CN109649964B - Parameter determination method for three-machine-driven sub-resonance self-synchronizing vibrating conveyor device - Google Patents

Parameter determination method for three-machine-driven sub-resonance self-synchronizing vibrating conveyor device Download PDF

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CN109649964B
CN109649964B CN201811540896.7A CN201811540896A CN109649964B CN 109649964 B CN109649964 B CN 109649964B CN 201811540896 A CN201811540896 A CN 201811540896A CN 109649964 B CN109649964 B CN 109649964B
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rotors
vibration
mass
unbalanced
exciter
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CN109649964A (en
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张学良
王志辉
徐金林
李超
高志国
马辉
闻邦椿
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Northeastern University China
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B65—CONVEYING; PACKING; STORING; HANDLING THIN OR FILAMENTARY MATERIAL
    • B65G—TRANSPORT OR STORAGE DEVICES, e.g. CONVEYORS FOR LOADING OR TIPPING, SHOP CONVEYOR SYSTEMS OR PNEUMATIC TUBE CONVEYORS
    • B65G27/00—Jigging conveyors
    • B65G27/10—Applications of devices for generating or transmitting jigging movements
    • B65G27/16—Applications of devices for generating or transmitting jigging movements of vibrators, i.e. devices for producing movements of high frequency and small amplitude
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B65—CONVEYING; PACKING; STORING; HANDLING THIN OR FILAMENTARY MATERIAL
    • B65G—TRANSPORT OR STORAGE DEVICES, e.g. CONVEYORS FOR LOADING OR TIPPING, SHOP CONVEYOR SYSTEMS OR PNEUMATIC TUBE CONVEYORS
    • B65G27/00—Jigging conveyors
    • B65G27/34—Jigging conveyors comprising a series of co-operating units

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Abstract

The invention discloses a parameter determination method of a three-machine driven sub-resonance self-synchronizing vibrating conveyor device, wherein a mass of the device is connected with a foundation through a spring and a guide rod; three vibration exciters are fixed on the mass, three unbalanced rotors are respectively driven by induction motors, and the rotation directions are all anticlockwise; the three unbalanced rotors respectively rotate around three central points of the rotating shaft, and the rotating central points are collinear; in addition, the expansion direction of the vibration spring is parallel to the movement direction of the device and is vertical to the guide rod; by utilizing the vibration self-synchronization principle, a dynamic model and a system differential equation are established, and synchronization conditions and stability conditions are deduced. And numerical analysis and simulation are performed. It is proved that the working point of the vibrating machine is only selected in the sub-resonance region during the design process of the vibrating machine, and the working efficiency is best when the three unbalanced rotors are the same. The invention adopts three machines to drive automatically and synchronously, has simple structure and strong stability, and can improve the production efficiency and save energy.

Description

Parameter determination method for three-machine-driven sub-resonance self-synchronizing vibrating conveyor device
Technical Field
The invention belongs to the technical field of vibratory conveying, and relates to a parameter determination method of a three-machine-driven sub-resonance self-synchronizing vibratory conveyor device.
Background
The vibration conveyer is a continuous conveying machine which utilizes a vibration exciter to vibrate a trough so as to enable materials in the trough to slide or throw in a certain direction. With the progress of vibration technology, the vibration conveyor is more economical and efficient. The conveying frequency of the vibrating conveyor is faster and faster, the required power is higher and higher, and the noise is larger. The invention belongs to a large-scale vibrating conveyor in vibrating conveyors. The principle of a common large-sized vibrating conveyor is the same as that of a small-sized vibrating conveyor, double-machine driving is adopted, and the double-machine driving can cause a plurality of problems:
1. in the working process of the double-motor driven conveyor, the requirement on the power of the motor is high, the size of the motor is large, the technical requirement on the motor is high, and the cost is greatly improved. And simultaneously, the utilization rate of electric energy is reduced. Does not meet the national requirements of energy conservation and emission reduction
2. The double-machine driven conveyor can only be used for conveying small equipment, a large-scale conveyor is needed in large-scale industrial production, and the double-machine drive cannot meet the actual requirement.
3. In the material conveying process, along with the reciprocating motion of the material, the amplitude generated by the double-machine driving to the conveyor is large, and for the sake of safety, the conveyor must be stopped for conveying, so that the vibrating conveyor cannot work continuously, and the automation degree is reduced.
To reduce this occurrence, some improvements to the vibration system are needed. A multi-machine driven sub-resonance self-synchronizing vibrating conveyor is designed, so that the production efficiency can be improved, energy sources can be saved, and meanwhile, the cost can be reduced.
Disclosure of Invention
The invention is realized by the following technical scheme:
a method for determining parameters of a three-machine driven sub-resonant self-synchronizing vibratory conveyor apparatus, a dynamic model of the apparatus comprising: the vibration exciter, the unbalanced rotor, the mass and the supporting device; the supporting device comprises a spring and a guide rod; the plastid is connected with the foundation through a spring and a guide rod; the three vibration exciters are fixed on the mass, the three unbalanced rotors are respectively driven by the induction motors, and the rotation directions of the three unbalanced rotors are all anticlockwise; the three unbalanced rotors respectively rotate around three central points of the rotating shaft, and the three rotating central points are collinear; in addition, the expansion direction of the vibration spring is parallel to the movement direction of the device and is vertical to the guide rod; the parameter determination method of the three-machine driven sub-resonance self-synchronizing vibrating conveyor device comprises the following steps:
step 1, establishing a dynamic model and a system motion differential equation:
the vibration transport dynamics model is shown in fig. 1, a coordinate system shown in the figure is established, and based on a lagrangian equation, a differential equation of system motion is deduced to be:
Figure GDA0002355369750000021
wherein the content of the first and second substances,
M＝m+m1+m2+m3；Joi＝miri 2；r1＝r2＝r3＝r
in the formula (I), the compound is shown in the specification,
m is the total mass of the system;
mi-exciter i mass (i ═ 1,2, 3);
Joithe moment of inertia of exciter i (i ═ 1,2, 3);
r-eccentricity of vibration exciter;
Tei-the motor i outputs torque electromagnetically;
fithe shaft damping coefficient of motor i (i ═ 1,2, 3);
kx,ky,kψ-spring stiffness in x, y and psi directions;
fx,fy,fψ-damping coefficients in x, y and psi directions;
Figure GDA0002355369750000031
and
Figure GDA00023553697500000310
d.dt and d.dt2
Step 2, deriving a synchronization condition:
from the dynamic model, three rotor masses can be assumed: m is1＝η1m0；m2＝η2m0；m3＝η3m0(wherein the mass ratio is η11); the phase relation of the three rotors is satisfied
Figure GDA0002355369750000032
Wherein 2 α1，2α2Respectively the phase difference between the rotors 1,2 and the rotors 2,3,
Figure GDA0002355369750000033
is the average phase of the three rotors when the system is synchronized, take 2 α1And 2 α2Median of integral of
Figure GDA0002355369750000034
And replace it with 2 α1And 2 α2Thereby obtaining
Figure GDA0002355369750000035
Assuming three unbalanced rotors to realize synchronous operation and synchronous angular speed
Figure GDA0002355369750000036
Its angular acceleration is approximately 0, i.e.
Figure GDA0002355369750000037
The equation (3) is substituted into the first differential equation in the equation (1) to obtain the response equation of the system
Figure GDA0002355369750000038
Solving the formula (4) by a transfer function method to obtain the system response
Figure GDA0002355369750000039
Wherein the content of the first and second substances,
Figure GDA0002355369750000041
in pair type(5) After the second derivative is obtained from the middle time t, the second derivative is substituted into the three equations after the formula (1), and the second derivative is simultaneously paired on 0-2 pi
Figure GDA0002355369750000042
After integration and arrangement, the balance equation of the three motors can be obtained as follows.
Figure GDA0002355369750000043
Wherein
Figure GDA0002355369750000044
Figure GDA0002355369750000045
Figure GDA0002355369750000046
In the formula (I), the compound is shown in the specification,
Figure GDA0002355369750000047
is the kinetic energy of the standard excitation,
Figure GDA0002355369750000048
the output electromagnetic torque of the motor.
Subtracting the formula (6) to obtain the difference of the output electromagnetic torque, delta T, between the motors 1 and 2 and between the motors 2 and 3012,ΔT023The expression is as follows
Figure GDA0002355369750000049
Is transformed by the formula (7)
Figure GDA0002355369750000051
Wherein
Figure GDA0002355369750000052
Figure GDA0002355369750000053
Then
Figure GDA0002355369750000059
Dimensionless coupling torque between motors 1 and 2 and between motors 2 and 3 respectively. Meanwhile, the left part of equation (8) corresponds to the difference between dimensionless residual torques between motors 1 and 2 and between motors 2 and 3, respectively. The constraint function is as follows
Figure GDA0002355369750000054
Figure GDA0002355369750000055
Therefore, the synchronization criterion of the three exciters is as follows:
Figure GDA0002355369750000056
Figure GDA0002355369750000057
equations (11) and (12) may be described as: the absolute value of the difference between the dimensionless residual torques of any two motors is less than or equal to the maximum value of the dimensionless coupling torque thereof.
Adding the formulas in the formula (6) and converting to obtain the average dimensionless load moment of the three motors
Figure GDA0002355369750000058
With a limiting function of
Figure GDA0002355369750000061
In order to better analyze the synchronous performance capability of the vibration system, therefore, a synchronous performance force coefficient between unbalanced rotors in the vibration system is defined, and the expression is as follows:
Figure GDA0002355369750000062
wherein the content of the first and second substances,
Figure GDA0002355369750000063
generally, the larger the synchronization capability coefficient of the vibration system is, the stronger the synchronization capability of the vibration system is, and the easier the system is to achieve synchronization.
And 3, deriving a stability condition:
from the formula (3)
Figure GDA0002355369750000064
Let Delta nui＝νi-νi0Can be used for (delta v)i) ' about
Figure GDA0002355369750000065
Is linearized to
Figure GDA0002355369750000066
Figure GDA0002355369750000067
Figure GDA0002355369750000068
Wherein
Figure GDA0002355369750000069
Integrate the median for the phase difference at steady state (·)0To represent
Figure GDA00023553697500000610
Figure GDA0002355369750000071
Time Δ α ═ Δ α1Δα2}TThe value of (c).
Let Δ α be { Δ α ═ e1Δα2}TSubtracting the formula (17) from the formula (18), subtracting the formula (18) from the formula (19), and finishing to obtain the final product
Figure GDA0002355369750000072
Wherein D ═ Dij)2×2And is provided with
Figure GDA0002355369750000073
Figure GDA0002355369750000074
Figure GDA0002355369750000075
Figure GDA0002355369750000076
Assuming that Δ α is a function of time t, Δ α ═ v exp (λ t) is substituted into equation (20), and the characteristic polynomial of D is obtained as
λ2-(d11+d22)λ+(d11d22-d12d21)＝0 (21)
The solution of equation (21) stabilizes if and only if all values of λ have a negative real part. According to the Router criterion, the condition that all eigenvalues of D have negative real parts is
-(d11+d22)＞0 (22)
d11d22-d12d21＞0 (23)
Equations (22) and (23) are the stability criterion of the vibration system in the synchronous state.
The invention has the beneficial effects that:
1) and a three-machine driving synchronous vibration mechanism is adopted, so that the working efficiency is improved and the energy is saved compared with the original two-machine driving vibration conveyor.
2) Innovations are made on the model, three motors are fixed on the mass body, three unbalanced rotors are driven by the motors respectively, and the rotating directions are the same. The mass is connected with the foundation through a spring and a guide rod. In addition, the vibration spring is parallel to the x-axis and perpendicular to the guide bar. The three unbalanced rotors rotate around the rotation axis centers, respectively, and the rotation centers are collinear. The induction motor drives the three unbalanced rotors to rotate together in a counterclockwise direction, which is closer to engineering practice on the model.
3) And (3) applying a vibration synchronization theory and realizing the synchronous work of the system by adopting three-machine driving. The working region is selected to be a sub-resonance region. In the sub-resonance state, the phase difference between every two three rotors is within (-pi/2, pi/2), the exciting forces are superposed, the mass motion state is linear reciprocating motion, and the working efficiency is the best.
Drawings
FIG. 1 is a schematic diagram of a dynamic model of a three-machine driven sub-resonance self-synchronizing vibrating conveyor.
In the figure: 1. spring 2, guide rod 3, vibration exciter 4, rotor 5 and mass.
The meaning of each parameter in the figure is as follows:
O1-left exciter centre of rotation;
O2-the center of rotation of the middle exciter;
O3-right exciter centre of rotation;
Figure GDA0002355369750000081
-left exciter rotational phase angle;
Figure GDA0002355369750000082
-the rotational phase angle of the medium exciter;
Figure GDA0002355369750000083
-right exciter rotational phase angle;
Figure GDA0002355369750000084
-left exciter rotation angular velocity;
Figure GDA0002355369750000085
-medium exciter rotation angular velocity;
Figure GDA0002355369750000086
-right exciter rotation angular velocity;
r1-left exciter eccentricity;
r2-the eccentricity of the middle exciter;
r3-right exciter eccentricity;
m1-left exciter mass;
m2-medium exciter mass;
m3-right exciter mass;
m- -mass of plastid;
kx-spring rate coefficient.
FIG. 2 shows a difference ηi(i ═ 1,2,3) by τcijmaxWith zxA variation diagram of (2).
FIG. 3 shows a difference ηi(i ═ 1,2,3) lower zeta potentialij(ij-12, 23,13) with zxη1＝η2＝η3＝1，(b)η1＝η2＝1，η3＝0.5，(c)η1＝1，η2＝0.75，η3＝0.5。
FIG. 4 shows a difference ηi(i-1, 2,3) stable retardation with zxη1＝η2＝η3＝1，(b)η1＝η2＝1，η3＝0.5，(c)η1＝1，η2＝0.75，η3＝0.5。
FIG. 5 shows a difference ηiZ along with lower synchronous stability performance coefficientxA variation diagram of (2).
FIG. 6 is a drawing when η1＝η2＝η3When the value is 1, a simulation result graph in a sub-resonance state; (a) motor speed, (b) phase difference between rotors 1 and 2, (c) phase difference between rotors 2 and 3, (d) phase difference between rotors 1 and 3, (e) displacement in x-direction.
FIG. 7 is a drawing when η1＝η2＝1,η3When the value is 0.5, a simulation result graph in a sub-resonance state; (a) motor speed, (b) phase difference between rotors 1 and 2, (c) phase difference between rotors 2 and 3, (d) phase difference between rotors 1 and 3, (e) displacement in x-direction.
FIG. 8 is a drawing when η1＝η2＝η3When the value is 1, a simulation result graph in a super-resonance state is obtained; (a) motor speed, (b) phase difference between rotors 1 and 2, (c) phase difference between rotors 2 and 3, (d) phase difference between rotors 1 and 3, (e) displacement in x-direction.
FIG. 9 is a drawing when η1＝η2＝1,η3When the value is 0.5, a simulation result graph in a super-resonance state is obtained; (a) motor speed, (b) phase difference between rotors 1 and 2, (c) phase difference between rotors 2 and 3, (d) phase difference between rotors 1 and 3, (e) displacement in x-direction.
Detailed description of the preferred embodiments
Example 1:
the three motors are set to be consistent in model, and the model parameters are as follows: three-phase squirrel-cage type, 50Hz, 380V, 6 grade, 0.75kW, and the rated rotating speed of 980 r/min. The internal parameters of the motor are as follows: rotor resistance Rr3.40 Ω, stator resistance Rs3.35 omega, mutual inductance Lm164mH, rotor inductance Lr170mH, stator inductance Ls170mH, coefficient of friction f1＝f2＝f30.05. The vibration system parameters were: m is0＝10kg，m＝1430kg，kx＝36000kN/m，fx7.6kN s/m, critical damping ratio ξ of the systemnx0.07 and 0.15m, to obtain the natural frequency omega of the systemnx≈157rad/s。
(a) Synchronization of vibration systems
According to the formulas (8) and (13), the dimensionless maximum coupling torque tau between any two motors can be obtained by combining the motor and the vibration system parameterscijmax(ij is 12,23,13) and three-motor average dimensionless maximum load moment τamax(ij 12,23,13) η at different mass ratiosi(i is 1,2,3) and (t)cijmaxValue of (a) with frequency ratio zxIs shown in FIG. 2, at a frequency ratio zxWhen 1, τcijmaxThe value of (a) is minimized, which shows that the dimensionless coupling torque between any two motors is very small. According to analysis, the vibration system performs energy distribution and redistribution among motors through a coupling mechanism to enable the vibration system to be synchronous, and for zxIn the case of 1, it is known that the synchronization capability of the vibration system is weak. To further analyze the synchronization capability, the synchronization capability coefficient of the system will be discussed next.
τcijmaxAnd τamaxZeta ratio ofijThe (ij is 12,23,13) is the synchronization capability coefficient between any two unbalanced rotors (also called the generalized dynamic symmetry coefficient of the vibration system), and the magnitude of the coefficient depends on the motor parameters of the three motors, according to the previous analysis, the larger the synchronization capability coefficient is, the stronger the synchronization capability of the system is, according to the equations (8) and (13), after substituting the parameters, the synchronization capability coefficient can be obtained in different quality ratios ηi(i-1, 2,3) with frequency ratio zxAccording to the variation curve of (a), the variation trend of which is shown in fig. 3. in fig. 3(a), the mass ratio η of the unbalanced rotor1＝η2＝η 31, i.e. three unbalanced rotors can be considered symmetrical, when the synchronism capability coefficient of the vibration system satisfies ζ12＝ζ23＝ζ13In FIG. 3(b), the mass ratio η of the unbalanced rotor1＝η2＝1＞η30.5, the unbalanced rotors 1 and 2 are therefore symmetrical, the synchronization capability then beingCoefficient ζ12＞ζ23＝ζ13In fig. 3(c), too, the mass ratio η1＝1，η2＝0.75，η30.5, the three unbalanced rotors are different, i.e. not symmetrical, in which case ζ12＞ζ23＞ζ13. Furthermore, according to FIG. 3, when z isxWhen the value is 1, the synchronization capability coefficient of the system is close to 0. According to analysis, when the frequency ratio zxWhen the synchronous speed is close to or equal to 1, namely the synchronous speed is close to or reaches a resonance point, the vibration system generates resonance phenomenon, and when z is close to or reaches the resonance pointxThe closer to 1, the easier the vibration system achieves resonance, and the weaker its ability to synchronize.
(b) Phase difference of unbalanced rotor
According to analysis, under the condition that the motor parameters of the three motors are consistent, when the vibration system stably runs, the electromagnetic output torques of the three motors are respectively equal to each other, namely delta T exists under the condition that the vibration system meets the synchronism criterion012＝ΔT 0230. At the same time, phase difference in steady state
Figure GDA0002355369750000111
And motor frequency conversion omegam0Meets the stability criterion in the synchronous state. The stable phase difference at different frequency ratios can be obtained by combining formula (7), and the result is shown in fig. 4. In FIG. 4, with the frequency ratio zxThe steady phase difference between the unbalanced rotors can be divided into two sections: sub-resonance state (i.e. z)x< 1); super resonance state (z)x＞1)。
When z isxBelow 1, there is only one balance point for the vibration system at the same frequency ratio, especially when the three unbalanced rotor masses are identical (i.e., η)1＝η2＝η31), the stable phase difference relationship satisfies
Figure GDA0002355369750000121
Also, according to fig. 4(b), when the unbalanced rotors 1 and 2 are of the same mass,
Figure GDA0002355369750000122
the results of these phase differences show that at zxIn case < 1, when the two unbalanced rotor masses in the vibration system are identical, the steady phase difference between them will be equal to 0. Therefore, according to analysis, the system runs in phase with the three rotors through mutual coupling, the stable phase difference interval is (-pi/2, pi/2), the excitation force is superposed, and the vibration system has large amplitude.
When z isxWhen the phase difference value is larger than 1, the vibration system has two balance points, and the stable phase difference values are opposite to each other. Furthermore, when z isxAt > 1, it can be found that with the frequency ratio zxAccording to FIG. 4, the constant value of the steady phase difference in the super-resonance state can be approximately obtained as (a) η1＝η2＝η3When the number is equal to 1, the alloy is put into a container,
Figure GDA0002355369750000123
(b)η1＝η2＝1，η3when the content is equal to 0.5,
Figure GDA0002355369750000124
(c)η1＝1，η2＝0.75，η3when the content is equal to 0.5,
Figure GDA0002355369750000125
according to analysis, the excitation forces generated by the three rotors reach a force balance, the system is in a state similar to a static state, and the state is not considered in engineering.
When ω ism0Near the resonance point (i.e. z)x1) when there are multiple equilibrium points or no equilibrium points of the vibration system, indicating that the vibration system is unstable in this state.
(c) Stability of vibration system in synchronous state
Based on the variation characteristics of the stable phase difference between the unbalanced rotors in the vibration system, the motion forms of the masses under different frequency ratios are obtained: in the sub-resonance region, the motion form of the mass body is linear reciprocating motion; in the super-resonance region, the unbalanced rotors reach a force balance with each other and the masses are at rest.
In the stability analysis process of the vibration system, the stability criterion of the vibration system in a synchronous state is obtained through a Router-Hurwitz criterion. To further analyze the synchronous stability capability of the vibration system, the negative real part λ of the eigenvalues of the matrix D is defined1And λ2The synchronous stability capability coefficient of the vibration system is set, and the closer the force coefficient with synchronous stability performance is to 0, the weaker the synchronous stability capability of the vibration system is. Lambda [ alpha ]1And λ2Value of with zxIs shown in fig. 5, comparing different ηiLower lambda1And λ2Can know when η1＝η2＝η3When 1 is compared with other conditions, λ1And λ2The smaller the value of (A), the strongest capability of synchronization stability. In addition, when the frequency ratio zxWhen equal to 1, λ1And λ2The value of (c) reaches a maximum and approaches 0, which indicates that the system has the weakest ability to synchronize stability when the synchronizing speed reaches the resonance point.
Example 2
To further validate the theoretical results, the vibrating system was subjected to the Runge-Kutta method at a different ηiThe sub-resonance and over-resonance states of the following were simulated. Mass m of standard unbalanced rotor 010 kg. During the simulation, the motor 2 in each set of simulations was given a pi/2 disturbance at 15 s.
As shown in fig. 6, when the spring rate k isxAt 36000kN/m, the synchronous speed of the motor is approximately 940r/min, thus obtaining zxAbout 0.62, the vibration system operates in a sub-resonance state, and the stable phase difference of the vibration system is 2 α1＝2α2When the motor runs synchronously and stably, the amplitude of the mass is about 10.8mm, and the displacement is in a sine state, comparing the simulation result with the characteristic analysis, η1＝η2＝η3The sub-resonance simulated steady phase difference at 1 corresponds to fig. 4(a)
Figure GDA0002355369750000131
Corresponding position. As can be seen from fig. 6, the three unbalanced rotors operate synchronously and stably at a stable phase difference in the sub-resonance state. At 15s, after a disturbance of pi/2 has been applied to the motor 2, the phase differences of the unbalanced rotors 1 and 3 fluctuate little and can therefore be considered as unchanged, while the phase differences between the unbalanced rotors 1 and 2,2 and 3 increase rapidly in a short time and the wave directions are opposite. It can also be seen in figure 6(e) that the displacement of the mass fluctuates at 15s, and then rapidly returns to its original state.
In FIG. 7, the same spring rate kx36000kN/m, the synchronous speed of the electric machine is likewise approximately 940r/min, so that zxApproximately equal to 0.62, the vibration system operates in a sub-resonance state, and the phase difference under the stable operation of the vibration system meets 2 α1≈0，2α2And the angle is approximately equal to-4.2 degrees. The amplitude of the mass is about 9.9 mm. The phase difference in the simulation result corresponds to that in FIG. 4(b)
Figure GDA0002355369750000143
The corresponding position. When the motor 2 is disturbed, the phase difference between the unbalanced rotors and the amplitude of the mass fluctuate in a short time, and then the system is restored to the original state, which shows that the system is stable.
Changing the spring rate so that kx624kN/m, two sets of simulation results were obtained as shown in fig. 8 and 9. The synchronous rotating speed of the two groups of simulation motors is about 983.2r/min, so that the frequency ratio z can be obtainedx5, the system runs under the super-resonance state, the stable phase difference is η1＝η2＝η 32 α when equal to 11＝2α2＝120°；η1＝η2＝1，η 32 α when equal to 0.51≈151.5°，2α2About 95.4. The results are shown in FIG. 4(a)
Figure GDA0002355369750000141
And in FIG. 4(b)
Figure GDA0002355369750000142
Approximately correspond. In addition, the amplitude of the steady motion of the mass in both sets of simulation results is 0. Also at the time of 15s, the time of the second,after the two sets of simulations give the motor 2 a pi/2 interference at the same time, the phase difference of the three unbalanced rotors is rapidly restored to the original state after the fluctuation occurs in a short time, and meanwhile, the displacement is also rapidly restored to 0 after the fluctuation occurs, which indicates that the system is stable in the state. In addition, before 7s, the phase difference of the vibration system fluctuates, and the system is unstable, and here, before 7s, a transition region which is experienced by the vibration system to achieve stable operation can be considered.
Example 3:
in the attached figure 1, a three-machine driven sub-resonance self-synchronizing vibrating conveyor device, three vibration exciters 3 are fixed on a mass body 5, three unbalanced rotors 4 are respectively driven by the vibration exciters 3, and the rotating directions are the same. The mass is connected to the foundation by means of springs 1 and guide rods 2. In addition, the vibration spring 1 is parallel to the x-axis and perpendicular to the guide bar. Three unbalanced rotors 4 each around the center o of the rotation axis1，o2And o3Rotate and the center o of rotation1，o2，o3Co-linear. The induction motor drives the three unbalanced rotors 4 to rotate together in the counterclockwise direction. In the model
Figure GDA0002355369750000151
Figure GDA0002355369750000152
Respectively the rotation angles of the three unbalanced rotors 4. Due to the action of the guide rods 2 the mass moves only in the x-direction.
The following are exemplary data parameters for one version of a vibratory conveyor designed using the present invention. The present invention is not limited to this design parameter.
The parameters are as follows: mass m of eccentric block of vibration exciter010kg, 1430kg mass m, spring rate kx36000kN/m, damping coefficient f in x directionx7.6 kN.s/m, the rotary radius r of the exciter is 0.15m, and z isx0.62. Operating at a natural frequency ω0Meets the stability requirement, and the phase difference under the stable operation meets 2 α1≈0，2α2And the angle is approximately equal to-4.2 degrees. The amplitude of the mass is about 9.9 mm. When in useAfter the motor 2 is interfered, the phase difference between unbalanced rotors and the amplitude of a mass fluctuate in a short time, and then the unbalanced rotors and the amplitude of the mass recover to the original state, which shows that the system is stable at the moment (three motors are set to be consistent in type, the type parameters are three-phase squirrel-cage type, 50Hz, 380V, 6 grade, 0.75kW and the rated rotation speed is 980R/min, and the internal parameters of the motor are rotor resistance Rr3.40 Ω, stator resistance Rs3.35 omega, mutual inductance Lm164mH, rotor inductance Lr170mH, stator inductance Ls170mH, coefficient of friction f1＝f2＝f3＝0.05)。

Claims (2)

1. A method for determining parameters of a three-machine driven, sub-resonant, self-synchronizing vibratory conveyor apparatus, the apparatus having a dynamic model comprising: the vibration exciter, the unbalanced rotor, the mass and the supporting device; the supporting device comprises a spring and a guide rod; the plastid is connected with the foundation through a spring and a guide rod; the three vibration exciters are fixed on the mass, the three unbalanced rotors are respectively driven by the induction motors, and the rotation directions of the three unbalanced rotors are all anticlockwise; the three unbalanced rotors respectively rotate around three central points of the rotating shaft, and the three rotating central points are collinear; in addition, the expansion direction of the vibration spring is parallel to the movement direction of the device and is vertical to the guide rod; the parameter determination method of the three-machine driven sub-resonance self-synchronizing vibrating conveyor device comprises the following steps:
step 1, establishing a dynamic model and a system motion differential equation
Establishing a coordinate system, and deducing a differential equation of system motion based on a Lagrange equation, wherein the differential equation comprises the following components:
Figure FDA0002355369740000011
wherein
M＝m+m1+m2+m3；Joi＝miri 2；r1＝r2＝r3＝r
In the formula (I), the compound is shown in the specification,
x is the displacement of the plastid;
m is the total mass of the system;
mithe mass of the exciter i, i being 1,2, 3;
m-mass of plastid;
Joithe moment of inertia of the exciter, i ═ 1,2, 3;
ri-eccentricity of a vibration exciter i;
r-eccentricity of vibration exciter;
Figure FDA0002355369740000012
-the phase of the exciter i;
Tei-the motor i outputs torque electromagnetically;
fithe shaft damping coefficient of motor i, i ═ 1,2, 3;
kx-spring rate in x-direction;
fx-a damping coefficient in the x-direction;
Figure FDA0002355369740000021
and
Figure FDA0002355369740000022
d/dt and d/dt2；
Step 2, deducing synchronization condition
According to the kinetic model, three rotor masses are assumed: m is1＝η1m0；m2＝η2m0；m3＝η3m0(ii) a Wherein m is0η for standard exciter qualityiMass ratio η for mass ratio between exciters11 is ═ 1; the phase relation of the three rotors is satisfied
Figure FDA0002355369740000023
Wherein 2 α1，2α2Respectively the phase difference between the rotors 1,2 and the rotors 2,3,
Figure FDA0002355369740000024
is the average phase of the three rotors, and when the system achieves synchronization, 2 α is taken1And 2 α2Median of integral of
Figure FDA0002355369740000025
And replace it with 2 α1And 2 α2Thereby obtaining
Figure FDA0002355369740000026
Figure FDA0002355369740000027
Figure FDA0002355369740000028
Three unbalanced rotors are arranged to realize synchronous operation and synchronous angular speed
Figure FDA0002355369740000029
Its angular acceleration
Figure FDA00023553697400000210
Substituting the formula (3) into the first differential equation in the formula (1) to obtain the response equation of the system:
Figure FDA00023553697400000211
solving equation (4) by a transfer function method to obtain a system response:
Figure FDA00023553697400000212
wherein
Figure FDA0002355369740000031
ξnxIs the critical damping ratio of the system
The balance equation of the three motors is as follows:
Figure FDA0002355369740000032
wherein
Figure FDA0002355369740000033
Figure FDA0002355369740000034
Figure FDA0002355369740000035
In the formula (I), the compound is shown in the specification,
Figure FDA0002355369740000036
is the kinetic energy of the standard excitation,
Figure FDA0002355369740000037
the output electromagnetic torque of the motor;
the synchronization criteria for the three exciters are:
Figure FDA0002355369740000038
Figure FDA0002355369740000039
equations (11) and (12) are described as: the absolute value of the difference between the dimensionless residual torques of any two motors is less than or equal to the maximum value of the dimensionless coupling torque;
step 3, deducing stability conditions
Figure FDA0002355369740000041
Let Delta nui＝νi-νi0Will (Δ ν)i) ' about
Figure FDA0002355369740000042
Is linearized to
Figure FDA0002355369740000043
Figure FDA0002355369740000044
Figure FDA0002355369740000045
Wherein
Figure FDA0002355369740000046
Figure FDA0002355369740000047
Integrate the median for the phase difference at steady state (·)0To represent
Figure FDA0002355369740000048
Figure FDA0002355369740000049
Time Δ α ═ Δ α1Δα2}TA value of (d);
Figure FDA00023553697400000410
wherein D ═ Dij)2×2And is provided with
Figure FDA00023553697400000411
Figure FDA00023553697400000412
Figure FDA00023553697400000413
Figure FDA00023553697400000414
A characteristic polynomial of D is
λ2-(d11+d22)λ+(d11d22-d12d21)＝0 (21)
The solution of equation (21) is stable if and only if all values of λ have negative real parts; according to the Router criterion, the condition that all eigenvalues of D have negative real parts is
-(d11+d22)＞0 (22)
d11d22-d12d21＞0 (23)
Equations (22) and (23) are the stability criterion of the vibration system in the synchronous state.
2. The method of claim 1, wherein the method further comprises the step of determining the parameters of the three-motor driven sub-resonance self-synchronizing vibratory conveyor device,
average dimensionless load moment of three motors:
Figure FDA0002355369740000051
the limiting function is:
Figure FDA0002355369740000052
defining the synchronous performance force coefficient between unbalanced rotors in the vibration system, and expressing the synchronous performance force coefficient as follows
Figure FDA0002355369740000053
Wherein
Figure FDA0002355369740000054
The larger the synchronization capability coefficient of the vibration system is, the stronger the synchronization capability of the vibration system is, and the easier the system is to realize synchronization.
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US4754870A (en) * 1982-08-12 1988-07-05 Litton Systems, Inc. Two mass vibrating feeder
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