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:

$$u = u_e + \Delta u$$

(The suffix $e$ stands for equilibrium.)

2.

Linearize the aerodynamic forces and moments, by writing them as first order Taylor-series expansions in state space. For example:

$$X \approx X_e + \left(\frac{\partial X}{\partial u}\right)_e\... ... \left(\frac{\partial X}{\partial w}\right)_e\Delta w + \cdots$$

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:

$$\left(\frac{\partial X}{\partial u}\right)_e \approx \frac{X(u_e+\delta u)-X(u_e-\delta u)}{2 \delta u}$$

3.

Linearize the Coriolis forces (terms such as $qw$) like this:

$$qw = (q_e+\Delta q)(w_e + \Delta w) \approx q_ew_e + q_e\Delta w + w_e\Delta q$$

4.

Assume that the Euler angles do not change much from their trim value. Wherever $\Phi$ and $\Theta$ appear in the equations of motion, replace with $\Phi_e$ and $\Theta_e$.

Once the equations are linearized, it is convenient to work in vector notation. Define the state vector as

$$\vec x = [\,u\;v\;w\;p\;q\;r\;\Phi\;\Theta\,]^T$$

and write it as a trim value plus a perturbation value:

$$\vec x = {\vec x}_e + \Delta\vec x$$

Then, if $A$ 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

$$\dot{\vec x} = A\,\Delta\vec x$$ (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 $\vec f$ is the function, the full (nonlinear) equations of motion can be written

$$\dot{\vec x} = \vec f(\vec x)$$

The linearization of vector function $\vec f$ is simply

$$\vec f(\vec x) \approx \vec f({\vec x}_e) + J\,\Delta\vec x$$

where $J$ is the Jacobian matrix of $\vec f$. 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:

$$\dot u \approx \left(\frac{\partial \dot u}{\partial u}\right... ...t(\frac{\partial \dot u}{\partial w}\right)_e\Delta w + \cdots$$

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 $A$ with various expressions. Programming a loop requires much less programmer time. On the other hand, the traditional method is more efficient computationally.

Eigenvalue Analysis

The solution to Equation 10 is

$$\vec x = \sum_{i=1}^8 C_i \vec\phi_i e^{\lambda_i t}$$

where $C_i$ is the $i$th arbitrary constant, $\vec\phi_i$ is the $i$th eigenvector, and $\lambda_i$ is the $i$th eigenvalue.

The stability characteristics of the solution are determined by the eigenvalues; they hold information 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. Figures 1-3 present 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 $p$, but a very small value for the other variables, is a roll mode because most of the motion 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 $-60$, a pitch mode between $-60$ and $-10$. Both modes are heavily damped. (In fact, they are so heavily damped I doubt their correctness.)

  

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.

  

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.