Multibody Modeling of Engage and Disengage
Operations of Helicopter Rotors∗
Carlo L. Bottasso
Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano
Milano, Italy.
Olivier A. Bauchau
Georgia Institute of Technology, School of Aerospace Engineering
Atlanta, GA, USA.
Abstract
This paper deals with the modeling of the engage and disengage operations of a
helicopter rotor during which blades can come in intermittent contact with the droop
or flap stops. The problem is formulated within the framework of finite element based
flexible multibody dynamics that allows the analysis of rotorcraft systems of arbitrary
topology with the potential of contact among various components of the system. Contact is assumed to be of finite duration, and the forces acting between the contacting
stops are explicitly computed during the simulation. The modeling of contact consists
of three parts: the evaluation of the relative distance between the stops, a unilateral
contact condition which is transformed into a holonomic constraint by the addition
of a slack variable, and a contact model which describes the relationship between the
contact force and the local deformation of the contacting bodies. The proposed analysis procedure is validated by comparison with experimental data, and the aeroelastic
analysis of shipboard engage and disengage operations of a H-46 rotor is presented.
Introduction
When operating in high wind conditions or from a ship-based platform, rotorcraft blades
spinning at low velocity during engage and disengage operations can flap excessively, to the
point of striking the helicopter fuselage. Several analysis tools have been developed in the
past to model this situation. Earlier efforts dealing with hingless rotor systems1, 2 were later
refined for the analysis of articulated rotors, including the effects of droop/flap stops on the
dynamics of rigid blades3–5 . More recently, transient finite element methods based on a modal
reduction of the system were developed to improve the modeling of this problem6–9 . In all
∗
Journal of the American Helicopter Society, 46, pp 290 – 300, 2001
1
these analyses, the flap/droop stops were represented by means of springs with high stiffness
constants to approximate the metal-to-metal contact. The simulation included time-domain
unsteady aerodynamics with non-linear separation and dynamic stall effects.
Using these analysis tools, low cost parametric studies can be quickly conducted in an
effort to identify the key variables governing the phenomenon. Among the key variables are:
the control settings, the flap damper characteristics, and the droop stop angle. In view of
the large number of variables involved in this problem, accurate numerical simulation tools
are particularly important to help reduce the number of costly experimental tests.
The approach to this problem seems to follow a trend typically used in rotorcraft modeling: simulation tools are developed when the need arises to deal with a specific problem. The
classical approach to rotor dynamics has been to use a modal reduction method, as pioneered
by Houbolt and Brooks10 . Early models were limited to a single articulated blade connected
to an inertial point, and the control linkages were ignored. The equations of motion were
specifically written for a blade in a rotating system, and ordering schemes were used to decrease the number of nonlinear terms11 . As the need arose, detailed models were developed
to deal with specific features of the system such as gimbal mounts, swash-plates, or bearingless root retention beams, among many others. The relevant equations of motion were
derived for the specific configurations at hand. In fact, the various codes developed in-house
by rotorcraft manufacturers are geared towards the modeling of the specific configuration
they produce.
While this approach has worked well in the past, it should be recognized that it limits
the generality and flexibility of the resulting analysis codes. Furthermore, the growing need
to simulate an ever-increasing number of aeromechanical problems causes a proliferation of
codes, each one geared towards the solution of a single problem. Developing a new simulation
tool for each novel configuration or each new problem is a daunting task. Finally, the resulting
software must be carefully validated, if the analyst is to use it with confidence.
Clearly, a more general and flexible paradigm for modeling rotorcraft systems is needed.
Finite element based multibody dynamics formulations provide such a paradigm. Multibody formulations can deal with complex flexible mechanisms of arbitrary topologies: a
given mechanism is modeled through an idealization process that identifies the mechanism
components from within a large library of elements implemented in the code. Each element
provides a basic functional building block, for example a rigid or flexible member, a hinge,
a motor, etc. After assembling the various elements, one can construct a mathematical description, now often called a virtual prototype, of the mechanism with the required level of
accuracy. In Ref. 12, the authors have proposed a multibody simulation procedure applicable to rotorcraft systems that provides a comprehensive, modular and expandable simulation
software.
The goal of this paper is to demonstrate that a study of the engage and disengage
operations of rotors can be conducted within the framework of finite element based multibody
dynamics formulations, including an appropriate model for intermittent contact among the
components of the virtual prototype. This means that this complex problem in rotorcraft
aeromechanics can be readily solved without having to resort to new analytical developments.
Therefore, there is no need to derive new equations of motion for the problem at hand, and
there is no need to validate the resulting software. Carefully validated elements taken from
an existing multibody element library are assembled to form the complete model. Hence,
2
the sole virtual prototype needs to be validated against experimental data or other available
information, and not its basic building blocks. It is this “incremental” validation approach
that is at the heart of the pervasive success of the finite element method in so many branches
of engineering.
The modeling of intermittent contact in multibody systems is a topic that has received
considerable attention in the literature. The results of this research can be readily applied to
the problem at hand. Two broad categories of models can be defined based on the assumed
duration of contact. In the first approach, contact is treated as a discontinuity, i.e. the
duration of contact is assumed to tend to zero. The configuration of the system is assumed
to be identical before and after impact, and the principle of impulse and momentum is used
to compute the momenta after impact. Energy transfer during impact can be modeled in
a heuristic manner using the concept of coefficient of restitution. This approach was first
proposed by Kane,13 then applied to rigid multi-body systems by Haug and Wehage,14 and
extended to flexible systems by Khulief and Shabana15 . The accuracy of this approach is
inherently limited by the assumption of a vanishing impact duration. Furthermore, energy
balance is not necessarily satisfied when the principle of impulse and momentum is applied16 .
In the second approach, contact is of finite duration, and the time history of the forces
acting between the contacting bodies, which can be either rigid or deformable, is explicitly
computed during the simulation. Clearly, a constitutive law describing the force-deformation
relationship for the contacting bodies is required if the bodies are deformable. This approach
was used by a number of researchers, as for example in Refs. 17–19. Various types of
constitutive laws were used, but the classical solution of the static contact problem presented
by Hertz20 has been used by many investigators. Energy dissipation can be added in an
appropriate manner, as proposed by Hunt and Crossley21 .
A methodology for the analysis of flexible multibody systems undergoing intermittent
contacts of finite duration was developed in Refs. 22,23. The overall approach is broken into
three separate parts: a purely kinematic part describing the configuration of the contacting
bodies, a unilateral contact condition, and an optional contact model. The first, purely
kinematic part of the problem uses the concept of candidate contact points,24 i.e. the points of
the bodies that are the most likely to come in contact. These points are defined by a number
of holonomic constraints that involve the kinematic variables defining the configuration of
the contacting bodies. The knowledge of the location of these candidate contact points leads
to the definition of the relative distance q between the bodies.
The second part of the model is the unilateral contact condition which is readily expressed
in terms of the relative distance as q ≥ 0. In previous work, e.g.,17 this condition has
been enforced by means of a logical spring-damper system, i.e. a spring-damper system
acting between the bodies when they are in contact, and removed when they separate. The
properties of the spring-damper system can be selected to model the physical characteristics
of the contact zone. In Ref. 19, the logical spring constant is taken to be a large number that
enforces the non-penetration condition through a penalty approach, and the logical damper
is added to control the spurious oscillations associated with this penalty formulation. In
this work an alternate route is followed: the contact condition is enforced through a purely
kinematic condition q − r2 = 0, where r is a slack variable used to enforce the positiveness
of q. This approach leads to a discrete version of the principle of impulse and momentum.
The last part of the model is the contact constitutive law which takes into account the
3
physical characteristics of the contacting bodies. When these bodies are perfectly rigid this
model is not necessary. When the bodies are deformable, their local penetration, or approach,
is defined, and the contact model relates the approach to the contact force.
The paper is organized in the following manner. The first Section reviews the approach
of finite element based multibody dynamics analysis applied to rotorcraft problems. The
modeling of intermittent contact in such systems is discussed in the next Section, and specific
attention is focused on droop stop contacts. The subsequent Section validates the model by
comparing numerical predictions with experimental measurements for non-rotating blade
drop tests involving blade-droop stop impacts. Finally, the aeroelastic analysis of shipboard
engage and disengage operations of a H-46 rotor8 is presented.
Finite Element Based Multibody Dynamics
Formulations
Finite element based multibody dynamics formulations provide a solid foundation for the
analysis of rotorcraft aeromechanics problems. A brief overview of the salient features of
this approach is presented here, while more details can be found in Ref. 12.
Element library
Multibody formulations allow the modeling of novel rotorcraft configurations of arbitrary
topology through the assembly of basic components chosen from an extensive library of
elements. The element library includes the basic structural elements such as rigid bodies,
composite capable beams and shells, and joint models. Although a large number of joint
configurations are possible, most applications can be treated using the simple lower pairs.
More advanced joints, such as contact and backlash elements, are also available in many
commercial codes. All elements are referred to a single inertial frame, and hence, arbitrarily
large displacements and finite rotations must be treated exactly. No modal reduction is
performed, i.e. the full finite element equations are used at all times. In fact, with today’s
advances in computer hardware, inexpensive PCs provide enough computational power to
run complex rotor systems. Hence, resorting to modal reduction in order to save CPU time
might no longer be an overwhelming argument, especially when considering the possible loss
of accuracy associated with this reduction25 . Furthermore, contact and impact problems are
inherently associated with high frequency motions and variable system topology, and hence,
modal reduction is clearly inadequate in this case. Indeed, the accurate modeling of the high
frequency content of the response to an impact would require a large number of modes in
the expansion. Furthermore, the topology of the system abruptly changes at impact, and
the eigenvalue spectrum changes accordingly. This means that new modal bases must be
computed for each topology change, a possible drawback in complex applications.
Rigid bodies can be used for modeling components whose flexibility can be neglected or
for introducing localized masses. For example, in certain applications, the flexibility of the
swash-plate may be negligible and hence, a rigid body representation of this component is
4
acceptable. Beams are typically used for modeling rotor blades, but can also be useful for
representing transmission shafts, pitch-links, or wings of a tilt rotor aircraft. In view of the
increasing use of composite materials in rotorcraft, the ability to model components made of
laminated composite materials is of importance. Specifically, it must be possible to represent
shearing deformation effects, the offset of the center of mass and of the shear center from
the beam reference line, and all the elastic couplings that can arise from the use of tailored
composite materials. Ref. 26 gives details and examples of application of the integration of
a cross-sectional analysis procedure with the multibody dynamic simulation.
A distinguishing feature of multibody systems is the presence of a number of joints that
impose constraints on the relative motion of the various bodies of the system. Articulated
rotors and their kinematic chains are easily modeled with the help of lower pair joints,
i.e. the revolute, prismatic, screw, cylindrical, planar and spherical joints. For example,
a conventional blade articulation can be modeled with the help of three revolute joints
representing pitch, lag and flap hinges. Another example is provided by the pitch-link, which
is connected to the pitch-horn by means of a spherical joint, and to the upper swash-plate
by a universal joint to eliminate rotation about its own axis.
All joints are formulated with the explicit definition of the relative displacements and
rotations as additional unknown variables. This allows the introduction of generic spring
and/or damper elements in the joints, as usually required for the modeling of realistic configurations. Furthermore, the time histories of joint relative motions can be driven according
to suitably specified time functions. For example, in a helicopter rotor, collective and cyclic
pitch settings can be obtained by prescribing the time history of the relative rotation at the
corresponding joints. Finally, for the problem at hand, the droop and flap stops limit the
relative angular motion at a revolute joint. Hence, the contact conditions will be formulated
as constraints on the relative rotation at such joints.
Robust integration of multibody dynamics equations
Special implicit integration procedures for non-linear finite element multibody dynamics have
been developed in Refs. 27, 28. These algorithms are designed so that a number of precise
requirements are exactly met at the discrete solution level. This guarantees robust numerical
performance of the simulation processes. In particular, the following requirements are met by
the schemes: nonlinear unconditional stability, a rigorous treatment of all nonlinearities, the
exact satisfaction of the constraints, and the presence of high frequency numerical dissipation.
The proof of nonlinear unconditional stability stems from two physical characteristics of
multibody systems that are reflected in the numerical scheme: the preservation of the total
mechanical energy and the vanishing of the work performed by constraint forces. Numerical
dissipation is obtained by letting the solution drift from the constant energy manifold in
a controlled manner in such a way that at each time step, energy can be dissipated but
not created. The use of these unconditionally stable schemes is particularly important in
intermittent contact problems whose dynamic response is very complex due to the large,
rapidly varying contact forces applied to the system and to the dramatic change in stiffness
when a contact condition is activated. More details on these non-linearly stable schemes can
be found in Ref. 28 and references cited therein.
5
Solution procedures
Once a multibody representation of a rotorcraft system has been defined, several types of
analyses can be performed on the virtual prototype. A static analysis solves the static
equations of the problem, i.e. the equations resulting from setting all time derivatives equal
to zero. The deformed configuration of the system under the applied static loads is then
computed. The static loads can be of various types such as prescribed static loads, steady
aerodynamic loads, or the inertial loads associated with prescribed rigid body motions.
Once the static solution has been found, the dynamic behavior of small amplitude perturbations about this equilibrium configuration can be studied. This is done by first linearizing
the dynamic equations of motion, then extracting the eigenvalues and eigenvectors of the
resulting linear system. Due to the presence of gyroscopic effects, the eigenpairs are, in
general, complex. Finally, static analysis is also useful for providing the initial conditions to
a subsequent dynamic analysis. For instance, a rotor run-down simulation would be started
with initial conditions corresponding to the static solution for the rotor rotating at nominal
speed under gravity and aerodynamic loads.
A dynamic analysis solves the nonlinear equations of motion for the complete multibody
system. The initial conditions are taken to be at rest, or those corresponding to a previously determined static or dynamic equilibrium configuration. Complex multibody systems
often involve rapidly varying responses. In such event, the use of a constant time step is
computationally inefficient, and crucial phenomena could be overlooked due to insufficient
time resolution. Automated time step size adaptivity is therefore an important part of the
dynamic analysis solution procedure. A simple but effective time adaptive procedure was
developed for the integrators used in this effort, as detailed in Ref. 29. This automated
procedure is crucial for the analysis of contact problems. Indeed, very small time steps must
be used during the short period when impact occurs, whereas much larger time steps can be
used when the stops are not in contact. Using the small time step for the entire duration of
the simulation would needlessly increase the required computational resources.
The finite element based, multibody dynamic analysis of rotorcraft generates massive
amounts of data that can be processed in a variety of ways. Apart from standard time history
plots of positions, velocities and stresses in any point of the model, objects of the multibody
system can be viewed in a symbolic manner to help model validation, or with associated
predefined geometrical shapes to improve the realism of the visualization. For static analyses,
step-by-step visualization is provided together with eigenmode animation. For dynamic
analyses, the time dependent system configuration is displayed, and different vector-type
attributes, such as linear or angular velocities, internal forces or moments, curvatures or
strains, and aerodynamic forces or moments can be added.
Modeling of Joints with Backlash
The droop stop is a mechanism that supports the blade weight at rest and at low speeds.
Usually, the device is located near the blade flap hinge. In its simplest version, the stop
6
merely provides a support when the blade comes to rest on it at low rotor speed. In a more
complex implementation, the stop is actuated by the centrifugal force caused by the rotor
angular velocity acting on a counterweight. At high rotor speed, the counterweight retracts
the stop, while below a specified rotor speed the stop extends. Therefore, the device in
this version provides a conditional stop, depending on rotor speed and flap angle. Excessive
upward motion of the blade is restrained by a second stop, called the flap stop. In a multibody
setting, the flap and droop stops for an articulated rotor can be modeled as a revolute joint
with backlash. For hingeless or bearingless configurations, two- or three-dimensional contact
elements could be used to model droop stops. In that case, friction and rolling in the stop are
expected to play an important role that could be properly modeled using appropriate contact
elements22, 23 . In the following, the modeling of joints in multibody systems is reviewed first,
then the formulation of backlash conditions in revolute joints is described in detail. It should
be noted that backlash conditions can be added to any joint provided that a suitable relative
distance has been defined.
Joints in multibody systems
A distinguishing feature of multibody systems is the presence of a number of joints that impose constraints on the relative motion of the various bodies of the system. Most joints used
for practical applications can be modeled in terms of the so called lower pairs 30 : the revolute,
prismatic, screw, cylindrical, planar and spherical joints, all depicted in Fig. 1. If two bodies
are rigidly connected to one another, their six relative motions, three displacements and
three rotations must vanish at the connection point. If one of the lower pair joints connects
the two bodies, one or more relative motions will be allowed. For instance, the revolute joint
allows the relative rotation of two bodies about a specific body-attached axis while the other
five relative motions are constrained. The constraint equations associated with this joint are
presented in the next section.
Modeling of revolute joints
Consider two bodies denoted with superscripts (.)k and (.)` , respectively, linked together by
a revolute joint, as depicted in Fig. 2. In the reference configuration, the revolute joint is
defined by coincident triads S0k = S0` , defined by three unit vectors ek10 = e`10 , ek20 = e`20 , and
ek30 = e`30 . In the deformed configuration, the orientations of the two bodies are defined by
two triads, S k (with unit vectors ek1 , ek2 , and ek3 ), and S ` (with unit vectors e`1 , e`2 , and e`3 ).
The kinematic constraints associated with a revolute joint imply the vanishing of the relative
displacement of the two bodies while the triads S k and S ` are allowed to rotate with respect
to each other in such a way that ek3 = e`3 . This condition implies the orthogonality of ek3 to
both e`1 and e`2 . These two kinematic constraints can be written as
`
C1 = ekT
3 e1 = 0
(1)
`
C2 = ekT
3 e2 = 0
(2)
and
Clearly, in the deformed configuration the origin of the triads is still coincident; this constraint is readily enforced within the framework of finite element formulations by Boolean
7
identification of the corresponding degrees of freedom. The relative rotation φ between the
two bodies is defined by adding a third constraint
`
kT `
C3 = (ekT
1 e1 ) sin φ + (e1 e2 ) cos φ = 0
(3)
The three constraints defined by Eqs. (1) to (3) are nonlinear, holonomic constraints that are
enforced by the addition of constraint potentials λi Ci , where λi are the Lagrange multipliers.
Details of the formulation of the constraint forces and their discretization can be found in
Ref.29 .
Revolute joints with backlash
A revolute joint with backlash is depicted in Fig. 3. The backlash condition will ensure
that the relative rotation φ, defined by Eq. (3), is less than the angle φ1 and greater than
the angle φ2 at all times during the simulation, i.e. φ2 ≤ φ ≤ φ1 ; φ1 and φ2 define the
angular locations of the stops. When the upper limit is reached, φ = φ1 , a unilateral contact
condition is activated. The physical contact takes place at a distance R1 from the rotation
axis of the revolute joint. The relative distance q1 between the contacting components of the
joint is readily found as
q1 = R1 (φ1 − φ)
(4)
When the lower limit is reached, φ = φ2 , a unilateral contact condition is similarly activated.
The relative distance q2 then becomes
q2 = R2 (φ2 − φ)
(5)
where R2 is the distance from the axis of rotation of the revolute joint.
Unilateral contact condition
If the stops are assumed to be perfectly rigid, the unilateral contact condition is expressed
by the inequality q ≥ 0, where the relative distance is given by Eq. (4) or (5). This inequality
constraint can be readily transformed into an equality constraint q − r2 = 0 through the
addition of a slack variable r. Hence, the unilateral contact condition is enforced as a
nonlinear holonomic constraint
C = q − r2 = 0
(6)
This constraint is enforced via the Lagrange multiplier technique. The corresponding forces
of constraint are
·
¸T ·
¸ ·
¸T
δq
λ
δq
(7)
δC λ =
=
Fc
δr
− λ 2r
δr
where λ is the Lagrange multiplier. To obtain unconditionally stable time integration
schemes29 for systems with contacts, these forces of constraint must be discretized so that
the work they perform vanishes exactly. The following discretization is proposed
¸
·
sλm
c
(8)
Fm =
− sλm 2rm
8
where s is a scaling factor for the Lagrange multiplier, λm the unknown mid-point value
of this multiplier, and rm = (rf + ri )/2. The subscripts (.)i and (.)f are used to indicate
the value of a quantity at the initial time ti and final time tf of a time step of size ∆t,
respectively. The work done by these discretized forces of constraint is easily computed as
W c = (Cf − Ci ) λm . Enforcing now the condition Cf = Ci = 0 will guarantee the vanishing
of the work done by the constraint forces. The Lagrange multiplier λm is readily identified
as the contact force.
Since the slack variable is not connected to any degree of freedom of the model, the
variation δr gives rise to the nonlinear equation −λm 2rm = 0 which possesses two solutions:
λm = 0 and rm = 0. The first solution, λm = 0, is associated with the no contact state and a
vanishing contact force. The second solution, rm = 0, indicates an active contact condition
and implies rf = −ri , which together with Cf = Ci = 0 results in qf = rf2 = ri2 = qi . In other
words, when the contact condition is active, a contact force will develop, i.e. λm 6= 0, and
the relative distance between the contacting bodies remains unchanged qf = qi . This clearly
corresponds to a discrete version of the principle of impulse and momentum.
For practical implementations, the introduction of the slack variable is not necessary. If
at the end of the time step qf ≥ 0, the unconstrained solution is accepted and the simulation
proceeds with the next time step. On the other hand, if qf < 0 at the end of the time step,
the time step is repeated with the additional constraint qf = qi and the Lagrange multiplier
associated with this constraint is the contact force.
The contact model
In general, the stops will present local deformations in a small region near the contact point.
In this case, the center of mass of the stops are allowed to approach each other closer than
what would be allowed for rigid stops. This quantity is defined as the approach and is
denoted a; following the convention used in the literature,20 a > 0 when penetration occurs.
For the same situation, q < 0, see Eqs. (4) and (5). When no penetration occurs, a = 0, by
definition, and q > 0. Combining the two situations leads to the contact condition q + a ≥ 0,
which implies q = −a when penetration occurs. Here again, this inequality condition is
transformed into an equality condition C = q + a − r2 = 0 by the addition of a slack variable
r. This nonlinear holonomic constraint can be dealt with in a manner identical to that
developed in the last section.
When the revolute joint hits a deformable stop, the contact forces must be computed
according to a suitable phenomenological law relating the magnitude of the approach to the
force of contact22, 31 . In a generic sense, the forces of contact can be separated into elastic
and dissipative components. As suggested in Ref. 21, a suitable expression for these forces
is
dV
dV dV d
+
f (ȧ) =
[1 + f d (ȧ)]
(9)
F contact = F elas + F diss =
da
da
da
where V is the potential of the elastic forces of contact, and f d (ȧ) accounts for energy dissipation during contact. In principle, any potential associated with the elastic forces can be
used; for example, a quadratic potential corresponds to a linear force-approach relationship,
or the potential corresponding to the Hertz problem. The particular form of the dissipative
force given in Eq. (9) allows the definition of a damping term that can be derived from the
9
sole knowledge of a scalar restitution coefficient, which is usually determined experimentally
or it is readily available in the literature for a wide range of materials and shapes21 .
Simulation of Rotor Blade-Droop Stop Impacts
The revolute joint with backlash was used for modeling blade droop stop impacts. A series
of non-rotating drop tests were reported in Ref. 9, and were here used for validating the
solution procedure. The tests were conducted using a Froude-scaled structural model of
the H-46 blade of length L = 3.3f t. The blade was rotated upward, given an initial flap
hinge angle, and held in place by an electromagnet located near this hinge. When the
electromagnet was shut off, the blade dropped under the effect of gravity. The motion of
the blade, its impact with the stop and the subsequent rebounds were recorded with a linear
motion potentiometer located at the flap hinge and an accelerometer mounted at the blade
tip. Furthermore, three strain gauges were attached at different stations along the span.
Details of the experimental setup, blade physical properties and a complete discussion of
the measured time histories are reported in Ref. 9 and references therein. A sketch of the
problem is given in Fig. 4, together with the corresponding multibody model employed for
the simulation. It consists of a beam modeling the flexible blade, a revolute joint with
backlash, and two rigid bodies to represent the rigid connections.
The problem is modeled by means of a revolute joint with backlash that represents
the flap hinge with droop stop, while the blade is modeled with five geometrically exact
cubic beam elements. The relative rotation at the revolute joint is denoted Φ, and the
beam vertical deflection Wtip . An automated time step refinement procedure29 was used
to accurately capture the impact events. Figure 5 shows the nondimensional, vertical tip
deflection Wtip /L of the beam, when it is dropped from an angle Φ = 9.7 deg. Figure 6
shows the hinge rotation Φ for the same test case. Finally, Fig. 7 gives the time history of
the strains at x1 /L = 0.40, when the beam is dropped from Φ = 5.2 deg. The strains are
initially positive, indicating that the beam is initially curved downward by the presence of
gravity. At about 0.04 sec. throughout the simulation, the beam reverses its curvature during
the fall. Considerable flexing in both directions follows the droop stop impact. Overall, the
numerical solution matches well the experimental values, and the full finite element numerical
simulations of Refs. 9, 32.
Aeroelastic Analysis of Shipboard Engage and
Disengage Operations
The multibody modeling technique described earlier was then used to simulate the engage
and disengage operations of a H-46 articulated rotor. The analysis was conducted in two
steps. First, a model validation was attempted, based on the available data for this rotor.
10
Then, the transient responses during engage and disengage operations were computed and
analyzed. In this effort, the aerodynamic model was based on an unsteady, two-dimensional
thin airfoil theory,33 and the dynamic inflow formulation developed by Peters34 . Note however that, given the low thrust generated by the rotor during the run-up and run-down
sequences, the use of an inflow model is not critical for obtaining accurate results. This was
verified by the fact that simulations with and without the inflow model yielded very similar
results.
H-46 model validation
The H-46 is a three-bladed tandem helicopter. The structural and aerodynamic properties
of the rotor can be found in Ref. 8 and references therein. Figure 8 depicts the multibody
model of the control linkages that was used for this study. The rotating and non-rotating
components of the swashplate are modeled with rigid bodies, connected by a revolute joint.
The lower swashplate is connected to a third rigid body through a universal joint. Driving
the relative rotations of the universal joint allows the swashplate to tilt in order to achieve the
required values of longitudinal and lateral cyclic controls. The collective setting is achieved by
prescribing the motion of this third rigid body along the shaft by means of a prismatic joint.
The upper swashplate is then connected to the rotor shaft through a scissors-like mechanism,
and controls the blade pitching motions through pitch-links. Each pitch-link is represented
by beam elements, in order to model the control system flexibility. It is connected to the
corresponding pitch-horn through a spherical joint and to the upper swashplate through
a universal joint to prevent pitch-link rotations about its own axis. Finally, the shaft is
modeled using beam elements. The location of the pitch-horn is taken from actual H-46
drawings, while the dimensions and topology of the other control linkages are based on
reasonable estimates. Figure 9 gives a graphical representation of the control linkages, as
obtained through the visualization module. Only one blade is shown, for clarity.
During the engage and disengage simulations, the control inputs were set to the following
values, termed standard control inputs: collective θ0 = 3 deg., longitudinal cyclic θs = 2.5
deg., lateral cyclic θc = 0.0693 deg. These values of the controls were obtained with the
proper actuations of the universal and prismatic joints that connect to the lower swashplate.
Figure 10 gives the pitch rotation as a function of blade azimuth as obtained through the
actuation of the control linkages, and compares it to the standard analytical expression
θ = θ0 + θs sin ψ + θc cos ψ. The two curves are nearly identical, as expected.
In this work, only the aft rotor system is modeled. The blades were meshed with 5 cubic
geometrically exact finite elements, while the droop and flap stops were modeled using the
revolute joint with backlash described previously. The stops are of the conditional type,
activated by centrifugal forces acting on counterweights. The droop and flap stop angles,
once engaged at rotor speed below 50% of the nominal value Ω0 = 27.61 rad/sec, are −0.54
and 1.5 deg, respectively.
Experimental data available for this rotor configuration include static tip deflections
under the blade weight and rotating natural frequencies. This data was used for a partial
validation of the structural and inertial characteristics of the model. As expected, static
tip deflections are in good agreement with Boeing average test data, within a 2% margin.
Figure 11 shows a fan plot of the first flap-torsion frequencies for the rotor considered in this
11
example, where quantities are nondimensionalized with respect to Ω0 . These modes are in
satisfactory agreement with the experimental data, and with those presented in Ref. 8.
Transient analysis of rotor engage operation
Next, complete rotor engage and disengage operations were simulated. A uniform gust
provides a downward velocity across the rotor disk, in addition to a lateral wind component.
The vertical wind velocity component was 10.35 kn, while the lateral one was 38.64 kn,
approaching from the starboard side of the aircraft. The situation is typical of a helicopter
operating in high wind conditions on a ship flight-deck. The run-up rotor speed profile
developed in Ref. 7 from experimental data was used in the analysis. The simulation was
conducted by first performing a static analysis, where the controls were brought to their
nominal values and gravity was applied to the structure. Then, a dynamic simulation was
restarted from the converged results of the static analysis.
Figure 12 shows a three dimensional view of the rotor multibody model at three different
time instants throughout the engage operation. Large flapping motions of the blades induced
by the gust blowing on the rotor disk are clearly noticeable even in this qualitative picture.
Figure 13 gives the out-of-plane blade tip deflection, positive up, for a complete run-up.
During the rotor engage operation, the maximum tip deflections are achieved during the
first 6 sec of the simulation. Then, as the rotor gains speed, the deflections decrease under
the effect of the inertial forces acting on the blade. Here and in the following figures, the thick
broken line shown in the lower part of the plot gives the time intervals when the revolute
joint stops are in contact. Because of the large downward gust blowing on the rotor disk,
only the droop stop is impacted by the blade, while the flap stop angle is never reached.
Figure 14 gives the time history of the flap hinge rotations. Multiple droop stop impacts
take place at the lowest rotor speeds, causing significant blade deflections and transfers of
energy from kinetic to strain energy. Furthermore, the intensity of the uniform vertical
gust component on the rotor disk causes large negative tip deflections even from the very
beginning of the analysis, when the blade angular velocity and resulting stiffening effect are
still small. After about 10 sec through the simulation, the droop stop is retracted and the
blade tip time history exhibits a smoother behavior. In order to simulate the conditional
nature of the particular droop stop mechanism used by this helicopter, the stop retraction
was modeled by changing the backlash angles of the flap revolute joint at the first time
instant of separation between the blade and its stops, once the activation rotor speed (50%
of Ω0 ) was reached.
The results are in reasonable agreement with the simulations of Refs. 6, 8. In particular,
the maximum negative tip deflections, that determine whether the blade will strike the
fuselage or not, are very similar, as well as the results at the higher speeds. Discrepancies
at the lower speeds might be due to the different aerodynamic models employed.
Transient analysis of rotor disengage operation
Figure 15 gives the blade tip deflection time history for a rotor run-down. Once again, the
rotor is subjected to a uniform gust of the same intensity and direction as in the previous
12
example. The speed profile is taken for this case from Ref. 6; it includes a freely spinning
phase after throttle cut, followed by a braking phase.
In order to simulate the disengage operation, a static analysis was conducted first in
which the controls were set to their nominal values and the gravitational and inertial loads
due to the constant rotor angular velocity were applied to the structure. Next, a dynamic
analysis was restarted from the static results to introduce the unsteady aerodynamic loads.
The rotor was allowed to stabilize at constant rotor speed, in order to reach a periodic
solution. Finally, the disengage sequence was initiated following the prescribed speed profile
and the droop stop was extended when its activation speed was reached. The stop extension
was modeled by changing the stop angles at that instant.
As shown in the figure, blade tip deflections steadily grow as the rotor speed decreases,
and maximum values are reached during the last 3 sec of the simulation. Multiple droop
stop impacts take place during the last 12 sec of the analysis causing the blades to bend
and vibrate; here again, contact events are depicted by thick broken lines in the lower part
of the plot. The time history of the flap hinge rotations is given in Fig. 16; the multiple
impacts after stop activation are clearly shown. For this case again, the results are in close
agreement with those of Refs. 6, 8.
Control linkage loads during impacts
The repeated contacts with the droop stops cause large bending of the blades, as described
in the previous section. Blade deflections can become excessive, to the point of striking the
fuselage. For less severe cases where such striking does not occur, significant over-loading
of the control linkages could still take place. The multibody formulation used in this work
readily allows the modeling of all control linkages, and the evaluation of the transient stress
they are subjected to during engage and disengage operations. In view of the multiple violent
impacts and subsequent large blade deflections observed in the two previous examples, the
loads experienced by the various components of the system during engage or disengage
operation in high winds could be significantly larger than during nominal flight conditions.
In order to demonstrate the potential of the proposed formulation in this respect, pitchlink loads were computed during the run-up and run-down sequences discussed earlier. Furthermore, the same engage and disengage operations were simulated for the case of vanishing
wind velocity, in order to provide “nominal” conditions for comparison. For the case of vanishing wind velocity, all other analysis parameters were identical to those used in the previous
simulations.
Figure 17 shows the axial forces at the pitch-link mid-point during the run-down sequence.
The uniform gust case is shown with a solid line, and contact events with the droop stop
are depicted by a thick broken line in the lower part of the plot. The vanishing gust velocity
case is shown with a dashed line. The blade impacts both droop and flap stops because there
is no downward wind component and the blade can then “fly” higher in this case. Contact
events are shown by two thick broken lines reported in the upper part of the plot; the higher
line depicts contact events with the flap stop while the lower one shows contacts with the
droop stop.
In both cases, large loads occur at the lowest rotor speeds due to the multiple contact
events. Both compression and tension loads can be observed. For vanishing gust velocity,
13
repeated contacts with both droop and flap stops generate high pitch-link loads in the first
seconds after stop activation. Since the blade repeatedly impacts both stops, this phase of the
run-down sequence seems to be more severe than in the presence of a uniform gust velocity,
where only the droop stop is impacted and lower over-loads are generated. However, the
situation is quite different during the last seconds of the simulation, before the rotor reaches
a complete stop. The downward gust blowing on the rotor disk presses the blade against
the droop stop, causing large downward bending of the blade, and in turn, large tension
loads in the pitch-link. In the absence of gust, far lower pitch-link load levels are observed.
Pitch-link overloads are observed in the presence of the downward gust.
The situation is markedly different for the rotor engage operation problem. Figure 18
shows the axial forces at the pitch-link midpoint during a time window between 2 and 10 sec
of the run-up sequence, for which the most violent blade tip oscillations where observed in
the previous analysis. The solid line corresponds to the uniform gust velocity case, while the
dashed line gives the “nominal”, vanishing wind velocity case. As in the previous figure, the
thick broken lines in the lower and upper parts of the plot indicate the contact events with
droop and flap stops. Here again, the pitch-links loads are far greater than those observed
at full rotor speed, owing to the large blade flapping motions and repeated impacts with
the stops. As for the run-down case, the vanishing gust velocity analysis predicts blade
impacts with both droop and flap stops. However, the uniform gust velocity case is far more
severe because of the large blade deflections and resulting compressive loads in the pitchlinks. Much larger loads are observed for this case. In terms of pitch-link loads, the run-up
sequence under a uniform gust velocity appears to be the most severe case.
Conclusions
This paper has described an analysis procedure for the intermittent contact problem that
arises in engage and disengage operations of helicopter rotors when the blades come in
intermittent contact with the droop or flap stops. A finite element based flexible multibody
dynamics formulation was used to solve the problem.
Rather than deriving specific procedures for this particular problem, existing tools developed for the modeling of contacts in multibody systems were directly applied. Backlash and
contact elements, a standard set of elements in multibody analysis packages, were shown to
provide the tools required for the analysis of this specific problem. In other words, this complex, intermittent contact problem in rotorcraft aeromechanics can be readily solved without
any new analytical development. There is no need to develop new equations of motion, no
need to validate the resulting model, and no need to interface it with an existing rotorcraft
aeromechanics tool. Carefully validated contact elements taken from an existing multibody
element library are assembled to form the complete model.
The flexibility and generality of the approach allows detailed analyses to be conducted
with great ease. For instance, the effect of contact forces on transient loads in the control
linkages were presented. Different properties or configurations of the control linkages could
be easily studied. For instance, the effects of lead-lag damper characteristics, pitch-horn con14
figuration and structural properties, or topology of the control linkages could be investigated
in a rational manner.
Finally, it should be noted that backlash and contact elements could also be used in the
modeling of other rotorcraft dynamics problems such as of the dynamic response of drive
trains.
Acknowledgments
The authors wish to acknowledge the help of G. Ghezzi and F. Soraci in the preparation of
the multibody models and the processing of the simulation results.
15
References
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18
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
The six lower pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revolute joint in the reference and deformed configurations. . . . . . . . . .
A revolute joint with backlash. The magnitude of the relative rotation φ is
limited by the stops so that φ2 < φ < φ1 . . . . . . . . . . . . . . . . . . . . .
Configuration of the drop test, and corresponding multibody model. . . . . .
Time history of the model blade tip deflection when dropped from a 9.7 deg
angle. Present solution: solid line; Refs. 9, 32: dashed line; experimental
values: ¤ symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time history of the model blade hinge rotations when dropped from a 9.7
deg angle. Present solution: solid line; Refs. 9, 32: dashed line; experimental
values: ¤ symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time history of the model blade strain at x/L = 0.40 when dropped from a 5.2
deg angle. Present solution: solid line; Refs. 9, 32: dashed line; experimental
values: ¤ symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multibody model of the rotor. . . . . . . . . . . . . . . . . . . . . . . . . . .
Graphical representation of the multibody model of the control linkages. One
single blade shown for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . .
Pitch rotation as a function of blade azimuthal position. Imposed through
the control linkages: solid line; analytical expression: dashed line. . . . . . .
H-46 fan plot. Present solution: solid line; Ref. 8: dashed line; experimental
values: ¤ symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Predicted configuration of the rotor system during an engage operation in a
uniform gust. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Out-of-plane blade tip response for a rotor engage operation in a uniform gust.
The thick broken line indicates the extent of the blade-stop contact events. .
Flap hinge rotation for a rotor engage operation in a uniform gust. . . . . . .
Blade tip response for a rotor disengage operation in a uniform gust. . . . .
Flap hinge rotation for a rotor disengage operation in a uniform gust. . . . .
Mid-point axial forces in the pitch-link for a rotor disengage operation. Uniform gust velocity: solid line; no gust velocity: dashed line. . . . . . . . . . .
Mid-point axial forces in the pitch-link for a rotor engage operation. Uniform
gust velocity: solid line; no gust velocity: dashed line. . . . . . . . . . . . . .
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
Cylindrical
Prismatic
Revolute
Spherical
Figure 1: The six lower pairs.
20
Screw
Planar
e 2l e2k
e = e
k
3
l
3
e 1l
e
u = u
R
l
k
l
e02k = e02
l
e = e
k
03
u0 = u0l
i3
R
k
1
k
Deformed
configuration k
i1
f
l
03
l
e01k = e01
R0k = R0l
Reference
configuration
i2
Figure 2: Revolute joint in the reference and deformed configurations.
21
k
2
e
e
l
2
e
Stop 1
f
1
R1
l
1
f
f
e
k
1
2
Stop 2
R2
Figure 3: A revolute joint with backlash. The magnitude of the relative rotation φ is limited
by the stops so that φ2 < φ < φ1 .
22
Electromagnet
Accelerometer
Potentiometer
Blade
Strain gages
Droop stop
i2
Revolute joint
with backlash
F
i1
Rigid body
Beam
Revolute joint
Ground clamp
Figure 4: Configuration of the drop test, and corresponding multibody model.
23
0.15
TIP DEFLECTION [W tip/L]
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
TIME [sec]
Figure 5: Time history of the model blade tip deflection when dropped from a 9.7 deg angle.
Present solution: solid line; Refs. 9, 32: dashed line; experimental values: ¤ symbols.
24
10
HINGE ANGLE [deg]
8
6
4
2
0
−2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
TIME [sec]
Figure 6: Time history of the model blade hinge rotations when dropped from a 9.7 deg
angle. Present solution: solid line; Refs. 9, 32: dashed line; experimental values: ¤ symbols.
25
800
600
−6
STRAIN (x10 )
400
200
0
−200
−400
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
TIME [sec]
Figure 7: Time history of the model blade strain at x/L = 0.40 when dropped from a 5.2 deg
angle. Present solution: solid line; Refs. 9, 32: dashed line; experimental values: ¤ symbols.
26
Flap, lag, and
pitch hinges
Hub
Blade
Pitchhorn
Scissors
Pitchlink
Swash-plate:
Rotating
Rigid body
Beam
Revolute joint
Spherical joint
Universal joint
Prismatic joint
Ground clamp
Non-rotating
Shaft
Figure 8: Multibody model of the rotor.
27
Hub
Pitchhorn
Pitchlink
Scissor
Upper and lower
swash-plates
Blade
Shaft
Figure 9: Graphical representation of the multibody model of the control linkages. One
single blade shown for clarity.
28
6
PITCH ROTATIONS [deg]
5
4
3
2
1
0
0
50
100
150
200
250
300
350
BLADE AZIMUTH [deg]
Figure 10: Pitch rotation as a function of blade azimuthal position. Imposed through the
control linkages: solid line; analytical expression: dashed line.
29
10
NONDIMENSIONAL FREQUENCY
9
8
4 th Flap Mode
7
6
5
4 1 st Torsion Mode
3 rd Flap Mode
3
2
2 nd Flap Mode
1 st Flap Mode
1
0
0
0.2
0.4
0.6
0.8
1
1.2
NONDIMENSIONAL ROTATIONAL SPEED
1.4
Figure 11: H-46 fan plot. Present solution: solid line; Ref. 8: dashed line; experimental
values: ¤ symbols.
30
Hub
T=0.46 sec.
T=4.24 sec.
T=6.24 sec.
Figure 12: Predicted configuration of the rotor system during an engage operation in a
uniform gust.
31
0.4
0.2
BLADE TIP DISPLACEMENT [m]
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−1.4
−1.6
0
2
4
6
8
10
12
14
TIME [sec]
Figure 13: Out-of-plane blade tip response for a rotor engage operation in a uniform gust.
The thick broken line indicates the extent of the blade-stop contact events.
32
2
FLAP ROTATION [deg]
1
0
−1
−2
−3
−4
0
2
4
6
8
10
12
TIME [sec]
Figure 14: Flap hinge rotation for a rotor engage operation in a uniform gust.
33
14
0.2
BLADE TIP DISPLACEMENTS [m]
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−15
−10
−5
TIME TO STOP [sec]
Figure 15: Blade tip response for a rotor disengage operation in a uniform gust.
34
0
1.5
1
FLAP ROTATION [deg]
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−15
−10
−5
TIME TO STOP [sec]
Figure 16: Flap hinge rotation for a rotor disengage operation in a uniform gust.
35
0
2500
2000
PITCHLINK AXIAL FORCE [N]
1500
1000
500
0
−500
−1000
−1500
−2000
−2500
−15
−10
−5
0
TIME TO STOP [sec]
Figure 17: Mid-point axial forces in the pitch-link for a rotor disengage operation. Uniform
gust velocity: solid line; no gust velocity: dashed line.
36
2000
1500
PITCHLINK AXIAL FORCE [N]
1000
500
0
−500
−1000
−1500
−2000
−2500
−3000
−3500
2
3
4
5
6
7
8
9
10
TIME [sec]
Figure 18: Mid-point axial forces in the pitch-link for a rotor engage operation. Uniform
gust velocity: solid line; no gust velocity: dashed line.
37