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8/12/2019 Helicopter Stability
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Introduction
Stability is one of the primary concerns in helicopter handling qualities. Helicopters must
have a level of stability to feel comfortable and easy to fly to the pilot.
This report aims to do describe a straightforward analysis of the stability of a helicopter. The
method is presented here is a mathematical black box analysis, that will determine stability
characteristics, but will fail to yield insight on how various helicopter parameters affect
stability.
Dynamic Modeling
Stability is a dynamics problem, and so we must begin by modeling the dynamics of the
helicopter. We do this using Newtonian mechanics. The equations of motion relate the
helicopter motion to the forces exerted on it. Thus, dynamic modeling boils down todetermination of the forces and moments.
Several assumptions can be made to simplify the analysis. The major assumptions are stated
here.
The helicopter is a rigid body, symmetric about its longitudinal plane. The transients of the rotor flapping motion are insignificant; when anything affecting the
blade flapping changes, the blades are assumed to react immediately. This is the quasi-
steady assumption.
The induced inflow approximately varies linearly across the disk. In particular, the inducedinflow is a linear function of distance downwind of the upwind tip.
For helicopters, the tip speed is fixed. For autogyros, the tip speed is set to zero the rotortorque.
Blade stall is ignored, as are tip effects and the effects of the reverse flow region. Blades are teetering (i.e. no hinge offset). This is perhaps not a good assumption, but the
unbelievable simplification it allowed made me willing to accept it.
Rigid-Body Equations of Motion
The derivation of the rigid body equations of motion are beyond the scope of this report. The
basis of the derivation is Newton's Second Law, in linear form ( ) and angular form
( ). Newton's law is applied to the force and moment resolved into body axes, taking
Coriolis and gravity forces into consideration.
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Simplifying, using the symmetry assumption, and solving for the accelerations, yield:
While the Coriolis and aerodynamic forces in Equations1-6are functions of the velocities
and angular velocities ( , , , , , ), the gravity terms are functions of two Euler
angles, and . Thus, the system is not yet closed.
Equations7and8relate the Euler angle rates to the angular velocities.
These make and state variables, forming a closed system of eight equations with eight
state variables.
In the above equations, only the aerodynamic forces and moments ( , , , , , )
are unknown. They are functions of the state variables, as well as the control variables. The
next two subsections detail their calculation.
Rotor Forces and Moments
The rotor forces are functions of the following:
State variables: Rotor collective and cyclic pitch: Induced inflow: Flap angles: External downwash
Only the state variables and the rotor collective and cyclic pitch are known. The others must be
calculated.
Rotor Axes
The calculation of rotor forces take place in shaft axes, where the -axis is collinear with the
shaft of the rotor. Although control axes use (much) simpler equations for in-plane forces, the
control axes are not fixed with respect to the helicopter fuselage. This introduces
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bookkeeping problems. The shaft axes are fixed, and so a fixed transformation can be used to
transform vectors from body axes to shaft axes.
The calculation of forces actually occurs in shaft-wind axes, which is a rotation of the shaft
axes so that the -axis points directly into the wind. In shaft-wind axes, the -component ofvelocity is zero by definition, which simplifies the equations somewhat. Variables in shaft-
wind axes are denoted by a subscript .
For example, to transform the angular velocity components from body axes to shaft-wind
axes, one would use the relation
where is the fixed body-to-shaft rotation matrix and is the angle between the shaft and
shaft-wind -axes. Velocity, angular velocity, cyclic pitch, and flapping angles must all be
transformed to shaft-wind axes for rotor calculations. The force and moment determined by the
calculation is transformed back to body axes afterwards by the inverse transformation.
Induced Flow
Calculation of the induced inflow requires the thrust of the rotor. Unfortunately, thrust is the
ultimate goal of this calculation, so it is unknown. An iterative calculation is required. In
Section3,we will see that the trim solution also requires iteration. Therefore, we calculate
based on the expected thrust at the trim condition, which, when the trim iteration converges,
will be the actual thrust produced by the rotor. For the main rotor, this is the helicopter's gross
weight. For the tail rotor, it is the amount of thrust needed to offset the main rotor torque.
Once the expected thrust, , is known, the induced downwash can be calculated using
Equation9.
(1)
http://www.aerojockey.com/papers/helicopter/node3.html#sec:trimhttp://www.aerojockey.com/papers/helicopter/node3.html#sec:trimhttp://www.aerojockey.com/papers/helicopter/node3.html#sec:trimhttp://www.aerojockey.com/papers/helicopter/node2.html#eq:downwashhttp://www.aerojockey.com/papers/helicopter/node2.html#eq:downwashhttp://www.aerojockey.com/papers/helicopter/node2.html#eq:downwashhttp://www.aerojockey.com/papers/helicopter/node2.html#eq:downwashhttp://www.aerojockey.com/papers/helicopter/node3.html#sec:trim8/12/2019 Helicopter Stability
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This relation is not accurate for airspeeds near ; furthermore, the function is not differentiable at
(which can possibly cause difficulties in the trim iteration). The relation can be improved by
``fairing'' the function near using empirical data.
Flapping
The quasi-steady assumption means that flapping transients are ignored. Thus flapping is a
function of the rotor velocity and angular velocity, the induced inflow, and the collective and
cyclic pitches.
To facilitate calculation of the flapping, the typical dimensionless ratios are defined:
There three flapping angles are the coefficients of a first harmonic expansion of the flapping
as a function of azimuth:
Note that the sign convention for and is the opposite as used in Glessow and Myers.
(Padfield was inconsistent about the flapping sign convention, leading to several implementation
headaches.)
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The relations for flapping are simplified (and corrected) versions of those given by Padfield(page 107):
Force and Moment
Once the flapping angles have been determined, the force and moment can be calculated. The
calculation of force is a straightforward, albeit long, calculation. The equations for force are
taken directly from Padfield, pages 110-111, and the moment equations come from pages
114-115. They are much too long to reproduce here.
Fuselage and Empennage Forces and Moments
Fuselage forces are difficult to calculate. Typically, empirical data is used. When less
accuracy is needed, such as for student projects, there are rough approximations based on
empirical testing in most aircraft aerodynamics textbooks.
For this report, I used rough estimation formulas from McCormick. The fuselage drag
coefficient (based on frontal area ) is a function of slenderness ( ) and for this
calculation is 0.0858. This is multiplied by to get the drag.
The other forces and moments on the fuselage are usually not great, as the fuselage is not
designed to produce these forces. For this report, I chose to neglect them. This is usually not a
good assumption at all, especially for pitching and yawing moments.
The empennages are lifting surfaces, aerodynamically optimized to produce a force
perpendicular to the incoming wind. This lift force is proportional to the angle of attack (or
angle of sideslip for the vertical stabilizer).
The lift coefficient of the horizontal stabilizer is given by
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and the coefficient of the vertical stabilizer by
where and are the lift curve slopes of the horizontal and vertical stabilizers, respectively.
The that appears in the angle of attack is the rotor downwash on the tail. The drag coefficient of
either stabilizer is approximated by
where is the aspect ratio of the empennage, and is an efficiency factor that accounts for
profile and induced drag effects at higher lift. The lift and drag forces are rotated into the body axes,
and converted to dimensional form by multiplying by , where is the planform area of the
empennage.
Moments produced by the empennages themselves are small and negligible; however, the
empennages are at long moment arms from the center of gravity. For example, the pitching
moment of the horizontal stabilizer is given by , where is the momentarm.
Trim
Trim is generally defined a condition in which none of the state variables change with time
(that is, all of the state variable rates are zero). This does not, however, preclude accelerating
conditions; the velocities must remain fixed in body axes, but will change direction in an
inertial reference frame if the helicopter rotates. Trim conditions include straight and level
flight, steady turns, and spins.
Determining the trim condition is an iterative process, as exact closed form solutions are not
possible. The four helicopter controls, the collective, the longitudinal and lateral cyclic, andthe tail-rotor cyclic, are adjusted along with some of the state variables until the state variable
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rates ( , , etc.) go to zero. Because there are 12 variables to adjust (eight state variables and
four control variables), while there are only eight state variable rates to zero, it follows that
some of the state variables are determined from the prescribed trim condition rather than
adjusted at each iteration.
For the calculations in this report, we prescribe a particular airspeed and horizontal,
nonturning flight. The angular velocities ( , , ) are zero. At a particular airspeed, the
helicopter will trim to a particular orientation, but we do not know this orientation
beforehand. Therefore, during the iteration, we will adjust Euler angles and . Given the
current guesses for and , the velocity components are given by [from Padfield, 277]:
Now, looking at Equations7and8, and are zero because , , and are zero. This
leaves , , , , , and to be zeroed.
There are six unknown variables ( , , , , , and ), and six variables to zero.
Solving this problem is the same as solving a system of nonlinear equations. I used Broyden's
secant method to do this [see Dennis and Schnabel], which is an secant interpolation method
for multivariable problems.
Stability Analysis
In stability analysis, we are interested in the helicopter's behavior near its trim condition. We
want to know whether small disturbances tend to converge back to the trim state, or diverge
and grow larger. We do this by linearizing the equations of motion at the trim condition. The
solution to a linear set of equations is an exponential. One can tell whether a particular mode
converges or diverges simply by looking at the exponents (eigenvalues): a positive real part
indicates divergence, while a negative real part indicates convergence.
Linearization of Equations of Motion
The traditional way to linearize the equations of motion is as follows:
1.
Write the state variables as a trim value plus a perturbation value:
(The suffix stands for equilibrium.)
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2.
Linearize the aerodynamic forces and moments, by writing them as first order Taylor-series
expansions in state space. For example:
The stability derivatives are constant and evaluated at the trim condition. Due to the
complex nature of the force equations, the stability derivatives are calculated using a central
difference:
3.
Linearize the Coriolis forces (terms such as ) like this:
4.
Assume that the Euler angles do not change much from their trim value. Wherever and
appear in the equations of motion, replace with and .
Once the equations are linearized, it is convenient to work in vector notation. Define the statevector as
and write it as a trim value plus a perturbation value:
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Then, if is a linear coefficient matrix for the linearized equations of motion, the homogeneous
equations (that is, the equations with no control inputs) can be written in vector form as
(2)
For this report, I chose to determine the values of the coefficient matrix in a nontraditional
way. The equations of motion can be thought of a multivalued function of the state variables,
returning the state variable rates. If is the function, the full (nonlinear) equations of motioncan be written
The linearization of vector function is simply
where is the Jacobian matrix of . The components of the Jacobian are calculated with finite
difference approximations. (The analogous scalar concept is to expand the state variable rates as
first-order Taylor series, using the equations of motion. For example:
The derivatives in the above formulation are the components of the Jacobian.)
There are several advantages to the second approach; the most important is the savings in
human time. Because it is vectorized, the Jacobian matrix can be generated with a double
loop. The traditional linearization method requires calculation of stability derivatives first,
and then filling in the components of the matrix with various expressions. Programming a
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loop requires much less programmer time. On the other hand, the traditional method is more
efficient computationally.
Eigenvalue Analysis
The solution to Equation10is
where is the th arbitrary constant, is the th eigenvector, and is the th eigenvalue.
The stability characteristics of the solution are determined by the eigenvalues; they holdinformation about the modes such as damping ratio, time constants, and frequency of
oscillation. Thus, the first look at stability characteristics looks at the eigenvalues.
The set of eigenvalues obtained while varying one parameter over a specific range is a root
locus. For the helicopter, a useful root locus plots the eigenvalues at different airspeed.
Figures1-3present the root loci of the example helicopter given in Prouty.
Eigenvectors are used to determine the nature of the motion in a particular mode. This
enables one to determine what the modes in a root locus are. An eigenvector that has a large
value for , but a very small value for the other variables, is a roll mode because most of themotion is in rolling.
Figure:Root locus plot for example helicopter given in Prouty, plotting eigenvalues at
airspeeds from 0 to 200 ft/s. There are two mode loci spanning the real axis: the roll mode at
values less than , a pitch mode between and . Both modes are heavily
damped. (In fact, they are so heavily damped I doubt their correctness.)
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Figure 2:Zoomed root locus plot of the example helicopter. This shows a curious mixture of
modes. The bow shaped mode locus on the left is a short-period pitch oscillation mode, and
occurs in forward flight but not hover. At low speeds, the mode locus intersects the real axis
and becomes two oscillatory modes: the pitch mode and a vertical motion mode. On the
right, there is another oscillatory mode, which is the Dutch roll mode. It only seems to exist
at high speeds; at low speeds the mode divides into two non-oscillatory modes. (This is very
unexpected and yet another reason to doubt my results.) Also on the real axis, intermeshed
with the other modes, is the spiral mode. It is not heavily damped, but is stable.
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Figure 3:The root locus plot zoomed even further. This plot shows the phugoid mode. It is
an unstable mode, because its real part is positive. At hover, the mode locus is at the right
side of the plot, and moves to the left at higher speeds. Even at high speed, the mode is still
slightly unstable.
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