Showing posts from 2010

Improve math skills of your kids - Learn step-by-step arithmetic from Math games

Math: Unknown - Step-by-step math calculation game for iOS.

Math: Unknown is much more than a math game. It is a step-by-step math calculation game which will teach users how to calculate in the correct order rather than just asking only the final calculated results.

The app consists of four basic arithmetic operations which are addition, subtraction, multiplication and division. In order to get started, users who are new to arithmetic can learn from animated calculation guides showing step-by-step procedures of solving each type of operation. It is also helpful for experienced users as a quick reference.

Generally, addition and subtraction may be difficult for users who just start learning math especially when questions require carrying or borrowing (also called regrouping). The app helps users to visualize the process of carrying and borrowing in the way it will be done on paper. Once users understand how these operations work, they are ready to learn multiplication and division.

For most students, division is considered as the most difficult arithmetic operation to solve. It is a common area of struggle since it requires prior knowledge of both multiplication and subtraction. To help users understand division, the app uses long division to teach all calculation procedures. Relevant multiplication table will be shown beside the question. Users will have to pick a number from the table which go into the dividend. Multiplication of selected number and divisor is automatically calculated, but the users have to do subtraction and drop down the next digit themselves. Learning whole calculation processes will make them master it in no time.

Math: Unknown is a helpful app for students who seriously want to improve arithmetic calculation skills.

Chebyschev Straight-line Mechanism

The Chebyschev linkage is a mechanical linkage that converts rotational motion to approximate straight-line motion.

It was invented by the 19th century mathematician Pafnuty Chebyschev who studied theoretical problems in kinematic mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight line motion. This was also studied by James Watt in his improvements to the steam engine. (Read more info about Watt Straight-line Mechanism)
The straight-line linkage of Chebyschev confines the point P — the midpoint on the link AB — on a straight line at the two extremes and at the center of travel. Between those points, point P deviates slightly from a perfect straight line. The proportions between the links are

O2O4 : O2A : AB = 200 : 250 : 100 = 4 : 5 : 2

Point P is in the middle of AB. This relationship assures that the link AB lies vertically when it is at the extremes of its travel.


Watt Straight-Line Mechanism

Watt's linkage (also known as the parallel linkage) is a type of mechanical linkage invented by James Watt to constrain the movement of a steam engine piston in a straight line. The idea of its genesis using links is contained in a letter he wrote to Matthew Boulton in June 1784.
"I have got a glimpse of a method of causing a piston rod to move up and down perpendicularly by only fixing it to a piece of iron upon the beam, without chains or perpendicular guides [...] and one of the most ingenious simple pieces of mechanics I have invented." This linkage does not generate a true straight line motion, and indeed Watt did not claim it did so.

Watt's straight-line mechanism is used in the rear axle of some car suspensions. It intends to prevent relative sideways motion between the axle and body of the car. Watt’s linkage approximates a vertical straight line motion more closely, and does so while locating the center of the axle rather than toward one side of the vehicle.


Roberts straight-line mechanism

Many engineering applications require things move in a linear fashion or "straight-line motion". We can use a linear motion guide that can guide a device accurately along a straight line. Manufacturing know-how of most linear guide manufacturers has let us keep expanding the range of linear guidance. The picture shown here is an example of commercially available linear guides from THK. This Linear Ball Slide is a lightweight, compact, limited stroke linear guide unit that operates with very low sliding resistance. It excels in high-speed responsive performance due to its very small frictional factor and low inertia.

In the late seventeenth century, before the development of the milling machine, it was extremely difficult to machine straight, flat surfaces. For this reason, good prismatic pairs without backlash were not easy to make. During that era, much thought was given to the problem of attaining a straight-line motion as a part of the coupler curve of a linkage having …

3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Part 3

In [3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Part 2], we've shown an example to do three-position synthesis of  a four-bar linkage using inversion method in Unigraphics NX4 sketch.

We can make a quick motion simulation using "animate dimension" command in Unigraphics (UG) NX4 sketch. Just draw lines as per a sketch and add one driving dimension as shown below. Then use animate dimension command to set the lower and upper limits, for this case they're minimum and maximum angles.

3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Part 2

In previous posts, the fixed pivot points were determined from the moving pivot points. We can get result that can't be fitted in our design due to space limit. The principle of inversion can be applied to solve this problem. The first step is to find the three positions of the ground plan that correspond to the three desired coupler positions.

We start with our desired positions of fixed pivot points.

1) Draw desired fixed pivots (O2 and O4) and moving pivot points. Red lines are three desired positions of links (moving pivots).

2) Draw lines to make fixed relations between the ground plane (O2O4) and the second coupler position.
3) Transfer the ground position to the first coupler position using same relations developed in previous step as shown in dashed lines. Name new ground positions as O'2 and O'4 respectively.

4) Draw lines to make fixed relations between the ground plane (O2O4) and the third coupler position (A3B3).

5) Transfer the ground position to the first cou…

3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Introduction

Example [3-Position Motion Generation Synthesis with Alternate Moving Pivots using Unigraphics NX4 Sketch] shows how to synthesize four-bar linkages according to required moving pivots. By doing this, we first define desired locations of moving pivots then get positions of fixed pivots O2 and O4. However, sometimes it may be more useful to define the location of fixed pivots O2 and O4 first, then find other linkages that can move according to 3 desired positions of moving pivots.

An Inversion method of four-bar linkage will be introduced in [3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Part 2]

3-Position Motion Generation Synthesis with Alternate Moving Pivots using Unigraphics NX4 Sketch

In the previous post [3-Position Motion Generation Four-Bar Linkage Synthesis using Unigraphics NX4 Sketch], the locations of fixed pivots O2 and O4 are fixed due to the fixed locations of moving pivots A and B. Sometimes, the location of O2 and O4 are undesirable with respect to your design constraints. More flexible method to get desirable locations of O2 and O4 will be shown in this post.
1) Draw link AB in its three design positions A1B1, A2B2, and A3B3 as shown above.

2) Draw new attachment points C1 and D1 and other lines to form the rigid link ABDC. Do the same for position 2 and 3. Use "Constraint" command in Unigraphics NX4 sketch to set the equal length constraint to all relevant lines e.g. A1C1, A2C2, and A3C3 have the same length, but don't need to specify the fixed value for it. Once we complete these settings at all positions, it means we have set the fixed relationship between our desired line AB and other moving pivots CD. 3) Draw construction lines from …

3-Position Motion Generation Four-Bar Linkage Synthesis using Unigraphics NX4 Sketch

In the real design problem, it will be more practical with 3 specified positions of a line in sequential order than 2-position synthesis. We can use a logical extension of the linkage synthesis technique described in [Four-bar linkage Synthesis using Unigraphics NX4 Sketch] to do the linkage synthesis with 3 positions.

Assume that we have to design the mechanism that move the link AB through 3 positions as shown below. The rectangle represents the limit of link AB, so link AB cannot interfere with the rectangle.

1) Draw link AB in its three design positions A1B1, A2B2, and A3B3 as shown above.
2) Draw construction lines from point A1 to A2 and A2 to A3

3) Bisect line A1A2 and line A2A3 and extend their perpendicular bisectors until they intersect with each other. Label the intersection O2.
4) Draw line O2A1. It's link 2.
5) Repeat the same for another end of the link. Draw construction lines from point B1 to B2 and B2 to B3

6) Bisect line B1B2 and line B2B3 and extend their perpen…

Four-bar linkage Synthesis using Unigraphics NX4 Sketch

In most design problems, we need to design the mechanism that moves between required positions. In this post, we will show the motion generation synthesis method to control a line in the plane to move in sequential positions. Any CAD software can be used to do this. But here we use Unigraphics NX4 sketch to draw and use "Animate Dimension" command to simulate the movement of all concerned links.

Assume that we need to design the mechanism to move line AB to A'B'. A simple and most commonly used four-bar linkage will be used here. Let's see how to find the fixed pivoting point at the ground.

1) Draw the link AB and A'B' in its desired positions in sketch of Unigraphics NX4 sketch (or any other CAD software) as shown in the above figure.

2) Draw construction lines from point A to A' and bisect line AA' and extend the perpendicular bisectors in convenience direction. (The choice is yours).

3) Select any convenient point on bisector as the fixed pivot…