Complete shaking force and shaking moment balancing of four types of six-bar linkages

The article presents design equations and techniques for the complete shaking force and shaking moment balancing of four types of six-bar linkages due to both linear and rotary inertia, but excludes external loads. The key components utilized are the geared inertia counterweight and planetary-gear-train-inertia counterweight. They are very useful for balancing of multi-bar linkages.     

Partial shaking moment balancing of fully force balanced linkages

This paper deals with a solution of the shaking force and shaking moment balancing of planar and spatial linkages. The conditions for balancing are formulated by the minimization of the root-mean-square value of the shaking moment. There are two cases considered: mechanism with the input link by constant angular velocity and mechanism with the input link by variable angular velocity. The method is realized by displacement of the axis of rotation of the input link connected with the counterweight. The efficiency of the suggested method is illustrated by two numerical examples: planar four-bar linkage and RSS'R spatial linkage.     

Determination of the complete set of shaking force and shaking moment balanced planar four-bar linkages

A mechanism is said to be force balanced if, for any arbitrary motion, it does not apply reaction forces on the base. Moreover, if it does not apply torques on the base, the mechanism is said to be moment balanced or dynamically balanced. In this paper, a new method to determine the complete set of force and moment balanced planar four-bar linkages is presented. Using complex variables to model the kinematics of the linkage, the force and moment balancing constraints are written as algebraic equations over complex variables and joint angular velocities. Using polynomial division, necessary and sufficient conditions for the balancing of planar four-bar linkages are derived.     

The quantitative influence of complete force balancing on the forces and moments of certain families of four-bar linkages

The effects of full force balancing on all the bearing forces, the input moments and the shaking moments of 39 families of crank and rocker four-bar linkages of standard configuration are investigated. Dimensionless results are presented in tabular form, and their use is demonstrated. To enable mechanism designers to make similar comparisons for their own specific applications, all necessary expressions are compiled.     

Multiobjective optimization for force and moment balance of a four-bar linkage using evolutionary algorithms

In this study, force and moment balance of a planar four-bar linkage is implemented using evolutionary algorithms. In the current problem, the concepts of inertia counterweights and physical pendulum are utilized to complete balance of all mass effects, independent of input angular velocity. A proposed multiobjective particle swarm optimization, and non-dominated sorting genetic algorithm II are applied to minimize two objective functions subject to some design constraints. The applied algorithms produced a set of feasible solutions called pareto optimal solutions for the design problem. Finally, a fuzzy decision maker is utilized to select the best solution among the obtained pareto solutions. The results show that optimal solutions minimize the weights of applied counterweights and eliminate both shaking forces and moments transmitted to the ground, simultaneously.     

Graphical angular jerk analysis of four-bar linkages

Graphical methods of analyzing angular jerks in a four-bar linkage are described. No jerk polygon or complicated calculations are involved. Only a few lines are needed in these methods. The technique is extended to four-bar linkages connected in series as well as to a slider-crank mechanism. Extremes of angular accelerations may be found whenever the angular jerks vanish.     

A complete classification of planar four-bar linkages

A system of classification of planar four-bar mechanisms is presented in which the three major categories are Grashof, non-Grashof and change point. The Grashof and non-Grashof categories are each organized to include four classes while the change point has six classes. The classification system is closely related to a partitioning of the positive octant of a three-dimensional space in which the coordinate axes are nondimensional ratios of the bar lengths. The positive octant is first reduced in size by four zero mobility planes which eliminate from consideration regions where the bar lengths would not permit assembly. The remaining volume is designated as the solution space, and it is partitioned by three change point planes into eight volumes each of which represent a class of Grashof and non-Grashof mechanism. All of the change point classes reside in the change point planes. This paper describes the classification system and the geometry of the solution space, and lays the foundation for future work to study the properties of planar four-bar mechanisms. Because all of the mechanisms within a particular volume share in common characteristics of the bar lengths which are unique to their class, they also share particular properties with regard to the maximum and minimum values of position angles, the range and average value of the position angles, the maximum and minimum value of angular velocity and angular acceleration, as well as time ratio. Hence all of the significant properties can be represented as characteristic surfaces within the solution space. The trace of the characteristic surfaces in the base plane can be used to construct design charts which will permit choosing mechanisms with desirable properties. The method can also be used to synthesize mechanisms by locating intersections of characteristic surfaces. These topics will be presented in forthcoming publications.     

General method for active balancing of combined shaking moment and torque fluctuations in planar linkages

The paper describes a novel method of active balancing designed to eliminate completely the shaking moment, the driving torque or both simultaneously. Active balancing is achieved by a single additional member: a balancerdisc revolving in a prescribed way related to the geometry and the mass characteristics of the original linkage. Noncircularr gear drive or other means may provide the necessary rotation. The synthesis of the balancing device is based on simple general principles equally applicable for any planar mechanism.     

On the coupler-great-circle envelopes of spherical four-bar linkages

With use of the techniques developed for coupler-point curves, the tangential equation is derived for the envelope of a great circle attached to the coupler of a general spherical four-bar linkage in the motion of this mechanism. The class of this envelope is determined to be 8. The dual relationships between the coupler-point curve and the coupler-great-circle envelope are discussed. With this concept, the techniques for designing a spherical four-bar coupler-great-circle envelope generator are introduced.     

曲柄滑块机构振动力和力矩的附加平衡机构

对机械中常用的曲柄滑块机构的振动力和振动力矩的平衡问题进行了探讨。提出了一种新的附加平衡机构。通过对机构的运动学和动力学分析，给出了振动力平衡的条件。振动力完全平衡后机构的振动力矩，可通过调整附加机构中偏心齿轮的回转中心得到部分平衡。根据振动力矩的均方根值最小原则，确定了偏心齿轮的回转中心的具体位置。提出的机构增加质量较小，不需要额外增加移动副。可以在工程中简化使用。     

General method for full force balancing of spatial and planar linkages by internal mass redistribution

In the present paper, a general method for full force balancing is developed. The method may be applied for any linkage both spatial or planar, single or multi-loop consisting of any kind of lower pairs. By the procedure, three sets of general force balancing conditions for the main directions X, Y and Z are derived. The conditions of identity or difference of these three sets are found the existence of particular balancing equations for some privileged directions are established. These equations do not follow from the Berkof and Lowen method. They could be usefully applied if the “Contour” theorem of Tepper and Lowen is violated or if the partial balancing along the existed privileged axes is preferred.     

平面四杆机构死点问题探讨及应用

主要以曲柄摇杆机构为例,分析了平面四杆机构死点实质、极限位置、形成死点条件等,为平面四杆机构解决死点问题、利用死点特性提供了理论依据。研究发现机构死点问题可以设法消除,而且存在着一定的应用和发展空间。     

平面四杆机构刚体导引综合的符号解法

本文应用的数学机械化方法和计算机符号处理技术,对平面四杆机构刚体与引综合问题进行了符号求妥,该法通过对变量的排序,分组,约化,构成基列,求余等运算,将一个非线性多基式方程组化简为一个同价的三角方程组,得到了原方程组的封闭形式的解析解。     

Reducible compositions of spherical four-bar linkages with a spherical coupler component

We use the output angle of a spherical four-bar linkage C as the input angle of a second four-bar linkage D where the two frame links are assumed in aligned position as well as the follower of C and the input link of D. We determine all cases where the relation between the input angle of the input link of C and the output angle of the follower of D is reducible and where additionally at least one of these components produces a transmission which equals that of a single spherical coupler. The problem under consideration is of importance for the classification of flexible 3 × 3 complexes and for the determination of all flexible octahedra in the projective extension of the Euclidean 3-space.     

Synthesis method based on solution regions for planar four-bar straight-line linkages

An analytical method for synthesizing and selecting desired four-bar straight-line mechanisms based on solution regions is presented. Given two fixed pivots, the point position and direction of the target straight-line, an infinite number of mechanism solutions can be produced by employing this method, both in the general case and all three special cases. Unifying the straight-line direction and the displacement from the given point to the instant center into the same form with different angles as parameters, infinite mechanism solutions can be expressed with different solution region charts. The mechanism property graphs have been computed to enable the designers to find out the involved mechanism information more intuitively and avoid aimlessness in selecting optimal mechanisms.     

Synthesis of four-bar linkages with a coupler point passing through 12 points

The symmetrical coupler curve of a four-bar linkage is made to pass through 12 points. Based on the graphical method analytical expressions can be developed easily and the linkages can be used to drive genevas. The resultant mechanisms have excellent dynamic characteristics.     

用广义逆矩阵进行平面四杆机构的近似函数综合

提出了平面四杆机构近似函数综合的一种新方法,其基本思路是：首先利用广义逆矩阵将四杆机构的近似函数综合方程组转化为一个二元二次方程组,然后采用消元法得到一个一元三次方程,由此可得机构近似运动综合方程组的所有解。为平面四杆机构近似函数综合提供了高效的实用方法。     

Spring equilibrator theory for static balancing of planar pantograph linkages

Static spring balancing theory for planar pantograph linkages is developed using a potential energy approach. Application of this theory achieves static balancing of pantograph linkages in a gravitational field throughout the full range of motion by guaranteeing perfect energy transfer between system gravitational and elastic potential energies. The general planar pantograph linkage is addressed by means of four pantograph types where each type represents a different revolute/prismatic joint arrangement. Equilibrator design for two of these pantograph types represents a first introduction of equilibrium theory to include linkages having prismatic pairs. Applications for walking machine leg design and for vertical plane selective compliance assembly robot design are discussed.     

Feasible parameter design spaces for force and root-mean-square moment balancing an in-line 4R 4-bar synthesized for kinematic criteria

The paper contains application oriented additions to the theory developed by Berkof and Lowen for root-mean-square (RMS) shaking moment optimization of a force balanced in-line 4R 4-bar linkage. The theory is extended to the problem of force-RMS moment balancing by mass redistribution when all link length ratios are fixed to satisfy kinematic design criteria. Feasible parameter design spaces and parameter inter-relationship monographs are presented. These aid in the selection of numerical values for five of the eight design parameters so that the three balancing equations can be solved for physically realistic values of the remaining three parameters.     

结合连续法与数值图谱法的四杆机构连杆轨迹的尺度综合

由连续法得到满足再现轨迹综合要求的平面四杆机构。利用该机构连杆轨迹的谐波特征参数,通过模糊识别的方法,从已建立的对应机构的“数值图谱库”中找到更多组符合设计要求的机构。最后给出算例进行了验证。