Friday, February 03, 2012

Inversions Of Four Bar or Quadratic Kinematic Chain

Inversions Of Four Bar or Quadratic Kinematic Chain

 1. Kinematics
Kinematics studies the geometric properties of the motion of points without regard to their masses or to the forces acting on them. A set of points with the property that the distances between any two of them never varies is called a rigid body or rigid link. The position of a rigid body in space is defined by six dimensions, three translations and three rotations. A kinematic chain is a set of links connected by joints that constrain their relative movement. An open chain is a set of links (such as a common industrial robot) with one end attached to a rigid base . A closed chain may be attached to a rigid base in more than one place. A single loop closed chain attached at each end is commonly called a mechanism.

The state of a mechanism is the set of properties needed to completely determine the positions of all of its constituent parts. These properties are called independent state variables, and their count determines a mechanism's degrees of freedom. Constraints are limitations on the motion of a mechanism or of its constituent parts. A number of dependent variables may be required to determine the relative positions of a mechanism's parts, and they are derived from the independent variables and the external constraints. A determinate mechanism has the property that the number of dependent variables is equal to the number of external constraint conditions.

The links of a planar mechanism are constrained to move in a plane parallel to a base plane, usually by hinged joints whose axes lie perpendicular to the base plane. The axes of hinged joints of a spherical mechanism all intersect at a point. Joints of spatial mechanisms have no special relationship to a common point or

2. The Scene Graph


The scene graph is the emergent object-oriented hierarchical data structure for describing geometric relationships, appearance, and behavior in computer generated virtual reality worlds. It is an acyclic tree, meaning that each branch has a single attachment point, and therefore, it cannot form closed loops. The Java 3D API Specification [3] describes the construction of the scene graph tree , which uses BranchGroup nodes and TransformGroup nodes to connect the branches and uses Leaf Nodes to contain information such as geometric shapes, appearance, lighting, and behavior.

Throughout, this paper uses a particular example from classical mechanics to illustrate the general methods. The four bar linkage, a particular class of planar, determinate, single loop closed kinematic chains, has been the focus of study by mathematicians, philosophers, royalty and engineers for centuries, and, respecting tradition, this paper will employ it as the concrete example. The following describes how the scene graph can be used to describe its geometry and connectivity.
Figure 1 shows a common configuration of a four bar linkage. Rigid links AB, BC, CD, DA are connected by hinged joints (A, B, C, D); one of the links (DA) is considered to be fixed to a foundation, one link is considered to be a driver or input link (AB), the adjacent link is a connector or drag link (BC), and the last, the driven link or the output. (CD).

Figure 2 shows how a scene graph could be used to represent the geometric arrangement of the links of a four bar linkage. The Shape Leaf Nodes represent the geometries and visual properties of the individual links themselves, the Branch and Transform Group Nodes describe the structure of the tree that defines the geometric relationship between them, and the Behavior Leaf Node gives motivation to the input link AB.

The central failings of the bare scene graph are apparent from the figure. While the Behavior node can set the transformation for joint A by setting the independent variable angle_A, and the lengths of the rigid links ab, bc, and cd determine the translations, the rotations at joints B and C are unknown, and the requirement that the 'D' end of the link CD engage the joint D fixed on the base remains unexpressed on the tree's acyclic topology.



3.APPROACH

This section describes a three-pronged approach to using inverse kinematics for solving external constraints on an animated mechanism model in Java 3D™. The first sub-section describes the mathematical background, the solution sequence, and the organization of the scene graph model, the second describes the approach to creating a Constraint class to organize and expedite geometric calculations in support of the solution, and the third sub-section describes the standard way that simple Java 3D™ models are animated, the inherent problems, and an approach to modifications necessary to permit stable and efficient computation.

Solving External Constraints

Typically, constraints can be expressed in a number of equations or inequalities that describe the relationship among machine parts or between machine parts and the foundation. This paper considers a subset of constraints that can be expressed in equations whose terms are time-independent. The example has constraints that set displacements of a mechanism part equal to zero relative to a fixed foundation point. Further, the determinate mechanism discussed in this paper has an equal number of dependent variables that can be set to satisfy those equations. The mathematical problem then reduces to solving N nonlinear equations in N unknowns.

Fi (x1, x2, x3, … xN) = 0 i = 1…N (1)

where F is a function of x, the dependent variables. For very simple mechanisms, the equations can be solved from quadratic or trigonometric closed forms, but numerical marching techniques are preferable for mechanisms of any complexity.

Press et al. [2] describe the Newton-Raphson method, simplest of the numerical techniques for solving sets of simultaneous nonlinear equations. The method begins with an initial guess for the dependent variables (xj), measures the resulting values of the functions F, calculates and applies a change to the dependent variables and repeats to convergence.

In the general case, the equation set (1) can be particularly troublesome when it represents kinematic constraints, because it may have many, one, or no solutions. According to Press, there are no good methods to find global solutions to the set, and accordingly, the methods described here are applicable only to mechanisms for which there are known solutions and only to stable regions for those mechanisms.

The key to stability and rapid convergence is an initial guess not too far from the final result. The changes required to the dependent variables are estimated by estimating the local derivatives of Fi with respect to xj, then using them to compute the new xj required to drive Fi to zero. The derivatives Fi/xj are measured by incrementing each xj in turn and measuring the changes to F. The NN matrix Fi/xj is called the Jacobian, J. In vector form:

F (x + x) = F (x) + J x + (higher order terms) (2)

Setting F (x + x) = 0 to satisfy the constraint equations, and neglecting higher order terms

J x = -F (xold),

which is a set of N linear equations in N unknowns. It may be solved for x by one of several different methods, such as direct inversion of the Jacobian

xnew = xold + x

xnew = xold + J-1 (-F (xold)), (3)

and the steps are repeated until F (xold) are sufficiently close to 0. For a high fidelity simulation, this procedure is executed and the dependent variables are set prior to the rendering of each frame.




A Simple Guide to Dimensions ...

This drawing (shown to the left) is symmetric about the horizontal centre-line.

Centre-lines are chain-dotted and are used for symmetric objects, and also for showing the centre of circles and holes.
Drawing dimensions should generally be done directly to the centre-line, as shown on the left. In many cases this method
can be clearer than just dimensioning
between surfaces.

Note again that the measurements show only numbers. The statement at the bottom of the drawing identifies that these numbers are the dimensions in Millimetres.

A Simple Guide to Dimensions

A Simple Guide to Dimensions

 

With the left side of the block composed solely of "radiuses" (radii) - as shown here, we break our rule that we should not duplicate dimensions. The total length is known because the radius of the curve on the left side is given. Then, for clarity, we add the overall length of 60 and we note that it is a reference (REF) dimension.
This means that it is not really required.

Somewhere on the paper, usually the bottom, there should be placed information on what measuring system is being used (e.g. inches and millimetres) and also the scale of the drawing.

Monday, January 09, 2012

“Assembly" Drawings

                                                   “Assembly" Drawings

1An isometric view of an "assembled" pillow-block bearing system is shown on the left below. 
It corresponds closely to what you actually see when viewing the object from a particular
angle. We cannot tell what the inside of the part looks like from this view.


                                                                                                         
We can also show isometric views of the pillow-block being taken apart or disassembled"
(above – right). This allows you to see the inner components of the bearing system.
Isometric drawings can show overall arrangement clearly, but not the details and the dimensions.



"Cross-Sectional Views"

A cross-sectional view portrays a cut-away portion of the object and is another way to show hidden components in a device. Imagine a plane that cuts vertically through the centre of the pillow block as shown in figure 12. Then imagine removing the material from the front of this plane, as shown in figure 13.

This is how the remaining rear section would look. Diagonal lines (cross-hatches)
show regions where materials have been cut by the cutting plane.

This cross-sectional view (section A-A, figure 14), one
that is ‘orthogonal’ to the
viewing direction, shows the relationships of lengths and
diameters in a much better way. These drawings are easier to make than isometric drawings. Experienced engineers can interpret orthogonal drawings without needing an isometric drawing, but this can take a bit of practice.

The top "outside" view of the bearing is shown in figure 15. It is an orthogonal perpendicular) projection.* Notice the direction of the arrows for the "A-A" cutting plane





Drawing Tools






To prepare a drawing, you can use manual drafting instruments (see below) or computer-aided drafting or design. ( CAD ) The basic drawing standards and conventions are the same regardless of what design tool you use to make the drawings. In learning ‘drafting‘ - the name given to drawing in architecture and engineering, we will approach it from the perspective of manual drafting. If the drawing is made without either instruments or CAD, it is called a freehand sketch.



It is important to use the set-square resting on the parallel motion bar - the horizontal motion bar - as
the accuracy of the lines drawn and the speed with which they can be added is significantly improved. Isometric drawings need the 30 / 60 set sqaure and Orthographics use the 45 / 90 degree. Always use a sharp pencil as the line is more accurate and can be mesured more easily. HB is the softest grade of pencil that should be used. Either H or 2H grades give lighter-looking lines but are far finer and cleaner looking.

A moments thought will also make you realise that unless the paper is attached to the surface with its edges aligned to the board - then the drawing will not be square to the edge and nor will replacing the drawing be quite as easy if for some reason it has to be removed and replaced later. Steel clips or masking tape can be used to attach the paper to the board’s surface. When peeling tape off drawings make sure it is pulled back almost along itself so that the paper surface is not damaged.

ISOMETRIC


ISOMETRIC (2 angles) TRIMETRIC (3 angles)
In Isometric drawings try to get used to ‘flipping’ the set-square over keeping the longest edge in contact with the horizontal motion bar.
Always keeping the ‘sharpest’ (narrow angle) part pointing either left or right. Only when a 90 degree angle is needed can the sharpest (narrow) angle of the set-square point to the top of the page. Trimetric drawing uses 3 angles.

Isometric Drawing

           "Engineering Drawing and Sketching"


  • 1 Isometric Drawing info
               One of the best ways to communicate one's ideas is through some form of picture or drawing. This is especially true for the engineer.
The purpose of this guide is to give you the basics of engineering sketching and drawing.

We will treat "sketching" and "drawing" as one. "Sketching" generally means freehand drawing. "Drawing" usually means using drawing instruments, from compasses to computers to bring precision to the drawings. As this is just an introduction, don't worry about understanding every detail immediately - just get a general feel for the language of graphics and drawings. Before starting on any technical drawings, let's get a good look at this block drawing, shown below, from several angles. Any engineering drawing should show everything - a complete understanding of the object should be possible from the drawing without any need for explanations. If the isometric drawing can show all details and all dimensions on one drawing, it is ideal
.





                                               "2 Isometric Drawing sempal"



The representation of the object seen
here is called an isometric drawing.
This is one of a family of three-dimensional views called pictorial drawings. In an
isometric drawing, the object's vertical
lines are drawn vertically, and the
horizontal lines in the width and depth planes are shown at 30 degrees to the horizontal.
When drawn under these guidelines, the
lines parallel to these three axes are at
their true (scale) lengths. Lines that are
not parallel to these axes will not be of
their true length.