Crank-rocker Linkage Research Articles

Adjoint Approach Between A Spatial Curve and A Ruled Surface Based On The Bishop Frame

The adjoint approach is usually used to study the crank-rocker linkages’ coupler curves and the geometry of rigid objects in spatial motion. In this paper, the adjoint approach between a spatial curve and a ruled surface based on the Bishop frame is presented. Also, a ruled surface by using the components of the Type-1 and Type-2 Bishop frames is expressed. Moreover, for a curve that adjoint to a ruled surface the fixed point conditions concerning the Bishop frame are determined. Finally, we presented four examples to show the relationship between the ruled surface and its adjoint curve.

Design and Development of an Indian Harmonium with Automatic Bellow Mechanism

Harmonium is a reed instrument widely used in Indian classical music, utilizing a hand-driven bellow to pump air into the soundbox. The pumped air is released through the metallic reeds producing musical notes. The hand-driven bellow action can be challenging and tiresome for beginners as well as professionals during long practice sessions. This problem can be resolved by automating the bellow. The design process is twofold – 1st is the synthesis of a motor-driven linkage capable of generating the rocker motion of the bellow, and 2nd is to obtain the required specification of the motor that can drive the mechanism. A 4-bar crank-rocker linkage has been synthesized using a modified Freudenstein equation, which allows the bellow to compress and expand in one complete rotation of the motor shaft. The novelty of this work is that the nonlinear dynamics of the harmonium bellow has been modeled using a spring-mass equivalence technique. This helps in accurately predicting the operating force for the bellow, the force transmission through the linkage, and estimating the required motor torque. The final design is simple and ergonomic. Based on the design, a working prototype has been developed at the Machine Elements Laboratory of Jadavpur University.

Novel design for the adjustable knock-out mechanism with a Roberts linkage

This article presents a new adjustable knock-out mechanism of a bolt former with a Roberts linkage and an adjustable crank-rocker linkage. The design requirements of the new mechanism are based on the analysis of an existing adjustable knock-out mechanism. The new adjustable knock-out mechanism mainly consisting of a Roberts linkage, and an adjustable crank-rocker linkage is proposed to meet the specified requirements. The path of the knock-out screw is approximately a straight line and the method presented is illustrated by an example.

Defect-free optimal synthesis of crank-rocker linkage using nature-inspired optimization algorithms

This paper proposes the synthesis of a human knee exoskeleton using the reduced number of necessary and sufficient constraints. The proposed human knee exoskeleton consists of a defect-free crank-rocker mechanism capable of accurately moving its coupler point along the prescribed trajectory. For synthesizing the crank-rocker mechanism based on proposed reduced number of constraints, an optimization problem is formulated to synthesize the mechanism. An algorithm based on teaching-learning-based-optimization (TLBO) is presented to solve this highly nonlinear optimization problem. The optimization minimizes the error between generated and prescribed trajectory and simultaneously avoids any defect in the synthesized mechanism. The penalty method is used to manage all the constraints. Besides a realistic nontrivial example of human knee flexion/extension, a straight line trajectory is also considered to demonstrate the effectiveness of the refinement scheme in the optimal syntheses of planar crank-rocker linkage free from all defects. The optimization problem is solved using well-established nature-inspired algorithms such as genetic algorithm (GA), particle swarm algorithm (PSO), and teaching-learning-based-optimization (TLBO) algorithm. It is found that TLBO is computationally more efficient than other algorithms used here. Additionally, the proposed human knee exoskeleton model is experimentally validated for one gait cycle.

From sliding–rolling loci to instantaneous kinematics: An adjoint approach

The adjoint approach has proven effective in studying the properties and distribution of coupler curves of crank-rocker linkages and the geometry of a rigid object in spatial motion. This paper extends the adjoint approach to a general surface and investigates kinematics of relative motion of two rigid objects that maintain sliding–rolling contact. We established the adjoint curve to a surface and obtained the fixed-point condition, which yielded the geometric kinematics of an arbitrary point on the moving surface. After time was taken into consideration, the velocity of the arbitrary point was obtained by two different ways. The arbitrariness of the point results in a set of overconstrained equations that give the translational and angular velocities of the moving surface. This novel kinematic formulation is expressed in terms of vectors and the geometry of the contact loci. This classical approach reveals the intrinsic kinematic properties of the moving object. We then revisited the classical example of a unit disc rolling–sliding on a plane. A second example of two general surfaces maintaining rolling–sliding contact was further added to illustrate the proposed approach.

Kinematic Analysis of Rectangular Path Generating Adjustable Four-Bar Linkage

This paper focuses on the kinematic analysis of rectangular path generating adjustable four-bar crank rocker linkage. One of the ground pivots of rocker side of the adjustable mechanism is subjected to continuous adjustment while the crank arm is given uniform angular velocity. Variations of coupler and rocker position, velocity and acceleration are computed for continuously changing phases of the adjustable four bar linkage. These kinematic parameters are compared with non-adjustable four-bar mechanism. Effect of the continuous adjustment on transmission angle and torque ratio are presented. The above procedures are implemented successfully in MATLAB environment and graphical results are presented.

Fourier series method for path generation of RCCC mechanism

The mathematical representation of the coupler curves of RCCC mechanism is established. The coupler curve of RCCC mechanism is analysed using the Fourier series theory. The internal relation between the coupler curve and the dimensional type of the RCCC mechanism in the frequency domain is discovered. Subsequently, a database comprising millions of basic dimensional types, taking the crank–rocker linkages as an example, is established. Based on the information of prescribed coupler and the database, the best suitable basic dimensional type can be determined by identification. The mathematical equations for calculating actual dimensions of mechanism and positional dimensions of installation are deduced. Thus, the path synthesis problem of RCCC mechanism can be solved by the mathematic equations and the basic dimensional type. The calculation illustration is given later in this article.

Effects of Time Ratio and Rocker Angular Stroke on Transmission Angle in Crank-Rocker Linkages

Effects of Time Ratio and Rocker Angular Stroke on Transmission Angle in Crank-Rocker Linkages

Relationship between Kinematics Characteristics and Link Lengths in Crank-Rocker Linkages

According to the relationships of link lengths in crank-rocker linkages, the effective areas of link lengths for type I, type II and type III of the linkages were analyzed firstly. The relationship between the minimum transmission angles, extreme position angles and the link lengths were derived theoretically and drawn visually as variation trend maps and some conclusions are summarized. The conclusions can be used as guidance for optimal synthesis of crank-rocker linkages.

Exo-Leg: An Active Single-Leg External Walking Assist Device

Lower extremity weakness is a serious problem afflicting people all over the world. Until recently, the mobility options for people with this condition have been confining and limit the individual’s functionality. Walking assist devices are presently in development to restore hands-free walking to people with lower extremity weakness. These devices provide the necessary support and power to enable the individual to restore normal ambulation. The proposed design of exoleg, a single leg external walking assist device, addresses the demographic of people with lower extremity weakness. The design includes replication of the gait cycle utilizing mechanical links and user control interface with emphasis on safety. The design couples the actuation of the knee and hip through the use of linkages connected to a single motor. The actuation of the hip is controlled by a 4 bars crank-rocker linkage system while the knee is actuated by corresponding linkages (designed in WORKING MODEL 2D, a commercial simulation software) that generates the knee kinematic profile. The angular profiles of the knee and hip actuations are compared with the actual knee and hip angular trajectories. The frame of the device incorporates a passive ankle stabilization system to compensate for the effects of foot drop. The system utilizes feedback from trigger points from pressure sensors on the foot and goniometers at the hip and knee joints to measure the angulations in gait to keep the device in synchronization with natural ambulation. An on-board microprocessor receives the feedback from the trigger points and sends the actuation signal to the motor. A conceptual design of electrostatic actuator motor is also proposed to keep the device light weight and compact.

THE SYNTHESIS OF MECHANISM SYSTEMS USING A MECHANISM CONCEPT LIBRARY

This paper presents an approach for synthesizing all possible mechanism systems of kinematic building blocks in a mechanism concept library. The kinematic building blocks are defined as SISO primitive mechanisms, and their serial and/or parallel combinations are expressed as corresponding out-trees based on graph representation. By representing the constructive building blocks as labeled vertices and their possible combination relationships as directed edges, the synthesis approach is developed by adopting graph enumeration theorem. An illustrative example of four kinematic building blocks, including two crank-rocker linkages and two slider-crank mechanisms, is provided to validate the presented approach. The result shows that all feasible mechanism systems can be obtained effectively by following the synthesis method and which provides more alternatives in the library during design or re-design of mechanisms.

Distribution of coupler curves for crank-rocker linkages

Based on a differential geometry approach, the basic equations for the coupler curve of the four-bar linkage are concisely derived. The geometric properties of coupler curves for the crank-rocker linkage are analyzed, and the distribution law of various shapes of coupler curves is revealed. The moving centrode, Ball's curve and the self-tangent curve divide the coupler plane into several areas. The points in each of these areas trace a specific shape of curves. Therefore, any shape of coupler curves with a cusp, inflection point, Ball's point, crunode, tacnode or oval shape, etc. for the first time, could be readily located.

Synthesis of Harmonic Motion Generating Linkages—Part II: Path and Motion Generation

This paper, a sequel to a companion paper on function generation, discusses the path and motion generation problems in the synthesis of linkages with relatively small input cranks. The point on the floating link (i.e., the coupler of a crank-rocker linkage or the connecting rod in a slider-crank linkage) traces an approximate ellipse. This fact serves as a major distinction between the method described herein and the conventional, more general, synthesis techniques. In other words, only elliptical paths may be generated by the path (or coupler) points in the synthesis of linkages with small cranks. Higher-order path and motion generation, in which velocity, acceleration, slope and the rate of change of slope of the coupler path may be specified, are also addressed in this paper.

Analysis and Design of Basic Linkages for Harmonic Motion Generation

This paper deals with basic planar linkages driven by relatively small cranks, and as a consequence generating approximate simple harmonic motion at the output members. A unique approach is presented to yield the kinematic equations, as exemplified by planar crank-rocker and offset slider-crank linkages. The method involves the presumption of a mean linkage configuration and a small motion excursion therefrom. Its validity is demonstrated by means of several error plots, generated for varying, nondimensional linkage parameters and a constant mean transmission angle. These plots also serve to illustrate the order of approximation involved for a particular combination of parameters. Finally, the synthesis of a crank-rocker linkage for a representative set of requirements is exemplified using the equations developed herein.

Proposals for finite 5-dimensional atlases for all planar 4-bar linkage coupler curves

The paper seeks to extend earlier work that makes proposals for an atlas of curves traced by coupler points on crank-rocker linkages. We now examine ways of providing atlases for coupler curves of other 4-bar linkages. Provided that every family of similar movable quadrilaterals are included it is found that, as a further consequence of Roberts' cognate theory, it is sufficient to confine attention to coupler points that lie on or within boundaries that demarcate half the area bounded by two arcs having centres at the coupler bearings. For non-Grashof linkages only one quarter of this area is needed. This leads to an alternative way of providing an atlas of curves traced by coupler points of crank-rocker linkages.

Proposals for a finite 5-dimensional atlas of crank-rocker coupler curves

The coupler curves traced by planar 4-bar crank-rocker linkages are examined with the objective of providing guidelines for a new awareness survey. It is found that the coupler points chosen by Hrones and Nelson[1] in the best known atlas of this kind leads to some avoidable duplication and omissions. It is concluded that coupler points should be confined to a circle on the coupler plane having centre on the coupler-rocker bearing and radius equal to the distance between coupler bearings. Five non-dimensional variables are proposed, three of which are discussed in more detail in earlier work[6].

The Analysis of an Elastic Four-Bar Linkage on a Vibrating Foundation Using a Variational Method

A variational method is employed to derive the equations of motion and the associated boundary conditions for a flexible crank-rocker linkage sited on a foundation which vibrates perpendicular to the plane of the mechanism. The links oscillate in axial, flexural and torsional modes, and the equations governing this behavior are systematically constructed using a variational theorem by permitting independent variations of the stress, strain, displacement and velocity parameters.

Triangular nomograms for symmetrical coupler curves

Triangular nomograms have been constructed as a simple means of finding symmetrical coupler curves of the crank-rocker linkage. A nomogram of this kind has been presented by the author of this article during a Conference of the Institution of Mechanical Engineers at London in 1972. Twenty-seven versions have now been developed and fully worked out. They are dimensionless and have then three angles as parameters to determine the type of curve with all its properties. The curve types are concerned with those characteristics which the crank-rocker linkage yields for symmetrical coupler curves such as: curves with one or two points of reversal; with internal contact; 8-shaped curves; curves with 2 equal radii of curvature; curves with one or two undulation points on the axis of symmetry; curves with two undulation points oriented symmetrically in relation to each other and a top angle; curves with Burmester points, where the second derivative of the radius of curvature is zero; and curves with one or two tangents which touch the curve at three different points each. The 27 triangular nomograms refer to the symmetrical coupler curves of the 4-bar linkage and are used for selection of the type of curve and corresponding data for the linkage. This article will deal with three of them which relate to coupler curves with the property that a tangent touches the curve at three separate points, while two more examples of the various verions of the nomogram will be presented. A example of the application of the nomogram is given by the design of a transfer mechanism, where use has been made of a curve with two cusps and further modification to a 6-bar linkage.