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.

Comments

Post a Comment

Popular posts from this blog

Linear Actuators and Linear Motion

Mechanical energy is an area of science that is making strides every day. The study of how actuators produce mechanical motion by converting various forms of energy into mechanical energy is a source of great exploration. Science finds new ways to make use of actuators every day including for medical purposes. Many scientists believe that the more they study these seemingly simple machines, the more they will discover ways of helping mankind. The way in which a linear actuator works is that there is a motor that rotates a drive screw using a synchronous timing belt drive. Some linear actuators can also use a worm gear drive or direct drive. Whichever the choice, the turning of the screw pushes a drive nut along the screw, which in turn pushes the rod out and the rotating the screw in the opposite direction will retract the rod. According to the Association of Sciences, the drive screw is either an ACME or ball thread or is belt-driven which is what gives the machine its motion. A cov
Read more about this Mechanical Design Article »

Ball Detent Torque Limiter: Overload Clutch

Image
A torque limiter is an automatic overload clutch that provides machine protection and reduces repair time during jamming load conditions. This is done to protect expensive machines and prevent physical injuries. A torque limiter may limit the torque by slipping (as in a friction plate slip-clutch), or uncouple the load entirely (as in a shear pin). Ideally the torque limiter should be placed as close as possible to the source of the jam. This will allow the system inertia and torque to be quickly and effectively disconnected from the jammed section. The system can then be allowed to stop without causing further machine damage. A mechanical torque limiter will provide faster response times and better protection than typical electronic methods at high crash rates. There are several disconnect types available, but we will focus at the Ball Detent type . A ball detent type torque limiter transmits force through hardened balls which rest in detents on the shaft and are held in pl
Read more about this Mechanical Design Article »

Watt Straight-Line Mechanism

Image
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
Read more about this Mechanical Design Article »