balancing mechanism ( fourbar linkage , fivebar linkage, crank machasim,vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv..................................................................................................)
THAI NGUYEN UNIVERSITY OF TECHNOLOGY FACULTY OF INTERNATIONAL TRAINING REPORT BALANCING AND SHAPE SYNTHESIS OF LINKS IN PLANAR MECHANISMS Hoang Mai Trung School of Mechanical Engineering, Thai Nguyen University of Technology, 3-2 Road, Tich Luong District, Thai Nguyen City, Thai Nguyen, Vietnam * e-mail: k175905218022@tnut.edu.vn * Instructor: MSc Nguyen Thi Thanh Nga Lecturer of Faculty of International Training, Thai Nguyen University of Technology Table of Contents and Figure I Introduction II Literature review Planar four-bar mechanism .5 Fig Four-bar mechanism detached from its frame.[1] Fig Variations of shaking force and shaking moment without converting link #2 into physical pendulum: Case (1), (2), (5) and (6); with link #2 as physical pendulum: Case (3) and (4).[1] Fig Variations of driving torque without converting link #2 into physical pendulum: Case (1), (2), (5) and (6); with link #2 as physical pendulum: Case (3) and (4).[1] Fig Performances using different balancing methods.[4,5] 13 Fig Performances using different radius of gyration limits.[4,5] 14 Fig Performance using different weight factors for same radius of gyration.[4,5] 15 Fig Dynamic performance of the Hoeken’s mechanism, case (a) normalized driving torque, (b) normalized shaking force, (c) normalized shaking moment to fixed point O1.[6] 17 Fig Balancing of planar four-bar mechanism using three counterweights (Tricamo and Lowen, 1983a,b).[7] 18 Planar five-bar mechanism 18 Fig Variations of shaking force and shaking moment for complete cycle for planar five-bar mechanism.[8] .19 Fig 10 Planar five-bar mechanism detached from its frame.[8] 20 Planar Stephenson six-bar mechanism 20 Fig 11 Original and optimized link shapes of five-bar mechanism [figure drawn on scale].[9] 21 Planar slider-crank mechanism .22 Fig 12 Original and optimized link shapes for planar slider-crank mechanism [figure drawn on scale].[11] 22 Fig 13 Variations of shaking force and shaking moment for complete crank cycle for slider-crank mechanism.[11] .23 Fig 14 Planar slider-crank mechanism.[12] 24 Fig 15 Comparison of different cases for shaking force and shaking moment.[12] 25 Fig 16 Variations in shaking force and shaking moment for complete cycle.[12] 26 III Conclusion .27 IV.References 28 ACKOWLEDGEMENTS I want to thank the teacher and everyone who guided me to complete this project Thank you for everything I would like to express the deepest appreciation to my supervisor, Nguyen Thi Thanh Nga, who has helped and guided in this subject and worked on this project Without her guidance and persistent help, this project would not have been easyly completed One time again, I thank to Nguyen Thi Thanh Nga from who have been guiding and helping in undertaking this project ABSTRACT The linkage balancing problem is an old problem to reduce amplitude of vibration of the frame due to shaking forces and moments which in turn cause noise, wear, fatigue, etc.; and to smoothen highly fluctuating input-torque needed to obtain nearly constant drive speed However, the problem has faced new challenges, particularly, in balancing the combined shaking forces, shaking moments, and input-torque fluctuations in the design of high-speed machinery The methods of balancing linkages are well developed and documented Most of the techniques are based on mass redistribution, addition of counterweights to the moving links ,and attachment of rotating disks or duplication of the linkages These methods have dealt with forces involved, or the momentum fluctuations in the linkages I Introduction A planar mechanism consists of links which are said to move only in parallel planes When planar mechanisms comprise only lower pair joints (revolute and prismatic pairs), they are called planar linkages The balancing is very important because it helps links in planar mechanisms to reduce shaking force and find the optimine shape synthesis of links Without considering the interface between a mechanism and its mounting frame, design of that mechanism cannot be completed During the time that an unbalanced linkage moves, it transmits shaking forces and moments to its surroundings These transmitted forces and moments may cause some serious and undesirable problems such as vibration, noise, wear, and fatigue These disturbances cause vibrations and therefore limit the full potential of many machines Therefore, several methods are developed to eliminate the shaking forces and shaking moments in planar mechanisms Several methods to reduce the shaking force and shaking moment based on various principles The complete force balancing can be achieved by making the total mass center of moving links stationary either using mass redistribution or by adding counterweights Force balancing and trajectory tracking are achieved in a five-bar real-time controllable mechanism using adjusting kinematics parameter approach Instead of complete balance of shaking force and shaking moment, some methods are developed to minimize them simultaneously through optimization The conventional optimization technique is used to optimally balance the planar four-bar mechanismand to analyze the sensitivity of shaking force and shaking moment to the design variables This research present literature review of some methods to balance the shaking force and shaking moment in the planar mechanism and we will take an example of the most common mechanism, the four-bar mechanism, five-bar mechanism, six-bar mechanism and the slide-crank mechanism to explain how to method application for theses II Literature review Planar four-bar mechanism H Chaudhary, S.K Saha el at.(1) using maximum recursive dynamic algorithm for balancing of four-bar linkages ,a numerical problem of planar four-bar mechanism [2,3] as shown in Fig is solved using the method proposed Fig Four-bar mechanism detached from its frame.[1] The link length, mass and other geometric parameters of the unbalanced mechanism are given in Table Table Original parameters of four-bar mechanism Link i Length (m) Mass mi (kg) Moment of inertia di (m) θi (°) 0.1 0.4 0.3 0.3 0.392 1.570 1.177 ∑moi= 3.139 0.0014 0.0841 0.0356 - 0.5 0.20 0.15 0 This Four-bar mechanism has different cases are investigated to balance : + In case (1), in order to make full force balancing, the weighting factors, w1 and w2, are taken as and 0, respectively, in Eq: Minimize Z=w1fsh,rms+ w2nsh,rms Link #1 and link #3 are considered for mass redistribution keeping link #2 intact + In case (2), the simultaneous minimization of force and moment is considered taking the weighting factors, w1 and w2, as 0.5 In this case, the point-masses of all three links are chosen to find their optimum mass distribution + Investigate the effect of this conversion, the cases (1) and (2) are repeated after making link #2 as the physical pendulum and reported as cases (3) and (4) + In Case (5), the mechanism balancing is achieved without changing the original link masses + Balancing using non-symmetric shapes is presented as Case (6) Comparison of original RMS values and peak values of shaking force and shaking moment with those of optimum values Fig Variations of shaking force and shaking moment without converting link #2 into physical pendulum: Case (1), (2), (5) and (6); with link #2 as physical pendulum: Case (3) and (4).[1] Fig Variations of driving torque without converting link #2 into physical pendulum: Case (1), (2), (5) and (6); with link #2 as physical pendulum: Case (3) and (4).[1] In original mechanism have shaking force have RMS equal 5.9604, peak equal 11.8837 ; shaking moment have RMS equal 10.7250, Peak equal 12.3939 and finally driving torque have RMS equal 3.0588 , Peak equal 5.0480 Base on the original mechanism divide values of dynamic quantities in the original and optimized mechanisms in six case In case : when w1 = 1, w2 = to keeping link #2 intact , it’s force balance have + shaking force have RMS equal 0.0087 (−99.85%) Peak equal 0.3395 (−97.14%) + shaking moment have RMS equal 16.1133 (+50.24%) Peak equal 16.3722 (+32.10%) + driving torque have RMS equal 5.4359 (+77.71%) Peak equal 9.4000 (+86.21%) In case : w1 = 0.5, w2 = 0.5, all links are considered It’s force and moment balance have + shaking force have RMS equal Peak equal 2.9620 (−50.31%) 4.4830 (−62.27%) + shaking moment have RMS equal 3.4484 (−67.85%) Peak equal + driving torque have RMS equal 5.1770 (−58.23%) 0.8769 (−71.33%) Peak equal 1.6990 (−66.34%) In case : w1 = 1, w2 = 0; Link #2 is made as physical pendulum and then kept intact It’s force balance have + shaking force have RMS equal Peak equal 0.0688 (−99.85%) 0.1861 (−98.43%) + shaking moment have RMS equal 27.1472 (+153.12%) Peak equal + driving torque have RMS equal Peak equal 30.6055 (+146.94%) 8.7761 (+186.91%) 15.7899 (+212.79%) In case : w1 = 0.5, w2 = 0.5: Link #2 is made as physical pendulum and then all links are considered It’s Force and moment balance have + shaking force have RMS equal Peak equal 4.0330 (−37.34%) 6.3998 (−46.15%) + shaking moment have RMS equal 5.9502 (−44.52%) Peak equal + driving torque have RMS equal Peak equal 8.5520 (−30.99%) 1.5747 (−48.52%) 3.0076 (−40.42%) In case : w1 = 0.5, w2 = 0.5: Keeping link masses unchanged It’s force and moment balance have + shaking force have RMS equal Peak equal 4.3761 (−27.66%) 7.2856 (−38.69%) + shaking moment have RMS equal 5.5974 (−48.12%) Peak equal 6.7374 (−45.64%) + driving torque have RMS equal Peak equal 1.5819 (−48.49%) 2.7111 (−46.29%) In case : w1 = 0.5, w2 = 0.5: Non-symmetrical shapes, it’s force and moment balance have + shaking force have RMS equal Peak equal 4.0172 (−32.95%) 7.4941 (−36.94%) + shaking moment have RMS equal 6.3747 (−40.59%) Peak equal + driving torque have RMS equal Peak equal 7.4163 (−40.16%) 1.7958 (−41.30%) 2.9809 (−40.95%) Fig Variations of shaking force and shaking moment for complete cycle for planar five-bar mechanism.[8] Fig.9 shows the variations of the shaking force and shaking moment over the complete crank cycle Both (a,b), the shaking force and shaking moment reduce a lot in second The optimized link parameters are then found by using the equimomental conditions and given in Table Table Parameters of original and balanced planar five-bar mechanism Link i Length (m) 0.02 0.10 0.10 0.04 Standard mechanism Mas Moment s of mi inertia (kg) 0.03 1.00e−6 0.15 1.25e−4 0.15 1.25e−4 0.06 8.00e−6 Balanced mechanism di (m) θi (°) Mass mi (kg) Moment di of inertia (m) θi (°) 0.01 0.05 0.05 0.02 0 0 0.015 0.052 0.038 0.021 2.93e−6 1.07e−4 8.72e−5 8.05e−6 180 0 0.0017 0.0146 0.0312 0.0061 Fig 10 Planar five-bar mechanism detached from its frame.[8] The resulting effect on shaking moment and driving torque was not considered For the same numerical problem, both shaking force and shaking moment are simultaneously minimized in this paper using proposed methodology and the genetic algorithm Planar Stephenson six-bar mechanism In the method , it is used to solve the balancing problem of a planar Stephenson six-bar mechanism as reported in [9] shown in Fig.11 for which parameters of original mechanism are given in Table For the constant angular velocity of 2π rad/s for link #3, both the shaking force and the shaking moment are minimized by redistributing the link masses as against the addition of counterweights as suggested in [9,10] Table Parameters of original and balanced planar Stephenson six-bar mechanism Link i Length (m) 0.0559 0.1206 0.0032 0.1397 0.0444 Standard mechanism Mass mi (kg) 0.060 0.082 0.075 0.173 0.039 Balanced mechanism Moment of inertia di (m) θi (°) 4.98e−5 3.27e−4 7.27e−7 1.21e−3 1.53e−5 0.0286 0.0630 0.0031 0.0836 0.0197 19 Mass mi (kg) 0.031 0.060 0.019 0.058 0.018 Moment of inertia di (m) θi (°) 3.72e−5 2.00e−4 2.27e−6 2.88e−4 8.12e−6 0.0249 0.0409 0.0057 0.0566 0.0566 0 180 0 Fig 11 Original and optimized link shapes of five-bar mechanism [figure drawn on scale].[9] The corresponding shapes of mechanism links are shown in Fig 11 The reductions of 80.21% and 84.75% were found in the RMS values of shaking force and shaking moment, respectively whereas reductions in peaks are 89% for both quantities Planar slider-crank mechanism The slider-crank mechanism is the most popular in application V Arakelian, S Briot el at.(11) is solved a numerical problem of planar slider-crank mechanism using the proposed method for which a cam mechanism with counterweight to simultaneously reduce the shaking force and shaking moment Fig 12 Original and optimized link shapes for planar slider-crank mechanism [figure drawn on scale].[11] The reductions of 48.45% and 44.14% were found in the RMS values of normalized shaking force and shaking moment, respectively and corresponding shapes of mechanism links are shown in Fig 12 The variations of shaking force and shaking moment over the complete crank cycle are shown in figure: Fig 13 Variations of shaking force and shaking moment for complete crank cycle for slider-crank mechanism.[11] For shaking force: The shaking force is maximum reducing in second - In second the shaking force reduce from 3.5 to - In 0.2 second the shaking force reduce from 1.3 to 0.5 - In 0.4 second the shaking force reduce from 1.2 to 0.7 - In 0.6 second the shaking force reduce from 1.6 to 0.6 - In 0.8 second the shaking force reduce from 1.2 to 0.4 - In second the shaking force reduce from 3.5 to 1.9 For the shaking moment , the shaking moment is the maximum stability in 0.4 second Kailash Chaudhary and Himanshu Chaudhary el at (12) proposed method using equimomental system of point-masses.This method can be effectively used to balance the mechanisms having revolute and prismatic joints while most of the methods available in the literature are for the mechanisms with revolute joints only A slider-crank mechanism is balanced in this paper by optimally distributing the link masses while a cam mechanism with counterweight was used to balance the same mechanism Therefore, it is advantageous to use the method proposed in this paper as compared to the method counterweight which increases the overall mass and complexity of the mechanism to start searching the optimum solution and likely to produce local optimum solution close to the start point In this paper, the formulation of optimization problem is simplified by modelling the rigid links of mechanism as dynamically equivalent system of pointmasses, known as equimomental system [15, 16] This optimization problem is solved by using genetic algorithm which doesn’t require a start point and searches the solution in the entire design space Therefore, it produces the global optimum solution for the problem Fig 14 Planar slider-crank mechanism.[12] The results corresponding to the different combinations of the weighting factors are presented in Table and shown in Fig 14 The case is complete shaking force balancing in which the rms value of shaking moment increases to four times of that the unbalanced mechanism Similarly, in case 3, shaking force increases while shaking moment reduces substantially Reduction in both the quantities can occur in case 2, in which equal weighting factors are assigned to them Table RMS values of dynamic quantities for different combinations of weighting factors Standard value Case 1: (w1=1.0;w2=0.0) Case 2: (w1=0.5;w2=0.5) Case 3: (w1=0.0;w2=1.0) Shaking force 3.6877 2.2247x10-4 2.9132 3.8605 Shaking moment 1.0047 4.1714 0.1883 7.7980 x10-5 Fig 15 Comparison of different cases for shaking force and shaking moment.[12] The comparison of the original rms values with the optimum rms values of the shaking force and shaking moment obtained using conventional and genetic algorithm are presented in Table and Fig The comparison of the original rms values with the optimum rms values of the shaking force and shaking moment obtained using conventional and genetic algorithm are presented in Table and Fig The comparison of the original rms values with the optimum rms values of the shaking force and shaking moment obtained using conventional and genetic algorithm are presented in Table and Fig 16 Table RMS values of dynamic quantities of normalized standard and optimized mechanisms Balancing method Standard mechanism fmincon Genetic algorithm Shaking force 3.6877 2.9132 (-21) 2.0051(-46) Shaking moment 1.0047 0.1883(-81) 0.0105 (-99) Fig 16 Variations in shaking force and shaking moment for complete cycle.[12] By using the conventional optimization method, the reduction of 21% and 81% was found in the rms values of shaking force and shaking moment, respectively The application of the genetic algorithm results in reduction of 46% and 99% in the values of shaking force and shaking moment, respectively It is observed that if the mass of slider is not at CG, shaking moment reduces in the mechanism The moment of inertia of slider about CG doesn’t affect the values of shaking force and shaking moment and hence not given for original and balanced mechanisms Moreover, the shaking force rises up to in the original unbalanced mechanism In the balanced mechanism, it goes up to as shown in Fig 16(a) However, the peak value of normalised shaking moment reduces from 1.8 to 0.25 as shown in Fig 16(b) One of the advantages of the proposed method is that the links of the balanced mechanism are of the uniform thickness while the force and inertia counterweights added to the original mechanism in traditional methods are of large thickness and radius compared to the original link parameters Also, it doesn't require any predefined shapes or design domain to start The percentage error of resulting inertia values were found within ±5% Also, the resulting stresses for all the links of balanced mechanism can be calculated at the weakest sections under external loads III Conclusion When an unbalanced linkage must run at high speeds, or contains massive links, considerable shaking force and shaking moment are transmitted to its surroundings These disturbances cause vibrations and therefore limit the full potential of many machines With some of methods in the reseach help links in planar mechanisms to reduce shaking force, shaking moment( noise,etc,…) and find the optimine shape synthesis of links.This is very important and its show in planar four-bar and slide-crank mechanism to explain these method The comparison of the original rms values with the optimum rms values of the shaking force and shaking moment obtained using conventional and genetic algorithm are presented in Table and Fig The comparison of the original rms values with the optimum rms values of the shaking force and shaking moment obtained using conventional and genetic algorithm are presented in Table and Fig The comparison of the original rms values with the optimum rms values of the shaking force and shaking moment obtained using conventional and genetic algorithm are presented in Table and Fig The comparison of the original rms values with the optimum rms values of the shaking force and shaking moment obtained using conventional and genetic algorithm are presented in Table and Fig References [1]H Chaudhary, S.K Saha, Balancing of four-bar linkages using maximum recursive dynamic algorithm, Mech Mach Theory 42 (2) (2007) 216–232 [2] R.S Berkof, Complete force and moment balancing of inline four-bar linkage, Mech Mach Theory (1973) 397–410 [3] M.R Farmani, A Jaamialahmadi, M Babaie, Multiobjective optimization for force and moment balance of a four bar linkage using evolutionary algorithms, J.Mech Sci Technol 25 (12) (2011) 2971–2977 [4] G.G Lowen, R.S Berkof, Determination of forced-balance four-bar linkages with optimum shaking moment characteristics, ASME Journal of Engineering for Industry 93 (1) (1971) 39–46 [5] T.W Lee, C Cheng, Optimum balancing of combined shaking force, shaking moment, and torque fluctuations in high speed linkages, ASME Journal of Mechanisms, Transmissions, and Automation in Design 106 (1984) 242–251 [6] G.G Lowen, F.R Tepper, R.S Berkof, The quantitative influence of complete force balancing on the forces and moments of certain families of four-bar linkages, Mechanism and Machine Theory (1974) 299–323 [7] Tricamo and Lowen (1983a, b), A partial force balancing method for a planar four-bar mechanism [8] D Ilia, R Sinatra, A novel formulation of the dynamic balancing of five-bar linkages with application to link optimization, Multibody Sys Dyn 21 (2009) 193–211 [9] M Verschuure, B Demeulenaere, J Swevers, J.D Schutter, “On The Benefits of Partial Shaking Force Balance in Six-Bar Linkages”, Proceeding of 12th IFToMM World [10] R.S Berkof, G.G Lowen, A new method for completely force balancing simple linkages, ASME J Eng Ind 91 (1) (1969) 21–26 [11] V Arakelian, S Briot, Simultaneous inertia force/moment balancing and torque compensation of slider-crank mechanisms, Mech Res Commun 37 (2010)265–269 [12] Kailash Chaudhary and Himanshu Chaudhary, balancing using equimomental system of point-masses ... principles The complete force balancing can be achieved by making the total mass center of moving links stationary either using mass redistribution or by adding counterweights Force balancing and trajectory... modeling and is solved by conventional optimization method It uses non-linear constraint optimization in which the center of mass parameters of moving links are chosen as the design variables Fig... is simplified by modelling the rigid links of mechanism as dynamically equivalent system of pointmasses, known as equimomental system [15, 16] This optimization problem is solved by using genetic