA CN201710657775A CN107469702B CN 107469702 B CN107469702 B CN 107469702B CN 201710657775 A CN201710657775 A CN 201710657775A CN 107469702 B CN107469702 B CN 107469702B
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chaotic
eccentric block
formula
rotating shaft
curve
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CN107469702A (en
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殷玉枫
郑瑶
李闯
姚德臣
高崇仁
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Taiyuan University of Science and Technology
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B01—PHYSICAL OR CHEMICAL PROCESSES OR APPARATUS IN GENERAL
    • B01F—MIXING, e.g. DISSOLVING, EMULSIFYING OR DISPERSING
    • B01F31/00—Mixers with shaking, oscillating, or vibrating mechanisms
    • B01F31/44—Mixers with shaking, oscillating, or vibrating mechanisms with stirrers performing an oscillatory, vibratory or shaking movement
    • B01F31/449—Stirrers constructions

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Abstract

The invention relates to a nonlinear chaotic vibrating spear, which belongs to a nonlinear chaotic vibrating spear used in the industries of buildings, foods, medicines and the like. The vibrating rod mainly solves the technical problems that the existing vibrating rod is difficult to achieve a material area comprehensively due to vibration excitation, is not compact in vibration and is not uniform in stirring. The technical scheme adopted by the invention is as follows: a nonlinear chaotic vibrating rod comprises a motor, a transmission flexible shaft and a chaotic rod head, wherein: the chaotic rod head comprises a motor, a chaotic rod head, a transmission flexible shaft, an eccentric block, a transmission flexible shaft, a rotating shaft, an eccentric block and a working cavity, wherein the motor is connected with the driving end of the transmission flexible shaft, the rotating shaft is arranged in the inner cavity of the chaotic rod head, one end of the rotating shaft is fixed at an opening of the inner cavity of the chaotic rod head, the other end of the rotating shaft is provided with the eccentric block, the tail end of the transmission flexible shaft is connected with the fixed end of the rotating shaft, the L-section curve of the: g (x) ═ acos (bx) + d, and then verifying whether the L-section curve meets the critical condition that the eccentric block achieves the chaotic motion.

Description

Nonlinear chaotic vibrating spear
Technical Field
The invention relates to a nonlinear chaotic vibrating spear, which belongs to a nonlinear chaotic vibrating spear used in the industries of buildings, foods, medicines and the like.
Background
The vibrating spear is widely used in the present society, so whether the selection of the vibrating spear is reasonable or not and whether the stirring degree is uniform or not directly influence the quality of products. The vibrating rod is a mechanical tamping tool, and bubbles in a substance are removed through vibration, so that gaps are eliminated. However, in the actual use process, the vibration rails of the existing vibrating spears are mostly round or regular patterns, the vibration excitation is difficult to reach the material area comprehensively, so that the air bubbles cannot be released completely, the vibration is not compact, the stirring is not uniform, and the vibration stirring quality is seriously influenced.
Chaos refers to unpredictable, random-like motion exhibited by a deterministic, macroscopic, nonlinear system under certain conditions. The mechanical method is adopted, the theory of nonlinear chaotic dynamics is introduced into the vibrating rod technology, and the vibrating rod is excited to reach each area of the substance to be stirred through chaos, so that stirring is particularly uniform, and the vibrating stirring quality is greatly improved.
Disclosure of Invention
The invention aims to solve the technical problems that the existing vibrating spear is difficult to achieve a material area comprehensively due to vibration excitation, is not compact in vibration and is not uniform in stirring, and provides a nonlinear chaotic vibrating spear.
In order to solve the technical problems, the invention adopts the technical scheme that:
a nonlinear chaotic vibrating rod comprises a motor, a transmission flexible shaft and a chaotic rod head, wherein: the chaotic rod head comprises a motor, a chaotic rod head, a transmission flexible shaft, an eccentric block, a transmission flexible shaft, a bearing, an eccentric block, a rotating shaft, an inner cavity of the chaotic rod head, a rotating shaft, an inner wall of the chaotic rod head, an L-shaped curve of the inner wall of the chaotic rod head from the fixed end of the rotating shaft to the working cavity of the eccentric block along the cross section of an axis, a parabolic section with small curvature and a small curvature, wherein the L-shaped curve is set firstly: g (x) acos (bx) + d, wherein g (x) is the distance from the contact position of the rotating shaft and the inner wall of the chaotic rod head to the axis of the chaotic rod head, and x is the distance from the contact position of the rotating shaft and the inner wall of the chaotic rod head to the fixed end; and verifying whether the L-section curve meets the critical condition that the eccentric block reaches the chaotic motion, if the L-section curve meets the critical condition of the chaotic motion, the curve is the section curve of the inner wall of the chaotic rod head from the fixed end of the rotating shaft to the working cavity of the eccentric block along the axis, otherwise, another curve needs to be additionally arranged, and the verifying step is repeated until the critical condition that the eccentric block generates the chaotic motion is met.
Further, the step of verifying whether the L-segment curve meets the critical condition of the chaotic motion of the eccentric block is as follows:
(1) calculating the stiffness of the shaft
Figure BDA0001369764730000021
In the formula: y-amplitude at the free end of the shaft; a is1-a taylor series expansion constant term; b1-expanding the first order coefficients of the taylor series; c. C1TaylorExpanding a quadratic term coefficient by series;
(2) establishing nonlinear power equation of eccentric block at free end of rotating shaft
Figure BDA0001369764730000022
In the formula: p ═ me ω2M-eccentric mass; e-eccentricity; omega-the angular velocity of the beam of light,
Figure BDA0001369764730000023
n-rotation speed, c-damping coefficient;
substituting formula (1) into formula (2)
Figure BDA0001369764730000024
In the formula:
Figure BDA0001369764730000025
epsilon is a small amount of,
Figure BDA0001369764730000026
(3) α judged by Melnikov function1、α2、α3Whether the critical condition that the eccentric block achieves the chaotic motion is met
Figure BDA0001369764730000027
In the formula η1-the imaginary part of the first pole,
Figure BDA0001369764730000028
η2-the imaginary part of the second pole,
Figure BDA0001369764730000029
(4) comprehensively judging whether the power system of the eccentric block generates chaos or not by combining the phase plane trajectory diagram, the time-course curve diagram and the Poincare mapping diagram
The eccentricity e and the damping coefficient c determined by the formula (3) of the critical condition are substituted into the formula (2-1), and the obtained power system is
Figure BDA00013697647300000210
Drawing a phase plane trajectory diagram, a time course curve diagram and a Poincare mapping diagram by utilizing matlab software according to the power system, and comprehensively judging whether the power system of the eccentric block is chaotic or not by combining the phase plane trajectory diagram, the time course curve diagram and the Poincare mapping diagram; and if the chaos occurs, verifying that the L-section curve is a section curve generating the chaos motion.
By adopting the technical scheme, the chaotic rod head inner wall curve is designed according to the critical condition that the chaotic motion occurs to the rotating shaft, the eccentric block is enabled to reach the chaotic motion state in the high-speed rotating process, and the rod body is driven to generate nonlinear chaotic vibration under the excitation action of the eccentric block. The non-linear chaotic motion is never repeated in the track in a limited area, and the characteristics of extremely complex motion state and ergodicity are utilized, so that the technical problems that the common vibrating stirring rod is difficult to completely reach a material area, the vibrating is not compact and the stirring is not uniform are solved.
Drawings
FIG. 1 is a schematic structural view of the present invention;
FIG. 2 is a schematic view of a chaotic stick head of the nonlinear chaotic vibratory stick of the present invention;
FIG. 3 is a phase plane trajectory diagram of the eccentric mass of the present invention;
FIG. 4 is a graph of the time course of the eccentric mass of the present invention;
fig. 5 is a Poincare map of the eccentric mass of the present invention.
Detailed Description
The invention is described in further detail below with reference to the figures and examples.
As shown in fig. 1 and fig. 2, the nonlinear chaotic vibration rod in the present embodiment includes a motor 1, a flexible transmission shaft 2, and a chaotic rod head 7, wherein: the chaotic stick comprises a motor 1, a transmission flexible shaft 2, a rotating shaft 4 is arranged in the inner cavity of a chaotic stick head 7 in the horizontal direction, one end of the rotating shaft 4 is fixed at an opening of the inner cavity of the chaotic stick head 7 through a bearing 3, an eccentric block 6 is arranged at the other end of the rotating shaft 4, the tail end of the transmission flexible shaft 2 is connected with the fixed end of the rotating shaft 4, an L-section curve of the inner wall 5 of the chaotic stick head from the fixed end of the rotating shaft to the working cavity of the eccentric block 6 along the cross section of an axis is a parabolic section with small curvature, and the L-section curve is firstly set: g (x) acos (bx) and d, wherein g (x) is the distance from the contact position of the rotating shaft 4 and the inner wall 5 of the chaotic rod head to the axis of the chaotic rod head 7, x is the distance from the contact position of the rotating shaft 4 and the inner wall 5 of the chaotic rod head to the fixed end, and a, b and d are constants and are more than 0; and verifying whether the L-section curve meets the critical condition that the eccentric block 6 achieves the chaotic motion, if the L-section curve meets the critical condition of the chaotic motion, the curve is the section curve of the inner wall 5 of the chaotic rod head from the fixed end to the working cavity of the eccentric block 6 along the axis, otherwise, another curve needs to be set, and the verifying step is repeated until the critical condition that the eccentric block 6 generates the chaotic motion is met.
The steps for verifying whether the L-section curve meets the critical condition that the eccentric block 6 reaches the chaotic motion are as follows:
(1) calculating the stiffness of the shaft 4
Figure BDA0001369764730000041
In the formula: y-amplitude at the free end of the shaft; a is1-a taylor series expansion constant term; b1-expanding the first order coefficients of the taylor series; c. C1-coefficients of taylor series expansion quadratic terms;
(2) establishing a nonlinear equation of power of the eccentric block at the free end of the rotating shaft 4
Figure BDA0001369764730000042
In the formula: p ═ me ω2M-eccentric mass; e-eccentricity; omega-the angular velocity of the beam of light,
Figure BDA0001369764730000043
n-rotation speed, c-damping coefficient;
substituting formula (1) into formula (2)
Figure BDA0001369764730000044
In the formula:
Figure BDA0001369764730000045
epsilon is a small amount of,
Figure BDA0001369764730000046
(3) α judged by Melnikov function1、α2、α3Whether the critical condition that the eccentric block achieves the chaotic motion is met
Figure BDA0001369764730000047
In the formula η1-the imaginary part of the first pole,
Figure BDA0001369764730000048
η2-the imaginary part of the second pole,
Figure BDA0001369764730000049
(4) comprehensively judging whether the power system of the eccentric block generates chaos or not by combining the phase plane trajectory diagram, the time-course curve diagram and the Poincare mapping diagram
The eccentricity e and the damping coefficient c determined by the formula (3) of the critical condition are substituted into the formula (2-1), and the obtained power system is
Figure BDA00013697647300000410
Drawing a phase plane trajectory diagram, a time course curve diagram and a Poincare mapping diagram by utilizing matlab software according to the power system, and comprehensively judging whether the power system of the eccentric block is chaotic or not by combining the phase plane trajectory diagram, the time course curve diagram and the Poincare mapping diagram; and if the chaos occurs, verifying that the L-section curve is a section curve generating the chaos motion.
The implementation verification process of the invention comprises the following steps:
in the working process of the vibrating rod, the inner wall of the chaotic rod head plays a supporting role, so that an L-section curve g (x) of the cross section of the inner wall of the chaotic rod head from the free end of the rotating shaft 4 to the working cavity of the eccentric block along the axial line is smaller than the curvature of a flexible line, the L-section curve equation is set to be g (x) ═ acos (bx) + d (a, b and d are constants), and then verification is carried out according to calculation.
(1) Determining the rigidity of the L-section curve g (x) and the rotating shaft 4 under the support of the inner wall of the chaotic rod head 7
Figure BDA0001369764730000051
Substituting y ═ g (x) ═ acos (bx) + d into formula (1-1), and expanding according to Taylor series to obtain:
Figure BDA0001369764730000052
will be provided with
Figure BDA0001369764730000053
Substituting the formula (1-2) to obtain:
Figure BDA0001369764730000054
order to
Figure BDA0001369764730000055
Obtaining:
Figure BDA0001369764730000056
so that it satisfies the following conditions:
Figure BDA0001369764730000057
during the operation of the vibrating rod, the inner wall of the rod head supports the rotating shaft, so that the curve g (x) has smaller curvature than the bending line of the rotating shaft without support, and the bending line equation of the rotating shaft without support is used
Figure BDA0001369764730000061
The conditions in (1) and (4) are such that a is 0.001, b is 0.5, and d is-0.0016. That is, g (x) is 0.001cos (0.5x) -0.0016.
Respectively substituting the values of a, b and d into a1，b1And c1Obtaining:
Figure BDA0001369764730000062
a is to1，b1And c1Substituting the value of (2) into the formula (1) to obtain:
Figure BDA0001369764730000063
(2) establishing a nonlinear equation of power of the eccentric block at the free end of the rotating shaft 4
Figure BDA0001369764730000064
In the formula: c-damping coefficient; p-excitation force, P ═ me ω2(ii) a e-eccentricity; omega-angular velocity; n-the rotation speed;
Figure BDA0001369764730000065
the mass m of the eccentric block is 0.1kg, the exciting force P is 50N, and the rotating speed N is 2840 r/min.
Substituting the expression (1-5) into the expression (2) to obtain the nonlinear dynamical equation of the eccentric block as follows:
Figure BDA0001369764730000066
wherein
Figure BDA0001369764730000067
Figure BDA0001369764730000068
(3) α judged by Melnikov function1、α2、α3Whether the critical condition that the eccentric block 6 achieves the chaotic motion is met
The Melnikov function is defined as:
Figure BDA0001369764730000069
wherein
Figure BDA00013697647300000610
Then
Figure BDA00013697647300000611
Calculated as follows:
Figure BDA00013697647300000612
Figure BDA0001369764730000071
in the formula:
Figure BDA0001369764730000072
Figure BDA0001369764730000073
ζ -the real part of the two poles of the integrand,
Figure BDA0001369764730000074
η1-the imaginary part of the first pole,
Figure BDA0001369764730000075
η2-the imaginary part of the second pole,
Figure BDA0001369764730000076
when the Melnikov function has a zero point, the perturbed system considered has a chaos in the sense of Smale horseshoe, i.e. M (τ) is 0:
Figure BDA0001369764730000077
in order to make equation (3-4) rooted, sin ω (t + τ) must be made less than or equal to 1
Figure BDA0001369764730000078
Thus, the eccentric mass is obtained to achieve the chaotic motion α1、α2、α3The critical conditions that must be met:
Figure BDA0001369764730000079
Figure BDA00013697647300000710
α will be mixed1、α2And α3The value of (3) is substituted into the formula (3), and from the formula (3-5), it is found that e and c exist so that the formula (3) holds, that is, α1、α2And α3The value of (A) can meet the critical condition of the eccentric block generating chaotic motion.
(4) And comprehensively judging whether the power system of the eccentric block 6 is chaotic or not by combining the phase plane trajectory diagram, the time-course curve diagram and the Poincare mapping diagram.
According to the critical condition of chaos occurrence, e is 0.005653m, c is 0.1, and the power system is obtained by substituting the formula (2-1):
Figure BDA0001369764730000081
according to the power system, matlab software is utilized, the initial condition is that y is-0.0006,
Figure BDA0001369764730000082
drawing a phase plane trajectory diagram, a time-course curve diagram and a Poincare mapping diagram, and comprehensively judging whether the power system of the eccentric block generates chaos.
It can be seen from the phase plane trajectory diagram shown in fig. 3, the time course graph shown in fig. 4 and the Poincare mapping diagram shown in fig. 5 that the time course curve is irregular, the phase plane trajectory is just inverted 8, the Poincare mapping is more chaotic, and the group of parameters fall in a chaotic region given by the Melnikov function, so that the power system corresponding to the judgment formula (2-1) enters a chaotic state, and the rotating shaft moves in a chaotic manner under the support of the inner wall of the rod shell. Therefore, the curve of the section of the inner wall (5) of the chaotic rod head along the axial line from the fixed end of the rotating shaft to the working cavity of the eccentric block (6) is determined to be L-shaped (x) ═ 0.001cos (0.5x) -0.0016.

Claims (2)

1. The utility model provides a nonlinear chaotic vibrating spear, includes motor (1), transmission flexible axle (2) and chaotic stick head (7), its characterized in that: motor (1) be connected with the drive end of transmission flexible axle (2), the inner chamber of chaos stick head (7) in the horizontal direction be equipped with pivot (4), the opening part at chaos stick head (7) inner chamber is fixed through bearing (3) in the one end of pivot (4), the other end of pivot (4) is equipped with eccentric block (6), the end of transmission flexible axle (2) is linked with the stiff end of pivot (4), chaos stick head inner wall (5) are followed pivot stiff end to eccentric block (6) working chamber and are the parabola section of a section of little camber along the L section curve of axis cross-section, establish earlier L section curve does: g (x) ═ a cos (bx) + d, wherein g (x) is the distance from the contact position of the rotating shaft (4) and the chaotic rod head inner wall (5) to the axis of the chaotic rod head (7), and x is the distance from the contact position of the rotating shaft (4) and the chaotic rod head inner wall (5) to the fixed end; and then verifying whether the L-section curve meets the critical condition that the eccentric block (6) achieves the chaotic motion, if the L-section curve meets the critical condition of the chaotic motion, the curve is the section curve of the inner wall (5) of the chaotic rod head from the fixed end of the rotating shaft to the working cavity of the eccentric block (6) along the axis, otherwise, another curve needs to be arranged, and the verifying step is repeated until the critical condition that the eccentric block (6) generates the chaotic motion is met.
2. A nonlinear chaotic vibratory rod in accordance with claim 1, wherein: the step of verifying whether the L-section curve meets the chaos motion critical condition of the eccentric block (6) is as follows:
(1) calculating the stiffness of the shaft (4)
Figure FDA0002370206650000011
In the formula: y-amplitude at the free end of the shaft; a is1-a taylor series expansion constant term; b1-expanding the first order coefficients of the taylor series; c. C1-coefficients of taylor series expansion quadratic terms;
(2) establishing a nonlinear equation of power of the eccentric block at the free end of the rotating shaft (4)
Figure FDA0002370206650000012
In the formula: p ═ me ω2M-eccentric mass, e-eccentricity, ω -angular velocity,
Figure FDA0002370206650000013
n-rotation speed, c-damping coefficient;
substituting formula (1) into formula (2)
Figure FDA0002370206650000014
In the formula:
Figure FDA0002370206650000015
epsilon is a small amount of,
Figure FDA0002370206650000016
(3) α judged by Melnikov function1、α2、α3Whether the critical condition that the eccentric block achieves the chaotic motion is met
Figure FDA0002370206650000021
In the formula:
η1-the imaginary part of the first pole,
Figure FDA0002370206650000022
η2-the imaginary part of the second pole,
Figure FDA0002370206650000023
(4) comprehensively judging whether the power system of the eccentric block generates chaos or not by combining the phase plane trajectory diagram, the time-course curve diagram and the Poincare mapping diagram
The eccentricity e and the damping coefficient c determined by the formula (3) of the critical condition are substituted into the formula (2-1), and the obtained power system is
Figure FDA0002370206650000024
Drawing a phase plane trajectory diagram, a time course curve diagram and a Poincare mapping diagram by utilizing matlab software according to the power system, and comprehensively judging whether the power system of the eccentric block is chaotic or not by combining the phase plane trajectory diagram, the time course curve diagram and the Poincare mapping diagram; and if the chaos occurs, verifying that the L-section curve is a section curve generating the chaos motion.
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