CN116992588A - Four-machine reverse driving self-balancing vibrating machine and parameter determination method - Google Patents
Four-machine reverse driving self-balancing vibrating machine and parameter determination method Download PDFInfo
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Abstract
The invention belongs to the technical field of vibration equipment, and discloses a four-machine back-drive self-balancing vibration machine and a parameter determination method. The four endoplasmids are symmetrically distributed on the exosomes; the four vibration exciters are respectively arranged at the mass center of the inner mass body, the rotation directions of the two vibration exciters in the same vertical direction are the same, and the rotation directions of the two vibration exciters in the same horizontal direction are opposite; by utilizing the vibration self-synchronization principle, a dynamic model and a motion differential equation are established, a synchronicity criterion for realizing reverse synchronous operation and a stability condition for keeping a synchronous state stable are deduced, and a stable area for realizing a self-balancing function of the system is finally defined. The invention realizes the circular motion trail of four endoplasma, and effectively improves the working quality and efficiency of the vibrator; the exciting forces received by the exosomes are mutually offset to achieve the minimum transmission to the basic load, and the vibration isolation problem of the system is effectively solved; operating in the first sub-resonance region, the motor power required by the vibration system is reduced under the same amplitude condition.
Description
Technical Field
The invention relates to the technical field of vibration equipment, in particular to a four-machine back-drive self-balancing vibration machine and a parameter determination method.
Background
The self-synchronizing vibration equipment is one of the core hosts for material processing in the industrial process flow, and can realize the functions of vibration crushing, conveying, feeding, screening, dewatering, ball milling, grinding, polishing, drying, cooling, forming, compacting, aging, separating, tamping, stirring, shakeout and the like of materials by utilizing the vibration effect. In the engineering field, the most typical material treatment modes are vibration screening, dehydration, ball milling and the like. The vibration screening device is widely applied to screening of materials such as coal, ore, metal, chemical industry and the like, and when the vibration screening device works, a single or a plurality of eccentric vibration exciters transmit exciting force to the screen box, so that the screen surface can realize circumferential, elliptical or linear motion and the like, further continuous throwing motion of the materials is realized, and the diameter of the screen holes is used as a limit when the materials fall onto the screen surface, so that classification of the materials is realized. The existing vibration ball milling equipment is mainly applied to industries such as cement, building materials and nonferrous metal ore dressing, a roller of the vibration ball mill rotates under the excitation force generated by an exciter, materials to be ground and grinding bodies in the roller rotate along with the roller body when the equipment runs, the materials and the grinding bodies are thrown down due to the action of gravity when rotating to a certain height, and collision and friction occur between the materials and the grinding bodies in the falling process, so that the functions of crushing the materials and finishing the surface are realized. The equipment such as vibration screening, dehydration, ball milling and the like has very high utilization rate and very large application market in industrial production. The conventional above-described vibration apparatus causes the following problems:
1. When the traditional vibration equipment meets the self vibration function requirement, excessive noise can be generated, the basic load is transferred to the equipment, and the defects can cause the vibration of the building in the peripheral range of the factory with the equipment as the center, influence the peripheral environment and finally influence the health and normal life of human beings.
2. When the traditional vibration equipment runs, parts of the traditional vibration equipment are damaged in an accelerating way due to vibration, so that the equipment is unstable in running and reduced in function, and the processing and maintenance cost of the equipment is increased.
3. The traditional vibration equipment is mostly single-machine and double-machine driven single-body or double-body, and has low working efficiency, low productivity, low energy consumption, poor technological effect and the like.
Along with the continuous and intensive research on the vibration exciter synchronization theory, an advanced vibration synchronization technology is applied, and a vibration machine which can meet the requirements on functions and performances of equipment and realize the self-balancing function of the equipment is designed, so that the vibration machine has the advantages of high productivity, energy conservation, vibration suppression, noise reduction, environmental protection requirement satisfaction and the like. The patent creatively proposes a new self-balancing theory of a self-synchronizing system for the first time, which is one of effective ways for realizing the functional advantages of the vibration equipment.
Disclosure of Invention
The invention belongs to self-synchronous driving and self-balancing vibration conveying/feeding/sieving/dewatering/ball milling/grinding/polishing/drying/cooling/separating/stirring/shakeout equipment and the like. In order to overcome the defects in the prior art, the technical scheme of the invention is as follows:
a four-machine back-drive self-balancing vibratory machine comprising: four vibration exciters, a plastid, a spring A11 and a spring B8; the plastids comprise four inner plastids and one outer plastid 1; each inner body is connected with the outer body 1 through a spring A11, and four inner bodies are symmetrically distributed on the outer body 1 to form a rectangle; the exosome 1 is connected with the foundation through a spring B8; the four vibration exciters are respectively arranged at the mass centers of the four inner plastids; the vibration exciter comprises an induction motor and an eccentric rotor; each exciter being driven by a respective induction motor, the eccentric rotor being respectively about a respective rotation axis centre o 1 、o 2 、o 3 、o 4 Rotating, wherein the rotation directions of two vibration exciters in the same vertical direction are the same, and the rotation directions of two vibration exciters in the same horizontal direction are the sameThe opposite direction. The four machines are driven reversely and automatically, so that the self-balancing function of the equipment and the circular track movement function of the working mass are realized; therefore, the working efficiency of the processes such as vibration screening, ball milling and the like is improved, and the working quality of the equipment is improved.
Two vibration exciters positioned on one side inside the outer body 1 rotate anticlockwise, and two vibration exciters on the other side rotate clockwise.
The excitation frequency of the vibration exciter is not more than the main natural frequency omega of the system 0 and ω4 When the system is in the sub-resonance region, the self-vibration suppression function can be realized, the output of the system is improved by realizing the circular motion track of four working bodies, the working region is selected in the first sub-resonance region, the exciting force required by the system for exciting the same amplitude in the working region is 1/5-1/3 of that required by the system under the ultra-far resonance condition, and the power of a driving motor required by the vibration system is reduced.
A parameter determining method of a four-machine back-drive self-balancing vibrating machine comprises the following steps:
step 1, establishing a dynamic model and a system motion differential equation;
establishing a coordinate system: four vibration exciters respectively around respective rotation central axes o 1 ，o 2 ，o 3 and o4 Rotating; the rotation angles of the four eccentric rotors respectively; connecting line o between four inner plastid centroids and outer plastid 1 centroids i The angles between the-O and the positive direction of the x axis are respectively beta 1 ，β 2 ，β 3 and β4 A representation; the four vibration exciters are respectively arranged at the mass centers of the inner mass bodies, the rotation centers of the vibration exciters are coaxial with the mass centers of the inner mass bodies, the swing angles of the four inner mass bodies are ignored, and only the swing response psi of the outer mass bodies is considered. The degree of freedom of the whole four-machine back-driving five-mass vibration system is as follows: the five plastids respond in the x-direction and the y-direction respectively, i.e. x i ，y i I=1, 2,3,4,5, oscillation of the exosome in the direction ψ, and rotational phase angles of the four exciters
According to Lagrange deducing, motion differential equation matrix expressions of the four-machine reverse driving five-mass vibration system in the x direction and the y direction are respectively as follows:
wherein ,
M 1 ＝m 1 +m 01 ，M 2 ＝m 2 +m 02 ，M 3 ＝m 3 +m 03 ，M 4 ＝m 4 +m 04 ，M 5 ＝m 5
in the formula ,Mx 、M y The mass matrix of the five-mass system in the x and y directions is respectively driven by four machines in a reverse way; f (F) x 、F y Damping matrixes of the four-machine reverse driving five-mass system in the x and y directions are respectively adopted; k (K) x 、K y Respectively driving stiffness matrixes of the five-mass system in the x and y directions in a four-machine reverse mode; x, Y are response matrices of the four-machine back-driven five-mass vibration system in the x direction and the y direction respectively;five-mass vibration system driven by four machines in opposite directionsUnifying velocity matrices in the x-direction and the y-direction;acceleration matrixes of the four-machine reverse driving five-mass vibration system in the x direction and the y direction are respectively adopted; q (Q) x and Qy The external excitation matrix of the five-mass vibration system in the x direction and the y direction is reversely driven by four machines;
simultaneously, the four-machine back-driving five-mass system is obtained about the degrees of freedom psi and phiThe motion differential equation of (2) is that,
wherein ,
J ψ ＝J m5 +(m 01 +m 02 +m 03 +m 04 )(l 0 2 +r 2 )；
in the formula ,l0 -the distance of each exciter centre of rotation from the system centroid O; r-the eccentric radius of the four vibration exciters; m is m 0n -eccentric rotor mass of exciter n, n=1, 2,3,4; m is m i -inner partMass of plastid i, i=1, 2,3,4; m is M i -mass sum of the inner mass i and the eccentric rotor mounted thereon, i = 1,2,3,4; m is M 5 -mass of the exosomes; j (J) 0n Moment of inertia of the induction motor of exciter n, n=1, 2,3,4, j 0n ＝m 0n r 2 ；J m5 -moment of inertia of the vibration isolation body; j (J) ψ -moment of inertia of the whole system; t (T) en -electromagnetic torque of the induction motor of the exciter n, n=1, 2,3,4; f (f) w Spring k w Damping coefficients in x and y directions, w=1, 2,3,4,5; k (k) w Spring k w Stiffness coefficients in x and y directions, w=1, 2,3,4,5; where w=1, 2,3,4 is the spring a, k 5 Is a spring B; the stiffness and damping coefficients of the springs A and B in the x and y directions are respectively equal; f (f) ψ -damping coefficient of the system in the direction ψ; k (k) ψ -stiffness coefficient of the system in the direction ψ; f (f) dn -motor shaft damping coefficient of induction motor n, n=1, 2,3,4; l (L) x1 -horizontal distance from the point of connection of the spring a connected to the left side of the endosome to the centre of the endosome; l (L) x2 -horizontal distance from the point of connection of the spring a connected to the left side of the inner mass to the outer mass center; l (L) x3 -horizontal distance from the point of connection of the spring B connected to the right side of the exosome to the centre of the exosome; l (L) y1 -vertical distance from the connection point of the spring a connected to the right side of the endosome to the centre of the endosome; l (L) y2 -vertical distance from the point of connection of the spring a connected to the left side of the endosome to the centre of the endosome; l (L) y3 -vertical distance from the connection point of the spring a connected below the endosome to the centre of the endosome; l (L) y4 -vertical distance from the connection point of the spring a connected below the inner mass and the outer mass to the centre of the inner mass; l (L) y5 -vertical distance from the connection point of the spring B connected below the exosome and the exosome to the centre of the exosome;
step 2, determining the response of a four-machine reverse driving five-mass vibration system;
in the stable running state of the system, the average value of the rotation phases of the four eccentric rotors is recorded asThe mean angular velocity transient is recorded as +.>The phase difference between adjacent eccentric rotors is marked as +.> The value of phi is the rotational phase matrix of the four eccentric rotors, which is as follows,
Φ＝Φ m +R α α (3)
wherein ,
in the formula ,Φm Is the average phase matrix of the eccentric rotor, alpha is the phase difference matrix, R α Is a phase coefficient matrix;
the four vibration exciters apply to the vibration exciting force of the four-machine reverse driving five-mass vibration system to periodically change, and the system vibration is also periodic; the least common multiple of four exciting force change periods is taken as T 0 Let T be 0 Inner omega m0 (t) average value of omega m ； andFor omega m The instantaneous fluctuation coefficients of (a) are respectively epsilon 0 ，ε h H=1, 2,3, there are,
ε＝[ε 1 ε 2 ε 3 ] T
wherein epsilon is a phase difference instantaneous fluctuation matrix;
substituting equation (3) into equation (4) to obtain instantaneous angular velocity matrix of four eccentric rotorsAnd instantaneous angular acceleration matrix->The following are provided;
wherein C= [ 11 11 ]] T As a constant matrix, v=ε 0 C+R α Epsilon is the instantaneous angular velocity fluctuation matrix of the four eccentric rotors,is the instantaneous angular acceleration fluctuation matrix of four eccentric rotors, V= [ V ] 1 ν 2 ν 3 ν 4 ] T ；
When the four-machine back-driven five-mass vibration system runs stably, the change of the instantaneous angular acceleration of the four eccentric rotors is small, in order to ensure that the system has good structural symmetry, the instantaneous angular acceleration of the four eccentric rotors is not considered, the designed four internal masses and four vibration exciters are all identical, the parameters of the spring A11 are kept consistent,
m 0n ＝m 0 ，f dn ＝f d0 ，M i ＝M 0 ，k w ＝k 0 ，f w ＝f 0 ，i,n,w＝1,2,3,4 (6)
according to the transfer function method, the response of the four-machine reverse driving five-mass vibration system in the x direction, the y direction and the psi direction is obtained
wherein ,
in the formula ,Fx and Fy Stress matrix of vibration response, C γ and Sγ Cosine and sine phase matrices of the vibration response, respectively;
wherein ,r m ＝m 0 /M，M＝4M 0 +M 5 +4m 0 ，z ψ ＝ω m /ω nψ r l ＝l 0 /l e ，
m-the total mass of the entire vibration system; l (L) e -equivalent radius of rotation of the whole vibration system about its centroid; r is (r) m -mass ratio of standard eccentric rotor to whole vibration system; omega nψ -natural frequency of the vibration system in the direction ψ; zeta type toy nψ Damping ratio of the whole vibration system in the psi direction; gamma ray i -is the hysteresis angle between the plastid response and the exciter, i= 1,2,5,21.
Step three, determining the synchronism condition of the four vibration exciters;
irrespective of system damping f 1 ，f 2 ，f 3 ，f 4 ，f 5 Substituting the formula (6) into the formula (1) to obtain a stiffness matrix K and a mass matrix M of the four-machine back-drive five-mass vibration system in the x and y directions as follows;
the characteristic equation of the four-machine back-driving five-mass vibration system is that,
when delta (omega) 2 ) When the vibration system is=0, the natural frequency of the four-machine back-drive five-mass vibration system in the x direction and the y direction is calculated as follows,
irrespective of natural frequency omega 5 For only the critical natural frequency omega of the system 0 and ω4 Unfolding the study; differential calculation is performed on the equation in the formula (7) to obtain andSubstituting them into formula (2) and combining formula (5), ignoring the instantaneous angular acceleration fluctuation matrix of four eccentric rotors when the four-machine back-drive five-mass vibration system is in steady operation>Higher-order terms of (2) and then putting them in +.>Integrating to obtain four deviations The single-cycle average differential equation for the cardiac rotor is,
wherein ,
W snn ＝2F 1 cosγ 1 -l 0 F 21 cosγ 21 ，W cnn ＝2F 1 sinγ 1 -l 0 F 21 sinγ 21 ，n＝1,2,3,4
in the formula ,Te0n Omega for four motors m Electromagnetic torque, k, output during steady-state operation e0n Omega for four motors m Stiffness coefficient at steady state operation; in the above integration process, the phase difference α is determined by the integrated median value thereofIn the alternative to this, the first and second,
the four induction motors selected by the four-machine reverse driving five-mass vibration system are identical, and the rotational inertia and the damping of the four induction motor shafts are respectively equal, namely
J 0n ＝m 0 r 2 ，f dn ＝f d0 ，n＝1,2,3,4 (12)
Dividing both sides of formula (11) by m 0 r 2 ω m Post-writing into matrix form
Wherein a= [ a ] nq ] 4×4 ，B＝[b nq ] 4×4 ，u＝[u 1 u 2 u 3 u 4 ] T ；
in the formula ,for the period of operation T 0 Instantaneous average angular velocity of the internal four exciter induction motors with respect to +.>The non-dimensional average disturbance parameter matrix A and the matrix B are a non-dimensional inertial coupling matrix and a non-dimensional rigidity coupling matrix of the four eccentric rotors respectively, and u represents non-dimensional load torque of the four eccentric rotors;
four-machine back-driving five-mass vibration system with frequency omega m During synchronous operation, four induction motors are operated in the operation period T 0 The average velocity within the range is kept consistent,the finishing type (13) is carried out to obtain,
in the formula ,is the kinetic energy of a standard vibration exciter, < >>Representing four induction motors at frequency omega m Payload torque during synchronous operation; in the formula (13) > andThe difference in the interaction between the two,
in the formula ,(Te0n -f dn ω m )-(T e0q -f dq ω m) and respectively representing the frequencies omega of motors n and q m During synchronous operation, the difference between the effective electromagnetic output torque and the effective load torque;
from (14) let DeltaT be known 0nq 1.ltoreq.n < q.ltoreq.4 is concerned andIs a function of (2); tidying (15) to obtain dimensionless type
in the formula ,is about-> andIs a bounded function of the sum of the dimensionless load torques between the four induction motors, [ (T) e0n -f dn ω m )-(T e0q -f dq ω m )]/T u Indicative senseThe difference of dimensionless effective electromagnetic output torque when the motor n and the induction motor q synchronously run is used; let τ be cnqmax Is a bounded function->Maximum value of (1)
The conditions for achieving synchronous operation of the four-machine back drive system according to formulas (16) and (17), i.e.
The meaning of formula (18) is: the four-machine reverse driving five-mass vibration system is used for realizing synchronous operation of four-machine reverse driving, and the absolute value of the difference of the dimensionless effective electromagnetic output torque between any two motors is smaller than or equal to the maximum value of the difference of dimensionless load torque of the two motors; when the formula (18) is satisfied, a synchronous solution of the phase difference and the operating frequency between the eccentric rotors under the synchronous operating condition of the system can be obtained by respectively andAnd (3) representing.
Determining a system stability condition;
synchronously solving the phase difference of a five-mass vibration system driven by a four-machine reverse direction in (14) andLinear expansion of the first order, neglecting the damping coefficient f of the motor shaft dn Taking into account formula (5)
wherein , in the formula ,Is a function in parentheses-> andA value at;
the finishing (19) is carried out to obtain,
consider (4) simultaneous use of integral medianReplacement α, the last three equations in equation (20) are replaced with respect toI.e., a phase difference perturbation parameter matrix equation,
wherein C= [ C ] nh ] 3×3 ，
The characteristic equation of the phase difference disturbance system is obtained according to det (C-lambda I) =0 and is as follows
λ 3 +d 1 λ 2 +d 2 λ+d 3 ＝0 (22)
Based on the Routh-Hurwitz criterion, formula (21) relates toWhen the zero-solution of (2) is stabilized, the parameter in the formula (22) satisfies the following condition
d 1 ＞0,d 3 ＞0,d 1 d 2 ＞d 3 (23)
wherein ,d1 ＝-c 11 -c 22 -c 33 ，d 2 ＝-c 12 c 21 -c 23 c 32 -c 13 c 31 +c 11 c 22 +c 22 c 33 +c 33 c 11 ；
d 3 ＝-c 11 c 22 c 33 -c 12 c 23 c 31 -c 13 c 21 c 32 +c 11 c 23 c 32 +c 22 c 13 c 31 +c 33 c 12 c 21
Because ofFrom formula (5)>Therefore, when the four-machine back-driving five-mass vibration system meets the formula (23), the stable and synchronous operation of the four-machine back-driving system is realized; adjusting the formula (23) to obtain a system stability capability coefficient H 1 ，H 2 and H3 The following are listed below
H 1 ＝d 1 ＞0,H 2 ＝d 3 ＞0,H 3 ＝d 1 d 2 -d 3 ＞0 (24)。
The inner plastid is a working plastid, the outer plastid is a vibration isolation plastid,
the invention has the beneficial effects that: 1) The invention innovates on a dynamic model, four motors are arranged at the mass centers of four inner plastids, the four inner plastids are symmetrically arranged on an outer plastid, and the four motors are driven in a reverse self-synchronous way, so that the productivity and the working quality of the system are effectively improved.
2) The invention can realize the circular motion track of four inner bodies, and simultaneously offset the exciting forces borne by the outer bodies, thereby achieving the effect of transmitting the minimum basic load, reducing noise pollution, protecting the environment, ensuring the performance requirement of equipment and realizing the self-balancing function of the system.
3) The dynamic model provided by the invention selects the first sub-resonance area as the working area, reduces the motor power required by the vibration system, simultaneously realizes the functions of equipment yield increase and self-balancing, reduces the amplitude of exosomes, reduces the energy consumption, improves the energy utilization rate and protects the environment.
Drawings
Fig. 1 is a diagram of a kinetic model of a four-machine back-drive five-mass vibration system.
In the figure: 1. an exosome; 2. a second endoplasmic body; 3. a second vibration exciter; 4. a first endoplasmic body; 5. a first vibration exciter; 6. a fourth vibration exciter; 7. a fourth endoplasmic body; 8. a spring B;9. a third vibration exciter; 10. a third endoplasmic body; 11. and a spring A.
Meaning of each parameter in the figure: oxy- -an absolute coordinate system; o- -the center of the entire system; o (O) 1 -a first exciter rotation center; o (O) 2 -a second exciter rotation center; o (O) 3 -a third exciter rotation center; o (O) 4 -a fourth exciter rotation center;-a first exciter rotational phase angle;-the second exciter rotational phase angle;-a third exciter rotational phase angle;-a fourth exciter rotational phase angle; m is m 01 -the mass of the first exciter; m is m 02 -the mass of the second exciter; m is m 03 -the mass of the third exciter; m is m 04 -fourth exciter mass; m is m 1 -a first endoplasmic body mass; m is m 2 -a second endoplasmic body mass; m is m 3 -a third endoplasmic body mass; m is m 4 -a fourth endoplasmic body mass; m is m 5 -a fifth exosome mass; r- -the eccentricity of the exciter; k (k) 1 Spring A stiffness coefficient, k 1 ＝k 2 ＝k 3 ＝k 4 ；k 5 -the stiffness coefficient of the spring B; beta 1 -line o between the centroids of the first endoplasmid and the exosome 1 -O at an angle to the positive x-axis; beta 2 -line o between the centroids of the second endoplasmid and the exosome 2 -O at an angle to the positive x-axis; beta 3 -line o between the centroids of the third endoplasmid and the exosomes 3 -O at an angle to the positive x-axis; beta 4 -line o between the centroids of the fourth endoplasmid and the exosomes 4 -O at an angle to the positive x-axis; l (L) 0 -the distance between the centre of rotation of each exciter and the centre of mass O of the system; l (L) x1 -the horizontal distance from the connection point of the spring A connected to the left side of the inner mass and the inner mass to the center of the inner mass; l (L) x2 -the horizontal distance from the connection point of the spring A connected to the left side of the inner mass and the outer mass to the center of the inner mass; l (L) x3 -the horizontal distance from the connection point of the spring B connected to the right side of the outer body and the outer body to the center of the outer body; l (L) y1 -the vertical distance from the connection point of the spring A connected to the right side of the inner mass to the center of the inner mass; l (L) y2 -the vertical distance from the connection point of the spring A connected to the left side of the inner mass and the inner mass to the center of the inner mass; l (L) y3 -the vertical distance from the connection point of the spring A connected below the inner mass to the center of the inner mass; l (L) y4 -the vertical distance from the connection point of the spring A connected below the inner mass and the outer mass to the center of the inner mass; l (L) y5 -spring B connected under the outer body and outer bodyThe vertical distance from the plastid connection point to the centre of the outer plastid; psi-the angle at which the exosomes oscillate about the central axis.
Fig. 2 is a graph of the stability capacity coefficient of a four-machine back-driven five-mass vibration system.
Fig. 3 is a graph of steady-state phase relationship of a four-machine back-drive five-mass vibration system:
FIG. 3 (a) shows the stable phase difference between four vibration exciters; fig. 3 (b) shows the phase difference between the exciter and the mass.
FIG. 4 is a graph of steady state response of a four-machine back-drive five-mass vibration system.
Fig. 5 is a simulation result of the resonance region I: FIG. 5 (a) is a displacement of a mass in the x-direction; FIG. 5 (b) is a displacement of the mass in the y-direction; FIG. 5 (c) is a displacement of the plastid in the direction ψ; FIG. 5 (d) is a plastid motion profile; fig. 5 (e) is the rotational speeds of four motors; fig. 5 (f) shows the phase difference between the vibration exciters.
Fig. 6 is a simulation result of the resonance region III: FIG. 6 (a) is a displacement of a mass in the x-direction; FIG. 6 (b) is a displacement of the mass in the y-direction; FIG. 6 (c) is a displacement of the plastid in the direction ψ; FIG. 6 (d) is a plastid motion profile; fig. 6 (e) is the rotational speeds of four motors; fig. 6 (f) shows the phase difference between the vibration exciters.
FIG. 7 shows the test results at an excitation frequency of 12.8 Hz: FIG. 7 (a) is a displacement of a mass in the x-direction; FIG. 7 (b) is a displacement of the mass in the y-direction; FIG. 7 (c) is a displacement of the plastid in the direction ψ; FIG. 7 (d) is an enlarged view of the plastid motion profile; fig. 7 (e) is the rotational speeds of four motors; fig. 7 (f) shows the phase difference between the vibration exciters.
FIG. 8 shows the test results at an excitation frequency of 16.4 Hz: FIG. 8 (a) is a displacement of a mass in the x-direction; FIG. 8 (b) is a displacement of the mass in the y-direction; FIG. 8 (c) is a displacement of the plastid in the direction ψ; FIG. 8 (d) is an enlarged view of the plastid motion profile; fig. 8 (e) is the rotational speeds of four motors; fig. 8 (f) shows the phase difference between the vibration exciters.
Fig. 9 shows the test results at a power supply frequency of 19.2 Hz: FIG. 9 (a) is a displacement of a mass in the x-direction; FIG. 9 (b) is a displacement of the mass in the y-direction; fig. 9 (c) is a displacement of the plastid in the direction ψ; FIG. 9 (d) is an enlarged view of the plastid motion profile; fig. 9 (e) is the rotational speeds of four motors; fig. 9 (f) shows the phase difference between the vibration exciters.
Detailed Description
Example 1: numerical qualitative analysis of a four-machine reverse-drive five-mass mechanical system;
assume parameters of the vibration system: m is m 0i ＝10kg(i＝1,2,3,4)，m 1 ＝m 2 ＝m 3 ＝m 4 ＝m 0 ＝1000kg，m 5 ＝10000kg，J m5 ＝1430kg·m 2 ，k 1 ＝k 2 ＝k 3 ＝k 4 ＝k 0 ＝15000kN/m，k 5 ＝200kN/m，k ψ ＝2000kN·m/rad，r＝0.15m，β 1 ＝π/4，β 2 ＝3π/4，β 3 ＝-3π/4，β 4 = -pi/4. The main natural frequency is easily obtained according to the parameters of the vibration system: omega 0 ＝122rad/s，ω 4 =144 rad/s. Type of motor: three-phase squirrel cage, 50Hz,380V,6-pole,0.75kW, rated rotation speed: 980r/min. Setting motor parameters: rotor resistance R r =3.40Ω, stator resistance R s =3.35Ω, mutual inductance L m Rotor inductance l=164 mH r =170 mH, stator inductance L s ＝170mH。
According to the ratio z of the operating frequency to the natural frequency 0 and z4 Dividing the entire frequency domain into three resonance regions, where z 0 ＝ω m /ω 0 ，z 4 ＝ω m /ω 4 . The three resonance regions are respectively: (1) zone I: omega 0 and ω4 Is the sub-resonance region of (z) 0 < 1 and z 4 < 1; (2) zone II: omega 0 And ω 4 Is the sub-resonance region of (z) 0 > 1 and z 4 < 1; (3) zone III: omega 0 and ω4 Is z 0 > 1 and z 4 ＞1。
(a) Stability capability of the system;
substituting system parameters into H 1 ，H 2 and H3 The expression is used for obtaining the numerical solution of the stability capacity of the four-machine reverse driving five-plastid system. FIG. 2 is a graph of the stability factor of the system, showing that the stability factor of the system is greater than 0 over the entire resonance region, satisfying equation (24), indicating that the system is capable ofSo that synchronous and stable operation can be realized in all three resonance areas. When omega m ＜ω 0 In this case, the system stability coefficient value is low, but the curve change is gentle and less affected by the frequency ratio, so that the frequency ratio is more stable in this section than in a system with frequent change. H i At resonance point omega 0 and ω4 Near the maximum value and change sharply, H i The larger the value of (c) indicates a stronger system stability.
(b) Phase relation when the system is in steady state operation;
obtaining steady-state phase differences among the four vibration exciters according to the formula (18) and the formula (23)With operating frequency omega m The curve of the change is shown in fig. 3 (a). In the resonance regions I and III, the stable phase difference between the vibration exciters is +.>At this time, the inner plastid can realize circular motion, and the outer plastid keeps a static state, and has better self-balancing characteristic. Within the resonance region II, < >>The vibration forces of the inner mass are mutually overlapped, the outer mass has larger motion, and vibration isolation of the system can not be effectively realized.
Fig. 3 (b) is a graph showing a change in the hysteresis angle of the response of the mass and the exciting force in the frequency domain. As can be seen from FIG. 3 (b), γ 1 Stable around 0 ° in the resonance region I at the resonance point ω 0 The vicinity starts to rise to about 180 DEG, at omega 4 The vicinity suddenly drops to about 2 pi/3, then reverts to 180 deg., and stabilizes at that value throughout zone III. Gamma ray 2 Respectively stabilized around 180 °, 0 ° and 180 ° in the resonance regions I, II, III, at the resonance point ω 0 and ω4 A significant change in the vicinity occurs. Likewise, gamma 5 and γ21 At omega m ＜ω 4 At approximately 180 DEG up to an operating frequency omega m Run to resonance point omega 4 In the vicinity of the point of the approach,γ 5 gradually decreasing to 0 deg. and maintaining this value all the way through zone III, while gamma 21 Not at resonance point omega 4 Changes occur nearby, stabilizing at 180 ° throughout the resonance region. Gamma ray i (i=1, 2, 5) have a pronounced resonance effect in the vicinity of the resonance point, so that in engineering applications the excitation frequency of the vibration device is generally set far from the natural frequency ω 0 and ω4 Where it is located.
(c) Steady state response of system
Substituting the system parameters into equation (7) to obtain the system amplitude in the x and y directions, as shown in FIG. 4, λ i Corresponding to the magnitudes of the five masses at steady state. In the figure, the four internal bodies have the same amplitude, and lambda is in the resonance region I i (i=1, 2,3, 4) with ω m Increasing and increasing, near omega 0 Suddenly rising to the maximum value; in zone II, four endoplasmic bodies have larger amplitudes up to ω m ＝ω 4 The time begins to fall; in zone III, with ω m Gradually increase lambda i (i=1, 2,3, 4) gradually decreases, but its value is always greater than 0. Amplitude lambda of exosomes 5 Is always 0 in zones I and III, which is the same as in zones I and III shown in FIG. 3 (a)The vibration force borne by the outer body is counteracted, the bearing force is 0, the amplitude of the vibration force is 0, the dynamic load transmitted to the foundation is 0, and the system has a good self-balancing function. The system is at natural frequency omega 0 and ω4 Resonance effect occurs at the location, the amplitude lambda of the exosomes 5 Rapidly increasing to a maximum value, and is within region II shown in FIG. 3 (a)>The conditions are consistent, under the conditions, the system is larger in transmitted base load, and the self-balancing function of the system is poorer.
Example 2: simulation analysis of a four-machine back-drive five-mass mechanical system;
for further analysis and verification of the results of the numerical qualitative analysis, a simulation of the different resonance regions was given by the range-Kutta methodAnd (3) carrying out numerical solution on parameters such as the motor rotating speed, the system response, the motion trail and the like of the true point, and analyzing the numerical solution. If the system parameters are selected in the resonance areas I and III, the vibration forces transmitted by each inner mass to the outer mass can be mutually offset while the equipment function requirements are met, the load of the system to the foundation is reduced, the self-balancing and effective noise reduction functions of the system are realized, and the function has stronger engineering application, so that simulation analysis is only carried out on simulation points in the resonance areas I and III. Vibration system parameters and motor parameters are given above. In order to obtain the motion state of the system in different areas, the spring rate k is generally changed 0 To adjust omega 0 and ω4 Is a value of (2).
(a) Simulation results for region I
Adjusting spring rate k 0 =20000 kN/m, and the other parameter settings are the same as those of example 1. The set of parameters corresponds to the natural frequency of the system as andThe corresponding frequency ratio is z 0 =0.64 and z 4 =0.54, and l in fig. 3 (a) 1 Corresponding to each other.
As can be seen in FIG. 5 (e), the synchronous steady rotational speed of the system is about 860r/min (90 rad/s). When the system operates for 150 seconds, pi/2 interference is applied to the second vibration exciter 3, and the rotation speed of the system is restored to the rotation speed before interference after short fluctuation, which indicates that the synchronous state of the system is stable. In FIG. 5 (f), the phase difference between the four exciters is 2α during steady state operation of the system 1 ＝2α 3 ＝0°，2α 2 ＝2α 4 The vibration forces of the inner mass cancel each other out at 180 °, the outer mass is stressed at 0, the response is 0, and the load transferred to the base is theoretically also 0, so the system under this set of parameters has a strong self-balancing function, consistent with the analysis in fig. 3 (a).
Fig. 5 (a) and (b) depict the response of five plastids in the x and y directions, respectively. From the graph, it can be seen that the response amplitudes of the four endosomes in the x and y directions are the same, about 7.6mm each, and the response amplitude of the exosomes is 0. The steady state response of the system remains consistent before and after the disturbance is applied at 150 s. In fig. 5 (c), the angle of oscillation of the outer mass is about 0 °, indicating that the outer mass is theoretically at rest. Fig. 5 (d) is a motion trace diagram of five plastids, in which four inner plastids perform circular motion with the centroid of the plastid as the center, and the outer plastids are in a stationary state. If the working area of the vibration equipment is selected in the resonance area I, the inner mass not only can realize effective high vibration intensity to meet the yield increasing requirement of the equipment, but also can reduce the dynamic load transmitted to the foundation by the equipment and reduce the negative influence on the surrounding environment.
(b) Simulation results for region III
Adjusting spring rate k 0 =6000 kN/m, the other parameters being unchanged. FIG. 6 (e) shows that the corresponding rotation speed is 980r/min when the four motors are synchronously and stably operated, and the frequency ratio corresponding to the set of parameters is z 0 ＝1.33，z 4 =1.12, and l of region III in fig. 3 (a) 2 Has the same dynamic characteristics. Fig. 6 (e) shows that the motor speed is consistent before and after the disturbance is applied, indicating that the system has strong disturbance rejection capability.
In FIG. 6 (f), the phase difference between the four exciters is stabilized to 2α due to the coupling torque and the load torque between the motors 1 ＝2α 3 ＝0°，2α 2 ＝2α 4 The dynamic characteristics of the system are similar to those of the region I, vibration forces transmitted by four inner bodies under the phase difference are mutually counteracted, the stress of the outer bodies is 0, and the load transmitted to the foundation by the system is 0 at the moment, so that the effective vibration isolation of the system is realized. In addition, FIGS. 6 (a) - (d) show that the response amplitudes of the endosomes in the x and y degrees of freedom are the same, both about 0.9mm, while the four endosomes are in circular motion and the outer body is stationary. Compared with the resonance area I, the system in the resonance area III can realize good vibration isolation and noise reduction functions, but the equipment in the area I has low running frequency and high vibration intensity, so that the yield is increased and the energy consumption is reduced. Therefore, in engineering practice, the working point of the vibration equipment can be selected in the I area, and the requirement of the equipment for material treatment can be met Basic function requirements, and can realize the functions of increasing yield, vibration isolation and noise reduction of the system.
Example 3: test analysis of a four-machine back-drive five-mass mechanical system;
in order to further verify the correctness of theory and numerical analysis, a test bed is built according to the model, and test research is carried out. The four selected motor types are YZS-2.5-2 (380V, 50Hz,0.22kW, excitation force 0-2.5kN, rated rotation speed 2720 r/min). The parameters of the vibration synchronous test system are as follows: m is m 1 ＝m 2 ＝m 3 ＝m 4 ＝m 0 ＝17.71kg，m 5 ＝128.70kg，m 0i ＝1.23kg，J m5 ＝2.34kg·m 2 ，k 1 ＝k 2 ＝k 3 ＝k 4 ＝k 0 ＝256.90kN/m，k 5 139.38kN/m, r=0.025 m. The main natural frequency is easily obtained according to the parameters of the vibration system: omega 0 ＝120.44rad/s，ω 4 = 151.28rad/s. Four internal bodies are symmetrically arranged on the external body, four motors are respectively arranged at the mass centers of the four internal bodies, wherein the eccentric rotors of the first vibration exciter 5 and the fourth vibration exciter 6 rotate clockwise, and the eccentric rotors of the second vibration exciter 3 and the third vibration exciter 9 rotate anticlockwise. In the test, the power supply frequency of the motor is adjusted through the frequency converter to obtain different motor rotating speeds, and the exciting force of the vibration exciter can be adjusted through adjusting the included angle of the eccentric block. The rotating speeds and phases of the four vibration exciters in the test bed are measured by Hall sensors, and the responses of the five plastids in all directions are measured by unidirectional acceleration sensors. The pulse signals measured by the sensor are collected and filtered through an intelligent signal collection and processing analyzer (INV 306 DF), required test data are obtained through matlab software, and finally a test result diagram is drawn. And respectively carrying out synchronous test on four-machine reverse-drive five-mass vibration test tables with power supply frequencies of 12.8Hz, 16.4Hz and 19.2 Hz.
(a) Test result of 12.8Hz of power supply frequency
Before the test, the power supply frequency of the four motors is adjusted to be 12.8Hz, and then the four vibration motors are simultaneously supplied with power, and the test result is shown in figure 7. As can be seen from fig. 7 (a) - (c), the vibration system vibrates severely within 50s after power-on, which is because the motor frequency will resonate through the lower-order natural frequency of the system during the increase of the motor speed, and the resonance response gradually decreases due to the damping of the system. The motor frequency is continuously increased to the power supply frequency, at the moment, the coupling moment between the motors enables the four vibration exciters to realize synchronous operation, and the vibration system is stably maintained in the synchronous state.
As shown in FIG. 7 (e), when the system is operating stably in a synchronous state, the motor speed is stabilized around 761r/min, i.e., 79.68rad/s, at which time z 0 ≈0.65，z 4 And approximately 0.52, corresponding to resonance region I (first sub-resonance region). FIG. 7 (f) shows the phase differences between the four vibration exciters after severe vibration during the motor start-up phase, which are measured by the test set, stabilized at 2α 1 ≈1.5°～10.5°，2α 2 ≈172.3°～180.2°，2α 3 ≈-4.4°～4.4°。
Fig. 7 (a) - (c) show the response of five plastids in the x and y directions and the angle of oscillation of the exosomes in the ψ direction, respectively. The response of the system when operating stably in synchronous mode can be seen as: in the x direction, the single amplitude maximum of the endosome is about 0.37mm, and the exosome is about 0.04mm; in the y direction, the single amplitude maximum of the endosome is about 0.38mm, and the exosome is about 0.12mm; in the direction phi, the maximum value of the swing angle of the outer body is smaller than 0.3 degrees. As can be seen from the enlarged view, the vibration response of each mass changes periodically, the responses of the first and third mass bodies 4 and 2 and the fourth mass body 7 in the x direction are close to the same phase change, respectively, while the vibration forces of the four mass bodies cancel each other out when the responses of the outer mass bodies are close to 0 and the load transferred to the foundation is also approximately 0 in the y direction. Fig. 7 (d) is an enlarged view of the movement trace of the plastid, and it can be seen that the four inner plastids in the test move in a nearly circular motion and the outer plastid in a nearly stationary state.
The test results show that the system in the group of tests can realize synchronous and stable operation, has stronger self-balancing capability and vibration strength, and basically keeps consistent with the simulation results of the resonance area I.
(b) Test results with a supply frequency of 16.4Hz
The power supply frequency of each motor was adjusted to 16.4Hz, and the results of the related test are shown in FIG. 8. Similar to the test result under the 12.8Hz power supply frequency, the resonance response of the vibration system in the starting stage is gradually reduced under the damping effect, and the system gradually reaches a stable synchronous running state under the effect of the motor coupling moment along with the increase of the motor rotating speed.
FIG. 8 (e) is a graph of motor speed showing motor synchronous speed 960r/min (100.48 rad/s), at which time z 0 ≈0.83，z 4 Approximately 0.66, corresponding to the resonance region I in fig. 3. FIG. 8 (f) is a phase difference diagram among the four vibration exciters, the phase difference among the four vibration exciters is 2α in the steady state of the system 1 ≈2.4°～8.4°，2α 2 ≈173.9°～183.3°，2α 3 Approximately equal to-5.8 degrees to 0 degrees. In test 2 alpha i The values of (i=1, 2, 3) all fluctuate around the theoretical value, but the difference from the theoretical value is smaller, and the synchronous and stable operation of the system is not affected.
Fig. 8 (a) - (c) can see that the system steady state response is: in the x direction, the amplitudes of the four inner plastids are approximately equal and are about 0.77mm, and the amplitude of the outer plastid is about 0.10mm; in the y direction, the amplitude of the four endosomes is about 0.72mm, and the amplitude of the exosomes is about 0.05mm; in the direction ψ, the response maximum of the exosomes is below 0.3 °. As can be seen from the enlarged view, the response delay angle difference in the x direction of the first and third internal bodies 4 and 10, the second and fourth internal bodies 2 and 7 is about 0 °, and the response delay angle difference in the y direction is about 180 °.
And the vibration force and the exciting force borne by the outer body are about 0 when the system synchronously and stably runs, the response and the load transmitted to the foundation are about 0, and the system can realize the stronger vibration intensity of the inner body and can also display stronger self-balancing capability. It can be seen in fig. 8 (d) that the four inner plastids in the trial were approximately circular in motion and the outer plastids were stationary. The set of test results are basically consistent with the numerical analysis results and simulation results of the resonance region I.
(c) Test result with a supply frequency of 19.2Hz
The test results for each motor at a power frequency of 19.2Hz are shown in figure 9. Similar to the test results when the power supply frequency is 12.8Hz and 16.4Hz, the vibration system generates strong vibration in the initial stage after being electrified, then the rotating speed of the motor is quickly increased to the synchronous operation speed, and the system realizes stable synchronous operation under the action of the coupling torque between the motors.
FIG. 9 (e) is a graph of motor speed at a supply frequency of 19.2Hz, with four motors having synchronous speeds of 1117r/min, i.e., 116.91rad/s, at which time z 0 ≈0.97，z 4 And approximately 0.77, corresponding to the first sub-resonance region (region I) of the system. FIG. 9 (f) is a graph showing the phase difference between four exciters under the test condition of 19.2Hz, the fluctuation range of the phase difference is 2α 1 ≈3.7°～10.7°，2α 2 ≈177.2°～185.2°，2α 3 Approximately-3.6-6.7 degrees. In test 2 alpha i The values of (i=1, 2 and 3) all fluctuate around the theoretical value obtained by numerical analysis, but the fluctuation error is smaller, and the system can still realize synchronous and stable operation.
Fig. 9 (a) - (c) are response diagrams of the system: the amplitude of the four endosomes was about 1.27mm in the x-direction and about 1.19mm in the y-direction; the amplitude of the exosomes in x-direction is about 0.09mm and in y-direction is about 0.09mm; the maximum pivot angle of the exosome in the direction ψ is 0.4 °. As can be seen from the enlarged view, the responses of the first and third internal volumes 4 and 10, the second and fourth internal volumes 2 and 7 in the x-direction are approximately in phase change, and the responses in the y-direction are approximately in phase change. Fig. 9 (d) is a motion trace diagram of the plastids, consistent with test results of 12.8Hz and 16.4Hz of power supply frequency, the four plastids approximately show circular motion, the vibration forces of the four plastids cancel each other, the outer plastids are in a static state, and the load transmitted to the foundation by the system is almost 0 at the moment, so that effective vibration isolation and noise reduction of the system are realized.
The stable phase differences of the system in the three groups of test results are slightly different from the simulation results, but are consistent in nature. The reason for the deviation may be that the included angles of the eccentric blocks of the four vibration exciters are deviated, so that the excitation forces generated by the four vibration exciters are different. It is also possible that the test bed structure is not perfectly symmetrical due to machining errors. Three groups of test results show that the system realizes the effective vibration intensity of four inner bodies in the resonance area I and simultaneously realizes the vibration suppression of the outer bodies, and effectively reduces the load transmitted to a foundation by the system, which is required in engineering, and can provide reference for the design of self-synchronous vibration screening and ball milling equipment with a self-balancing function.
Claims (7)
1. The four-machine back-driving self-balancing vibrating machine is characterized by comprising: four vibration exciters, a plastid, a spring A (11) and a spring B (8); the plastids comprise four inner plastids and one outer plastid (1); each inner body is connected with the outer body (1) through a spring A (11), and four inner bodies are symmetrically distributed on the outer body (1) to form a rectangle; the exosome (1) is connected with the foundation through a spring B (8); the four vibration exciters are respectively arranged at the mass centers of the four inner plastids; the vibration exciter comprises an induction motor and an eccentric rotor; each exciter being driven by a respective induction motor, the eccentric rotor being respectively about a respective rotation axis centre o 1 、o 2 、o 3 、o 4 The two vibration exciters in the same vertical direction rotate in the same direction, and the two vibration exciters in the same horizontal direction rotate in opposite directions.
2. The four-machine back-drive self-balancing vibrator according to claim 1, wherein the two exciters inside the outer body (1) on one side rotate counterclockwise and the two exciters on the other side rotate clockwise.
3. The four-machine back-drive self-balancing vibrator according to claim 1 or 2, wherein the excitation frequency of the exciter is not greater than the main natural frequency ω 0 。
4. A method of determining parameters of a four-machine back-drive self-balancing vibratory machine as recited in claim 3, comprising the steps of:
step 1, establishing a dynamic model and a system motion differential equation;
establishing a coordinate system: four vibration exciters respectively around respective rotation central axes o 1 ，o 2 ，o 3 and o4 Rotating;the rotation angles of the four eccentric rotors respectively; connection o between four centroids of inner plastid and centroids of outer plastid (1) i The angles between the-O and the positive direction of the x axis are respectively beta 1 ，β 2 ，β 3 and β4 A representation; the degree of freedom of the whole four-machine back-driving five-mass vibration system is as follows: the five plastids respond in the x-direction and the y-direction respectively, i.e. x i ，y i I=1, 2,3,4,5, oscillation of the exosome in the direction ψ, and rotation phase angles of the four exciters +.>
According to Lagrange deducing, motion differential equation matrix expressions of the four-machine reverse driving five-mass vibration system in the x direction and the y direction are respectively as follows:
wherein ,
M 1 ＝m 1 +m 01 ，M 2 ＝m 2 +m 02 ，M 3 ＝m 3 +m 03 ，M 4 ＝m 4 +m 04 ，M 5 ＝m 5
in the formula ,Mx 、M y The mass matrix of the five-mass system in the x and y directions is respectively driven by four machines in a reverse way; f (F) x 、F y Damping matrixes of the four-machine reverse driving five-mass system in the x and y directions are respectively adopted; k (K) x 、K y Respectively driving stiffness matrixes of the five-mass system in the x and y directions in a four-machine reverse mode; x, Y are response matrices of the four-machine back-driven five-mass vibration system in the x direction and the y direction respectively; The speed matrixes of the four-machine reverse driving five-mass vibration system in the x direction and the y direction are respectively adopted;acceleration matrixes of the four-machine reverse driving five-mass vibration system in the x direction and the y direction are respectively adopted; q (Q) x and Qy The external excitation matrix of the five-mass vibration system in the x direction and the y direction is reversely driven by four machines;
simultaneously, the four-machine back-driving five-mass system is obtained about the degrees of freedom psi and phiThe motion differential equation of (2) is that,
wherein ,
J ψ ＝J m5 +(m 01 +m 02 +m 03 +m 04 )(l 0 2 +r 2 )
in the formula ,l0 -the distance of each exciter centre of rotation from the system centroid O; r-the eccentric radius of the four vibration exciters; m is m 0n -eccentric rotor mass of exciter n, n=1, 2,3,4; m is m i Mass of endoplasmid i, i=1, 2,3,4; m is M i -mass sum of the inner mass i and the eccentric rotor mounted thereon, i = 1,2,3,4; m is M 5 -mass of the exosomes; j (J) 0n Moment of inertia of the induction motor of exciter n, n=1, 2,3,4, j 0n ＝m 0n r 2 ；J m5 -moment of inertia of the vibration isolation body; j (J) ψ -moment of inertia of the whole system; t (T) en -electromagnetic torque of the induction motor of the exciter n, n=1, 2,3,4; f (f) w Spring k w Damping coefficients in x and y directions, w=1, 2,3,4,5; k (k) w Spring k w Stiffness coefficients in x and y directions, w=1, 2,3,4,5; where w=1, 2,3,4 is the spring a, k 5 Is a spring B; the stiffness and damping coefficients of the springs A and B in the x and y directions are respectively equal; f (f) ψ -damping coefficient of the system in the direction ψ; k (k) ψ -stiffness coefficient of the system in the direction ψ; f (f) dn Motor shaft damping of induction motor nCoefficients, n=1, 2,3,4; l (L) x1 -horizontal distance from the point of connection of the spring a connected to the left side of the endosome to the centre of the endosome; l (L) x2 -horizontal distance from the point of connection of the spring a connected to the left side of the inner mass to the outer mass center; l (L) x3 -horizontal distance from the point of connection of the spring B connected to the right side of the exosome to the centre of the exosome; l (L) y1 -vertical distance from the connection point of the spring a connected to the right side of the endosome to the centre of the endosome; l (L) y2 -vertical distance from the point of connection of the spring a connected to the left side of the endosome to the centre of the endosome; l (L) y3 -vertical distance from the connection point of the spring a connected below the endosome to the centre of the endosome; l (L) y4 -vertical distance from the connection point of the spring a connected below the inner mass and the outer mass to the centre of the inner mass; l (L) y5 -vertical distance from the connection point of the spring B connected below the exosome and the exosome to the centre of the exosome;
step 2, determining the response of a four-machine reverse driving five-mass vibration system;
Step 3, determining the synchronism condition of the four vibration exciters;
and 4, determining system stability conditions.
5. The method for determining parameters of a four-machine back-drive self-balancing vibratory machine according to claim 4, wherein the determining four-machine back-drive five-mass vibratory system response is specifically;
in the stable running state of the system, the average value of the rotation phases of the four eccentric rotors is recorded asThe mean angular velocity transient is recorded as +.>The phase difference between adjacent eccentric rotors is sequentially recorded asLet phi be the rotational phase of the four eccentric rotorsThe matrix, whose values are as follows,
Φ＝Φ m +R α α (3)
wherein ,
in the formula ,Φm Is the average phase matrix of the eccentric rotor, alpha is the phase difference matrix, R α Is a phase coefficient matrix;
the four vibration exciters apply to the vibration exciting force of the four-machine reverse driving five-mass vibration system to periodically change, and the system vibration is also periodic; the least common multiple of four exciting force change periods is taken as T 0 Let T be 0 Inner omega m0 (t) average value of omega m ； andFor omega m The instantaneous fluctuation coefficients of (a) are respectively epsilon 0 ，ε h H=1, 2,3, there are,
ε＝[ε 1 ε 2 ε 3 ] T
wherein epsilon is a phase difference instantaneous fluctuation matrix;
substituting equation (3) into equation (4) to obtain instantaneous angular velocity matrix of four eccentric rotorsAnd instantaneous angular acceleration matrix- >The following are provided;
wherein C= [ 11 11 ]] T As a constant matrix, v=ε 0 C+R α Epsilon is the instantaneous angular velocity fluctuation matrix of the four eccentric rotors,is the instantaneous angular acceleration fluctuation matrix of four eccentric rotors, V= [ V ] 1 ν 2 ν 3 ν 4 ] T ；
When the four-machine back-driven five-mass vibration system runs stably, the designed four internal mass and four vibration exciters are identical without considering the instantaneous angular acceleration of the four eccentric rotors, the parameters of the spring A (11) are kept consistent,
m 0n ＝m 0 ，f dn ＝f d0 ，M i ＝M 0 ，k w ＝k 0 ，f w ＝f 0 ，i,n,w＝1,2,3,4 (6)
according to the transfer function method, the response of the four-machine reverse driving five-mass vibration system in the x direction, the y direction and the psi direction is obtained
wherein ,
in the formula ,Fx and Fy Stress matrix of vibration response, C γ and Sγ Cosine and sine phase matrices of the vibration response, respectively;
wherein ,
r m ＝m 0 /M，M＝4M 0 +M 5 +4m 0 ，z ψ ＝ω m /ω nψ
r l ＝l 0 /l e ，
6. the method for determining parameters of a four-machine back-drive self-balancing vibrator according to claim 5, wherein the determining conditions for the synchronism of the four-vibration exciter are specifically as follows;
irrespective of system damping f 1 ，f 2 ，f 3 ，f 4 ，f 5 And substituting formula (6) into formula (1) to obtain four-machine reversalThe stiffness matrix K and the mass matrix M of the five-mass vibration system in the x and y directions are driven as follows;
the characteristic equation of the four-machine back-driving five-mass vibration system is that,
When delta (omega) 2 ) When the vibration system is=0, the natural frequency of the four-machine back-drive five-mass vibration system in the x direction and the y direction is calculated as follows,
irrespective of natural frequency omega 5 For only the critical natural frequency omega of the system 0 and ω4 Unfolding the study; differential calculation is performed on the equation in the formula (7) to obtain andSubstituting them into formula (2) and combining formula (5), ignoring the instantaneous angular acceleration fluctuation matrix of four eccentric rotors when the four-machine back-drive five-mass vibration system is in steady operation>Higher-order terms of (2) and then putting them in +.>The upper integral, the single-period average differential equation of the four eccentric rotors is obtained,
wherein ,
W snn ＝2F 1 cosγ 1 -l 0 F 21 cosγ 21 ，W cnn ＝2F 1 sinγ 1 -l 0 F 21 sinγ 21 ，n＝1,2,3,4
in the formula ,Te0n Omega for four motors m Electromagnetic torque, k, output during steady-state operation e0n Omega for four motors m Stiffness coefficient at steady state operation; in the above integration process, the phase difference α is determined by the integrated median value thereofFor replacement of->
The four induction motors selected by the four-machine reverse driving five-mass vibration system are identical, and the rotational inertia and the damping of the four induction motor shafts are respectively equal, namely
J 0n ＝m 0 r 2 ，f dn ＝f d0 ，n＝1,2,3,4 (12)
Dividing both sides of formula (11) by m 0 r 2 ω m Post-writing into matrix form
wherein ,
A＝[a nq ] 4×4 ，B＝[b nq ] 4×4 ，u＝[u 1 u 2 u 3 u 4 ] T
in the formula ,for the period of operation T 0 Instantaneous average angular velocity of the internal four exciter induction motors with respect to +. >The non-dimensional average disturbance parameter matrix A and the matrix B are a non-dimensional inertial coupling matrix and a non-dimensional rigidity coupling matrix of the four eccentric rotors respectively, and u represents non-dimensional load torque of the four eccentric rotors;
four-machine back-driving five-mass vibration system with frequency omega m During synchronous operation, four induction motors are operated in the operation period T 0 The average velocity within the range is kept consistent,the finishing type (13) is carried out to obtain,
in the formula ,is the kinetic energy of a standard vibration exciter, < >>Representing four induction motors at frequency omega m Payload torque during synchronous operation; in the formula (13)> andThe difference in the interaction between the two,
in the formula ,(Te0n -f dn ω m )-(T e0q -f dq ω m) and respectively representing the frequencies omega of motors n and q m During synchronous operation, the difference between the effective electromagnetic output torque and the effective load torque;
from (14) let DeltaT be known 0nq 1.ltoreq.n < q.ltoreq.4 is concerned andIs a function of (2); finishing type (1)5) Obtain dimensionless type
in the formula ,is about-> andIs a bounded function of the sum of the dimensionless load torques between the four induction motors, [ (T) e0n -f dn ω m )-(T e0q -f dq ω m )]/T u Representing the difference between the dimensionless effective electromagnetic output torques of the induction motor n and the induction motor q when the induction motors are operated synchronously; let τ be cnqmax Is a bounded function->Maximum value of (1)
The four-machine back-drive five-mass system according to formulas (16) and (17) achieves the condition of synchronous operation, namely
The meaning of formula (18) is: the four-machine reverse driving five-mass vibration system is used for realizing synchronous operation of four-machine reverse driving, and the absolute value of the difference of the dimensionless effective electromagnetic output torque between any two motors is smaller than or equal to the maximum value of the difference of dimensionless load torque of the two motors.
7. The method for determining parameters of a four-machine back-drive self-balancing vibrator according to claim 6, wherein the system stability condition is determined specifically as follows;
synchronously solving the phase difference of a five-mass vibration system driven by a four-machine reverse direction in (14) andLinear expansion of the first order, neglecting the damping coefficient f of the motor shaft dn Taking into account formula (5)
wherein , in the formula ,Is a function in parentheses-> andA value at;
the finishing (19) is carried out to obtain,
consider (4) simultaneous use of integral medianReplacement α, the last three equations in equation (20) are replaced with respect toI.e., a phase difference perturbation parameter matrix equation,
wherein ,
C＝[c nh ] 3×3 ，
the characteristic equation of the phase difference disturbance system is obtained according to det (C-lambda I) =0 and is as follows
λ 3 +d 1 λ 2 +d 2 λ+d 3 ＝0 (22)
Based on the Routh-Hurwitz criterion, formula (21) relates toWhen the zero-solution of (2) is stabilized, the parameter in the formula (22) satisfies the following condition
d 1 ＞0,d 3 ＞0,d 1 d 2 ＞d 3 (23)
wherein ,
d 1 ＝-c 11 -c 22 -c 33 ，d 2 ＝-c 12 c 21 -c 23 c 32 -c 13 c 31 +c 11 c 22 +c 22 c 33 +c 33 c 11
d 3 ＝-c 11 c 22 c 33 -c 12 c 23 c 31 -c 13 c 21 c 32 +c 11 c 23 c 32 +c 22 c 13 c 31 +c 33 c 12 c 21
because ofFrom formula (5)>Therefore, when the four-machine reverse driving five-mass vibration system meets the formula (23), the stable and synchronous operation of the four-machine reverse driving five-mass system is realized; adjusting the formula (23) to obtain a system stability capability coefficient H 1 ，H 2 and H3 The following are listed below
H 1 ＝d 1 ＞0,H 2 ＝d 3 ＞0,H 3 ＝d 1 d 2 -d 3 ＞0 (24)。
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