机械设计外文翻译英文原文1概要

Study on cascaded whole-leaf spring oscillation mechanism for mould in continuous casting

L.-P.Zhang*1,X.-K.Li2,Y.-F.Yao2and L.-D.Yang3

A design method of a cascaded whole-leaf spring mechanism is proposed,which is a new oscillation guidance device for the mould in continuous casting.Then its prototype designed in this paper is produced in the lab,of which kinematics and dynamics simulations are carried out based on the rigid–flexible coupling virtual model.Simulation curves of the displacement and velocity of the mould are almost consistent with the ideal ones,which verifies the model built in this paper is rational.Furthermore,natural frequencies and mode shapes of the mechanism are calculated by dynamics simulation,and forces applied on leaf springs and revolute joints are analysed and effects of the basic parameters on these forces are also studied,which establish the basis for further studies and next application of this mechanism.

Keywords:Continuous casting,Mould,Cascaded whole-leaf spring oscillation mechanism,Design method,Dynamics analysis

Introduction

The oscillation system for the mould is the key equipment to the modern continuous casting technol-ogy,of which the technical performance and reliability directly affect the quality and production of continuous casting slabs.The oscillation system is composed of the oscillation generating device and guiding mechanism, and the latter is studied in this paper.

During continuous casting,the oscillation guiding mechanism plays a guidance role in the motion of the mould.Only when the mould vibrates along the correct track,the quality of the strand can be ensured.So the strand requires very high guidance accuracy of the oscillation guiding mechanism for the mould.But for inevitable gap and wear of the

bearings,the four-eccentric axes and four-bar linkage oscillation mechanisms widely used in modern casting will cause uncontrolled deviation in motions of the mould,which badly affects the quality of the strand.1Therefore,the semi-and whole-leaf spring mechanisms are gradually used as guidance mechanisms for the mould in billet and slab continuous casting.2,3In recent years,with further development of the semi-and whole-leaf spring oscilla-tion mechanisms,the cascaded whole-leaf spring oscilla-tion mechanism is developed abroad,4which has a longer life,higher lateral rigidity and reliability and so on.

However,until now,reports on this oscillation mechanism are few.5,6Its working principle has been proposed by the authors,7based on which design method of the cascaded whole-leaf spring oscillation mechanism is proposed in this paper and its experi-mental prototype is manufactured.Furthermore,the rigid–?exible coupling virtual model of the cascaded whole-leaf spring oscillation mechanism is built and its kinematics and dynamics simulations are analysed using many types of software,such as ANSYS.System modes and the forces applied on leaf springs and revolute joints of the mechanism are analysed,which establish the basis for further studies and application of this mechanism.

Working principle of cascaded whole-leaf spring guidance mechanism

The structure of the cascaded whole-leaf spring oscilla-tion mechanism is shown in Fig.1.It mainly consists of the cascaded leaf spring,vibration table and frame, which can be driven by machinery(Fig.1a,or hydraulics servo(Fig.1b,generating device of sinusoi-dal or non-sinusoidal oscillation.And the cascaded leaf spring is composed of four steel plate springs divided into two sets.All leaf springs’extension lines join to the circular ar c centre of the continuous caster and their ends are separately connected to the vibration table and the frame,as shown in Fig.2,and then two sets of leaf springs,the vibration table and the frame form two four-bar linkage guidance devices.During the mould vibrat-ing,?exible leaf springs produce elastic deformations, which make two leaf spring four-

bar linkage guidance devices alternately play a guidance role in the mould without any interference by the generating device.7

1The Mechatronics and Information Engineering School,Foshan

University,Foshan528000,China

2College of Mechanical Engineering,Yanshan University,Qinhuangdao 066004,China

3No.2Research Development Department,Xi’an Heavy Machinery Research Institute,Xi’an710032,China

*Corresponding author,email zhanglpok@https://www.360docs.net/doc/a81187293.html,

204?2010Institute of Materials,Minerals and Mining

Published by Maney on behalf of the Institute

Received14November2008;accepted16September2009

DOI10.1179/030192309X12573371383677Ironmaking and

Steelmaking2010VOL37NO3

Design of cascaded whole-leaf spring oscillation mechanism

From working principle of the cascaded whole-leaf spring oscillation mechanism,it can be seen that two leaf spring four-bar linkages carry on the guidance to the mould.If design is unreasonable,the motion of the two leaf spring four-bar linkages will interfere during vibration,which will block the mould.Therefore,how to design these two leaf spring four-bar linkages becomes the key problem.

Design method of cascaded whole-leaf spring oscillation mechanism

According to the characters of the cascaded whole-leaf spring mechanism developed from the short-arm four-bar linkage,its design method is proposed as following:(ibased on the design principles of the short-arm

four-bar linkage simulating arc,8design two rigid four-bar linkages under the same basic para-meters of the mould by optimum design,which fulfil the requirement for guiding accuracy of the mould simulating arc

(iifor settling interference of the two four-bar

linkages,rigid linkages optimised in step (iare substituted by steel plate springs with elastic deformation and then form two leaf spring four-bar linkages

(iiibased on the basic principles of the short-arm

four-bar linkage,arrange two leaf spring four-bar linkages according to Fig.2.That is,end-points A 1,C 1and A 2,C 2are fixed on the frame;endpoints B 1,D 1and B 2,D 2are connected to the vibration table of the mould.In this way,two leaf spring four-bar linkages are cascaded,so a cascaded whole-leaf spring mechanism is designed.

Calculation example

Calculations of parameters and guidance accuracy of rigid four-bar linkages

According to the geometry relations and the movement relationships of the four-bar linkage guidance devices of the arc caster (see Fig.2,the position of any point on the mould can be calculated during the mould vibration.In this paper,taken the bottom point E on the outer arc of the mould as an example,formulas of the track and the guiding accuracy of point E are deduced,as listed in Table 1,with only one rigid four-bar linkage guiding for the mould.

To calculate conveniently,it is assumed that l 1,l 2,l 3and l 4are respectively the lengths of linkages of A 1C 1,A 1B 1,B 1D 1and C 1D 1,and A 2C 2,A 2B 2,B 2D 2and

C 2

D 2;S is the amplitude of the mould;D Q is the swing angle of the linkage of B 1D

1and B 2D 2under the max displace-ment of the mould downward;R is the caster’s basic radius;R 1and R 2are radius respectively from the arc centre to two endpoints of linkages A 1C 1,B 1D 1,A 2C 2and B 2D 2;H 1and H 2are the heights of both sides to the horizontal centerline of the mould;a is the included angle between OE and the horizontal centreline of the arc caster when the mould is at equilibrium position;h 1,h 2,r 1and r 2are respectively the included angles between linkages of A 1C 1,A 2C 2,B 1D 1,B 2D 2and the horizontal centreline of the caster;D R E is the simulated arc error of the mould.

Mathematical model and optimisation of linkages

Oscillation parameters of the mould and the installation positions of linkages are the primary designing terms of the guidance mechanism.Taking arc caster for

example,outside penalty function optimisation method is adopted in optimum design of the two rigid four-bar linkages.In four-bar linkages design,it is the goal that the trajectory error of the mould meets the requirement for guiding accuracy.So the objective function of the optimisation mathematical model can be written by f(x ~D R E ~OE ’{R ?0:02

(1

According to Table 1,it can be seen that the design optimisation variables are x

~R ,S ,h ,r ,R 1,R 2? T

(2

By experience,geometric dimensions and installation positions of two four-bar linkages must meet the following

constraints

2Layout of cascaded whole-leaf

spring

a mechanic driven;

b hydraulics servodriven

1Cascaded whole-leaf spring oscillation mechanism

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30D ?R ?50D 2?S j j ?10R 1w 0,h w 0R 2w R 1,r w h 0:91R ?R 2v R 0:82R ?R

1?0:91R

(3

where D is the thickness of billet;h and r are included angles between linkages and horizontal centreline of the caster.Based on the objective function (equation

(1,optimisation variables (equation (2and constraints established in equation

(3,optimisation procedure of outside penalty function is compiled by C language and two four-bar linkages are separately optimum designed.

Optimisation results

In optimisation,it is assumed that the mould is arranged symmetrically about its equilibrium position and its height H 1z H 25900mm;the thickness of the billet D

5150mm;the guiding accuracies of two rigid four-bar linkages D R E

50?02mm.Therefore,the geometrical parameters and the guiding accuracies of the two rigid four-bar linkages are optimum designed (Table 2.By the optimised results,rigid linkages are substituted by leaf springs arranged according to Fig.2;then a cascaded whole-leaf spring mechanism is designed.

Experimental prototype of cascaded whole-leaf spring oscillation mechanism

Based on the optimised results,the experimental proto-type of the cascaded whole-leaf spring oscillation mechan-ism for the mould is manufactured,as shown in Fig.3,in which leaf spring four-bar linkage A 1C 1B 1D 1is composed of leaf springs 1and 4;and leaf spring four-bar linkage A 2C 2B 2D 2is made up of leaf springs 2and 3.Leaf spring four-bar linkages A 1C 1B 1D 1and A 2C 2B 2D 2are located at both sides of the mould in the vertical direction of casting.And parameters of the prototype are rounded numbers of Table 2and guidance accuracy is calculated after round-ing,as listed in Table 3.

Flexible multibody theory

Vectors of location,velocity and acceleration of point on flexible body

Based on the small deformation theory,complicated motion of the ?exible body can be decomposed to several

Table 2Optimised results of two rigid four-bar linkages

Four-bar linkage A 1B 1C 1D 1

Four-bar linkage A 2B 2C 2D 2R ,mm 5974.695902.74S,mm ?3.33?3.19h ,rad

0.050.08r ,rad 0.170.14R 1,mm 5219.795233.23R 2,mm 5490.995505.41

D R ,mm

20.02

0.

019

3Experimental prototype

Table 1Formulas of parameters of two rigid four-bar linkages and error of bottom point E on mould Four-bar linkage A 1B 1C 1D 1

Four-bar linkage A 2B 2C 2D 2

l 15l 35R 22R 1l 15l 35R 22R 1

a 5arcsin(H 2/R

a 5arcsin(H 2/R

D Q 52arcsin(S /2l 1cos r 1D Q 52arcsin(S /2l 1cos r 1l 252R 1sin[(r 12h 1/2]l 252R 1sin[(r 12h 1/2]l 452R 1sin[(r 12h 1/2]

l 452R 1sin[(r 22h 2/2]

D 1

E ~?R 2

2z R 2{2R 2R cos (r 1{a 1=2D 2E ~?R 2

1z R 2{2R 1R cos (r 2{a 1=2

C 1E ~?R 2

2z R 2{2R 2R cos (h 1{a 1=2

C 2E ~?R 2

1z R 2{2R 1R cos (h 2{a 1=2

A 1D 1’~f l 21z l 2

2z 2l 1l 2sin ?(r 1{h 1=2z D Q g 1=2

A 2D 2’~f l 21z l 2

2{2l 1l 2sin ?(r 2{h 2=2{D Q g 1=2

c ~arccos ?(l 2

4z D 1E 2{C 1E 2 (2l 4:D 1E

c ~arccos ?(l 2

4z D 2E 2{C 2E 2 (2l 4:D 2E

v ~arccos ?(l 24

z AD 1’2

{l 23.(2l 4:AD 1’ v ~arccos ?(l 24z AD 2’2

{l 2

3.(2l 4:AD 2’

b ~arccos ?(l 21z AD 1’2

{l 2

2.(2l 1:AD 1’

b ~arccos ?(l 21z AD 2’2

{l 2

2.(2l 1:AD 2’

OD 1’~?R 21

z l 2

1z 2R 1l 1cos D Q 1=2OD 2’~?R 22z l 21{2R 2l 1cos D Q 1=2 w ~arccos ?(OD 1’2

z l 21{R 2

1.(2OD 1’:l 1

w ~arccos ?(OD 2’2

z l 21{R 2

2.(2OD 2’:l 1

d 5w z c z v z b

d 5w z c 2(v z b

OE ’~?D 1E 2z OD 1’2

{2D 1E :OD 1’:cos d 1=2OE ’~?D 2E 2z OD ’22

{2D 2E :OD 2’:cos d 1=2D R E 5OE 92R D R E 5OE 92R

Table 3Parameters of prototype

Leaf spring four-bar linkage A 1B 1C 1D 1

Leaf spring four-bar linkage A 2B 2C 2D 2R ,mm 6000.006000.00S ,mm

?3.00?3.00h ,u 3.005.00r ,u 10.008.00R 1,mm 5220.005235.00R 2,mm 5490.005505.00

D R ,mm

20.016

0.018

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simple motions.So the location vector of any point on the ?exible body can be expressed as equation (4.9

r p ~r 0z A (s P z u P

(4

where A is the matrix of direction cosine;r P is the vector of point P in the inertial coordinate system;r 0is the vector of the origin of moving coordinate in the inertial coordinate system;s P is the vector of point P in moving coordinate system when the

?exible body is undeformed;and u P is the relative de?ec tion of point P expressed by modal coordinates namely u P 5W P q (where W P is the assumed modal matrix and q is the generalised coordinate of deformation.

Differentiating equation (4with respect to time,vectors of velocity and acceleration are calculated

:

r p ~:r 0z :A (s P z u P z A W P :q

::r p ~::r 0z ::A (s P z u P z 2:A W P :q z A W P ::

q

(5Flexible multibody dynamic equation

Considering the location,direction and mode of point P on the ?exible body,the generalised coordinate is selected,as in equation (6.

j ~x y z y h w q i (i ~1,:::,M ? T ~r y q ? T

(6

The motion equation of ?exible body is deduced from d L L

: {L L z l L y T {Q ~0

y ~08><>:

(7where y is restraint equation;l is Lagrange multiplier corresponding to restraint equation;Q is generalised force projected to generalised coordinate j ;L is Lagrange item with L 5T 2W ,and T and W respectively denote kinetic energy and potential energy.

The kinetic energy of ?exible body is calculated by T ~12

:

j T M (j :j

(8

where M (j is mass matrix and M (j ~M tt M tr M tm M T tr M rr

M rm M T tm M T

rm M mm

2

435;subscripts t ,r and m respectively denote translation,revolution and modal freedom.

The potential energy of ?exible body includes the gravitational potential energy and the elastic potential energy,that is

W ~W g (j z 1

2

j T K j

(9where K is generalised stiffness matrix corresponding to modal coordinates and is a constant.

Because the mass of the ?exible leaf springs is very small comparing to other parts of the oscillation mechanism with cascaded whole-leaf spring,its potential energy could be ignored.So substituting equations (8and (9into equation (7,differential equation of motion of the ?exible body is written as follows M ::j z :M :j {

12L M L j :j T :j z K j z l L y L j

j

T

~Q (10

Kinematics simulation of cascaded

whole-leaf spring oscillation mechanism

Based on the ?exible multibody theory,the virtual design and then kinematics simulation of the experi-mental prototype of the cascaded whole-leaf spring oscillation

mechanism for the mould with non-sinusoi-dal oscillation are carried out using many types of softwares,such as ANSYS,10,11and curves of displace-ment and velocity of the mould are obtained and compared with the ideal ones.

Simulation model

Based on the structural characteristic of the cascaded whole-leaf spring oscillation mechanism,it is assumed as follows:

(ithe vibration table is symmetric about x –y plane

and direction of –y is the casting direction,with the coordinate as shown in Fig.4

(iileaf springs with elastic deformation are regarded

as flexible bodies.Eccentric shaft,connecting rod,vibration table and frame with big stiffness are taken for rigid bodies.

So the simulation model of the experimental prototype is built,in which leaf springs 1and 4are composed of two groups of leaf spring four-bar linkage A 1C 1B 1D 1;and leaf springs 2and 3are formed by two groups of leaf spring four-bar linkage A 2C 2B 2D

2(Fig.4.

Kinematics simulation and results

To simulate the actual non-sinusoidal motion law of the

mould with the guidance of the cascaded whole-leaf spring mechanism,a non-uniform rotational speed is imposed on the eccentric shaft,as shown in Fig.4,in which the frequency f 52Hz,the de?ection ratio of oscillation a 530%and the amplitude of the mould h 53mm.

The displacement and velocity curves of the vibration table (i.e.movement curves of the mouldare shown in Fig.5,of which the errors compared with the ideal curves are shown in Fig.6.

From Fig.5,it can be seen the virtual vibration table can move along the given non-sinusoidal rule.Although the oscillation waveform of the mould in simulation has error compared with the ideal curves (Fig.6,the maximum errors of the displacement and the velocity are very small (0?0068mm and 0?1287mm s 21respec-tively,and can be ignored.Therefore,it is concluded that the virtual model is rational and can be used

for

4Simulating model

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further study on the cascaded whole-leaf spring mechanism.

Dynamics simulation of oscillation mechanism

Based on the dynamic simulation of the cascaded whole-leaf spring oscillation mechanism,system modal and the forces applied on leaf springs and revolute joints are analysed.

Modal analysis of oscillation mechanism

Using the instantaneous freezing method of mechan-ism,12modal analysis of the cascaded whole-leaf spring

guidance mechanism is carried out,from which system modals and natural frequencies are obtained.For the frequency of non-sinusoidal oscillation in continuous casting is not very high,13natural frequencies and mode shapes from the ?rst to the ?fth are emphasised in this paper with related information of the modals listed in Table

4.Limited by space,only the ?rst to th e third mode shapes are shown in Fig.7.

From modal analysis of the mechanism above,it is known that the ?rst and the second natural frequencies are lower and resonance may occur on the mould in continuous casting.14When resonance occurs,the mould will de?e ct from the correct trajectory,which badly impacts on the quality of strand.So it must be ensured that the working frequency in continuous casting is far from the ?rst and the second natural frequencies of the cascaded whole-leaf spring oscillation mechanism.

Analysis of forces applied on leaf springs

Owing to the symmetry about the x –y plane,leaf springs in both sides of the cascade whole-leaf spring oscillation mechanism have the same forces and deformations,only the one at the positive axis of z is analysed.For the cascade whole-leaf spring mechanism,leaf

spring four-bar linkage A 1C 1B 1D 1and A 2C 2B 2D 2alternately play the guidance role on the mould and forces applied on the leaf spring and their deformations are very complicated.So it is dif?cult to analyse leaf springs well only by experiments and computer simulation has important signi?cance.In this paper,based on the virtual proto-type technology,dynamics simulation is carried out,by which forces curves of leaf springs are obtained during the mould movement.Furthermore,effects of basic oscillation parameters on these forces are analysed,which establish the basis for further study on the reliability of the

mechanism.

5a displacement and b velocity curves of vibration

table

Table 4Natural frequ encies and modes shapes from ?rst to ?fth order Modality order Natural frequency,Hz Mode shapes

172058Vertical vibration of the vibration table along the casting direction (–y 211.4723Deviated swing of the vibration table about the axis of x 323.7038Vibration of leaf spring 1

423.70685

24.0872

Transverse vibration of the vibration table along the z axis z vibration of leaf spring