Design of New Extrusion Equipment in Mechanical Vibration and Simulated Analysis of Vibration Performance

WU Shang-sheng (吴上生), HU An-tao (胡安涛)， JIN Peng (金 鹏)

School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640， China

Design of New Extrusion Equipment in Mechanical Vibration and Simulated Analysis of Vibration Performance

WU Shang-sheng (吴上生), HU An-tao (胡安涛)*， JIN Peng (金 鹏)

SchoolofMechanicalandAutomotiveEngineering,SouthChinaUniversityofTechnology,Guangzhou510640，China

A new extrusion equipment in mechanical vibration was put forward. Mathematical equations of the undulant surface of vibration excitation were built up by analyzing the principle of the screw axial vibration. 3D model of the vibration equipment was created. Through force analysis of the vibration parts, vibration model of the extrusion equipment was established, and key parameters were theoretically deduced. Using the ADAMS software, dynamics simulation experiment of the vibration performance of the screw was carried out. And simulated results prove that this new vibration extrusion equipment is feasible and valid.

extrusion;vibration;design;AMAMS;simulation

Introduction

The development of traditional extrusion equipment focuses on in-depth study and continuous improvement of the screw structures. For example, in order to improve the mixing effect of polymer materials and the quality of extrusion products, many researchers used the method of increasing the length to diameter ratio of a screw, or innovating the design of a screw. However, such designs often lead to increased equipment size, more severe friction and wear between the charging barrel and the screw, increasing equipment noise, as well as the shortening of service life. Besides, the innovated screws are always difficult to manufacture[1-4]. Thanks to the development of the technique of vibration, it becomes possible to introduce the vibration force field to extrusion process. Predecessors applied vibration force to some parts of the extruder[5-6], which has creatively innovated the extrusion characteristics of the screw, and greatly improved the quality of extrusion products. In addition to lower noise, better adaptability to materials, lower plasticizing temperature, and shorter cooling time, manufacture cost and energy consumption of the vibration type extrusion equipment have reduced nearly by 50% respectively compared with the traditional non-vibrant extrusion equipment[7-10].

The vibrational excitation of current vibration extrusion equipments are mainly based on electromagnetic force. The structures of these equipments are always complicated, and the stability as well as the reliability of the vibration needs to improve. In this article, a new vibration extrusion equipment, which uses mechanical rollers and undulant surface as the vibration excitation, has been put forward[11]. The structure of this new extrusion equipment is greatly simplified, and the screw can axially vibrate as it rotates.

1 Operating Principle of the Vibration Extruder

The primary structure of the mechanical vibration extrusion equipment, which is shown in Fig.1, is mainly composed of power input components (including parts 1, 2, 10),vibration components (including parts 7, 8, 9, 10), and vibration output components (including parts 10, 12, 17). Part 10, the half-coupling of vibration excitation, which has an undulant surface for motivating vibration, is the major part of this vibration extrusion equipment. Part 10 connects to part 1, the drive shaft, through part 11, the rolling spline. Making sure that part 10 and part 1 can rotate synchronously as well as move relatively along the axial direction.

1-drive shaft; 2-drive gear; 3-angular contact bearing; 4-bearing bush; 5-thrust bearing; 6-box structure; 7-roller positioning plate; 8-roller; 9-rollers’ retainer;10-half-coupling of vibration excitation; 11-rolling spline;12-ordinary half coupling; 13-check ring; 14-preloaded spring; 15-the Funnel; 16-charging barrel; 17-the screw; 18-the support; 19-the support; 20-foundation bed; 21-angular contact bearingFig.1 Main structure of the extrusion equipment in vibration

Part 7, the roller positioning plate, which has a center hole, fixedly connects with part 6, the box structure, through stud bolts, and it meanwhile connects to part 1 by a roller bearing. Part 5，the thrust bearing, is an important force-bearing component, which bears the axial force, unloading the force exerted on part 7 to part 6, avoiding part 7 working under periodic shear force as well as axial force, and ensuring the strength and reliability of part 7. Part 8, the rollers of vibration excitation seriatim install in the half conical roller slots, which uniformly distribute in the underside of part 7, the roller positioning plate. Part 12, the ordinary half coupling, fixedly connects to part 10 through stud bolts, and it also connects with part 17, the screw, by a spline. The shaft ends of part 1 and part 17, where they respectively connect with part 10 and part 12, are fixed by shaft end rings. So, part 10, part 12, and part 17 become an organic whole. They can rotate together as well as move axially and synchronously.

In the initial state, the rollers of vibration excitation mesh with the undulant surface of vibration excitation of part 10 at the lowest point of the undulant surface, as shown in Fig.2. Rotational torque is transmitted from part 2, the drive gear, to part 1, the drive shaft. Then the power makes part 10 rotating, and because of this rotation, there comes a contact movement between the rollers of vibration excitation and the undulant surface of vibration excitation. The axial component of the contact reaction force from rollers of vibration excitation exerting on the undulant surface of vibration excitation, is the exciting force for axial vibration of the organic whole of part 10, part 12, and part 17. When the rollers mesh with the undulant surface at the lowest point of the undulant surface, the screw reaches its minimum point of the vibration amplitude, and when they mesh at the highest point, screw reaches its maximum amplitude of vibration[11].

During the course of extrusion processing, the backpressure from molten polymer exerted on the screw can ensure the consistent contact between the rollers and the undulant surface. But, on restarting of the vibration extrusion equipment as well as before the extrusion processing, no backpressure exerts on the screw, and the undulant surface will possibly leap because of the inertia. In order to avoid the inertial leaps of the undulant surface, and keep the contact between the rollers and the undulant surface consistent, a preloaded spring, part 14, is set, as shown in Fig.1.

2 Design of the Undulant Surface

2.1 The determination of vibration equation

The simplified model of meshing state of rollers and the undulant surface is shown in Fig.2. In this model, the undulant surface intersects with a basic cylinder of radiusR. From the intersecting surface, we can see the meshing state of the rollers and the undulant surface. Assuming that the undulant surface is fixed, and rollers rotate around the drive shaft, the rollers will roll circumferentially along the undulant surface, and generatrices of the rollers are tangent to the undulant surface all the time.

Fig.2 Simplified model of meshing state

Fig.3 Meshing state at radius R in unfolding XY plane

To simplify the research, take a single roller for study, and unfold the intersecting surface of the meshing state at radiusRintoXYplane, as shown in Fig.3. According to enveloping theory of single parameter[12], the undulant curve of vibration excitationC1which is shown in Fig.3 is the envelope curve of the circle cluster of the intersecting surface of a single roller.When the roller rolls tangently along the undulant curveC1, the center track of the intersecting surface of a single rollerC0which is determined by Eq.(2) is just the vibration curve. To avoid impact and stress concentration between the rollers and the undulant surface during contact movement, we determine that the form of vibration is simple harmonic oscillation, and the equation of which is

(1)

whereS0is vibration displacement of the rollers’ center,Ais the amplitude,λis a proportionality coefficient,his the displacement of the center of the roller’s intersecting surface alongXaxis, andη0is the initial phase. Here givenη0=0, and the simplified equation is

(2)

From Eq. (2), we obtain

(3)

(4)

From Eqs. (3) and (4), we see that the vibration velocity and the vibration acceleration are all continuous. So the vibration mode which is determined by vibration Eq. (2) is stable and reliable. Therefore the chosen vibration Eq. (2) is rational.

2.2 Mathematical model of the undulant curveC1

Supposing that the radius of the roller’s cross section at radiusRof the undulant surface’s intersecting surface isr. From Eq.(2) and the circle cluster of radiusrin Fig.3, we obtain

(x-h)2+[y-S0(h)]2=r2.

(5)

Assuming that

F(x, y, h)=(x-h)2+[y-S0(h)]2-r2.

(6)

Solving Eqs. (2) and (6), we obtain

(7)

According to the classical enveloping theory[12-13], we obtain

(8)

Equation(8) is the equation of the undulant curve of vibration excitation,C1in Fig.3.

Solving Eqs.(6), (7), and (8), the undulant curve of vibration excitationC1, can be also expressed by the parameterized simultaneous equations as follows.

(9)

2.3 Mathematical model of the undulant surface

Based on Eq.(9), the mathematical model of the undulant surface in Fig.2 will be deduced. In order to map the curve based on Eq.(9) to three dimensional space, and get the mathematical model of the undulant surface, we will make the following definitions for the mapping parameters.

his the arc length of the cross section of the undulant surface and the basic cylinder at radiusR. Assuming the rotation angle of the undulant surface during working isθ,hcan be expressed as follows

h=θR, θ∈(0, 2π).

(10)

nis the ratio of vibration period and the rotation period.

n=λR.

(11)

βis cone angle of the rollers, where,

(12)

Besides, from the geometrical relationship shown in Fig.2 we obtain

(13)

Solving Eqs.(2), (10), and (11), the vibration equation of the undulant surface of part 10 can be expressed as follows

(14)

whereS2is the axial vibration displacement of the screw.

Here we take Eqs.(10)-(13) into Eq.(9), and map the curve based on Eq.(9) to three dimensional space. Then we can get the mathematical model of the undulant surface, which can be expressed by the following simultaneous equations.

(15)

2.4 The boundary conditions

According to Fig.3, the minimum radius of curvatureρminof the curveC0must meet the following condition

ρmin≥r.

(16)

Radius of curvatureρof the curveC0in Fig.3 can be expressed as

(17)

Solving Eqs.(2), (10), (11), (12), (13), (16), and (17), we obtain that the minimum radius of the basic cylinder,Rf, is

(18)

Therefore, the scope ofRis

R∈(Rf, Rf+L),

(19)

whereLis the axial length of the rollers.

3 Geometrical Parameter and 3D Model of the Undulant Surface as Well as the Screw

According to Eqs.(15) and (19), and selecting the appropriate geometric parameters, as shown in Table 1, we can build the 3D model of the undulant surface of vibration excitation in UG software. The built model is shown in Fig.4.

Table 1 Geometric parameters of the undulant surface and rollers

A/mmnL/mmRf/mmρmin/mmrmax/mmrmin/mm212707027.218.49.2

Fig.4 The undulant surface of vibration excitation

According to the standard of screw diameter, we can figure out the other structural parameters of the screw. Commonly, to simplify the analysis, we translate the damping upon the feeding section and the melting section of the screw into the equivalent viscous damping of the metering section. So here we just determined the structural parameters of metering section of the screw，which are shown in Table 2， whereDSis the maximum diameter of the screw,H3is the depth of the channel of the metering section,φis the helix angle,eis the normal width of the screw flight,Wis the normal width of the screw channel,δFis the fit clearance between the screw flight and the charging barrel, andL3is the length of the metering section of the screw. The model of the metering section of the screw is shown in Fig.5.

Table 2 Structural parameters of metering section of the screw

DS/mmH3/mmφ/(°)e/mW/mmδF/mmL3/mm30217.653260.2150

Fig.5 Model of metering section of the screw

4 Establishment of Vibration Model of the Screw and Determination of Key Parameters

4.1 Establishment of a dynamic vibration model

The vibration parts of this new vibration extruder are mainly composed of the half-coupling of vibration excitation, the ordinary half coupling, and the screw,etc. In order to simplify the study, we make the following assumptions[14-15]:

(1) taking the half-coupling of vibration excitation, the ordinary half coupling, the screw, and other auxiliary components as a organic whole, which is named as the generalized screw, with a piece of mass block substituting for it;

(2) assuming that the screw is a rigid body, which is never out of shape;

(3) converting the friction damping and viscous damping that polymer exerts on the screw into an equivalent viscous damping;

(4) ignoring the quality and damping of the preloaded spring.

According to the vibration principle of the screw, and combining with above basic assumptions, we can build the dynamical model of screw vibration, which is shown in Fig.6.

Fig.6 Dynamical model of screw vibration

In Fig.6,Kis stiffness coefficient of the preloaded spring,mis the quality of the generalized screw,Cis the equivalent viscous damping coefficient that polymer exerts on the screw,FTis the force that rollers exert on the undulant surface of vibration excitation,FTZwhich is the axial component ofFTis the exciting force for axial vibration,FZis the axial force on the screw exerted by polymer during extrusion molding process,x(t) is axial displacement of the vibration screw, andVis the tangential velocity of the half-coupling of vibration excitation.

4.2 Mathematical equations of the vibration model and the key parameters

According to the dynamical model in Fig.6, we can get the mathematical vibration model of the screw, which is as follows:

(20)

(1) The exciting force for axial vibrationFTZ

Assuming that the input power of the half-coupling of vibration excitation isP(kW), rotation speed of the drive shaft isN(r/min), and we obtain the input torque is

(21)

According to the pressure angleαin Fig.6, the curveC0in Fig.3 and Eq. (2),we can obtain

(22)

Then,

(23)

(2) The axial force on the screwFZ

During the extrusion processing of molten polymer, the axial forceFZincludes two parts: the backpressure exerted by the melt on the nose of the screw head end, and additional axial force produced by the dynamic load. Generally it can be calculated by the following formula[16]

(24)

(3) Equivalent viscous damping coefficientC

Considering the friction on the screw exerted by solid particles and solid bed of the melt is complex, in order to simplify the analysis, here we convert the damping upon the feeding section and the melting section of the screw into the equivalent viscous dampingCof the metering section[17]. Namely,

C=KcC3,

(25)

whereKcis the conversion coefficient which is determined through experiments, andC3is the viscous damping coefficient of the metering section, which can be expressed by the following equation[18]

(26)

whereμis the internal friction coefficient of polymer melt.

From Eqs. (25) and (26), we obtain

(27)

(4) Spring stiffnessK

For cylindrical coil spring, the stiffness of it can be worked out using material mechanics formula as follows

(28)

whereGis shear modulus of the spring material,dis the diameter of the spring wire,Dis the spring diameter, andnKis the effective number of turns of the spring.

(5) Pre-loaded length of the compression springl0

On restarting of the vibration extruder as well as before the extrusion processing, the equivalent viscous resistance of polymer upon screw is 0, so the axial force exerted on the screw is 0, too. According to these conditions, we can simplify Eq.(20) as

(29)

Because the vibration equation of the screw is

(30)

solving Eqs.(29) and (30), and simplifying the result, we obtain

(31)

(32)

In order to ensure that the rollers and the undulant surface of vibration excitation don’t isolate from each other, the following conditions should be met

(FTZ)min≥0.

(33)

Taking Eq.(33) into Eq.(32), and simplifying it, we obtain the condition thatl0must meet.

(34)

5 Simulation for the Vibration Performance

5.1 Establishment of the simulation model

On the basis of the 3D model of the undulant surface in Fig.4 and the structure parameters in Tables 1 and 2, we can build the 3D model of the primary structures of this vibration extrusion equipment in UG software. Through appropriate format conversion, we import it into ADAMS which is the dynamics simulation software. In order to facilitate the analysis, we equivalently substitute “ground”, the virtual part in ADAMS simulation environment, for the stationary solid parts of the vibration extruder such as the charging barrel, the box structure, and the supports. In the same way, the solid preloaded spring is replaced with the equivalent virtual spring in ADAMS. At last, with motions, constraints, loads, and contacts added, and we get the simulation model that shown in Fig.7.

Fig.7 Simulation model of the axial vibration extruder

5.2 Simulation and analysis for the result

According to the corresponding equations from Eq. (21) to Eq. (34), key simulation parameters were figured out, which are shown in Table 3. When calculating the parameterC, we use low density polyethylene (LDPE)as the sample.

Table 3 Vibration simulation parameters

N/(r·min-1)l0/mmK/(N·mm-1)FZ/NC/(N·s·mm-1)409.552145.40.693

Output waveforms of the simulated experiment are shown in Figs.8-13, whereA(f) is the vibration amplitude (including displacement amplitude, velocity amplitude, and acceleration amplitude) changing with the frequency, andAccis the vibration acceleration.

In Figs.8 and 9, the displacement curve and the velocity curve of the axial vibration are well coincident with Eqs. (2) and (3), of which the output waveforms are smooth. However, in Fig.10, the acceleration waveform has certain waveform distortion.

From Figs.11-13 we know that the axial vibration frequency of the screw is only related to the rotation speed of the drive shaft in the case that the ratio of vibration period and the rotation periodnis sure. In practice, it is reliable and convenient to control the vibration frequency of the screw by adjusting the motor speed.

The frequency spectra in Figs.11 and 12 show that the breadths of spectral lines within 17-100 Hz are almost zero, which means that there are almost no higher harmonics added to the fundamental waves, that’s why the waveforms in Figs.8 and 9 are without waveform distortion. However, in Fig.13, the spectral line within 17-100 Hz fluctuates in a narrow range above the zero value, which means that large amounts of high frequency random waves are superimposed on the fundamental wave, so there is certain waveform distortion in the acceleration waveform. Combining with working practice, the waveform distortion in Fig.10 is easy to understand. Due to the nonlinear contact between the rollers and the undulant surface, as well as the impact between different mechanical parts, the stress state of the undulant surface is not only complicated but also random to a certain extent. As a result, there are often higher harmonics and random waves superimposed on the fundamental wave of the vibration acceleration, so the output waveform of the vibration acceleration is impossible to be ideal waveform[19].

Fig.8 Displacement curve of the vibration screw

Fig.9 Velocity curve of the vibration screw

Fig.10 Acceleration curve of the vibration screw

Fig.11 Frequency spectrum of the displacement

Fig.12 Frequency spectrum of the velocity

Fig.13 Frequency spectrum of the acceleration

But in a word, on the basis of simplifying the structure and using a new mode of vibration, this mechanical vibration extrusion equipment provides a new piece of equipment as well as a new method, with which the screw can vibrate axially as it rotates. And the output waveforms and the frequency spectra prove that the new vibration extrusion equipment is feasible and valid.

6 Conclusions

(1) In this article, a new screw extruder in mechanical vibration is put forward, and we analyze the working principle of this new machine, of which the screw can axially vibrate as it rotates. Through the analysis of the vibration mode, the mathematical model of the undulant surface is deduced. On the basis of this, we build the 3D model of the mechanical vibration extruder.

(2) The dynamic vibration model as well as the mathematical vibration model is built up. And the key parameters of the vibration model are derived.

(3) On the basis of the 3D model and the vibration model, vibration performance of the screw is simulated in the simulation environment of ADAMS software. The output waveforms of the vibration displacement, the vibration velocity, and the vibration acceleration, as well as the frequency spectrums are obtained.

(4) According to the analysis of simulation results, the screw vibration characteristics well conform to the anticipated target. The axial vibration of the screw is relatively stable and reliable. Its vibration displacement curve and the vibration speed curve of the simulation experiment are well consistent with theoretical analysis. But the acceleration curve has certain nonlinear distortion, which is the normal phenomenon. Through this simulation experiment, the feasibility and effectiveness of this new mechanical vibration extrusion equipment is proved. And a new type of molding equipment as well as a new technology is put forward, which can be a support of the development of equipments in the polymer molding field.

[1] Sakharov A S, Kolosov A E, Sivetskii V I,etal. Modeling of Polymer Melting Processes in Screw Extruder Channels [J].ChemicalandPetroleumEngineering, 2013, 49(5/6): 357-363.

[2] Qu J P. New Concept of Plasticizing Extrusion of Polymer [J].JournalofSouthChinaUniversityofTechnology:NaturalScienceEdition, 1992(4): 1-8. (in Chinese)

[3] Deng J, Li K, Harkin-Jones E,etal. Energy Monitoring and Quality Control of a Single Screw Extruder [J].AppliedEnergy, 2014, 113: 1775-1785.

[4] Khametova M G. Analysis of Steady Non-isothermal Flow of Polymer Melts in a Single-Screw Extruder [J].ChemicalandPetroleumEngineering, 2012, 48(7/8): 488-493.

[5] Ibar J P. Control of Polymer Properties by Melt Vibration Technology [J].PolymerEngineeringandScience, 1998, 38(1): 1-20.

[6] Casulli J, Clermont J R. The Oscillating Die: a Useful Concept in Polymer Extrusion [J].PolymerEngineeringandScience, 1990, 30(23): 1551-1556.

[7] Hu H J. Extrusion Molding Method of Plastics and the Breakthrough of Equipment Research [J].ChineseScienceFund, 1997(3): 75. (in Chinese)

[8] Qu J P, Zeng G S, Feng Y H,etal. Effect of Screw Axial Vibration on Polymer Melting Process in Single Screw Extruders [J].JournalofAppliedPolymerScience, 2006, 100(5): 3860-3876.

[9] Fu L H, Wang Z W, Yan S M,etal. The Influence of Vibration Extrusion on the Rheological Behavior of the Polyolefin [J].PlasticsScienceandTechnology, 2013(2): 41-43. (in Chinese)

[10] Liu T L, Du Y X. The Influence of Vibration Force Field on Mixing Process of Pin-Screw Extruder [J].EngineeringPlasticsApplication, 2011, 39(3): 80-84. (in Chinese)

[11] Wu S S, Lu Z W, Hu A T. Axial Thrust Rotating Roller Exciting Type Mechanical Vibration Device: CN, 201310064147.2 [P]. 2013-02-28. (in Chinese)

[12] Chang Z Y, Zhang C, Yang Y H.Generation of the Meshing Surface of Spatial Indexing Cam according to the Envelope Theory [J].MachineDesign, 2000(2): 10-11. (in Chinese)

[13] Churchill F T, Hansen R S. Theory of Envelopes Provides New Cam Design Equation [J].ProductofEngineering, 1962, 35: 45-55．

[14] Yan S W, Qu J P. The Influence of Rotation on the Axial Vibration of Single Screw Extruder in Vibration [J].ChinaMechanicalEngineering, 2006(9): 941-945. (in Chinese)

[15] Qu J P, Wang S R, Yan S W. Research of Relationship between Rotation and Axial Vibration of the Electromagnetic Dynamic Extruder [J].MachineryDesignandManufacture, 2009(1): 141-143.

[16] Zhang J M, Yang Z F, Chen F,etal. Plastic Mechanical Design[M]. 2nd ed. China Light Industry Press, 1995. (in Chinese)

[17] Qu J P. Theory and Technology of Dynamic Plasticizing Processing for Polymer(Volume2) [M]. Beijing: Science Press, 2005. (in Chinese)

[18] Wei J Z, Qu J P.Polymer Damping Analysis for Screw Vibration System of the Electromagnetic Dynamic Extruder [J].NoiseandVibrationControl, 2004(4): 39-41. (in Chinese)

[19] Yang S M. Technical Requirements and Verification of the Mechanical Vibration Table [J].AviationStandardizationandQuality, 1978, 4: 25-29. (in Chinese)

Foundation item: Science and Technology Planning Project of Guangdong Province, China (No. 2012B011300006)

TQ320.5 Document code: A

1672-5220(2015)01-0073-06

Received date： 2014-05-15

* Correspondence should be addressed to HU An-tao, E-mail: jiangyetaotao@163.com