Kinematic synthesis of linkages /​ Richard S. Hartenberg, Jacques Denavit. Author. Hartenberg, Richard S. (Richard Scheunemann). Other Authors. Denavit . Kinematic synthesis of linkages. Front Cover. Richard Scheunemann Hartenberg, Jacques Denavit. McGraw-Hill, – Technology & Engineering – pages. linkage is known to be a sextic, i.e., a curve described by an implicit function ( Hartenberg and Denavit, ) of the form. F(x, y)=0 in which F(x, y) is a linear.

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Skip to kinematjc content. Log In Sign Up. Software Packages Relevant to Linkage Syn- thesis. Transmission Angle and Transmission Quality. Broadly speaking, the main function of a structure is to be capable of withstanding the anticipated loads without exhibiting major deformations that would hamper the integrity of the structure or the safety of its occupants. Likewise, the main function of a machine is to be able to perform the intended task, usually involving finite displacements of its parts, without major deformations that would hamper the integrity of the machine or the safety of its users.

In the above preamble we have introduced concepts of engineering design as pertaining to mechanical systems at large. Of these, we have focused on structures and machines. For this reason it is necessary to dwell on this concept. Kinemztic of its importance, the engineering design process has been the subject of study over the centuries, starting with Marcus Vitruvius Polio ca.

Modern engineering design theory owes its origins, to a great extent, to Franz Reuleaux —who first proposed a grammar to describe the kinematic chain of a machine Moon, A modern model of the design process, due to Frenchis depicted in Fig. In this model, four stages are distinguished: In the first stage, analysis of the problem, the functions required from the object under design, in our case, a machine, are clearly defined, in general, but precise terms.

In the second stage, the design team produces a set of design variants, as rich as possible, after several sessions, structured or unstructured, which are part of the creative aspect of the design process.

In this stage, the task of the team is to produce a preliminary model of the design solution, with a clear identification of the main parameters defining a specific design variant.

At this stage, then, a parametric model of each of the short-listed candidate variants is produced, which is then subject to optimization with the aim of deavit the specific fundamental dimensions that either maximize a profit or minimize a cost of the design solution, or even do both at the same time, in a process that is known as multiobjective optimization.

The final stage involves materials selection, manufacturing issues, and production-cost liknages. As a result of this stage, a set of manufacturing drawings is produced that is then sent out for prototype manufacturing, when the design job so requires, or directly to production.

Kinematic Synthesis Of Linkages (Mechanical Engineering Series) – R.S. Hartenberg, J. Denavit

Kinematic synthesis plays a key role in the first three stages of the foregoing design process, as pertaining to machine design. In fact, in the first stage, analysis of the prob- lem, more than kinematic synthesis, what is required is kinematics knowledge, as design functions and specifications are to be understood by any engineer trained in the discipline.

It is in the second and third stages where kinematic synthesis plays a fundamental role, as explained below. As pertaining to machine design or, more specifically, to mechanism design, Denavit and Hartenberg proposed three phases of kinematic synthesis: Both type and number syntheses pertain to the conceptual design phase, as the former refers to choosing the type of mechanism to perform the required function, namely, a linkage, a cam-follower mechanism, a belt-pulley transmission, or a gear train, for example.

Number synthesis refers to the numbers of links and joints in a linkage, along with the type of joints to be used—kinematic joints, or lower kinematic pairs, are studied in Ch. The conceptual phase of the design process is fundamental. Moreover, this phase is the one that has posed the major challenges to those attempting to automate the design process.


In the realm of kinematic synthesis, we introduce a methodology, termed quali- tative synthesis, in Ch. Qualitative synthesis focuses on the synthesis of linkages. Chapters 3—5 are devoted to what Denavit and Hartenberg call dimensional synthesis, as the main objective here is to find the dimensions defining the geometry of the various links and joints of the kinematic chain underlying the mechanism under design. In these chapters, we assume that a preliminary layout of the conceptual design—obtained as a result of the type and number syntheses of the kinematic chain at hand—is available, our main job being to contribute to the production of the embodiment of this design.

Catalog Record: Kinematic synthesis of linkages | Hathi Trust Digital Library

The synthexis embodiment is the realization of a kinematic chain as a table of what is known as the Denavit-Hartenberg parameters—to be introduced in Ch. Going back to the more general machine-design process, dimensioning involves two phases: It is the functional dimensioning where kinematic synthesis plays a major role. Mechanical dimensioning pertains to the dimensioning of the machine elements for stress, strength, heat capacity, and dynamic-response requirements.

Before we embark on the details of the course, a review of the glossary is in order. The branch of mechanics that studies motion, independent of its rela- tion with forces.

The branch of mechanics that studies the equilibrium of forces and moments acting on particles, rigid bodies, and flexible bodies.

The physical prevention of relative motion—rotation and translation—between two bodies in one or more jinematic. The term also denotes the algebraic or differential relations representing the physical constraint. The branch of mechanics that studies the interplay of forces and linkags with motion variables under static, conservative conditions.

The concepts of machine and hartenherg are frequently interchanged as if they were equivalent, but they are not. We give below linkagez definitions from various sources, with added comments: Different definitions of machine have been given by scholars for more than two millennia, starting with Vitruvius in 28 BCE, namely, — A machine is a combination system, assemblage of moving material bodies Vitruvius, 28 B.

An apparatus for applying mechanical power, having several parts, each with a definite function Comment: An apparatus consisting of interrelated parts with separate functions, used in the performance of some kind of work.

Any system in which a specific correspondence exists between an input form of energy or hartenberrg and the corresponding ones at the output loosely translated. Could be much terser and more informative. Kinmatic Body A continuum whose points remain equidistant under ddenavit possible motion. Rigid-Body Pose The position of linkagess landmark point of the body and the orientation of a coordinate frame fixed to the body with respect to a reference frame. Rigid-Body Twist The velocity of one landmark point of the body and the angular velocity of the body.

Kinematic Synthesis The fundamental problems in mechanism kinematics can be broadly classified into: Given a linkage, find the motion of its links, for a prescribed motion of its input joint s.

Given a task to be produced by a linkage, find the linkage that best performs the task. The task at hand can be one of three, in this context: Kinematic synthesis being a quite broad concept, it involves various aspects Denavit and Hartenberg, Given a task to be produced by a mechanism, find the type that will best perform it, e.

Given a task to be produced by a mechanism of a given type, find the number of links and joints that will best execute the task. Given a task to be produced by a mechanism, find its rele- vant geometric parameters. We have, further, two types of dimensional synthesis: Number of linkage parameters ot is sufficient to pro- duce exactly the prescribed motion. Problem leads to—linear or, most frequently, nonlinear—equation solving. Number of linkage parameters available is not sufficient to produce exactly the prescribed motion.

Optimum dimensions are sought that approximate the prescribed motion with the minimum error.

Once the problem is formulated as one of dimensional synthesis, the geometric relations of the task at hand are manipulated, by means of drafting in- struments alone, including CAD software, to produce the desired linkage parameters as intersections of circles, of lines, or of circles and lines.


Once the problem is formulated as one of dimensional synthesis, the geo- metric relations of the task at hand are manipulated, by algebraic means, supported with computer hardware and software, to produce the desired linkage parameters as the solutions to the underlying synthesis equations.

Purely algebraic methods entail some drawbacks, like algebraic sin- gularities, which are conditions under which some solutions cannot be found for reasons other than kinematic. Semigraphical methods reduce the system of alge- braic equations to a subsystem of bivariate equations, i. The bivariate equations defining a set of contours in the plane of those two variables, the solutions to the problem at hand are found as the intersections of all those contours. Harteenberg merits of graphical methods as educational tools cannot be overstated.

However, as practical means of solving engineering problems, their scope is rather limited. On the one hand, only lines and circles can be traced with standard drafting instruments— paper, pencil, ruler, square, and compass—the tracing of other geometric shapes requiring specialized instruments, for example, templates or CAD software. On the other hand, kinematic synthesis problems, as pertaining to lower-pair linkages, usually lead to alge- braic systems of equations1i.

Each equation is a linear combination of products of integer powers of several unknowns, e. The sum of the exponents of each product, n1 pkis known as the degree of the product; the highest product-degree of the ith linkagees is termed the degree di of the equation. Using a suitable elimination procedure, it is conceptually possible, synthhesis not really possible all the time, to reduce the system of n equations in n unknowns to one single monovariate polynomial equation.

The number of possible solutions, thus, synghesis be as high as P. Now, with ruler, square and com- pass one cannot find but the intersection of circles and lines, and hence, one cannot solve graphically but linear and quadratic equations.

While some simple kinematic synthesis 1 The exception here occurs when a linkage involves at least one screw pair, to be introduced in Chapter 2, where transcendental equations occur.

Some synthesis problems of planar four-bar linkages lead to resolvent polynomials of a degree lying in the billions! Chen and Angeles, Because of the above reasons, we stress in this course semigraphical methods of kine- matic synthesis.

An alternative to the use of this matrix for the same purpose is the use of complex numbers. The problem with complex numbers is that they are quite useful to represent two-dimensional vectors, their application to three and higher dimensions being still unknown. On the contrary, the two-dimensional representation of the cross product is just a particular case of three- dimensional vector algebra.

Methods for the numerical solution of linear systems of equations are also included. Here we describe how to represent in two dimensions the cross product, without resorting to three- dimensional vectors. Also note that E is skew-symmetric: We assume that r and f both lie in a plane normal to the unit vector k. We have thus shown that the cross product of two two-dimensional hartehberg reduces to a scalar, i. This rotation is represented algebraically by means of a proper orthogonal matrix Q.

We have Fact 1. Determined Systems Consider solving for x the system below: Never compute—unless instructed to do so! A matrix inverse is seldom needed and incurs a waste of precious CPU time!

Instead, find a good numerical approximation to the solution, while taking into account linlages A and b are usually known only up to a certain roundoff error. In computing the solution of system 1. The definitions of these norms are given below: Not only this; with the foregoing value of W, kAkF is the rms value of the singular values3 of A.