Lecture Notes: Rotational Dynamics of Wind Turbines
Introduction
- Focus on rotational dynamics of wind turbines.
- Maintenance scenario: wind turbine stops over 2 minutes.
- Angular speed expressed as a time-dependent equation.
- Aim to find angular velocity, angular acceleration (average and instantaneous).
Angular Velocity and Acceleration
Direction Determination
- Right Hand Rule: Determines direction.
- Curl fingers in direction of rotation.
- Thumb points to direction of angular speed.
- Angular Acceleration:
- Opposite to angular speed direction.
- Occurs because angular speed is decreasing (turbine is stopping).
Intuition
- Analogous to a ball thrown upwards:
- Velocity is upwards, acceleration (gravity) downwards.
- Opposite directions indicate slowing down.
Calculating Average Angular Acceleration
Definition of Average
- Considers final and initial states.
- Requires final and initial angular speed values.
Calculation Steps
- Initial Angular Speed: At time
t=0, calculated as 1.44 rad/s.
- Final Angular Speed: At full stop, speed is 0 rad/s.
- Formula:
- Average Angular Acceleration (( \bar{\alpha} )) = ( \frac{\Delta \omega}{\Delta t} )
- ( \Delta \omega ): Change in angular speed (final - initial).
- ( \Delta t ): Time interval (120s).
- Result: -0.012 rad/s² (negative sign indicates decrease in speed).
Instantaneous Angular Acceleration
Concept
- Differs from average acceleration.
- Measures rate of change at a specific moment.
Calculation Method
- Requires derivative of angular speed with respect to time.
- Formula: ( \frac{d}{dt}[\omega(t)] )
Example Calculation
- Given angular speed function: ( \omega(t) = 2t - 240 )
- Derivative results in a linear function for instantaneous acceleration.
- Calculate for specific times:
- At 30 seconds: Plug into derived equation.
- At 120 seconds: Verifies stop (zero acceleration).
Conclusion
- Derive both average and instantaneous angular acceleration from angular velocity.
- Practical implications for turbine maintenance and dynamics understanding.
Additional Notes
- Practice derivative calculations if unfamiliar.
- Encouraged to ask questions and engage with additional resources.
Note: These notes should be supplemented with practice problems and further reading to fully grasp the concepts of rotational dynamics and their applications in wind turbines.