Derivation for modified sine curve equations

In summary, the derivation for modified sine curve equations involves adjusting the standard sine function to accommodate changes in amplitude, frequency, and phase shift. This process typically includes applying transformations to the sine function, such as scaling and shifting, to model various phenomena more accurately. The modified equations allow for capturing the nuances of real-world applications, such as wave patterns and oscillations, by providing a more flexible representation of the sine curve.
  • #1
balaji19991
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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.
 
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  • #2
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?
 
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  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
  • #6
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 .
 
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  • #7
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.
 

FAQ: Derivation for modified sine curve equations

What is a modified sine curve?

A modified sine curve is a sine wave that has been altered in some way, such as by changing its amplitude, frequency, phase, or by adding additional terms to the equation. These modifications allow the sine curve to better fit specific data or to model more complex phenomena.

How do you derive the equation for a modified sine curve?

To derive the equation for a modified sine curve, you start with the basic sine function, y = A * sin(Bx + C) + D, where A is the amplitude, B affects the period, C is the phase shift, and D is the vertical shift. By adjusting these parameters, you can fit the sine curve to your specific requirements or data points. You may also add additional terms or use Fourier series for more complex modifications.

What is the significance of each parameter in the modified sine curve equation?

In the modified sine curve equation y = A * sin(Bx + C) + D, the parameter A represents the amplitude, which determines the height of the peaks and the depth of the troughs. The parameter B affects the frequency, which determines how many cycles occur over a given interval. The parameter C is the phase shift, which moves the curve left or right along the x-axis. The parameter D is the vertical shift, which moves the entire curve up or down along the y-axis.

How can you fit a modified sine curve to experimental data?

To fit a modified sine curve to experimental data, you can use curve fitting techniques such as least squares regression. This involves adjusting the parameters A, B, C, and D to minimize the difference between the experimental data points and the values predicted by the sine curve. Software tools like MATLAB, Python's SciPy library, or specialized curve fitting software can be used to perform this optimization.

Can a modified sine curve represent non-periodic data?

While a basic sine curve represents periodic data, a modified sine curve can be adapted to approximate non-periodic data by adding additional terms or using piecewise functions. However, for highly non-periodic data, other types of functions or models might be more appropriate. Fourier series, which represent a function as a sum of sine and cosine terms, can also be used to model more complex, non-periodic behaviors.

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