hover flight helicopter modelling and vibrations analysis · hover flight helicopter modelling and...

Hover Flight Helicopter Modelling and Vibrations Analysis

Salvador Castillo-Rivera

The School of Engineering and Mathematical Sciences. The City University London. United Kingdom,

email: [email protected]

M. Tomas-Rodriguez

The School of Engineering and Mathematical Sciences. The City University London. United Kingdom,

email: [email protected]

Abstract

In this work, different modelling aspects of helicopter

aerodynamics are discussed. The helicopter model is

on Sikorsky configuration, main rotor in

perpendicular combination with a tail rotor. The

rotors are articulated and their blades are rigid. The

main rotor implementation takes into account flap,

lag and feather degrees of freedom for each of the

equispaced blades as well as their dynamic

couplings. The model was built by using VehicleSim,

software specialized in modelling mechanical

systems composed by rigid bodies. Appearing

vibrations due to the rotating behaviour of the rotors

are studied in here. This work presents an

aerodynamic model that allows to simulate hover

flight. The aerodynamic model has been built up

using blade element theory. The aerodynamic load

creates vibrations on the helicopter and these are

analyzed on the fuselage by using short time Fourier

transform processing to study the vibrations

spectrum. The results of the simulations are

presented and compared to existing mechanical

vibrations generated by a dynamic model developed

by the authors as well as existing theory in the

specialist literature.

Keywords: Vibrations, helicopter, dynamics,

aerodynamics.

1 INTRODUCTION

Helicopter vibration is common a problem which

involves complex interactions between the inertial,

structural loads and aerodynamic loads. The major

source of vibrations in helicopters is the main rotor,

whereas in fixed-wing aircraft, the vibrations are

mostly originated by the engines or are caused by

atmospheric turbulences. In aircraft's design,

vibrations have remained one of the major problems

affecting helicopter development for years. In fact,

the maximum speed and manoeuvring capabilities for

most of the modern helicopters are limited by

excessive vibration. Vibrations frequencies are either

equal to the rotors frequencies or multiple of them.

The rotors frequencies are a function of the angular

speeds at which they rotate [5].

Vibrations affect ride comfort, adds to the fatigue of

pilots, crew, and passengers and increases

maintenance time and cost. High vibration levels

experienced by a helicopter could in many cases pose

a limitation to the vehicle’s forward speed and

manoeuvring capabilities. In addition to this,

vibrations affect the helicopter handling qualities,

contribute to the fatigue of structural components,

reduce the reliability of on board electronic

equipment, and influence the precision of equipment

such as cameras, measure devices, etc. [7], [11].

The reduction of helicopter vibrations has

traditionally been a difficult task to achieve.

Vibrations should be analysed and study in order to

identify their main frequencies. This information can

be used to design vibration control methods for

helicopters [8].

The structure of the article is as follows: in sections 2

and 3, the aspects of the dynamic helicopter model

are considered. Section 4 describes VehicleSim

environment, used to implement the model of the

helicopter. In section 5, a hover aerodynamic model

is provided. Vibrations analysis on the fuselage, as a

consequence of the aerodynamic and dynamic loads

are given in Section 6. Section 7 contains the

conclusions of the article.

2 DYNAMIC MODEL

The helicopter under study is on Sikorsky

configuration i.e., main rotor in perpendicular

combination with a tail rotor. Both systems are

mounted on the fuselage. The helicopter model

consists of fuselage, main rotor and tail rotor, both

articulated. The main rotor consists of four equally

spaced blades joined to a central hub and the tail

rotor consists of two equally spaced blades joined to

a secondary hub. The blades are rigid in both rotors.

Actas de las XXXVI Jornadas de Automática, 2 - 4 de septiembre de 2015. Bilbao ISBN 978-84-15914-12-9 © 2015 Comité Español de Automática de la IFAC (CEA-IFAC) 473

mailto:[email protected]

The helicopter has six degrees of freedom: three

translations along the (X, Y, Z) axes and three

rotations around the same axes. The model presented

in this paper is based in previous works developed by

the authors (see [2], [3], [4], [10], [14]).

2.1 FUSELAGE

The fuselage is the rotorcraft's main body section that

holds crew and passengers, amongst other. Its

degrees of freedom are the lateral and longitudinal

translation in the horizontal plane X-Y axis, vertical

translation (Z axis) and rotation about these same

axes corresponding to yaw, pitch and roll.

2.2 MAIN ROTOR

The role of the main rotor is to support the aircraft's

weight, as it generates the lift force. It allows to keep

the helicopter suspended in the air and provides the

control that allows to follow a prescribed trajectory

in the various spatial directions by changing altitude

and executing turns. It transfers prevailingly

aerodynamic forces and moments from the rotating

blades to the non-rotating frame (fuselage). The

blades are kept in uniform rotational motion

(rotational speed ), by a shaft torque from the

engine. A common design solution adopted in the

development of the helicopter is to use hinges at the

blades roots that allow free motion of the blade

normal to and in the plane of the disc. The most

common of these hinges is the flap hinge which

allows the blade to flap, this is, to move in a plane

containing the blade and the shaft, of the disc plane,

about either the actual flap hinge or in some other

cases, the flap hinge is substituted by a region of

structural flexibility at the root of the blade. The flap

hinge is more frequently designed to be a short

distance from the centre line. This is termed an

"offset" (eR), and it offers the designer a number of

important advantages.

A blade which is free to flap, experiences large

Coriolis moments in the plane of rotation and a

further hinge (called lag) is provided to relieve these

moments. This degree of freedom produces blade

motion on the same plane as the disc. In presence of

aerodynamic loads this degree of freedom generates

the blade’s drag force.

A blade can also feather around an axis parallel to the

blade span. Blade feather motions are necessary to

control the aerodynamic lift developed and, in

forward motion of the helicopter, to allow the

advancing blade to have a lower angle of incidence

than the retreating blade and thereby to balance the

lift across the craft. In order to be able to climb up,

the feather angle needs to be increased. On the other

hand, in order to descend, the blade's feather angle is

decreased. Because all blades are acting

simultaneously in this case, or collectively, this is

known as collective feather and allows the rotorcraft

to rise/fall vertically. Additionally to this control, for

achieving forward, backward and sideways flight, a

different additional change of feather is required. The

feather on each individual blade is increased at the

same selected point on its circular pathway. This is

known as cyclic feather or cyclic control. Blade

feather control is achieved through linkage of the

blade to a swashplate.

2.3 TAIL ROTOR

The tail rotor is mounted in perpendicular to the main

rotor. It counteracts the torque and the yaw motion

that the main rotor disc naturally produces. In

accordance to Newton's third law of action and

reaction, the fuselage tends to rotate on the opposite

direction to the main rotor's blades as a reaction of

the torque that appears (see Figure 1).

Figure 1: Tail rotor counteracting the torque induced

by the main rotor rotation.

This torque must be counteracted and/or controlled

before any type of flight is possible. Two anti-torque

pedals allow the pilot to compensate for torque

variance by providing a means of changing pitch

(angle of attack) of the tail rotor blades. This

provides heading and directional control in hover and

at low airspeeds. Driven by the main rotor at a

constant ratio, the tail rotor produces thrust in a

horizontal plane opposite to torque reaction

developed by the main rotor. Since the main rotor

torque varies during flight when power changes are

made, it is necessary to vary the thrust of the tail

rotor. A significant part of the engine power is

required to drive the tail rotor, especially during

operations when maximum power is used. Any

change in engine power output produces a change in

the torque effect.


3 VEHICLESIM AS MODELLING

TOOL

VehicleSim is a multibody modelling software. It has

been used over a wide range of mechanical dynamic

studies, mainly in connection to vehicle dynamics

[13] and it has provided the basis for commercial

simulation codes such as TruckSim, CarSim and

BikeSim [6]. The syntactic rules of VehicleSim are

straightforward. The output from VehicleSim can

take one of several forms: (a) a rich text format file

containing the symbolic equations of motion of the

system described; (b) a ”C” language simulation

program with appropriate data files containing

parameter values and simulation run control

parameters or (c) linear state space equations in a

MATLAB ”M‐file” format that contains symbolic

state‐space A, B, C, D matrices that can be used for

linear analysis. Once the model has been built, it

becomes independent of VehicleSim and can be

executed at any time.

4 PROGRAM STRUCTURE

VehicleSim Lisp is used to develop the multibody

system code that represents a helicopter system. The

multibody system is subdivided into its constituent

bodies for the purpose of writing the VehicleSim

code. The bodies are arranged as a parent-child

relationship. The first body is the inertial frame and it

has the fuselage as its only child. The fuselage is

located at the origin of the inertial coordinates system

and it is the parent of both the main and tail rotors.

The main rotor rotates around its vertical axis, Z axis.

The main rotor is the parent of the flap hinge that

rotates around the corresponding X axis, the lag

hinge is the child of the flap hinge. The lag hinge

rotates around the Z axis. The feather hinge is the

child of the lag hinge. The feather hinge rotates

around the Y axis. Finally, the blade is added to the

program structure as the child of the feather hinge.

The tail rotor is built up following this same parent-

child structure.

4.1 DYNAMIC MODEL DESCRIPTION

The parameters used for the dynamic simulation

carried out in this work are shown in table 1 ([14]).

For the purpose of dynamical modelling only, the

action of external forces not considered, for example:

gravity. The system can be considered in the vacuum.

An unbalance of masses is considered on the main

and the tail rotors blades in order to simulate a source

of vibrations in the helicopter with an amplitude to be

detected in the spectrum. On the other hand, in order

to simulate the engine effects on each rotor, the main

and tail rotors angular speeds are modelled using PID

controllers.

Table 1: Parameter’s values.

Parameters Symbol Value

Helicopter Mass mh 2064 (kg)

Blade one mass mbl1 31.06 (kg)

Blade two mass mbl2 31.26 (kg)

Blade three mass mbl3 29.96 (kg)

Blade four mass mbl4 31.16 (kg)

Spring flap hinge kfj 46772 (Nm/rad)

Spring lag hinge klj 314938(Nm/rad)

Damping lag hinge dlj 349.58(Nms/rad)

Main Rotor Speed Ω 44.4 (rad/s)

5 AERODYNAMIC MODEL IN

HOVER

5.1 AIR DENSITY

For every flight condition, the air density changes

with the height (h), for the lower atmosphere where

helicopters fly below 6000 m, the standard value of

air density can be approximated as:

where h is expressed in meters and ρo is 1.225 kg/m3

(air density at sea level) [9].

5.2 INDUCED HOVER SPEED

In hovering flight, the induced velocity can be

obtained as vi = vio, vio is the hover induced velocity,

which can be considered constant in hover, the

traction force, T, becomes equal to the disc loading

(weight of the helicopter), see Figure 2 [9]:

Figure 2: Main forces acting on a helicopter in hover

flight.


5.3 BLADE ELEMENT ANALYSIS IN HOVER

FLIGHT

Blade element theory forms the basis of most modern

analyses in helicopter rotor aerodynamics as it

estimates the radial and azimuthal distributions of

blade aerodynamic forces (and moments). In addition

to this, the rotor performance can be obtained by

integrating the sectional airloads at each blade's

elements over the length of the blade and averaging

the result over a rotor revolution [9].

Figure 3 is a plan view of the rotor disc, viewed from

above. The blade radius is R and the tip speed is

given by ΩR. An elementary blade section is

considered at radius y, of chord length c and

spanwise width dy.

Figure 3: Main rotor disc viewed from above.

The velocity components on the blade's section are

shown in Figure 4. The flow seen by the section has

velocity components Ωy in the disc plane and (vi +

Vc) (vi is the induced velocity and Vc is the upward

velocity) perpendicular to it [12].

The resultant local flow velocity at any blade element

at a radial distance y from the rotational axis has an

out of plane component Up = (vi +Vc) normal to the

rotor plane as a result of climb and induced inflow

and an in plane component UT = Ωy parallel to the

rotor due to blade rotation, relative to the disc plane.

The resultant velocity at the blade element is

therefore the composition of both [9]:

The blade’s feather angle θ, is imposed by the pilot's

collective control input. The angle between the flow

direction and the plane of rotation, known as the

inflow angle ϕ, is therefore [12]:

Figure 4: Velocity components UT and Up.

If the feather angle at the blade element is θ, then the

aerodynamic or effective angle of attack is:

The resultant incremental lift, dL, and drag dD per

unit span on a blade element are:

where ρ is the air density, Cl and Cd are the lift and

drag coefficients, c is the local blade chord. The lift

dL and drag dD act perpendicular and parallel

respectively to the resultant flow velocity.

5.4 THRUST COEFFICIENT

The thrust coefficient approximation for hover flight

can be written as:

θ is the feather angle, a is the lift slope, λ = vih/ΩR,

vih is the induced hover velocity, R is the rotor radius

and σ is the solidity factor which for a constant blade

chord is given by:

N is the number of main rotor blades and c is the

blade chord [12].

6 VIBRATIONS ANALYSIS

The analysis of the vibration modes on a helicopter is

a difficult task due to the complexity of the structure,


but reasonable accuracy is achievable with modern

techniques.

This work is carried out by estimating the vibrations

appearing in hover flight when the aerodynamical

model has been implemented as well as a comparison

to dynamic vibrations generated under the excitation

conditions described previously. In order to develop

the proposed analysis, it is necessary to follow the

next steps: a) detection of vibration signals on the

fuselage as a consequence of the hover aerodynamic

load and with unbalance of masses, separately b)

analysis of the spectrogram for identification of the

detected vibrations.

6.1 VIBRATIONS IN HOVER FLIGHT

In here, the vibrations appearing on the fuselage's

axes X (roll) and Y (pitch) during hover flight are

studied. Only two axes are analysed as these are

enough to validate the expected behaviour. It is

known that vibrations are generally low in hover

flight [7]. The aerodynamic model satisfies the

following structural characteristics: the flap and lag

degrees of freedom do not have springs fitted

although the lag maintains a damper. In addition to

this, it does not exist an unbalance of masses on the

main and tail rotors.

There is a consequence of the use of a rotating frame

of reference that affects vibration frequencies created

in the rotor and transmitted to the fuselage. The

frequencies generated in the rotor may contain the

rotational frequency of the rotor and the external

perturbation frequency [15].

In order to analyze the vibrations appearing on the

fuselage, the Short Time Fourier Transform (STFT)

is used.

A hover flight simulation is done; the simulations are

carried out for 50 seconds, although the results

plotted in figures 5, 7, 9 and 11 show the first 5

seconds only, for clarity of the view. The height is h

= 250 m and the main rotor's collective feather angle

is 0.175 rad and tail rotor's collective feather angle is

also 0.175 rad. Figure 5, shows the fuselage's

oscillations (vibrations) on the X axis during hover

flight for 5 seconds. Figure 6, shows the

corresponding spectrogram obtained for this

simulation. Various predominant frequencies which

come from the main rotor loads (approximately 7.6

Hz) can be seen, this is the flap frequency. There is a

second predominant frequency (approximately 15.2

Hz) this is twice the flap frequency and a third

predominant frequency is found at around 37.1 Hz,

this frequency coincides with the tail rotor blades’

flap frequency.

Figure 5: Vibrations on the fuselage's X axis for

hover flight.

Figure 6: Fuselage's X axis spectrogram in hover

flight conditions.

Similar analysis is carried out for the oscillations

appearing on the fuselage's Y axis (see Figure 7).

The three predominant frequencies of these

vibrations appear in the spectrogram in Figure 8.

These are approximately 7.6 Hz which are caused by

the main rotor blades’ flap, the second frequency is

approximately 15.2 Hz, this value is twice the main

rotor flap frequency and the third frequency is around

37.1 Hz, it is associated to the tail rotor flap motion.

Figure 7: Vibrations on the fuselage's Y axis for

hover flight.


Figure 8: Fuselage's Y axis spectrogram in hover

flight.

6.2 DYNAMIC VIBRATIONS

In this section, the vibrations when both flap and lag

modes are coupled will be studied when the main and

tail rotor have a constant angular speed. In this case,

the vibrations are introduced in the model by

unbalance of masses in both rotors; the mass of the

blades on the main rotor are mbl1 = 31.06 kg, mbl2 =

31.26 kg, mbl3 = 29.96 kg and mbl4 = 31.16 kg and

on the tail rotor they are; mbltl1 = 6.212 kg and

mbltl2 = 6.222 kg. The collective and cyclic feather

angles are zero in the main rotor.

The tail rotor flap dynamics are modelled as a

Fourier series: βtl=ao - a1 cos(Ωt) -b1 sin(Ωt) where ao

= 0.01 rad, a1 = 0.01 rad and b1 = 0.01 rad are the

corresponding coefficients, and the collective angle is

0.01 rad. The main rotor angular speed is constant at

44.4 rad/s. Figure 9, shows the fuselage's oscillations

(vibrations) around the X axis under the previous

conditions.

Figure 9: X axis fuselage vibrations for mbl1 = 31.06

kg, mbl2 = 31.26 kg, mbl3 = 29.96 kg and mbl4 =

31.16 kg and on the tail rotor they are; mbltl1 = 6.212

kg and mbltl2 = 6.222 kg.

From the spectrogram of frequencies shown in Figure

10 (the window length is reduced in order to check

the noise and other additional frequencies are not

presented in this case), the oscillation on the

fuselage's X axis shows two predominant frequencies

which are originated at the main rotor (7 Hz

approximately) and at the tail rotor, in this case, the

value is around of 37 Hz.

Figure 10: Fuselage vibrations spectrogram on the X

axis for mbl1 = 31.06 kg, mbl2 = 31.26 kg, mbl3 =

29.96 kg and mbl4 = 31.16 kg and on the tail rotor

they are; mbltl1 = 6.212 kg and mbltl2 = 6.222 kg.

Similar analysis can be carried out for the

spectrogram of frequencies in fuselage’s Y axis (see

figures 11 and 12). Two predominant frequencies

appear, these vibrations are originated at the main

and tail rotors', similarly to the X axis.

Figure 11: Y axis fuselage vibrations for mbl1 =

31.06 kg, mbl2 = 31.26 kg, mbl3 = 29.96 kg and mbl4

= 31.16 kg and on the tail rotor they are; mbltl1 =

6.212 kg and mbltl2 = 6.222 kg.

A comparison can be carried out for the vibrations

presented in these sections. Clearly, the aerodynamic

load is a source of vibrations on the helicopter's

fuselage. The obtained spectrograms for hover flight

agree with theoretical predictions: a) there is an

important source of helicopter vibrations, the

aerodynamics forces [7], b) the aerodynamic loads on

a helicopter rotor blade vary considerably as it moves

around the rotor disc. These loads arise from the

aerodynamic forces on the rotor blades, together with


the inertial forces produced by the flap and lag

motions of the blade [1].

Figure 12: Fuselage vibrations Spectrogram on the Y

axis for mbl1 = 31.06 kg, mbl2 = 31.26 kg, mbl3 =

29.96 kg and mbl4 = 31.16 kg and on the tail rotor

they are; mbltl1 = 6.212 kg and mbltl2 = 6.222 kg.

7 CONCLUSIONS

The helicopter model under study is on Sikorsky

configuration, the model reproduces the dynamic

behaviour of a helicopter, which is capable to

transmit perturbations from the main rotor to the

fuselage in form of vibrations. The model has been

implemented in VehicleSim, a program allows to

define the systems as composition of several bodies

and ligatures by using a parental relationship

parent/child structure.

This work has presented a helicopter aerodynamic

model in which the blade element theory has been

used for implementation hover flight. This is

important in order to study and analyze the vibrations

appearing in the fuselage as a consequence of

aerodynamic load under these flying conditions.

Various tests under the action of these conditions

were carried out in order to study pure vibrations

appearing on the fuselage roll and pitch axes.

In hover flight and under unbalance of masses on

both rotors, fuselage vibrations spectrograms were

obtained and analyzed by using a short time Fourier

transform process, various cases were considered,

when the aerodynamic load for hover flight was

included in the dynamic helicopter model, the

obtained results match those predicted by theoretical

approaches. In addition to this, these vibrations were

compared to dynamic vibrations generated with the

same helicopter model in absence of the aerodynamic

load. As a consequence, the spectrograms were also

studied and these showed a reasonable discrepancy

according to the expected behaviour introduced by

the aerodynamic model. These results are still on an

early stage and the authors expect to develop further

analogies with experimental results in future work.

In summary, a full helicopter model has been

modelled using VehicleSim. Spectrograms analysis

on the fuselage’s roll and pitch axes captured the

appearing vibrations as a consequence of the

aerodynamic load in hover flight, these results have

been compared to dynamic vibrations obtained with

identical helicopter model under the action of

unbalance of masses on the rotors. As a result of this,

the vibrations in hover have been shown to satisfy the

predicted behaviour.

References

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