Derivation for modified sine curve equations

Hello,
I am looking for a detailed derivation of the equations used to generate the modified sine curve. I found one in Cam design handbook by Harold A. Rothbart but I didn't understand how we get certain equations. My end goal is to combine the modified sine curve with constant velocity and get equations for those to generate the necessary displacement,velocity and acceleration graphs.

Welcome to PF.

The modified sine curves are usually generated by adding small amounts of odd harmonics to the fundamental.
You will need to specify how modified you want the curves to be.
Given sufficient harmonics, you could synthesise a square or a triangle wave. But sharp corners and vertical edges are impossible in the engineering world because they require infinite forces, and infinite energy, to create infinite acceleration .

Which one in the Cam Design Handbook did you like?

Baluncore said:

Welcome to PF.

The modified sine curves are usually generated by adding small amounts of odd harmonics to the fundamental.
You will need to specify how modified you want the curves to be.
Given sufficient harmonics, you could synthesise a square or a triangle wave. But sharp corners and vertical edges are impossible in the engineering world because they require infinite forces, and infinite energy, to create infinite acceleration .

Which one in the Cam Design Handbook did you like?

I wanted the standard modified sine curve where two sine waves of frequencies Beta/2 and 3Beta/2 are combined. The modified sine curve inside the handbook is what I am looking for. I used those final equations and got the result i wanted. But I want to understand the derivation.
Once I understand the derivation probably I could derive equations to generate a constant velocity curve with sinusoidal acceleration.

balaji19991 said:

I wanted the standard modified sine curve where two sine waves of frequencies Beta/2 and 3Beta/2 are combined.

There are 600 pages of detail in the Cam Design Handbook, please give me a section, or the equation number in the text.

Baluncore said:

There are 600 pages of detail in the Cam Design Handbook, please give me a section, or the equation number in the text.

Oh sorry I didn't know you have the handbook.
Chapter 3 modified cam curves.
The derivation starts on page 73 (3.7 modified sine curve) and equation 3.15 on page 74 is what I am unable to figure out.

balaji19991 said:

Oh sorry I didn't know you have the handbook.

It just so happens, that I found a copy, that had just fallen off the back of a fibre-optic truck.

The quadrants have different generation equations, that blend at the transitions. You may need to go back to look at the referenced paper to find the origin of equation 3.15 .

Baluncore said:

It just so happens, that I found a copy, that had just fallen off the back of a fibre-optic truck.

The quadrants have different generation equations, that blend at the transitions. You may need to go back to look at the referenced paper to find the origin of equation 3.15 .

That was exactly my first thought but to my surprise i couldn't find any research papers related to this topic on the internet .If you know where it can be found please help me out.