### 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.

### Timing Diagram (Part 2 - Maximum acceleration calculation)

Let's calculate the acceleration of the die from previous post [Timing Diagram (Part 1 - No Overlap Movement)]

The die moves using Cycloid cam profile. So first we have to get the formula to calculate the maximum acceleration of cycloid cam profile.

If the machine speed is N (pcs/h) and the indexing angle (degree) is Bm, the indexing time (second) tm can be calculated as follows.

Cycle time (sec) = 3600/N
Indexing time tm (sec) = (Bm/360) x Cycle time = (Bm/360) x (3600/N)

Hence,
Indexing time tm (sec) = 10Bm/N

Cycloid cam profile has the equation of displacement as follows.

h = hm x [t/tm - 1/(2 x pi) x sin(2 x pi x t/tm)]

where:
hm: Maximum displacement (m)
tm: Indexing time (s)
pi: 3.141592654

We can get velocity equation by differentiation.

v = dh/dt = hm x [1/tm - (2 x pi)/(2 x pi x tm) x cos(2 x pi x t/tm)]
v = hm/tm x [1 - cos(2 x pi x t/tm)]

Then, the acceleration is as follows.
a = d2h/dt2 = dv/dt = hm/tm x [0 - (-2 x pi/tm) x sin(2 x pi x t/tm)]
a = 2 x pi x hm/tm2 x sin(2 x pi x t/tm)

The maximum acceleration (amplitude) occurs when sin(2 x pi x t/tm) = 1 or -1
Therefore the amplitude of of maximum acceleration is as follows.

amax = 2 x pi x hm/tm2

We can clearly see from the above derivations that the acceleration is inversely proportional to the square of indexing time. Since the indexing time (tm) is proportional to the indexing angle (Bm), then the maximum acceleration is also inversely proportional to the square of the indexing angle.

That means if we can increase the indexing angle by a factor of two, the maximum acceleration will reduce by a factor of four!!

And we can do this by putting more overlapped motion in the timing diagram design.

From previous post, the die moves 50 mm in 55 degrees using cycloid cam profile.
Let's assume we calculate the maximum acceleration of the die at the machine speed (N) of 2000 pcs/h
So, we have

N = 2000 pcs/h
Bm = 55 deg
hm = 50 mm = 0.05 m

tm = 10Bm/N = 10(55)/2000 = 0.275 s
amax = 2 x pi x hm/tm2 = 2 x 3.141592654 x 0.05 / 0.2752
amax = 4.154 m/s2

Let see in the next post when we put more overlap motion, the indexing angle will be increased as well as the indexing time, thus reduce the maximum acceleration.