The Blades Of A Windmill Turn On An Axis That Is 40 Feet From The Ground. The Blades Are 15 Feet Long And Complete 3 Rotations Every Minute. Write A Sine Model, $y = A \sin (b T) + K$, For The Height (in Feet) Of The End Of One Blade As A

Introduction

Windmills are a crucial source of renewable energy, and understanding the motion of their blades is essential for optimizing their performance. In this article, we will explore the mathematics behind the motion of a windmill blade, specifically the height of the end of one blade as it rotates. We will develop a sine model to describe this motion, which will provide valuable insights into the behavior of windmill blades.

The Problem

The blades of a windmill turn on an axis that is 40 feet from the ground. The blades are 15 feet long and complete 3 rotations every minute. We want to write a sine model, $y = a \sin (b t) + k$, for the height (in feet) of the end of one blade as a function of time.

Understanding the Motion

To develop a sine model, we need to understand the motion of the blade. The blade rotates around the axis, which is 40 feet from the ground. As it rotates, the end of the blade moves in a circular path. We can think of this motion as a combination of a vertical motion (up and down) and a horizontal motion (around the axis).

The Sine Model

A sine model is a mathematical function that describes a periodic motion. In this case, the motion of the blade is periodic, as it completes 3 rotations every minute. The sine model has the form:

$$y = a \sin (b t) + k$$

where:

• $y$ is the height of the end of the blade (in feet)
• $a$ is the amplitude of the motion (the maximum height of the blade)
• $b$ is the frequency of the motion (the number of rotations per minute)
• $t$ is time (in minutes)
• $k$ is the vertical shift of the motion (the height of the axis)

Determining the Parameters

To determine the parameters of the sine model, we need to analyze the motion of the blade. We know that the blade is 15 feet long and completes 3 rotations every minute. This means that the frequency of the motion is 3 rotations per minute.

We also know that the axis is 40 feet from the ground, which means that the vertical shift of the motion is 40 feet.

To determine the amplitude of the motion, we need to consider the length of the blade. As the blade rotates, the end of the blade moves in a circular path. The amplitude of the motion is the maximum height of the blade, which is equal to the length of the blade.

Calculating the Amplitude

The amplitude of the motion is equal to the length of the blade, which is 15 feet. However, this is not the maximum height of the blade, as the blade is rotating around the axis. To calculate the maximum height of the blade, we need to consider the angle of rotation.

As the blade rotates, the end of the blade moves in a circular path. The maximum height of the blade occurs when the blade is at a 90-degree angle to the vertical axis. At this angle, the height of the blade is equal to the length of the blade plus the height of the axis.

Calculating the Maximum Height

The maximum height of the blade is equal to the length of the blade plus the height of the axis:

$$\text{Maximum height} = \text{Length of blade} + \text{Height of axis}$$

$$\text{Maximum height} = 15 \text{ feet} + 40 \text{ feet}$$

$$\text{Maximum height} = 55 \text{ feet}$$

Determining the Amplitude

The amplitude of the motion is equal to the maximum height of the blade, which is 55 feet.

Determining the Frequency

The frequency of the motion is equal to the number of rotations per minute, which is 3.

Determining the Vertical Shift

The vertical shift of the motion is equal to the height of the axis, which is 40 feet.

The Sine Model

Now that we have determined the parameters of the sine model, we can write the equation:

$$y = 55 \sin (3 t) + 40$$

where:

• $y$ is the height of the end of the blade (in feet)
• $t$ is time (in minutes)

Conclusion

In this article, we developed a sine model to describe the height of the end of a windmill blade as a function of time. We determined the parameters of the model, including the amplitude, frequency, and vertical shift. The sine model provides a valuable tool for understanding the motion of windmill blades and optimizing their performance.

References

• [1] "Wind Turbine Blade Design and Optimization" by J. A. J. van Kuik, et al.
• [2] "Mathematical Modeling of Wind Turbine Blades" by M. A. A. El-Sayed, et al.

Appendix

A.1 Derivation of the Sine Model

The sine model is derived from the equation of motion for a circular path:

$$x = r \cos (\theta)$$

$$y = r \sin (\theta)$$

where:

• $x$ and $y$ are the coordinates of the end of the blade
• $r$ is the radius of the circular path (equal to the length of the blade)
• $\theta$ is the angle of rotation (in radians)

The equation of motion for the sine model is:

$$y = a \sin (b t) + k$$

where:

• $y$ is the height of the end of the blade (in feet)
• $a$ is the amplitude of the motion (the maximum height of the blade)
• $b$ is the frequency of the motion (the number of rotations per minute)
• $t$ is time (in minutes)
• $k$ is the vertical shift of the motion (the height of the axis)

The parameters of the sine model are determined by analyzing the motion of the blade. The amplitude of the motion is equal to the maximum height of the blade, which is equal to the length of the blade plus the height of the axis. The frequency of the motion is equal to the number of rotations per minute. The vertical shift of the motion is equal to the height of the axis.

A.2 Calculating the Maximum Height

The maximum height of the blade is equal to the length of the blade plus the height of the axis:

$$\text{Maximum height} = \text{Length of blade} + \text{Height of axis}$$

$$\text{Maximum height} = 15 \text{ feet} + 40 \text{ feet}$$

$$\text{Maximum height} = 55 \text{ feet}$$

A.3 Determining the Amplitude

The amplitude of the motion is equal to the maximum height of the blade, which is 55 feet.

A.4 Determining the Frequency

The frequency of the motion is equal to the number of rotations per minute, which is 3.

A.5 Determining the Vertical Shift

Introduction

In our previous article, we developed a sine model to describe the height of the end of a windmill blade as a function of time. We determined the parameters of the model, including the amplitude, frequency, and vertical shift. In this article, we will answer some frequently asked questions about the mathematics of windmill blades.

Q: What is the purpose of the sine model?

A: The sine model is a mathematical function that describes the periodic motion of a windmill blade. It provides a valuable tool for understanding the motion of windmill blades and optimizing their performance.

Q: How is the amplitude of the motion determined?

A: The amplitude of the motion is determined by analyzing the motion of the blade. The amplitude is equal to the maximum height of the blade, which is equal to the length of the blade plus the height of the axis.

Q: What is the frequency of the motion?

A: The frequency of the motion is equal to the number of rotations per minute. In the case of the windmill blade, the frequency is 3 rotations per minute.

Q: What is the vertical shift of the motion?

A: The vertical shift of the motion is equal to the height of the axis. In the case of the windmill blade, the vertical shift is 40 feet.

Q: How is the sine model used in practice?

A: The sine model is used in practice to optimize the performance of windmill blades. By analyzing the motion of the blade, engineers can determine the optimal angle of attack and the optimal speed of rotation to maximize energy production.

Q: What are some common applications of the sine model?

A: The sine model has many common applications in the field of wind energy. Some examples include:

• Wind turbine design: The sine model is used to design wind turbines that are optimized for maximum energy production.
• Wind farm optimization: The sine model is used to optimize the performance of wind farms by analyzing the motion of individual blades.
• Wind energy forecasting: The sine model is used to predict wind energy production and optimize energy storage and distribution.

Q: What are some limitations of the sine model?

A: The sine model has some limitations, including:

• Assumes a simple harmonic motion: The sine model assumes a simple harmonic motion, which may not accurately represent the complex motion of a windmill blade.
• Does not account for turbulence: The sine model does not account for turbulence, which can affect the performance of windmill blades.
• Requires accurate data: The sine model requires accurate data on the motion of the blade, which can be difficult to obtain.

Conclusion

In this article, we have answered some frequently asked questions about the mathematics of windmill blades. The sine model is a powerful tool for understanding the motion of windmill blades and optimizing their performance. However, it has some limitations, including the assumption of a simple harmonic motion and the lack of consideration for turbulence.

References

• [1] "Wind Turbine Blade Design and Optimization" by J. A. J. van Kuik, et al.
• [2] "Mathematical Modeling of Wind Turbine Blades" by M. A. A. El-Sayed, et al.

Appendix

A.1 Derivation of the Sine Model

The sine model is derived from the equation of motion for a circular path:

$$x = r \cos (\theta)$$

$$y = r \sin (\theta)$$

where:

• $x$ and $y$ are the coordinates of the end of the blade
• $r$ is the radius of the circular path (equal to the length of the blade)
• $\theta$ is the angle of rotation (in radians)

The equation of motion for the sine model is:

$$y = a \sin (b t) + k$$

where:

• $y$ is the height of the end of the blade (in feet)
• $a$ is the amplitude of the motion (the maximum height of the blade)
• $b$ is the frequency of the motion (the number of rotations per minute)
• $t$ is time (in minutes)
• $k$ is the vertical shift of the motion (the height of the axis)

A.2 Calculating the Maximum Height

The maximum height of the blade is equal to the length of the blade plus the height of the axis:

$$\text{Maximum height} = \text{Length of blade} + \text{Height of axis}$$

$$\text{Maximum height} = 15 \text{ feet} + 40 \text{ feet}$$

$$\text{Maximum height} = 55 \text{ feet}$$

A.3 Determining the Amplitude

The amplitude of the motion is equal to the maximum height of the blade, which is 55 feet.

A.4 Determining the Frequency

The frequency of the motion is equal to the number of rotations per minute, which is 3.

A.5 Determining the Vertical Shift

The vertical shift of the motion is equal to the height of the axis, which is 40 feet.