﻿ Casyopee - Modelling with respect to an angle

What they say:   I used Casyopée to help students solve optimisation problems. It allowed me to present students very open tasks. Students explored and found results that they had to prove afterward. Casyopée also helped students reuse basic strategies for solving.
What they say:   Casyopée is faster and more convenient than a calculator.... We have the geometric and algebraic side of the problem at the same time. It is easier to see how a function "reacts." It's useful and interesting.
What they say:   Casyopée makes it easy to calculate a derivative, to factor, to calculate zeros... and have a graph of the function next to it in the same window. It allows on a geometric problem to be able to establish variables that can then be used to study the problem by way of functions...
What they say:   Casyopee is a powerful application that can prove useful to both students and teachers It allows you to use various exploration and modeling tools, with the purpose of studying or teaching mathematical functions.
What they say:   Casyopee comes with lots of features. One of these features is the help provided for proving a function. There is also a feature for writing HTML reports that include the mathematical functions. Casyopee is guaranteed to improve the mathematical knowledge of its users.
What they say:   Besides the concept of number, the concept of function is the most important one in mathematics
What they say:   The notion of function is present in all scientific disciplines, and also in everyday life. Our experience as a teacher shows every day that it is a problem for many students. Situations with Casyopée can also be used outside of a technological environment and everyone will be able to reflect on her professional practice.

## Connecting rod and crank mechanism

Consider M a mobile point on the circle centred in o with radius one and P the point on the half-line [ox) such as MP is constant (equal to a, a being a positive parameter). We want to model the dependency between the distance oP and the position of M over the circle. For mechanics, [oM]  is part of the crank and [MP] is the connecting rod.

101 - Connecting rod and crank mechanism

Follow the construction in the NotePad Figure 101. You may assign a speed to point M in order to animate the mechanism (see p. 11). Stop by Action Stop Mobile Points.

To model the mechanism, OP is obviously the dependent variable to choose. But, what might we choose an independent variable? We can create the angle between the x-axis and OM that varies while M is moving on the circle.  Click the angle button  and then a point on the axis (choosing fixed point in the menu), O and M.

Figure 102 - Creating calculations (distance and angle)

Figure 103 - domain and formula of the model function

After asking Casyopée to compute the function by way of the model menu, you evaluate the domain and confirm (OK). If you asked to check existence, you get a message warning that Maxima cannot guaranty that the definition is mathematically correct.

Figure 104 - Casyopée's warning

Note that such warning can happen whereas the definition is correct; it is because of symbolic computation limits. That is why you can ignore the warning. After a new warning the function can be created.

Figure 105 - Rod and crank, model function

Note also that the reverse dependency (c0 -> c1) does not work because for one value of OP correspond two values of the angle. The diagnostic will be: The calculation depends on M. Casyopée cannot compute a model function with the independent variable c0.

## Connecting rod and crank mechanism

Consider M a mobile point on the circle centred in o with radius one and P the point on the half-line [ox) such as MP is constant (equal to a, a being a positive parameter). We want to model the dependency between the distance oP and the position of M over the circle. For mechanics, [oM]  is part of the crank and [MP] is the connecting rod.

101 - Connecting rod and crank mechanism

Follow the construction in the NotePad Figure 101. You may assign a speed to point M in order to animate the mechanism (see p. 11). Stop by Action Stop Mobile Points.

To model the mechanism, OP is obviously the dependent variable to choose. But, what might we choose an independent variable? We can create the angle between the x-axis and OM that varies while M is moving on the circle.  Click the angle button  and then a point on the axis (choosing fixed point in the menu), O and M.

Figure 102 - Creating calculations (distance and angle)

Figure 103 - domain and formula of the model function

After asking Casyopée to compute the function by way of the model menu, you evaluate the domain and confirm (OK). If you asked to check existence, you get a message warning that Maxima cannot guaranty that the definition is mathematically correct.

Figure 104 - Casyopée's warning

Note that such warning can happen whereas the definition is correct; it is because of symbolic computation limits. That is why you can ignore the warning. After a new warning the function can be created.

Figure 105 - Rod and crank, model function

Note also that the reverse dependency (c0 -> c1) does not work because for one value of OP correspond two values of the angle. The diagnostic will be: The calculation depends on M. Casyopée cannot compute a model function with the independent variable c0.

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## Connecting rod and crank mechanism

Consider M a mobile point on the circle centred in o with radius one and P the point on the half-line [ox) such as MP is constant (equal to a, a being a positive parameter). We want to model the dependency between the distance oP and the position of M over the circle. For mechanics, [oM]  is part of the crank and [MP] is the connecting rod.

101 - Connecting rod and crank mechanism

Follow the construction in the NotePad Figure 101. You may assign a speed to point M in order to animate the mechanism (see p. 11). Stop by Action Stop Mobile Points.

To model the mechanism, OP is obviously the dependent variable to choose. But, what might we choose an independent variable? We can create the angle between the x-axis and OM that varies while M is moving on the circle.  Click the angle button  and then a point on the axis (choosing fixed point in the menu), O and M.

Figure 102 - Creating calculations (distance and angle)

Figure 103 - domain and formula of the model function

After asking Casyopée to compute the function by way of the model menu, you evaluate the domain and confirm (OK). If you asked to check existence, you get a message warning that Maxima cannot guaranty that the definition is mathematically correct.

Figure 104 - Casyopée's warning

Note that such warning can happen whereas the definition is correct; it is because of symbolic computation limits. That is why you can ignore the warning. After a new warning the function can be created.

Figure 105 - Rod and crank, model function

Note also that the reverse dependency (c0 -> c1) does not work because for one value of OP correspond two values of the angle. The diagnostic will be: The calculation depends on M. Casyopée cannot compute a model function with the independent variable c0.

Creation date : 06/10/2014 - 16h14
Category : - help