Rotor

Rotor dynamics

Since R2023a

Libraries:
Aerospace Blockset / Propulsion

Description

The Rotor block computes the aerodynamic forces and moments generated by a rotating propeller or rotor in all three dimensions. The block lets you include the effect of the tilt in rotor disc due to flap motion, while in forward flight, on the generated force and moments.

Examples

Mars Helicopter Simulink-Based System Level Design

Model a helicopter with coaxial rotors suitable to fly on Mars.

Open Live Script

Limitations

The block follows a simplified approach with the option to include steady state flap effects. It does not model dynamic flap, lag, or feathering motion of blade.
The block outputs have been verified against the algorithms used, for input and parameter values within reasonable limits (as observed from literature).
The block uses Twist distribution to model only linear or ideal twist distributions. The block assumes that the blade chord and lift curve slope are constant.
The effect of collective pitch input and drag coefficient is incorporated in the block through the thrust and torque coefficients alone. Hence, the collective input port and the drag coefficient is active only when CT and CQ Source dropdown is set to Compute using BEMT.

Ports

Input

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Ω (rad/sec) — Rotor speed
scalar

Rotor speed, specified as a scalar in rad/sec in body frame.

Data Types: double

ρ — Air density
positive scalar

Air density, specified as a positive scalar in specified units.

Data Types: double

V_b — Body velocity of rotor
3-by-1 vector | 1-by-3 vector

Velocity of rotor, specified as a 3-by-1 or 1-by-3 vector in body frame. To perform a multisystem analysis, consider connecting output from a State-Space or Integrator block to this port.

Dependencies

This enable this input, set Modeling to With flap effects.

The unit of velocity depends on the value of the Units parameter.

Data Types: double

ω_b — Angular velocity
3-by-1 vector | 1-by-3 vector

Angular velocity of entire vehicle, specified as a 3-by-1 or 1-by-3 vector in rad/s in body frame.

Dependencies

This enable this input, set Modeling to With flap effects.

Data Types: double

θ₀ — Collective blade pitch angle
scalar

Collective blade pitch angle, specified as a scalar in radians.

Dependencies

To enable this input port, select the Include pitch angle inputs check box and set the CT and CQ Source dropdown to Compute using BEMT.

Data Types: double

θ_c — Lateral cyclic pitch angle
scalar

Lateral cyclic pitch angle, specified as a scalar in radians.

Dependencies

To enable this input port,

Select the Include pitch angle inputs check box.
Set Modeling to With flap effects.

Data Types: double

θ_s — Longitudinal cyclic pitch angle
scalar

Longitudinal cyclic pitch angle, specified as a scalar in radians.

Dependencies

To enable this input port,

Select the Include pitch angle inputs check box.
Set Modeling to With flap effects.

Data Types: double

Output

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F_xyz(N) — Total force
three-element vector

Total force, returned as a three-element vector in body frame. Units depend on the Units parameter.

Data Types: double

M_xyz(Nm) — Net moment
three-element vector

Net moment in the x-y-z direction, returned as a three-element vector in body frame. Units depend on the Units parameter.

Data Types: double

C_T — Computed thrust coefficient
positive scalar

Computed thrust coefficient, returned as a positive scalar.

Dependencies

To enable this output port, select the Output computed CT and CQ check box and set the CT and CQ Source dropdown to Compute using BEMT.

Data Types: double

C_Q — Computed torque coefficient
positive scalar

Computed torque coefficient, returned as a positive scalar.

Dependencies

To enable this output port, select the Output computed CT and CQ check box and set the CT and CQ Source dropdown to Compute using BEMT.

Data Types: double

Parameters

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Main

Units — Input and output units
`Metric (MKS)` (default) | `English (Velocity in ft/s)` | `English (Velocity in kts)`

Units	Density	Force	Moment	Velocity	Radius	Chord	Hinge offset
`Metric (MKS)`	kg/m³	Newtons	Newton-meter	Meters per second	Meters	Meters	Meters
`English (Velocity in ft/s)`	slug/ft³	Pound force	Pound force-feet	Feet per second	Feet	Feet	Feet
`English (Velocity in kts)`	slug/ft³	Pound force	Pound force-feet	Knots	Feet	Feet	Feet

Programmatic Use

Block Parameter: units

Type: character vector

Values: 'Metric (MKS)' | 'English (Velocity in ft/s)' | 'English (Velocity in kts)'

Default: 'Metric (MKS)'

Modeling — Rotor thrust calculation method
`Without flap effects` (default) | `With flap effects`

Rotor thrust calculation method, specified as:

Without flap effects — Model rotor thrust using force and moment calculations. For more information, see Force and Moment.
With flap effects — Effect of tilt in rotor disc due to flap motion, while in forward flight, is included. The steady state lateral and longitudinal flap angles are calculated using the equations from [1] and used in the computation of forces and moments [2]. For more information, see Force and Moment.

Programmatic Use

Block Parameter: modelMode

Type: character vector

Values: 'Without flap effects' | 'With flap effects'

Default: 'Without flap effects'

Include pitch angle inputs — Enable pitch angle inputs
`off` (default) | `on`

Select this check box to enable the pitch angle (swash plate) input ports.

Dependencies

To enable this checkbox

Set the CT and CQ Source dropdown to Compute using BEMT

and/or

Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: controlInput

Type: character vector

Values: 'on' | 'off'

Default: 'off'

Rotor

CT and CQ Source — Dropdown to select the source of thrust and torque coefficient values
`'Dialog'` (default) | `'Compute using BEMT'`

Select the source of the thrust coefficient and torque coefficient values, specified as a character vector of value 'Dialog' or 'Compute using BEMT'. The block assumes the aerodynamic and structural parameters to be constant.

Select 'Dialog' to manually enter the thrust and torque coefficient values.

Select 'Compute using BEMT' to compute the values using Blade Element Momentum Theory (BEMT). For more information on BEMT, see Thrust and Torque coefficient.

Programmatic Use

Block Parameter: CTCQMode

Type: character vector

Values: 'Dialog' | 'Compute using BEMT'

Default: 'Dialog'

Thrust coefficient (CT) — Thrust coefficient
`0.0107` (default) | nonzero positive scalar

Thrust coefficient, specified as a nonzero positive scalar.

Dependencies

To enable this parameter, set the CT and CQ Source dropdown to 'Dialog'.

Programmatic Use

Block Parameter: CT

Type: character vector

Values: '0.0107' | nonzero positive scalar

Default: '0.0107'

Torque coefficient (CQ) — Torque coefficient
`7.8263e-4` (default) | nonzero positive scalar

Torque coefficient, specified as a nonzero positive scalar.

Dependencies

To enable this parameter, set the CT and CQ Source dropdown to 'Dialog'.

Programmatic Use

Block Parameter: CQ

Type: character vector

Values: '7.8263e-4' | nonzero positive scalar

Default: '7.8263e-4'

Number of blades — Number of blades per rotor
`2` (default) | nonzero positive scalar

Number of blades, specified as a nonzero positive scalar.

Dependencies

To enable this parameter,

Set the CT and CQ Source dropdown to Compute using BEMT

and/or

Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: Nb

Type: character vector

Values: '2' | nonzero positive scalar

Default: '2'

Output computed CT and CQ — Output computed thrust coefficient and torque coefficient
off (default) | on

Select this check box to output the calculated thrust coefficient and torque coefficient to C_T and C_Q output ports. For more information on these calculations, see Thrust Coefficient and Torque Coefficient Calculations. The block assumes the aerodynamic and structural parameters to be constant.

Otherwise, clear this check box.

Dependencies

To enable this parameter, set the CT and CQ Source dropdown to 'Compute using BEMT'.

Programmatic Use

Block Parameter: CTout

Type: character vector

Values: 'on' | 'off'

Default: 'off'

Blade

Radius (m) — Rotor radius
`0.0330` (default) | nonzero positive scalar

Rotor radius, specified as a nonzero positive scalar.

Programmatic Use

Block Parameter: radius

Type: character vector

Values: '0.0330' | nonzero positive scalar

Default: '0.0330'

Chord (m) — Blade chord
`0.0080` (default) | nonzero positive scalar

Blade chord, specified as a nonzero positive scalar.

Dependencies

To enable this checkbox,

Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: chord

Type: character vector

Values: '0.0080' | nonzero positive scalar

Default: '0.0080'

Hinge offset (m) — Hinge offset
`0` (default) | positive scalar

Hinge offset, specified as a positive scalar. This value is typically 0 for propellers.

Dependencies

To enable this checkbox,

Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: hingeOffset

Type: character vector

Values: '0' | positive scalar

Default: '0'

Lift curve slope (per rad) — Lift curve slope
`5.5` (default) | nonzero positive scalar

Lift curve slope, specified as a nonzero positive scalar. The block assumes the aerodynamic and structural parameters to be constant. The block does not consider variation with respect to angle of attack.

Dependencies

To enable this checkbox,

Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: clalpha

Type: character vector

Values: '5.5' | nonzero positive scalar

Default: '5.5'

Mean drag coefficient — Mean drag coefficient
0 (default) | positive scalar

Mean drag coefficient, specified as a positive scalar. The block does not consider variation with respect to angle of attack.

Dependencies

To enable this parameter, set the CT and CQ Source dropdown to 'Compute using BEMT'.

Programmatic Use

Block Parameter: cd0

Type: character vector

Values: '0' | positive scalar

Default: '0'

Lock number — Lock number
`0.6051` (default) | nonzero positive scalar

Lock number, which is the ratio of aerodynamics forces to inertial forces, specified as a nonzero positive scalar.

Dependencies

To enable this parameter, set Modeling to With flap effects.

Programmatic Use

Block Parameter: gamma

Type: character vector

Values: '0.6051' | nonzero positive scalar

Default: '0.6051'

Twist distribution — Rotor blade twist distribution
`Linear` (default) | `Ideal`

Rotor blade twist distribution, specified as:

Linear — Close approximation of blade twist distribution.

$θ (r) = θ_{r o o t} + θ_{t w i s t} r$
Ideal — Optimal approximation of blade twist distribution.

$θ (r) = \frac{θ_{t i p}}{r}$

where r is the nondimensional radial location along the blade.

Dependencies

To enable this checkbox,

Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

Set Modeling to 'With flap effects'

Programmatic Use

Block Parameter: twistType

Type: character vector

Values: 'Linear' | 'Ideal'

Default: 'Linear'

Blade root angle (rad) — Blade root pitch angle
`0.2548` (default) | real scalar

Blade root pitch angle θ_root, specified as a real scalar.

Dependencies

To enable this parameter:

Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

Set Modeling to 'With flap effects'

Set Twist distribution to Linear.

Programmatic Use

Block Parameter: thetaRoot

Type: real scalar

Values: '0.2548' | nonzero positive scalar

Default: '0.2548'

Blade twist angle (rad) — Blade twist angle
`-0.1361` (default) | real scalar

Blade twist angle θ_twist, specified as a real scalar.

Dependencies

To enable this parameter:

Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

Set Modeling to 'With flap effects'

Set Twist distribution to Linear.

Programmatic Use

Block Parameter: thetaTwist

Type: real scalar

Values: '0.1018' | positive scalar

Default: '0.1018'

Blade tip angle (rad) — Blade tip pitch angle
`0.1018` (default) | real scalar

Blade tip pitch angle θ_tip for ideal twist distribution, specified as a real scalar.

Dependencies

To enable this parameter:

Set the CT and CQ Source dropdown to 'Compute using BEMT'.

and/or

Set Modeling to 'With flap effects'

Set Twist distribution to Ideal.

Programmatic Use

Block Parameter: thetaTip

Type: real scalar

Values: '0.06' | nonzero positive scalar

Default: '0.06'

Algorithms

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Force and Moment

Force and moment calculated using these equations:

Without flap effects

$\begin{array}{l} F_{x y z} = [\begin{matrix} 0 \\ 0 \\ - C_{T} ρ π R^{2} {(Ω R)}^{2} \end{matrix}] \\ M_{x y z} = [\begin{matrix} 0 \\ 0 \\ - C_{Q} ρ π R^{2} Ω | Ω | R^{3} \end{matrix}] \end{array}$
With flap effects

$\begin{array}{l} F_{x y z} = C_{T} ρ π R^{2} (^{Ω R) 2} [\begin{matrix} - \cos β_{s} \sin β_{c} \\ \sin β_{s} \\ - \cos β_{c} \cos β_{s} \end{matrix}] \\ M_{x y z} = [\begin{matrix} 0 \\ 0 \\ - C_{Q} ρ π R^{2} Ω | Ω | R^{3} \end{matrix}] \end{array}$

where:

Ω is the rotor speed in rad/s.
β_c and β_s are the steady state flap angles computed using the equations in Chapter 7 of [1].

Thrust Coefficient and Torque Coefficient Calculations

When the Compute CT and CQ check box is selected, the Rotor block calculates the thrust coefficient and torque coefficient using these equations.

With the inclusion of Prandtl’s tip loss function, the incremental thrust coefficient using blade element momentum theory is:

$d C_{T} = 4 F λ^{2} r d r$

where F is the correction factor:

$\begin{array}{l} F = \frac{2}{π} \cos^{- 1} e x p - f \\ f = \frac{1}{2} \frac{N_{b} (1 - r)}{r φ} \\ φ = \frac{λ}{r} \end{array}$

For the equations in this section:

r is the nondimensional radial location.
λ is the inflow through the rotor disc.
c_lα is the lift curve slope.
σ is the rotor solidity defined as $σ = \frac{N_{b} c}{π R}$ , where c is the chord, R is the radius and N_b is the number of blades.

According to blade element theory, the incremental thrust coefficient is:

$d C_{T} = \frac{1}{2} σ c_{l α} (θ (r) r - λ) r d r$

Here, θ(r) will depend on the blade twist distribution and the collective pitch input θ₀. Equating the two expressions for dC_T returns 4Fλ² = 0.5 σc_ıα (θ(r)r − λ), which can be iteratively solved for λ.

Sum dС_T = 4Fλ²r dr across the blade span, to find the net thrust coefficient C_T.

For the torque coefficient C_Q, the profile drag component is approximately included as $\frac{σ c_{d 0}}{8}$ . The net torque coefficient can be calculated by summing dС_Q = λdС_T across the blade span and adding the profile drag component.

Rotor and Propeller Comparison

Propellers and rotors are both mechanical devices designed to move fluid in order to generate thrust. Despite their similar function, they are optimized for different applications based on their design characteristics.

Propellers are typically used in airplanes, drones, and boats. Propellers move an aircraft or vessel forward by rotating and pushing air or water behind them.
Rotors are mainly found in helicopters and some types of drones. Rotors provide forward thrust and enable vertical lift and control over the pitch, roll, and yaw of the aircraft.

You can use the Rotor block for both rotors and propellers, however, the conventions differ based on the application.

Thrust and Torque coefficient

Thrust — The equation for the thrust in propellers is ${T = k}_{T} ρ n^{2} D^{4}$ , where:

k_T is the thrust coefficient.
ρ is the density.
n is the propeller speed in rev/s.
D is the propeller diameter.

In the Rotor block, the basic thrust equation is defined as ${T = C}_{T} ρ π R^{2} {(Ω R)}^{2}$ , where:

C_T is the thrust coefficient.
ρ is the density.
Ω is the rotor speed in rad/s.
R is the rotor radius.

The two constants are related by the equation $k_{T} = 0.25 C_{T} π^{3}$ .

Torque — In the Rotor block, torque is defined as $Q_{r o t} = C_{Q} ρ π R^{2} {(Ω R)}^{2} R$ .

In propellers, torque is defined by $Q_{p r o p} = k_{Q} ρ n^{2} D^{5}$ .

The two constants are related by the equation $k_{Q} = 0.125 C_{Q} π^{3}$ .

Advance Ratio

The advance ratio is the nondimensional forward velocity, defined as the ratio of the forward speed of the vehicle to the product of the rotational speed of the propeller or rotor and the diameter of the propeller or rotor.

The thrust and torque coefficient parameters may be available as functions of advance ratio. The definition of advance ratio differs based on the application of the rotating device, specifically whether the device is being used for propulsion or vertical flight.

References

[1] Prouty, Raymond W. Helicopter Performance, Stability, and Control Krieger Publishing Company, 1995.

[2] Riether, F. (2016). Agile quadrotor maneuvering using tensor-decomposition-based globally optimal control and onboard visual-inertial estimation (Doctoral dissertation, Massachusetts Institute of Technology).

Version History

Introduced in R2023a

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R2024b: Rotor block now includes swashplate as input

Starting R2024b, the Rotor block has been updated to support swashplate inputs. This addition allows for the inclusion of collective and cyclic blade pitch angles in the calculation of Forces and Moments through new optional input ports.
Rotor speed, Ω, also now accepts negative inputs.
Starting R2024b, the Rotor block has been updated to include the dropdown CT and CQ Source with two options: 'Dialog' and 'Compute using BEMT'.
- Select 'Dialog' to manually enter the thrust and torque coefficient values.
- Select 'Compute using BEMT' to compute the values using Blade Element Momentum Theory (BEMT). For more information on BEMT, see Thrust and Torque coefficient. This option is equivalent to the Compute CT and CQ checkbox from previous releases.

Rotor

Description

Examples

Mars Helicopter Simulink-Based System Level Design

Limitations

Ports

Input

Ω (rad/sec) — Rotor speed scalar

ρ — Air density positive scalar

Vb — Body velocity of rotor 3-by-1 vector | 1-by-3 vector

Dependencies

ωb — Angular velocity 3-by-1 vector | 1-by-3 vector

Dependencies

θ0 — Collective blade pitch angle scalar

Dependencies

θc — Lateral cyclic pitch angle scalar

Dependencies

θs — Longitudinal cyclic pitch angle scalar

Dependencies

Output

Fxyz(N) — Total force three-element vector

Mxyz(Nm) — Net moment three-element vector

CT — Computed thrust coefficient positive scalar

Dependencies

CQ — Computed torque coefficient positive scalar

Dependencies

Parameters

Units — Input and output units Metric (MKS) (default) | English (Velocity in ft/s) | English (Velocity in kts)

Programmatic Use

Modeling — Rotor thrust calculation method Without flap effects (default) | With flap effects

Programmatic Use

Include pitch angle inputs — Enable pitch angle inputs off (default) | on

Dependencies

Programmatic Use

CT and CQ Source — Dropdown to select the source of thrust and torque coefficient values 'Dialog' (default) | 'Compute using BEMT'

Programmatic Use

Thrust coefficient (CT) — Thrust coefficient 0.0107 (default) | nonzero positive scalar

Dependencies

Programmatic Use

Torque coefficient (CQ) — Torque coefficient 7.8263e-4 (default) | nonzero positive scalar

Dependencies

Programmatic Use

Number of blades — Number of blades per rotor 2 (default) | nonzero positive scalar

Dependencies

Programmatic Use

Output computed CT and CQ — Output computed thrust coefficient and torque coefficient off (default) | on

Dependencies

Programmatic Use

Radius (m) — Rotor radius 0.0330 (default) | nonzero positive scalar

Programmatic Use

Chord (m) — Blade chord 0.0080 (default) | nonzero positive scalar

Dependencies

Programmatic Use

Hinge offset (m) — Hinge offset 0 (default) | positive scalar

Dependencies

Programmatic Use

Lift curve slope (per rad) — Lift curve slope 5.5 (default) | nonzero positive scalar

Dependencies

Programmatic Use

Mean drag coefficient — Mean drag coefficient 0 (default) | positive scalar

Dependencies

Programmatic Use

Lock number — Lock number 0.6051 (default) | nonzero positive scalar

Dependencies

Programmatic Use

Twist distribution — Rotor blade twist distribution Linear (default) | Ideal

Dependencies

Programmatic Use

Blade root angle (rad) — Blade root pitch angle 0.2548 (default) | real scalar

Dependencies

Programmatic Use

Blade twist angle (rad) — Blade twist angle -0.1361 (default) | real scalar

Dependencies

Programmatic Use

Blade tip angle (rad) — Blade tip pitch angle 0.1018 (default) | real scalar

Dependencies

Programmatic Use

Algorithms

Force and Moment

Thrust Coefficient and Torque Coefficient Calculations

Ω (rad/sec) — Rotor speed
scalar

ρ — Air density
positive scalar

V_b — Body velocity of rotor
3-by-1 vector | 1-by-3 vector

ω_b — Angular velocity
3-by-1 vector | 1-by-3 vector

θ₀ — Collective blade pitch angle
scalar

θ_c — Lateral cyclic pitch angle
scalar

θ_s — Longitudinal cyclic pitch angle
scalar

F_xyz(N) — Total force
three-element vector

M_xyz(Nm) — Net moment
three-element vector

C_T — Computed thrust coefficient
positive scalar

C_Q — Computed torque coefficient
positive scalar

Units — Input and output units
`Metric (MKS)` (default) | `English (Velocity in ft/s)` | `English (Velocity in kts)`

Modeling — Rotor thrust calculation method
`Without flap effects` (default) | `With flap effects`

Include pitch angle inputs — Enable pitch angle inputs
`off` (default) | `on`

CT and CQ Source — Dropdown to select the source of thrust and torque coefficient values
`'Dialog'` (default) | `'Compute using BEMT'`

Thrust coefficient (CT) — Thrust coefficient
`0.0107` (default) | nonzero positive scalar

Torque coefficient (CQ) — Torque coefficient
`7.8263e-4` (default) | nonzero positive scalar

Number of blades — Number of blades per rotor
`2` (default) | nonzero positive scalar

Output computed CT and CQ — Output computed thrust coefficient and torque coefficient
off (default) | on

Radius (m) — Rotor radius
`0.0330` (default) | nonzero positive scalar

Chord (m) — Blade chord
`0.0080` (default) | nonzero positive scalar

Hinge offset (m) — Hinge offset
`0` (default) | positive scalar

Lift curve slope (per rad) — Lift curve slope
`5.5` (default) | nonzero positive scalar

Mean drag coefficient — Mean drag coefficient
0 (default) | positive scalar

Lock number — Lock number
`0.6051` (default) | nonzero positive scalar

Twist distribution — Rotor blade twist distribution
`Linear` (default) | `Ideal`

Blade root angle (rad) — Blade root pitch angle
`0.2548` (default) | real scalar

Blade twist angle (rad) — Blade twist angle
`-0.1361` (default) | real scalar

Blade tip angle (rad) — Blade tip pitch angle
`0.1018` (default) | real scalar