CHEN Xiulong, GAO Wenhua, JIANG Shuai, SONG Hao,GOU Yanyan

(1.College of Mechanical and Electronic Engineering, Shandong University of Science and Technology,Qingdao,Shandong 266590, China； 2.Department of Mechanical and Electronic Engineering, Taishan Vocational and Technical College, Taian, Shandong 271000, China)

Static and Dynamic Analysis of a Novel Single-DOF Six-bar Mechanical Press Mechanism

CHEN Xiulong1, GAO Wenhua1, JIANG Shuai1, SONG Hao1,GOU Yanyan2

(1.College of Mechanical and Electronic Engineering, Shandong University of Science and Technology,Qingdao,Shandong 266590, China； 2.Department of Mechanical and Electronic Engineering, Taishan Vocational and Technical College, Taian, Shandong 271000, China)

In this paper, a novel single-DOF six-bar mechanical press mechanism is proposed. In order to realize the static and dynamic analysis of this six-bar mechanism, a kinematic model was established by complex number vector method. The kinetostatic equation and kinetic equation of the mechanism were established by D’Alembert Principle and Lagrange Method respectively. Finally, theoretical computation and simulation were carried out by using MATLAB and ADAMS virtual prototype respectively to obtain the curves ofthe position, velocity andacceleration of the slider, and the changing curves of the balance moment and driving torque of the articulate. The research can not only provide theoretical basis for kinematic analysis, the solution and physical design of balance moment and driving torque of the six-bar mechanism, but it can also provide an effective method for the static and kinematic analyses of other multi-link mechanisms.

six-bar mechanism;kinematic analysis; statics; dynamics

As a new type of forging equipment,multi-link mechanical press has the advantages of high working accuracy, good operation, excellent dynamic performance and high production efficiency[1-3]. Nowmulti-link mechanical press has been applied in many important fields[4-5], such as automotive, military, aerospace, etc.

Static and dynamic analyses of multi-link mechanical press mechanism, which is the foundation of study on stiffness and control system design, can also provide an important theoretical basis for structure design of the mechanism. Up to now, domestic and overseas scholars have done a large number of researches on multi-link mechanism and achieved a series of research results in these areas[6-10]. Du et al.[11]designed a new type of hybrid driven seven bar press mechanism, which realized the journey and velocity control of the working table of the press. Guo et al.[12]studied on the trajectory planning and optimization of the variable velocity servo motor drive crank, using hybrid servo presses of feedback control, to achieve different stamping movement. Li et al.[13]put forward the idea of hybrid driven press, and introduced the characteristics of the hybrid drive machine, and the feasibility of the research of hybrid drive machine is prospected. Zhou et al.[14]studied the optimization of main driving mechanism for servo-punch press. However, there have been few efforts made oriented to analytical modeling which contains kinematic model, kinetostatics model and dynamic model at the same time. Then the static and dynamic analyses have not been sufficiently considered.

This paper takes a six-bar mechanism as an example, the analytical models of the mechanism are established, and kinematics and force analysis of the mechanism are carried out. Based on the kinematic analysis, the displacement, velocity and acceleration equations of the press are derived by using the complex vector method.Considering the gravity and inertia force of each component, the dynamic and statics analysis of the six-bar mechanical press is carried out.And the dynamic model of the press is established by using Lagrange method. Finally, by numerical computation and virtual prototype simulation, the correctness of the analytical models is verified. According to this, the structure model of the six-bar mechanical press is designed, as shown in Fig.1.

1 Kinematics model of the six-bar mechanical press mechanism

1.1 Architectural feature of the six-bar mechanism

As shown in Fig.1, this novel six-bar mechanism which is proposed in this paper includes Frame 1, Crank 2, Linkage 3, Quarter Panel 4, Linkage 5 and Slider 6. Crank 2 and Quarter Panel 4 are respectively connected with a machine frame through a revolute joint;Crank 2 is connected with servo motor and driven by the motor; Linkage 3 is connected with Crank 2 through a revolute joint; Linkage 3 and Linkage 5 are respectively connected with Quarter Panel 4 through a revolute joint; the end of Linkage 5 is connected with Slider 6 through a revolute joint;Slider 6 is in the fixed guide rail, which can move only in the vertical direction. The degree of freedom of the mechanism is one, and the motion of the mechanism can be driven by Crank 2. The structure chart of six-link mechanical press is shown in Fig.2.

1.2 Displacement model of the mechanism

(1)

Fig.1 Structure diagram of six bar mechanism

Fig.2 Structure chart of six-link mechanical press

From Formula(1), we can obtain

(2)

In the Formula(2), the motion law of Crank 2 and the length of each member are known. Therefore, three unknown direction anglesθ3,θ4,θ5and the displacement of the sliderS6are given by

(3)

Where,L1,L2,L3,L41,L42,L5,L,S6,αare the dimension parameters of the mechanism as shown in Fig.1.

1.3 Velocity model of the mechanism

Taking first derivative of Formula(2) with respect to time, we can get

(4)

Where,ωiis the angular velocity of the linki,V6is the velocity of Slider 6.

1.4 Acceleration model of the mechanism

Taking the second derivative of equation (2) with respect to time, the acceleration of the six-bar mechanism can be expressed as

(5)

Where,αiis the angular acceleration velocity of the linki,a6is the acceleration of Slider 6.

2 Kinetostatics model of the six-bar mechanical press mechanism

2.1 Kinetostatics modeling of the mechanism

Fig.3 Schematic diagram of single DOF six-bar mechanism

The force analysis of Crank 2, Link 3, Link 4, Link 5 and Slider 6 are shown in Fig.4, Fig.5, Fig.6, Fig.7, Fig.8 respectively.

Fig.4 Force analysis of crank

Fig.5 Force analysis of Link 3

Fig.6 Force analysis of quarter panel

Fig.7 Force analysis of linkage

Fig.8 Force analysis of slider

Through the analysis of the above five components, we can make a list of the fourteen balance equations. Because the Fourteen equations are linear equations, it can be sorted into the following matrix in accordance with the order of unknown forces on the members 2, 3, 4, 5, 6, and so on. Then we can get

CF=D．

(6)

3 Dynamics model of the six-bar mechanical press mechanism

3.1Generalizedcoordinatesofthemechanism

FromFig.2,Crank2isthedrivinglinkinthesingledegreeoffreedomsix-barmechanism.WhentheangulardisplacementofCrank2isgiven,thepositionofthewholemechanismisalsodetermined,andthus,theangulardisplacementofthecrankcanbetakenasthegeneralizedcoordinates,thatisq=θ2.

TheLagrangeformulaforasingledegreeoffreedomsystemisexpressedas

(7)

Where，E，V，Q，qare the kinetic energy, potential energy, generalized force and generalized coordinates of the system, respectively.

3.2 The Lagrange dynamic equation of the mechanism

3.2.1 Motion analysis of the mechanism

To calculate the kinetic energy of the mechanism, we must know the angular velocity and the velocity of the center of mass of each member.By kinematic analysis of the mechanism, we can get

(8)

Whereθiis the angular displacement of componenti,xSiandySiare the coordinates of the centroid of the component in thexaxis andyaxis.

(9)

(10)

The velocity of the center of massSican be given by

(11)

3.2.2 Kinetic energy of the mechanism

The kinetic energy can be expressed as

(12)

By substituting equation (9) and (11) into equation (12), the kinetic energy of the system can be given by

(13)

3.2.3 Potential energy of the mechanism

The zero potential energy position is located at the origin of the coordinate (see Fig.3), the positive direction of the Y axis is the direction of gravity, the potential energy of the system is written as

V=m2gys2+m3gys3+m4gys4+m5gys5+m6gys6．

(14)

3.2.4Generalizedforceofthemechanism

Thegeneralizedforcecanbedeterminedaccordingtothevirtualworkprinciple.Forasingledegreeoffreedomsix-barmechanism,therelationshipbetweenthevirtualworkδ Wofactiveforceandthegeneralizedvirtualdisplacementcanbegivenby

δW=Qδq ．

(15)

Where,thecoefficientQinfrontofgeneralizedvirtualdisplacementisgeneralizedforce.

Forthesingledegreeoffreedomsix-barmechanism,accordingtothevirtualworkprinciple,wecanget

(16)

From the equation (16), the generalized force of the system is given by

(17)

Where, M2isthetorqueofthemotoractingonthecrank,S6isthedisplacementoftheslider.

3.2.5Motiondifferentialequationofthemechanism

(18)

(19)

According to the equation(14) we can get

(20)

Take formula(17), (18), (19) and (20) into Lagrange equation (7) and simplify it, the dynamic equation of the novel six-bar mechanism is expressed as

(21)

4 Example of kinematic and force analysis of the six-bar mechanical press mechanism

4.1 System parameters of the mechanism

System parameters of the six-bar mechanism are shown in Tab.1.

Tab.1 System parameters of the six-bar mechanism

4.2 Example of kinematic analysis

When Crank 2 of the six-bar mechanism runs three cycles, the displacement curve of Slider 6 obtained by the numerical calculation of MATLAB and the displacement curve of Slider 9 obtained by the virtual proto type of ADAMS are shown in Fig.9. The velocity curve of Slider 6 obtained by the numerical calculation of MATLAB and the velocity curve of Slider 6 obtained by the virtual prototype of ADAMS are shown in Fig.10. The acceleration curve of Slider 6 obtained by the numerical calculation of MATLAB and the acceleration curve of Slider 6 obtained by the virtual prototype of ADAMS are shown in Fig.11.

From Fig.9 to Fig.11, the trend and the peak value of the displacement curves, velocity curves, and acceleration curves are basically similar; the maximum error of the displacement, which occurs in 1.58 s, is 0.000 4 m; the maximum error of the velocity, which occurs in 1.98 s, is 0.000 3 m/s; the maximum error of the acceleration, which occurs in 1.64 s, is 0.000 5 m/s2.

Fig.9 The displacement of six-bar mechanism’s slider

4.3 Example of kinetostatics analysis

When Slider 6 of the six-bar mechanism moves with no-load, that isFr=0, the balance moment curves of crank of mechanism obtained by the numerical calculation of MATLAB and the balance moment curves of crank of mechanism obtained by the virtual prototype of ADAMS are shown in Fig.12. When Slider 6 of the six-bar mechanism moves with load, that isFr=100 kN, the balance moment curves of crank of mechanism obtained by the numerical calculation of MATLAB and the balance moment curves of crank of mechanism obtained by the virtual prototype of ADAMS are shown in Fig.13. The numerical calculation of the bearing reaction between Linkage 5 and Slider 6 and the virtual prototype simulation of the constraint force between Linkage 5 and Slider 6 are shown in Fig.14.

Fig.10 The velocity of six-bar mechanism’s slider

Fig.11 The acceleration of six-bar mechanism’s slider

Fig.12 The balance moment of the six-bar mechanism when the slider has no load

From Fig.12 to Fig.14, the trend of MATLAB calculation result is consistent with the ADAMS virtual simulation result; when Slider 6 of the six-bar mechanism runs with no-load, the maximum error of the balance moment, which occurs in 1.28 s, is 0.198 2 N·m; when Slider 6 of the six-bar mechanism runs with load, the maximum error of the balance moment, which occurs in 1.98 s, is 14.332 4 N·m; when Slider 6 of the six-bar mechanism runs with load, the maximum error of the constraint force between Linkage 5 and Slider 6, which occurs in 1.58 s, is 30 N.

Fig.13 The balance moment of the six-bar mechanism when the slider has load

Fig.14 The constraint force between Linkage 5 and Slider 6 when the slider has load

4.4 Example of dynamics analysis

When Slider 6 of the six-bar mechanism moves with no-load, that isFr=0, the numerical calculation of the driving torque of the crank obtained by MATLAB and the virtual prototype simulation of the driving torque of the crank obtained by ADAMS are shown in Fig.15. When Slider 6 of the six-bar mechanism runs with load, that isFr=100 kN, The numerical calculation of the driving torque of the crank obtained by MATLAB and the virtual prototype simulation of the driving torque of the crank obtained by ADAMS are shown in Fig.16.

Fig.15 The driving torque of the six-bar mechanism when the slider has no load

Fig.16 The driving torque of the six-bar mechanism when the slider has load

From Fig.15 to Fig.16, the trend of numerical calculation result is consistent with the virtual simulation result.When Slider 6 of the six-bar mechanism runs with no-load, the maximum error of the driving torque, which occurs in 1.26 s, is 0.058 8 N·m.When the slider 6 of six-bar mechanism runs with load, the maximum error of the driving torque, which occurs in 1.98 s, is 10.332 4 N·m.

5 Conclusions

1) A novel single-DOF six bar mechanical press mechanism is proposed.The kinematic model of this novel six-bar mechanism is established by using the complex vector method; the kinematic analysis including slider position, slider velocity, and slider acceleration is realized and verified by numerical calculation and virtual simulation.

2) The kinetostatics model of the six-bar mechanism is established by using the matrix method, the constraint force of joints and balance moment are studied and verified.

3) The dynamic model of the six-bar mechanism is established by using the Lagrange Method. The driving torque on crank is solved and verified.

4) This study can not only provide theoretical basis for kinematic analysis, force analysis and structure design of the six-bar mechanism, but also suggest a way of static and dynamic analysis for other multi-link mechanisms.

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(責任編輯：李 磊)

2017-03-02

國家自然科學基金項目(51005138)；山東省優(yōu)秀中青年科學家獎勵基金項目(BS2012ZZ008)

陳修龍(1976—)，男，河北滄州人，副教授，博士，博士生導師，主要從事機構學理論和應用技術的研究. E-mail:cxldy99@163.com

新型單自由度六連桿機械壓力機機構的靜力學及動力學分析

陳修龍1，高文花1，姜 帥1，宋 浩1，茍巖巖2

(1.山東科技大學 機械電子工程學院，山東 青島 266590; 2.泰山職業(yè)技術學院 機電工程系，山東 泰安 271000)

提出了一種新型的單自由度六連桿機械壓力機機構，為了實現(xiàn)機構的動態(tài)靜力學以及動力學分析，采用復數(shù)矢量法建立了機構的運動學模型，基于達朗貝爾原理建立了機構的動態(tài)靜力學方程，利用拉格朗日法建立了機構的動力學方程；分別利用MATLAB理論計算與ADAMS虛擬樣機仿真得到壓力機機構中滑塊的位置曲線、速度曲線、加速度曲線，以及曲柄的平衡力矩和驅動力矩變化曲線。研究不僅為六連桿壓力機運動學分析、平衡力矩和驅動力矩的求解和結構設計提供了理論依據，也為其他壓力機機構的靜力學和動力學分析提供了有效方法。

六連桿機構；運動學分析；靜力學；動力學

TH112

A

1672-3767(2017)05-0080-11

10.16452/j.cnki.sdkjzk.2017.05.012