钢锥锥轮式的无级变速器的传动与设计钢锥式无级变速器设计.doc
钢锥锥轮式的无级变速器的传动与设计许勇[含CAD高清图纸和说明书]
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含CAD高清图纸和说明书
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钢锥锥轮式的无级变速器的传动与设计许勇[含CAD高清图纸和说明书],含CAD高清图纸和说明书,钢锥锥,轮式,无级,变速器,传动,设计,CAD,图纸,说明书
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译文滑动块转动曲柄机构的设计 第一部分:多阶段动作产生摘要设计滑动块曲柄机构到完成多阶段运动产生应用代表性地完成可通过可调节的平面的四杆运动加速器,这一方法是被提出来了的。这个方法的好处有两点：第一,多阶段的规定刚体位置是可完成的利用一个机构同较少数活动部分,它用的活动部分比那平面的四杆机构要少。第二，在这阶段滑动块曲柄运动加速器可以完成阶段的规定刚体位置不需任何人工的或自动调整的它的运动副。一滑块路径启动曲柄运动加速器到完成刚体位置的二阶段是设计者利用第7项命令多项式去联接那那可调节的平面四杆运动产生器推杆连接的运动副。这个多项式产生平滑平稳径向位移、速度、加速度和带有转角轮廓的边界条件,这些是可以被呈现的。在本研究中例子问题是考虑一个二阶段的运动副转动平面四杆的机构装置的调整。 2004 Elsevier 公司版权所有。1.介绍平面的四连杆机构广泛的被被用于机械系统和装置。由于平面四杆机构的平面运动学,平面的类型和连接轴方向,它可以是实际的设计并且实现这些机构（与大部分四杆空间机构相比较）。 除此之外, 平面的四连杆机构有一广范系列的图解式和解析设计和分析方法。机构分析中产生的问题要求一个刚体是通过一系列规定位置而被控制的.图.1显示了四连杆机构可用于生产这个运动通过制造那刚体作为它的耦合器连接的一部分。图.2显示了一部装配机器的三个位置的运动产生.理想耦合器的运动只能由个别的离散的精确位置近似表示。由于一连接点只有一有限数的有效的尺寸,设计师可以只规定一有限数的精确点。一个四连杆机构可以满足直到五个规定位置由那运动产生问题。然而,一个可调节的四连杆机构可以满足超过五个给定的位置用这一样的硬件。一四杆机构的运动副可以用二种不同的方法来调整:可调曲柄推杆长度（图.3）和安装曲柄推杆联杆调整（图.4）。那可调节的传动机构可以供应解决一般平面运动（图.5）两个阶段的方法。 如果在调整之后,一四连杆机构在第一阶段是被设计能达到达位置1，2和3,同相地、这同样的接合在第二阶段可以到达三个新的位置4,5和6。两个阶段的运动可以利用一样部件通过校准一个或多个接合叁数来完成,接合可以在这些位置精确地产生运动并且近似表示在其他的位置的运动。连接器的真实运动是精密位置被用愈较多,对理想的运动也愈靠近。 图.1 平面四杆机构 图.2 平面四杆卸栽机构 图.3 可调节长度的曲柄机构 图.4 固定长度的曲柄机构关于运动的产生在可调节的传动机构的区域内,在已出版的作品里1-19 略微被限制。上述的工作包括包括 Ahmad和 Waldron的工作1,他们发展一方法关于综合处理一四连杆机构同可调传动装置安装。他们解决二个阶段的问题用一最大量总数的五个位置。Tao和Krishnamoorthy2发明了绘图的合成程序用尖头产生可变耦合器弯曲。图.5 规定刚体位置的两阶段Mcgovern和Sandor 3,4提出了综合处理可调节的机构的功能和路径生成利用合成物耦合器的方法。Funabashietal.介绍一般方法到设计平面,球体和空间机构哪个可以校准的调整输入/输出关系。Shoup设计可调节的存在于空间的滑动块曲柄机构被当作可变的换置使用泵。 Cheun-chom 和 Kota7介绍了一般的方法关于合成可调节的机构利用可调节的二数。Wilhelm呈现了为可调节的四杆机构的二相运动产生问题的合成方法。Wangand Sodhi9 呈现了解决为那在每二时期中的二个阶段的恰当的移动铰链的三个位置的问题。Russell和Sodhi10,11 最近有耐心的介绍这些方法为综合处理可调节的空间的机构对于多阶段运动产生,空间的RRSS机构可以是综合处理到完成阶段的规定精确的刚体位置。最近Chang12 呈现了可调整四杆机构用指定的切线速度产生圆形的弧。如果存在过任何性能有关限制到那可调节的平面的四杆机构, 人工控制或自动控制是被要求完成所有的规定阶段在多阶段的申请。人工控制可能是耗费时间的尤其是如果那调整过程处于被涉及到的收上位置及机件控制被经常地运用。实现自动化调整能力可能使机制不实用从财务的立场来说-尤其当操作和维护开支被考虑的时候。对于一可调节的平面的四杆运动加速器它包含移动副和连接长度一起控制推杆连接而曲柄连接只能用移动副控制,一等效的滑动块曲柄运动加速器可以被设计成能完成多阶段的规定刚体的位置。这种方法的好处是规定刚体位置的多个阶段是可利用一机构与较少数活动件就能实现的,它与那平面的四连杆机构和那滑动块曲柄运动加速器相比较只用少数活动件就可以完成阶段的规定刚体位置而不需要任何实际的或自动操作控制的它的移动副在这些阶段中。在这个一工作中,一种方法设计偏置曲柄运动加速器实现一般地多阶段运动产生点样可利用可调节的平面的四杆运动加速器来完成是已经被提出来了的。一滑块路径启动曲柄运动加速器到完成刚体位置的二阶段是设计者利用第7项命令多项式去连接那那可调节的平面四杆运动产生器推杆连接的移动副。推杆连接的移动副的径向位移、速度、加速度和参数也被规定利用这个多项式的界限条件的情况2.刚体规则和多阶段运动链锁反应存在于这个工作中的滑块曲柄运动加速器设计法可适应事实上任何多阶段运动链锁反应可利用的方法,那方法含有移动副的控制与安装和可分别地调整曲柄和推杆长度。作者10,11发展了他们整个运动阶段链锁反应在这一个研究中被利用的方法。那平面的四杆运动加速器在图图.6是图解说明了的。在本研究中、连线a0-a1的是表示曲柄而连线b0-b1表示摇杆。平面的四连杆机构的杆a0-a1和杆b0-b1必须满足那固定长条件因为它的安装和移动副的连接轴要保持平行。给一固定支点b0和一移动的铰链b1它们的长度条件等价于公式(1)当用合成法合成平面的四连杆机构的曲柄和从动件时20,21必须被满足。等式（1）可以被重新写成等式（3）。在等式（3）里，变量R表示曲柄或从动件连杆的长度。这一个工作的一个目的是设计一个等效的滑动块-曲柄运动加速器作为一可调节的平面的四杆运动机构。虽然平面的四连杆机构中的曲柄和从动件连杆两者的运动铰链是可调整的,但只有动件连杆的长度可被调整（非那曲柄连杆)。 通过做这些,这个等效滑动块曲柄运动机构是被设计成将会有一个固定曲柄连杆长度和一滑动块路径这就相当于从动件连杆的调解。 图.6 平面四杆运动加速器及它刚体上的p、q、r点 （3）方程（2）是一刚体位移矩阵,它是存在于空间的刚体位移矩阵20,21的矩阵与逆矩阵之乘积。为一刚体在适当的位置“i”和那之后的位置“j”制定坐标,矩阵 Dij是一个变换矩阵要求变换坐标从位置“i”到位置“j”变量 p,q和r在等式方程（2）中表示那刚体在二维空间的位置。虽然这一位置的二维空间位置是通常被描述为单个点和一位移角（例如: p和 ),作家选择描述刚体使用三个点作为计算的目的。如果用户喜欢描述那刚体利用传统的的标记,这个位移矩阵在方程等式（2）将被替换为简单的平面刚体位移矩阵20,21。因为有四个变量（b0x、b0y、b1x,and b1y）,一个五个刚体位置的最大值可以被确定,不需要任意的选择一参数作为其中的一阶段（看表1）。点p、q和r 将不会全部的落在各刚体位置的同一直性上.拿这个预防措施防止那些在刚体位移矩阵（方程等式（2）中的排变成成比例项的。有比例项的排,这些矩阵不能被倒置的。在表1里、给出了为可调节的平面的四杆运动机构规定的刚体位置的最大极限数目转动曲柄和从动件连杆的固定和移动的铰链的数目确定了刚体位置的最大极限数目。 这个例子问题在这个工作中，一个等效滑动块曲柄是被设计成能完成一二相移动铰链控制请求为一可调节的平面的四杆运动加速器。在这二相中,可调整的移动铰链例子的问题在这一个工作中,需要的未知数是 a0,a1,a1n,b0,b1和b1n，未知数a0 和 b0表示平面的四杆机构繁荣固定支点。未知数a1,a1n,b1和b1n表示那移动铰链在平面四杆机构中的1阶段和阶段。由于这些未知数的中间每一个有二组成物,则总共由12个变量来确定。表1 可调的平面四杆机构的规定刚体位置和各阶段的变化方程等式 (4)-(8), 是用来计算六个未知者中的五个在a0, a1和a1n 。这个变量a0x 和连杆长度R是确定了的。方程等式 (9)-(13), 是用来计算六个未知者中的五个在b0, b1和b1n 。这个变量b0x 和连杆长度R1和R2是确定了的。3.轨道链锁反应级在前一单元描述了那多阶段运动链锁反应级方法之后,用户可以用合成法合成一个平面的四杆运动加速器和确定移动副的路径。这个从动件连杆的运动副的轨道必须以一种方式被连接的,这种方式以允许平滑的变位速度，加速度和 变换在确定运动副的轨道之间。突然的或不连续的变化将最终导致滑动块转动曲柄机构的过度磨损。这等效滑动块曲柄运动加速器的滑动块路径将由该从动件连杆的运动副的路径和连接他们的轨道组成。在一可调节的四连杆机构的操作期间由于在一个特别的阶段,这转动曲柄和从动件连杆的运动副的半径位置是固定的及这个运动副半径的速度,加速度和转角是零。这同样的适用在可调节的平面的四连杆机构的运动副,固定连杆长度调整的期间 ,当运动副和连杆长度调整被考虑的时候,这运动副的径向位置，速度,加速度和转角进行从这连杆参数在前阶段到这后阶段连杆参数的变化。如果转变曲线的产生是因为这从动件连杆,及这个曲线图是分段的连接到这个从动件的移动副的曲线图上的,这就相当于这个阶段前后的变化, 一个单一的滑动块轨道的形成说明在这些阶段之间的变化（或从动件连杆移动副的调整）。一7次顺序多项式22,23是要求确定这可调整的平面的四杆运动加速器中从动件的运动副的径向位置，速度，加速度和转角在这些阶段的变化。这径向变位，速度，加速度和转角边界条件关于这个多项式是在这一个工作中, R0是从动件连杆在阶段一的长度（连杆b0 - b1)而Rf是从动件连杆在阶段二的长度（连杆b0 -b1n）。这些约束确定了一线性集的八个方程等式与八个数,它们的解答式是4. 例问题带有固定转动曲柄的可调整的平面四杆运动加速器的两个阶段的运动副的调节和从动件从动件长度在这一个断面是被例证了的。在表2里是列出关于七个规定刚体位置的点p, q和r在X Y 座标系中的座标。表2可调整的平面四杆运动加速器规定刚体的位置等式方程（4）-（8）用来计算六个未知者中的五个在a0、a1和a1n中。这可变的a0x和连杆长度R1是确定了的（a0x = 0而R1 = 1）利用下列初始值:这平面的四连杆机构解答表示为 图.7 可调整的平面四杆运动加速器和相应规定刚体位置 图.8 用合成法合成可调整的平面的四杆运动加速器的运动副的轨迹等式方程.（9）-（13）是用来计算在 b 0 ， b 1 和 b 1n 中六个未知数中的五个。这可变的b0x和连接长度R1而R2是确定了的（b0x = 1.5、R1 = 1.5、R2 = 1.3）。利用下列初始估计：这平面的四连杆机构解答表示为:利用这已计算了的值和运动副的参数,可调整的平面四杆机构运动加速器的结果在图.7里被说明.在本研究中一等效滑动块曲柄加速器是被设计成平面的四杆运动加速器的。用合成法合成可调整的平面的四杆运动加速器在阶段一和阶段二的开始和结束位置是被说明的在图.8。 由于这曲柄连杆由一固定的长度的连杆的运动副来控制, 这个图.9 等效滑动块曲柄加速器和刚体的初始位置 滑块径向位移曲柄的角位移图.10 合成法合成滑动块曲柄运动加速器中滑动块相对曲柄转角径向位移连杆的运动副的全部位置（a1经过a4而a1n经过 D57 a1n）是位于同一条圆弧上的。而从动件连杆的运动副的位置（b1穿过b4和b1n穿过 D57 b1n）落在两条不同的弧上（一个为一个阶段）。为了完成等效滑动块曲柄运动加速器的滑动块轨道,等式方程（14）是用来计算一连接从动件运动副如在图.8.的路径。使用方程（14）和这规定边界条件,滑动块轨道在图.9是可以被设计的。滑动块轨道产生这径向变位，速度，加速度和转角轮廓这在图.1013中被说明了。 在表 3 列出是等效平面滑动块转动曲柄机构的七个规定刚体位置点p ,q 和r在X Y 座标系中的值。为了达到位置2,3 和4在表3中,连杆a0a1需绕X轴分别旋转到130,125和120。为了要在表 3 中达成位置 5,6 和 7,连杆a0-a1需绕X轴分别旋转到100,95和90。在这两个阶段,曲柄转角最初 135 是相对 X轴和刚体点坐标在这个转动曲柄的位置是表格3中位置 1 中是坐标。 滑块的径向速度 曲柄的角位移图.11 合成法合成滑动块曲柄运动加速器中滑动块相对曲柄转角径向速度 滑块的径向速度曲柄的角位移图.12 合成法合成滑动块曲柄运动加速器中滑动块相对曲柄转角径向加速度曲柄的角位移图.12 合成法合成滑动块曲柄运动加速器中滑动块相对曲柄转角径向加速度表3图.1013说明了等效滑动块曲柄运动机构径向这径向变位，速度，加速度和转角(相对于转动曲柄位移角）在阶段1到阶段2这其中的变化。这径向速度，加速度和转角的边界条件（等式方程( 16 ( 18)和等式方程( 20) ( 22)是指定到零的,这是为了产生的速度，加速度和转角轮廓与那外在变化的轮廓是相连的。这径向位移轮廓边界条件(等式方程。( 15)和( 19)是表示阶段1和阶段2从动件连杆的长度( R0 = 1.5和Rf = 1.3)，这也是为了形成的位移与那在外面变化的轮廓是连续的。5.讨论由于被综合的机械装置所需要，在这一个工作中被呈现的滑件路径设计方法因只有固定的和运动副对可调整平面的四杆机械装置对大部份的现有动作是可适用方法。虽然一个二个阶段的移动副问题在这一个工作被例证,规定的刚体位置的另外时期能被插入。 通过计算那平面四杆机构的坐标和移动副分别为附加的阶段,和每个另外的阶段另外的转变路径 (等式方程 (14)-(30) 等效滑动块曲柄运动加速器可以被设计成能达到这个附加的阶段。 虽然二维的空间的刚体的位置普遍被描述被一点和一个变位角 (p 和 h 举例来说),作家选择描述刚体使用三个点作为计算的目的。如果使用者偏爱描述使用的刚体传统的标记,位移矩阵在方程(2) 将会替换为这传统的平面刚体被放置成矩阵20,21。 计算机辅助设计软件将规定在这一个工作和数学软件的机械装置叁数用来计算机械装置。这一个软件使那能够作成表规定和有计划的叁数被四位有效数来表示。6.结论为滑件曲柄机构的一个设计方法达成多阶段动作链锁反应级请求的完成通过平面的四连杆机构带有可调整的运动副是被呈现在这项研究中的。这种方法的好处是那使用它,规定刚体使用一机构与较少数活动部分可完成的多阶段位置,它用到的活动部分部分与平面的四连杆机构相比较是少的。这一个方法的另一种利益是使用它,滑块曲柄机构能被设计达成规定刚体的各阶段不需任何的人工的或自动操作控制它的运动副在这些阶段中。一滑块路径启动曲柄运动加速器到完成刚体位置的二阶段是设计者利用第7项命令多项式去连接那那可调节的平面四杆运动产生器推杆连接的移动副。例子问题在本研究中认为一二阶段的移动副调整可调整平面的四杆机械装置。33On the design of slider-crank mechanisms. Part I:multi-phase motion generationKevin Russella, Raj S. Sodhib,*aArmaments Engineering and Technology Center, US Army Research, Development and Engineering Center,Picatinny Arsenal, NJ 07806-5000, USAbDepartment of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982, USAReceived 24 February 2003; received in revised form 12 July 2004; accepted 12 July 2004Available online 28 September 2004AbstractA method for designing slider-crank mechanisms to achieve multi-phase motion generation applicationstypically accomplished by adjustable planar four-bar motion generators is presented. The benefit of thismethod is twofold. First, multiple phases of prescribed rigid body positions are achievable using a mech-anism with fewer moving parts than the planar four-bar mechanism. Second, the slider-crank motion gen-erator can achieve phases of prescribed rigid body positions without any physical or automatedadjustments of its moving pivots between phases. A slider path that enables the slider-crank motion gen-erator to achieve two phases of prescribed rigid body positions is designed by using 7th order polynomialsto connect the moving pivot paths of the follower link of the adjustable planar four-bar motion generator.This polynomial generates smooth radial displacement, velocity, acceleration and jerk profiles with bound-ary conditions that can be prescribed. The example problem in this work considers a two-phase movingpivot adjustment of a planar four-bar mechanism.? 2004 Elsevier Ltd. All rights reserved.0094-114X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2004.07.009*Corresponding author. Tel.: +1 973 596 3362; fax: +1 973 642 4282.E-mail address: sodhi (R.S. Sodhi)./locate/mechmtMechanism and Machine Theory 40 (2005) 285299MechanismandMachine Theory1. IntroductionPlanar four-bar mechanisms are widely used in mechanical systems and devices. Due to the pla-nar kinematics, joint type and joint axis orientations of the planar four-bar mechanism, it can bepractical to design and implement these mechanisms (compared to most four-bar spatial mecha-nisms). In addition, an extensive array of graphical and analytical design and analysis methodsexists for planar four-bar mechanisms.Motion generation problems in mechanism synthesis require that a rigid body be guidedthrough a series of prescribed positions. The four-bar linkage shown in Fig. 1 can be used to pro-duce this motion by making the rigid body as a part of its coupler link. Fig. 2 shows motion gen-eration for the three positions in an assembly machine. An ideal motion of the coupler can only beapproximated by several discrete precision positions. Since a linkage has only a finite number ofsignificant dimensions, the designer may only prescribe a finite number of precision points. Afour-bar linkage can satisfy up to five prescribed positions for the motion generation problem.However, an adjustable four-bar linkage can satisfy more than five given positions with the sameCouplerFig. 1. Planar four-bar mechanism.Fig. 2. Planar four-bar loading mechanism.286K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299hardware. The moving pivots of a four-bar linkage can beadjusted in two different ways: withadjustable crank/follower lengths (Fig. 3) and with fixed crank/follower link adjustments (Fig. 4).The adjustable linkages can provide solution for two phases of general plane motion (Fig. 5). Ifa four-bar linkage is designed to reach positions 1, 2 and 3 in phase 1, after the adjustments, theFig. 3. Adjustable crank length.Fig. 4. Fixed crank length.PHASE ONEPHASE TWO412356Fig. 5. Two phases of prescribed rigid body positions.K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299287same linkage can reach three new positions 4, 5 and 6 in the second phase 2. Both phases of mo-tion can be accomplished using the same hardware by adjusting one or more of the linkage param-eters. The linkage can create the motion precisely at these positions and will approximate themotion at other positions. The more precision positions are used, the closer to the ideal motionis the actual motion of the coupler.In the area of adjustable linkages for motion generation, published work is somewhat limited119. Previous work includes the work of Ahmad and Waldron 1 who developed a techniquefor synthesizing a four-bar linkage with adjustable driven fixed pivot. They solved two-phaseproblems with a maximum total number of five positions. Tao and Krishnamoorthy 2 developedgraphical synthesis procedures to generate variable coupler curves with cusps. McGovern andSandor 3,4 presented methods to synthesize adjustable mechanisms for function and path gen-eration using complex variables. Funabashi et al. 5 presented general methods to design planar,spherical and spatial mechanisms which can adjust inputoutput relationships. Shoup 6 designedadjustable spatial slider-crank mechanism to be used as a variable displacement pump. Cheun-chom and Kota 7 have presented general methods for the synthesis of adjustable mechanismsusing adjustable dyads. Wilhelm 8 developed synthesis techniques for two-phase motion gener-ation problems of adjustable four-bar linkages. Wang and Sodhi 9 developed solutions for thetwo-phase adjustable moving pivot problems with three positions in each of the two phases. Rus-sell and Sodhi 10,11 recently presented methods for synthesizing adjustable three-dimensionalmechanisms for multi-phase motion generation with tolerances. Using these methods, spatialRRSS mechanisms can be synthesized to achieve phases of prescribed precise rigid body positionsand rigid body positions with tolerances. Recently Chang 12 presented synthesis of adjustablefour-bar mechanisms generating circular arcs with specified tangential velocities.If there is any performance-related limitation to the adjustable planar four-bar mechanism, it isthat manual or automated adjustments are required to achieve all of the prescribed phases inmulti-phase applications. Manual adjustments can be time consumingespecially if the adjust-ment procedure is involved and the mechanism adjustments must be performed frequently. Imple-menting automated adjustment capabilities may make the mechanism impractical from a financialstandpointespecially when operations and maintenance expenditures are considered.For an adjustable planar four-bar motion generator that incorporates both moving pivot andlink length adjustments for the follower link and only moving pivot adjustments for the cranklink, an equivalent slider-crank motion generator can be designed to achieve multiple phases ofprescribed rigid body positions. The benefits of the method are that multiple phases of prescribedrigid body positions are achievable using a mechanism with fewer moving parts than the planarfour-bar mechanism and the slider-crank motion generator can achieve phases of prescribed rigidbody positions without any physical or automated adjustments of its moving pivots betweenphases.In this work, a method to design slider-crank motion generators to achieve multi-phase motiongeneration applications typically accomplished by adjustable planar four-bar motion generators ispresented. A slider path that enables the slider-crank motion generator to achieve two phases ofrigid body positions is designed by using 7th order polynomials to connect the moving pivot pathsof the follower link of the adjustable planar four-bar motion generator. The radial displacement,velocity, acceleration and jerk parameters of the moving pivot of the follower link are also pre-scribed using the boundary conditions of these polynomials.288K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 2852992. Rigid body guidance and multi-phase motion generationThe slider-crank motion generator design method presented in this work is adaptable to virtu-ally any multi-phase motion generation method available that incorporates moving pivot adjust-ments with fixed and adjustable crank and follower lengths respectively. The authors 10,11developed the multi-phase motion generation method utilized in this work.The planar four-bar motion generator is illustrated in Fig. 6. In this work, link a0a1is the des-ignated crank link and link b0b1is the designated follower link. Links a0a1and b0b1of the pla-nar four-bar mechanism must satisfy the constant length condition only since its fixed and movingpivot joint axes remain parallel. Given a fixed pivot b0and a moving pivot b1the constant lengthcondition in Eq. (1) 20,21 must be satisfied when synthesizing the crank and follower links of theplanar four-bar mechanism.bj? b0Tbj? b0 b1? b0Tb1? b0j 2;3;.;n1whereb0 b0x;b0y;1b1 b1x;b1y;1bj Dij?b1andDij? pjxqjxrjxpjyqjyrjy111264375pixqixrixpiyqiyriy111264375?12Eq. (1) can be rewritten as Eq. (3). In Eq. (3), the variable R represents the length of the crank orfollower link.One objective of this work is to design an equivalent slider-crank motion generator for anadjustable planar four-bar motion generator. Although the moving pivots of both the crankjajjpYbpXqrrqjjiii01a0a1bbFig. 6. The planar four-bar motion generator with rigid body points p, q and r.K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299289and follower link of the planar four-bar mechanism are adjustable, only the length of the followerlink will be adjusted (not the crank link). By doing this, the equivalent slider-crank motion gen-erator to be designed will have a fixed crank link length and a slider path that accounts for theadjustment of the follower link.bj? b0Tbj? b0 R2j 2;3;.n3Eq. (2) is a rigid body displacement matrix. It is a derivative of the spatial rigid body displacementmatrix 20,21. Given the coordinates for a rigid body in position i and the subsequent j, ma-trix Dij is the transformation matrix required to transform coordinates from position i to posi-tion j. Variables p, q and r in Eq. (2) represent the position of the rigid body in two-dimensionalspace. Although the position of a rigid body in two-dimensional space is commonly described by asingle point and a displacement angle (p and h for example), the authors chose to describe the rigidbody using three points for computational purposes. If the user prefers to describe the rigid bodyusing conventional notation, the displacement matrix in Eq. (2) will be replaced with the conven-tional plane rigid body displacement matrix 20,21. Since there are four variables (b0x, b0y, b1xand b1y), a maximum of five rigid body positions can be prescribed, with no arbitrary choiceof parameter for one phase (see Table 1).Points p, q and r should not all lie on the same line in each rigid body position. Taking thisprecaution prevents the rows in the rigid body displacement matrix (Eq. (2) from becoming pro-portional. With proportional rows, this matrix cannot be inverted.In Table 1, the maximum numbers of prescribed rigid body positions for the adjustable planarfour-bar motion generator for several phases are given. The number of fixed and moving pivotcoordinates for the crank and follower links determine the maximum number of rigid body posi-tions. In the example problem in this work, an equivalent slider crank is designed to achieve atwo-phase moving pivot adjustment application for an adjustable planar four-bar motiongenerator.In the two-phase, adjustable moving pivot example problem in this work, the required un-knowns are a0, a1, a1n, b0, b1and b1n. The unknowns a0and b0represent the fixed pivots of theplanar four-bar mechanism. The unknowns a1, a1n, b1and b1nrepresent the moving pivots inphase 1 and phase 2 of the planar four-bar mechanism. Since each of these unknowns has twocomponents, there are a total of 12 variables to determine.a0 a0x;a0ya1 a1x;a1ya1n a1nx;a1nyb0 b0x;b0yb1 b1x;b1yb1n b1nx;b1nyTable 1Prescribed rigid body position and phase variations for the adjustable planar four-bar mechanismNumber of phasesMaximum number of rigid body positionsCrank or follower linksNumber of unknownsNumber of free choices1540286031180m5 + 3(m ? 1)2 + 2m0290K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299Eqs. (4)(8), were used to calculate five of the six unknowns in a0, a1and a1n. The variable a0xandthe link length R1are specified.D12?a1? a0TD12?a1? a0 ? R21 04D13?a1? a0TD13?a1? a0 ? R21 05D14?a1? a0TD14?a1? a0 ? R21 06D56?a1n? a0TD56?a1n? a0 ? R21 07D57?a1n? a0TD57?a1n? a0 ? R21 08Eqs. (9)(13) were used to calculate five of the six unknowns in b0, b1and b1n. The variable b0xand the link lengths R1and R2are specified.D12?b1? b0TD12?b1? b0 ? R21 09D13?b1? b0TD13?b1? b0 ? R21 010D14?b1? b0TD14?b1? b0 ? R21 011D56?b1n? b0TD56?b1n? b0 ? R21 012D57?b1n? b0TD57?b1n? b0 ? R21 0133. Trajectory generationAfter incorporating the multi-phase motion generation method described in the previous sec-tion, the user can synthesize a planar four-bar motion generator and determine the paths of itsmoving pivots. The moving pivot paths of the follower link must be connected in a manner thatwill allow smooth displacement, velocity, acceleration and jerk transitions between the determinedmoving pivot paths. Abrupt or discontinuous transitions will ultimately result in excessive wearon the slider-crank mechanism. The slider path of the equivalent slider-crank motion generatorwill be comprised of the moving pivot paths of the follower link and the trajectories that connectthem.During the operation of an adjustable four-bar mechanism within a particular phase, the radialpositions of the moving pivots of the crank and follower links are constant and the radial veloc-ities, accelerations and jerks of these moving pivots are zero. The same holds true during movingpivot, constant link length adjustments of the adjustable planar four-bar mechanism. When mov-ing pivot and link length adjustments considered, the radial positions, velocities, accelerations andjerks of the moving pivots undergo a transition from the link parameters in the former phase tothe link parameters in the latter phase. If transition curves are generated for the follower link, andthese curves are connected piecewise to the moving pivot curves of the follower corresponding toK. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299291the phases before and after this transition, a single slider path is generated that will account for thetransition between phases (or follower link moving pivot adjustment).A 7th order polynomial 22,23 is required to specify the radial position, velocity, accelerationand jerk parameters of the moving pivot of the follower link of the adjustable planar four-bar mo-tion generator during the transition between phases.Rh a0 a1h a2h2 a3h3 a4h4 a5h5 a6h6 a7h714The radial displacement, velocity, acceleration and jerk boundary conditions for this polynomialareRh0 R015dRh0dh0_R016dR2h0dh20R017dR3h0dh30Rv018Rhf Rf19dRhfdhf_Rf20dR2hfdh2fRf21dR3hfdh3fRvf22In this work, the term R0is the length of the follower link in phase one (link b0b1) and the term Rfis the length of the follower link in phase two (link b0b1n). The constraints specify a linear set ofeight equations with eight unknowns whose solutions area0 R023a1_R024a2R0225a3Rv0626292K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299a435h4fRf? R0?_R0hf?12R0h2f?16Rv0h3f?15h3f?_R0?R0hf?12Rv0h2f?52h2f?R0?Rv0hf 16hfRv027a5?84h5fRf? R0?_R0hf?12R0h2f?16Rv0h3f?39h4f?_R0?R0hf?12Rv0h2f?72h3f?R0?Rv0hf 12h2fRv028a670h6fRf? R0?_R0hf?12R0h2f?16Rv0h3f?34h5f?_R0?R0hf?12Rv0h2f?132h4f?R0?Rv0hf 12h3fRv029a7?20h7fRf? R0?_R0hf?12R0h2f?16Rv0h3f?10h6f?_R0?R0hf?12Rv0h2f?2h5f?R0?Rv0hf 16h4fRv0304. Example problemTwo-phase moving pivot adjustments of the adjustable planar four-bar motion generator withfixed crank and adjustable follower lengths are exemplified in this section. Listed in Table 2 arethe X- and Y-coordinates of p, q and r for seven prescribed rigid body positions.Table 2Prescribed rigid body positions for the adjustable planar four-bar motion generatorpqrPhase 1Position 1?0.5175, 0.9640?0.2148, 1.50490.3551, 1.3103Position 2?0.4502, 1.0207?0.1413, 1.55810.4263, 1.3570Position 3?0.3786, 1.0720?0.0645, 1.60640.5011, 1.3997Position 4?0.3030, 1.11730.0152, 1.64920.5792, 1.4382Phase 2Position 5?0.0583, 1.20420.4515, 1.68340.9782, 1.3914Position 60.1449, 1.21550.5383, 1.69451.0648, 1.4023Position 70.2319, 1.21950.6249, 1.69881.1517, 1.4070K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299293Eqs. (4)(8) were used to calculate five of the six unknowns in a0, a1and a1n. The variable a0xand the link length R1were specified (a0x= 0 and R1= 1). Using the following initial guesses:a0y 0:1a1 ?0:5;0:5a1n ?0:5;0:5the planar four-bar mechanism solutions converged toa0y 0:0761a1 ?0:7049;0:7859a1n ?0:1739;1:0608:a0b0a1b1b1nXYa1nr1p1p2p3p4p5p6p7q1q2q3q4q5q6 q7r2r3r4r5r6r7Fig. 7. Adjustable planar four-bar motion generator and corresponding prescribed rigid body positions.ab41n57a01ab01bba1n4D57D1na1nbYXFig. 8. Moving pivot paths of the synthesized adjustable planar four-bar motion generator.294K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299Eqs. (9)(13) were used to calculate five of the six unknowns in b0, b1and b1n. The variable b0xand the links length R1and R2were specified (b0x= 1.5, R1= 1.5 and R2= 1.3).Using the following initial guesses:b0y 0:1b1 0:6;1:2b1n 0:6;1:2the planar four-bar mechanism solutions converged tob0y ?0:1064b1 0:6821;1:1505b1n 1:2964;1:1775:Using the calculated fixed and moving pivot parameters, the resulting adjustable planar four-bar motion generator is illustrated in Fig. 7. An equivalent slider-crank motion generator was de-signed for the planar four-bar motion generator in this work.In Fig. 8, the starting and ending positions for the synthesized adjustable planar four-bar mo-tion generator in phases one and two are illustrated. Since the crank link underwent a constant0aa1b4b11nbYXq1r1p1phase 1phase 2transitionFig. 9. Equivalent slider-crank motion generator and initial rigid body position.1.251.301.351.401.451.501.550.000.000.250.300.35crank displacement angle radslider radial displacementFig. 10. Radial slider displacement versus crank angle for synthesized slider-crank motion generator.K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299295link length moving pivot adjustment, all of the moving pivot positions (a1through a4and a1nthrough D57a1n) for this link lie on the same circle. The moving pivot positions (b1through b4and b1nthrough D57b1n) for the follower link lie on two different circles (one for each phase).To complete the slider path of the equivalent slider-crank motion generator (the path betweenb1and b1n), Eq. (14) was used calculate a path to link both of the follower moving pivot arcsin Fig. 8. Using Eq. (14) and the prescribed boundary conditions, the slider path in Fig. 9 wasdesigned. This slider path produces the radial displacement, velocity, acceleration and jerk profilesillustrated in the Figs. 1013.Listed in Table 3 are the X- and Y-coordinates of p, q and r for seven prescribed rigid bodypositions generated by the equivalent planar slider-crank mechanism. To achieve positions 2, 3and 4 in Table 3, link a0a1is rotated to 130?, 125? and 120? with respect to the X-axis. To achievepositions 5, 6 and 7 in Table 3, link a0a1nmust rotated to 100?, 95? and 90? with respect to theX-axis. In both phases, the crank angle is initially 135? with respect to the X-axis and the rigidbody point coordinates at this crank position are the coordinates in position 1 of Table 3.-1.2-1.0-0.8-0.6-0.4-0.20.00.000.000.250.300.35crank displacement angle radslider radial velocityFig. 11. Radial slider velocity versus crank angle for synthesized slider-crank motion generator.-15-10-50510150.000.000.250.300.35crank displacement angle radslider radial accelerationFig. 12. Radial slider acceleration versus crank angle for synthesized slider-crank motion generator.296K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299Illustrated in Figs. 1013 are the radial displacement, velocity, acceleration and jerk profiles(with respect to crank displacement angles) during the transition from phase 1 to phase 2 forthe equivalent slider-crank motion generator. The radial velocity, acceleration and jerk profileboundary conditions (Eqs. (16)(18) and Eqs. (20)(22) were specified to zero in order to generatevelocity, acceleration and jerk profiles that are continuous with those profiles outside of the tran-sition. The radial displacement profile boundary conditions (Eqs. (15) and (19) were specified tothe phase 1 and phase 2 lengths of the follower link (R0= 1.5 and Rf= 1.3) in order to generatedisplacement profiles that are continuous with those profiles outside of the transition also.5. DiscussionThe slider path design method presented in this work is applicable for most of the existing mo-tion generation methods for adjustable planar four-bar mechanisms since only the fixed and mov-ing pivots for the synthesized mechanism are required. Although a two-phase moving pivotproblem was exemplified in this work, additional phases of prescribed rigid body positions can-250-200-150-100-500501001502002503000.000.000.250.300.35crank displacement angle radslider radial jerkFig. 13. Radial slider jerk versus crank angle for synthesized slider-crank motion generator.Table 3Rigid body positions generated by equivalent slider-crank motion generatorpqrPhase 1Position 1?0.5175, 0.9640?0.2148, 1.50490.3551, 1.3103Position 2?0.4508, 1.0205?0.1417, 1.55770.4258, 1.3564Position 3?0.3797, 1.0715?0.0654, 1.60570.5002, 1.3988Position 4?0.3046, 1.11670.0138, 1.64850.5778, 1.4372Phase 2Position 50.0794, 1.20840.4747, 1.68581.0001, 1.3915Position 60.1493, 1.25780.5510, 1.72981.0724, 1.4285Position 70.2232, 1.30290.6302, 1.77041.1482, 1.4632K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299297be incorporated. By calculating the fixed and moving pivots of the planar four-bar mechanism foreach additional phase, and additional transition paths for each additional phase (using Eqs. (14)(30) the equivalent slider-crank motion generator can be designed to achieve additional phases.Although the position of a rigid body in two-dimensional space is commonly described by a singlepoint and a displacement angle (p and h for example), the authors chose to describe the rigid bodyusing three points for computational purposes. If the user prefers to describe the rigid body usingconventional notation, the displacement matrix in Eq. (2) will be replaced with the conventionalplane rigid body displacement matrix 20,21. Computer aided design software was to prescribethe mechanism parameters in this work and mathematics software was used to compute the mech-anism solutions. This software enabled the tabulated prescribed and calculated parameters to beexpressed in four significant figures.6. ConclusionA design method for slider-crank mechanisms to achieve multi-phase motion generation appli-cations accomplished by planar four-bar mechanisms with adjustable moving pivots is presentedin this work. The benefit of the method is that using it, multiple phases of prescribed rigid bodypositions are achievable using a mechanism with fewer moving parts than the planar four-barmechanism. Another benefit of this method is that using it, slider-crank mechanisms can be de-signed to achieve phases of prescribed rigid body positions without any physical or automatedadjustments of its moving pivots between phases. A slider path that enables the slider-crank mech-anism to achieve two phases of prescribed rigid body positions is designed by using 7th order pol-ynomials to connect the moving pivot paths of the follower link of the adjustable planar four-barmechanism. The example problem in this work considers a two-phase moving pivot adjustment ofthe adjustable planar four-bar mechanism.References1 Ahmad, Anees, K.J. Waldron, Synthesis of adjustable planar 4-bar mechanisms, Mechanism and Machine Theory14 (19
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