Manufacturing Technology (ME461) Lecture4

8/12/2019 Manufacturing Technology (ME461) Lecture4

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Manufacturing Technology

ME461

Instructor: Shantanu Bhattacharya


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Review of previous lecture

Concatenation and compounding for series of

transformations.

Practical problem of compounding. Non-parametric representation (implicit and

explicit cases).

Example problem of straight line.


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Non parametric representation

(Explicit case)

Non parametric equations of curves can be further dividedinto explicit and implicit nonparametric equations. The explicit

non parametric representation of general two dimensional

and three dimensional curves taken the form:


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Non parametric representation

(Implicit case)

The implicit non parametric representation of a general n-dimensionalspace curve takes the form:

This equation must b solved analytically to

obtain the explicit form, which is not easily done

by computer.


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Example Problem

Develop the non

parametric

equation for the

ellipse as shownin the figure on

the left.

2A


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Solution


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Parametric Representation of

Synthetic curves

Analytic curves, are usually well defined and studied.

However, in the geometric design of mechanical

parts, some other curves are required.

One familiar example is the car body which contains

curves which not only envelop inside the mechanical

part but also help reduce air resistance.

Other examples might include fuselage, wings, andpropeller blades of airplanes, whose shape is based

on aerodynamic and fluid flow simulation.


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Parametric representation of synthetic curve

Generally speaking, a goodrepresentation ofengineering objects has to

have the followingproperties:

1. Easy to control thecontinuity of the curvesto be designed.

2. Requires less storage torepresent a curve.

3. Less computation timeand no computationalproblems.

4. Easy to input by user.


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Concept of continuity Intuitively, continuity means the smoothness of the connection of two

curves or surfaces at the connection of two curves or surfaces at the

connection points or edges.

Normally, three types of continuity, called C0, C1, C2 are defined to

characterize the smoothness of connection of two curves. C0continuity

implies simply connecting two curves. That means that the gradients of

these curves at the point of joining

In C1continuity, the gradients

at the point of joining must be

same.

However, C2 implies curvature

continuity; that is , not onlygradient but also the center of

curvature is the same.

Curves that are constructed by many curve segments are called synthetic curves.

Different types of curve segments can be used to construct synthetic curves, with certain

continuity requirements. They are easier in controlling their continuity requirements and

plotting them is computationally less intensive.


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Types of synthetic curves

Hermite cubic spline

Bezier cruves

B-Spline curves

Rational B-Splines

Nonuniform rational B-Spline


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Hermite Cubic Spline The main idea of the Hermite cubic spline is that a curve is divided into segments. Each

segment is approximated by an expression, namely a parametric cubic function.

The reasons for using cubic functions to approximate the segments are:

1. A cubic polynomial is the minimum order polynomial function that generates C0, C1,

C2 continuity curves.

2. Higher order polynomials have some drawbacks, such as oscillation about control

points, and are uneconomical in terms of storing information and computation.

D i ti f th t


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Derivation of the parameter

equation

This form can be adjusted to yield

various shapes of curve segment by

altering one or more of V(0), V(1),V(0) and V(1) appropriately.

The Hermite form of a cubic spline

is determined by defining positions

and tangent vectors at the data

points.

There are some disadvantages of

this method. For example,

1. First order derivatives are

needed; it is not convenient fora designer to provide first order

derivative.

2. The order of the curve is

constant regardless of the no,.

of data points.


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Example Problem

Determine and plot the equation of Hermite

form of a cubic spline form given position

vectors and slopes at the data points with

vector magnitude equal to 1.

Point 1: A= [1,2]T, slope A= 60 deg.

Point 2: B= [1,3]T, slope B = 30 deg.

Manufacturing Technology (ME461) Lecture4

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