Interactive Animation: Rotor Forces – Educational-Animation.org

The following animation illustrates the forces acting on the rotor of a wind turbine in the rotor plane. The various force components and their vector addition are displayed dynamically during rotation.

Description of the Animation

The animation shows a simplified wind turbine rotor with three blades in a top-down view. Pressing the Start button initiates the rotation, with the forces acting on the rotor blades being continuously updated.

The following forces are illustrated:

Gravitational Force F_G (blue): The weight of the rotor blade, always acting vertically downward
Tangential Force F_U (gray): Tangential component of the aerodynamic lift force. In the animation, the gray arrow always points tangentially along the circular path of the rotor blade – it rotates with the blade, constantly changing its direction in space while remaining consistently oriented relative to the blade.
Centrifugal Force F_F (purple): Radially outward inertial force caused by the rotation
Resultant Force F_Res (green): Vector sum of gravitational force and tangential force
Total Force F_Total (orange): Vector sum of the resultant force and centrifugal force
Driving Force F_Drive (yellow): Tangential component of the total force

Interactive Controls

The checkboxes on the right side allow individual forces and auxiliary elements to be shown or hidden:

Individual force vectors (F_G, F_U, F_F, F_Res, F_Total, F_Drive)
Parallelograms: Show the graphical vector addition using the force parallelogram method. The parallelograms only become visible when the corresponding component forces are also enabled – F_G and F_U for the green parallelogram, and F_Res and F_F for the orange parallelogram.
Connection Lines: Dashed auxiliary lines between the total force and the driving force

The parallelograms visualize the step-by-step addition:

F_G + F_U = F_Res (green parallelogram)
F_Res + F_F = F_Total (orange parallelogram)

Observation Tip

To best understand the alternating loads, start by enabling only the gravitational force (blue) and pressing Start. You will see how the blue arrows at all three blades point rigidly downward while the blades rotate. Next, enable the tangential force (gray) as well – now you can observe how the interplay between the two forces changes depending on blade position. Finally, enable the remaining forces and the parallelograms step by step to trace the complete force decomposition.

Physical Background

The tangential force is the effective component of aerodynamic lift in the rotor plane. It depends on wind speed, pitch angle, and rotational speed. In this animation, the tangential force is assumed to be constant.

The gravitational force acts due to the weight of the rotor blades and is always directed downward. During rotation, the position of the rotor blades relative to the direction of gravity changes continuously. This is clearly visible in the animation: the blue arrows (F_G) always point straight down regardless of blade position, while the rotor blades rotate around them. As a result, the gravitational force sometimes acts in the same direction as the tangential force (blade moving downward) and sometimes opposes it (blade moving upward).

The centrifugal force acts radially outward and arises from the rotation of the rotor. It is proportional to the square of the angular velocity and the mass of the rotor blade. In the animation, the purple arrow (F_F) always points away from the rotor hub. Unlike the gravitational force, which maintains its direction in space, the centrifugal force rotates with the blade – just like the tangential force. Enabling all three forces simultaneously makes this difference especially clear: gray and purple rotate with the blades, while blue always points downward.

The total force F_Total is the vector sum of all forces. Its magnitude and direction change continuously during one revolution. This is a critical parameter for rotor blade design. By enabling the F_Total checkbox (orange) and the parallelograms in the animation, you can observe how the orange arrow continuously changes in both length and direction during rotation. The step-by-step construction of the total force is visualized through two parallelograms: first F_G + F_U = F_Res (green), then F_Res + F_F = F_Total (orange).

The driving force F_Drive is the tangential component of the total force — the projection of the total force onto the direction of rotation. It determines the acceleration and the torque of the rotor. The magnitude of the driving force also varies during one revolution. In certain positions, the driving force may temporarily oppose the direction of rotation. This effect can be observed directly in the animation: enable F_Drive (yellow) and the connection lines to see how the yellow arrow represents the projection of the total force onto the tangential direction. At certain positions during the revolution, the yellow arrow reverses direction and points against the direction of rotation – at that moment, the force decelerates the rotor instead of driving it.

Mathematical Description of the Forces

The forces shown in the animation can be described mathematically as follows:

Gravitational force (constant magnitude, always directed vertically downward):

\[ F_G = m \cdot g \]

Centrifugal force (constant magnitude, directed radially outward from the hub):

\[ F_F = m \cdot \omega^2 \cdot r \]

\( m \) – mass of the rotor blade
\( g \) – gravitational acceleration (9.81 m/s²)
\( \omega \) – angular velocity of the rotor (rad/s)
\( r \) – distance from the rotor axis to the center of mass of the blade

The tangential force \( F_U \) is the component of aerodynamic lift that acts in the rotor plane. It depends on the relative wind speed, the blade profile, and the angle of attack. In this animation it is assumed constant; in reality, it varies with blade position.

The resultant, total, and driving force are obtained by vector addition, as shown by the parallelograms in the animation. The driving force is the projection of the total force onto the tangential direction – it is the component that actually accelerates (or decelerates) the rotor.

How a Rotor Blade Generates Lift

A wind turbine rotor blade works on the same aerodynamic principle as an aircraft wing. But unlike a wing, it moves through the air by rotating, not by translating. This creates a special situation: the blade experiences two simultaneous air flows:

The ambient wind blowing from the front (axial direction)
The headwind from rotation caused by the blade’s own motion through the air (tangential direction)

The vector sum of these two flows is the relative wind (also called apparent wind). The blade profile is angled so that this relative wind strikes it at a suitable angle of attack, generating an aerodynamic lift force perpendicular to the relative wind and a drag force parallel to it.

The tangential component of the lift force – the component in the rotor plane – is the tangential force F_U shown in this animation. It is this force that drives the rotation. The related animation Forces on an Airfoil illustrates these aerodynamic principles in detail.

Tip Speed Ratio

The tip speed ratio \( \lambda \) is one of the most important parameters in wind turbine design. It describes the ratio of the blade tip speed to the wind speed:

\[ \lambda = \frac{\omega \cdot R}{v_{wind}} \]

\( \omega \) – angular velocity (rad/s)
\( R \) – rotor radius (m)
\( v_{wind} \) – wind speed (m/s)

Modern three-blade rotors reach their maximum efficiency at a tip speed ratio of approximately \( \lambda \approx 6 – 8 \). This means the blade tips move six to eight times faster than the wind. At a wind speed of 12 m/s and \( \lambda = 7 \), the blade tip speed is about 84 m/s – over 300 km/h.

The tip speed ratio directly affects the angle of attack along the blade. This is why rotor blades are twisted: the inner sections, which move more slowly, need a steeper angle, while the fast-moving tips require a flatter one. The optimal tip speed ratio also depends on the number of blades – rotors with fewer blades need to spin faster to capture the same amount of energy.

Simplifications in the Animation

The animation depicts only the forces that are visible in the rotor plane when viewed from above. In reality, the rotor blades must also withstand the force exerted by the wind flowing from the front (thrust force). This axial force component is not shown in this animation.

The tangential force is assumed to be constant in this animation. In reality, it varies depending on the wind speed, turbulence, and the position of the rotor blade.

Practical Significance

Rotor Blade Design: The alternating loads determine material selection and dimensioning
Fatigue: Cyclic loading from periodically changing forces can lead to material fatigue
Performance Optimization: The driving force determines the torque and thus the power output
Control: Modern wind turbines adjust the pitch angle of the rotor blades to optimize loading conditions

Load Cycles and Material Fatigue

The animation clearly shows how the direction and magnitude of the total force change continuously during every revolution. This is not merely an academic observation – it is the central challenge of rotor blade engineering.

A rotor blade typically rotates at 10–15 rpm. Over a design lifetime of 20 years, this adds up to roughly 10⁸ load cycles (one hundred million). In each cycle, the gravitational component alternates between supporting and opposing the tangential force. This can be traced in the animation: enable F_G (blue) and F_U (gray) simultaneously to see how the blue arrow points in a similar direction to the gray arrow at the bottom of the circle (gravity supports the rotation), while at the top of the circle it opposes the tangential force (gravity decelerates). This periodic alternating load is the cause of edgewise bending and leads to cyclic material fatigue.

Engineers distinguish two principal bending directions:

Edgewise bending (in the rotor plane): Caused primarily by the gravitational force, which acts downward regardless of blade position. This is the load visible in this animation. The blade bends back and forth in its own plane once per revolution.
Flapwise bending (perpendicular to the rotor plane): Caused by the thrust force of the wind pushing the blade forward. This axial load is not shown in this animation but is typically even larger than the edgewise load.

Material selection and structural design must ensure that the blade survives these enormous cycle counts without developing cracks. This is described by S–N curves (stress vs. number of cycles), which define the allowable stress amplitude for a given fatigue life.

Pitch Control: How Modern Turbines Adapt

The brief mention of pitch control in the practical significance section above deserves a closer look, as it directly connects to the forces shown in the animation.

Pitch refers to the angle of the rotor blade around its longitudinal axis. By rotating the blade, the angle of attack of the relative wind changes, which in turn changes the aerodynamic forces – including the tangential force F_U shown in the animation.

Modern wind turbines use pitch control in several regimes:

Below rated wind speed: The blade pitch is set to maximize the tangential force and thus power capture. The tip speed ratio is kept near its optimum.
Above rated wind speed: The blades are pitched to reduce the angle of attack, limiting the aerodynamic forces so that the generator does not exceed its rated power. This also reduces the mechanical loads on the rotor.
Storm shutdown (feathering): In extreme winds, the blades are pitched to 90° (parallel to the wind), reducing the aerodynamic forces to near zero and bringing the rotor to a safe stop.

Real-World Dimensions

The animation shows a simplified schematic. The following figures give a sense of the real forces and dimensions involved in modern wind turbines:

Parameter	Typical values (modern onshore/offshore)
Blade length	60–120 m
Blade mass	15–25 tonnes per blade
Rotor diameter	130–240 m
Rotational speed	5–15 rpm
Blade tip speed	Up to 90 m/s (≈ 324 km/h)
Centrifugal force at blade root	Up to 1,000 kN and more
Materials	Glass fiber reinforced plastic (GRP), carbon fiber in spar caps, balsa or foam core
Design lifetime	20–25 years (≈ 10⁸ load cycles)

A centrifugal force of 1,000 kN corresponds to roughly the weight of 100 tonnes pulling outward at the blade root – and this force acts continuously during operation. The blade root bolts, typically 100–150 high-strength bolts per blade, must withstand this load for the entire lifetime of the turbine.

Further Resources

For those who wish to explore the topic further:

Erich Hau: Wind Turbines – Comprehensive reference textbook on wind turbine technology (Springer)
WindEurope – windeurope.org – European wind energy association with technical publications
Wind turbine aerodynamics – Wikipedia – Overview of the aerodynamic principles
Tip speed ratio – Wikipedia – Detailed explanation with derivations