Chapter 7 Flashcards
Aircraft dynamic stability focuses on the _____ of aircraft motion after the aircraft is disturbed from an equilibrium or trim condition.
time history
Yeckout Ch 7, Pg 331
A first-order differential equation typically has what type of response?
An exponential response.
Yeckout Ch 7, Pg 331
A second-order differential equation typically has what type of response?
Oscillatory response.
Yeckout Ch 7, Pg 331
What is the equation for the damping force provided by a dampener?
The damping constant multiplied by the velocity.
F=CV
Yeckout Ch 7, Pg 332
How many aircraft dynamic modes are there? What are the longitudinal modes? What are the lateral modes?
5 total.
Long: Short period and Phugoid
Lat: Roll, Spiral, and Dutch Roll
Yeckout Ch 7, Pg 334
The homogeneous solution to a differential equation is often called_______.
The transient solution.
Yeckout Ch 7, Pg 335
The solution to a non-homogeneous differential equation is commonly called ______.
The particular or steady-state solution.
Yeckout Ch 7, Pg 336
Define the time constant conceptually.
It is a measure of the time that it takes to achieve 63.2% of the steady-state value.
Yeckout Ch 7, Pg 338
For a first-order system what are the equations for time to half amplitude, and time to double amplitude?
T_1/2 = ln(2)*tau T_2 = ln(2)/tau
Yeckout Ch 7, Pg 339
For a second-order system, if there are two unequal real roots the system is referred to as _______.
Overdamped.
Yeckout Ch 7, Pg 339
If the roots of a second-order system are real and identical, the system is referred to as ______.
Critically Damped
Yeckout Ch 7, Pg 340
For a second order system that has two complex conjugate roots the system is referred to as _______.
Underdamped
Yeckout Ch 7, Pg 340
For an underdamped system, the real part of the root determines what behavior of the system in terms of the time response behavior?
The exponential decay (damping portion) of the time response.
Yeckout Ch 7, Pg 340
For an underdamped system, the complex part of the root determines what behavior of the system with respect to the time response?
The complex part of the root helps to determine the frequency of the oscillation.
Yeckout Ch 7, Pg 340
For a second-order, case three response, what will be the output range of the damping ratio? What will be the output range for a stable system in terms of the damping ratio?
-1 to 1
0 to 1
Yeckout Ch 7, Pg 341
Define the natural frequency (conceptually).
Their frequency in (radians per second) that the system would oscillate if there were no damping present.
Ture or False
The natural frequency is the highest frequency that the system is capable of, but it is not the frequency that the system actually oscillates at if damping is present.
True
Yeckout Ch 7, Pg 341
Conceptually define the damped frequency.
The damped frequency represents the frequency in (radians per second) that the system actually oscillates at with damping present.
Yeckout Ch 7, Pg 341
For a second-order system, how is the time constant related to the damping ratio and the natural frequency?
The time constant for a second-order system is related to one over the product of the damping ratio and the natural frequency.
tau = 1/(wn*zeta)
Yeckout Ch 7, Pg 342
Define the period of oscillation for a second-order system both conceptually and in equation form.
The period of oscillation for a second-order system is the time it takes between consecutive peaks of an oscillation.
T = 2pi/wd
Yeckout Ch 7, Pg 342
Explain how the stability of a system can be determined from the equation of motion.
Determine the characteristic equation and if the real part of the root(s) is/are negative the system is stable.
Yeckout Ch 7, Pg 346
How does stability relate to the complex plane?
If the roots occur to the left of the imaginary axis (the left half of the complex plane) the system is stable.
Yeckout Ch 7, Pg 346
How is neutral stability represented on the complex plane? What is the damping ratio?
If the roots are on the imaginary access the system is neutrally stable and the damping ratio is zero.
Yeckout Ch 7, Pg 346
How is instability represented in the complex plane? What is the damping ratio?
If the roots are to the right of the imaginary axis the system is unstable and the damping factor is less than zero.
Yeckout Ch 7, Pg 346
What is the conceptual definition of a transfer function?
A transfer function is defined as a ratio of laplace transforms of output over input.
Yeckout Ch 7, Pg 352
How is the characteristic equation obtained from an arbitrary transfer function?
The characteristic equation of the transfer function is obtained by setting the polynomial in the denominator of the transfer function equal to zero.
Yeckout Ch 7, Pg 353
With respect to transfer functions, what determines the dynamic stability characteristics of the response?
The sign on the real parts of the poles (roots of the equation in the denominator).
Note: The denominator of the transfer function set equal to zero is the characteristic equation whose roots (called poles) determine the dynamic stability characteristics of the response.
Yeckout Ch 7, Pg 358
True or False
All airplanes have longitudinal Pugoid and short period modes.
True
Yeckout Ch 7, Pg 358
What parameters identify the short period mode?
Complex conjugate roots with a moderate to high damping ratio and relatively high (as compared to the phugoid) natural frequency and damping frequency.
The time constant and period are short compared to the phugoid mode.
Yeckout Ch 7, Pg 358
In terms of the short period mode, how do variations in the angle of attack, pitch attitude, and x-axis velocity change over time?
The x-axis velocity (u) remains constant while the angle of attack and pitch attitude vary with time.
Yeckout Ch 7, Pg 358
The short period mode is considered _________.
Stable.
Yeckout Ch 7, Pg 359
What parameters identify the phugoid mode?
Complex conjugate roots with a relatively low damping ratio and natural/damped frequencies as compared to the short period mode. The time constant and period are large compared to the short period mode.
Yeckout Ch 7, Pg 359
Describe what a pilot would experience while testing the phugoid mode.
The response is determined by complex conjugate roots. The response is usually oscillatory with significant variations in pitch attitude and airspeed while the angle of attack remains relatively constant.
Yeckout Ch 7, Pg 359
True or False
The period for the phugoid mode is typically quite long, usually lying somewhere between 30 and 120 seconds.
True
Yeckout Ch 7, Pg 359
What is the input for the longitudinal linearized transfer functions with respect to the various aircraft control surfaces?
Elevator deflection: delta_e
Yeckout Ch 7, Pg 356
If the short period and phugoid roots are plotted on the complex plane how could you tell the difference between the two?
The short period roots are further out from the origin and have higher damping ratios than the phugoid roots.
Yeckout Ch 7, Pg 367
For the lateral directional linearized equations of motion, what are the two inputs with respect to the various aircraft control surfaces? What are the primary response variables?
Aileron and rudder deflection (delta_a, delta_r)
Sideslip (beta), bank (phi), and yaw (psi)
Yeckout Ch 7, Pg 369 & 370
What are the three modes for lateral directional motion?
The Dutch roll, the roll mode, and the spiral mode.
Yeckout Ch 7, Pg 369
Describe the lateral directional Dutch role mode.
The Dutch roll mode is a second-order response with complex conjugate roots. It is usually characterized by concurrent oscillations in the three lateral directional motion variables (beta, phi, and psi).
Yeckout Ch 7, Pg 372
Describe the lateral directional role mode.
The role mode has a real root in a first-order, non-oscillatory response that involves almost a pure rolling motion about the X stability axis. It’s time constant is relatively short.
Yeckout Ch 7, Pg 372
With respect to the lateral directional roll mode, it is usually ____ at low and moderate angles of attack but can be _____ at high angles of attack.
stable
unstable
Yeckout Ch 7, Pg 372
How can the lateral directional role mode be excited by a test pilot (excluding unplanned disturbances in flight)?
Aileron input.
Yeckout Ch 7, Pg 372
Describe the lateral directional spiral mode.
The spiral mode is a first-order response with a real root that involves a relatively slow roll and yawing motion of the aircraft.
Yeckout Ch 7, Pg 373
True or False
The lateral directional spiral mode is always unstable.
False.
It can be stable or unstable.
Yeckout Ch 7, Pg
How is the spiral usually initiated? What does it look like from the pilot’s perspective if it’s unstable/stable?
The spiral is usually initiated by a displacement in roll angle and appears as a descending turn with increasing roll angle if unstable. If the spiral is stable the aircraft simply returns to wings level after a roll angle displacement.
Yeckout Ch 7, Pg 373
What are the primary motion variables during a spiral?
phi and psi (beta is usually close to zero)
Yeckout Ch 7, Pg 373
What stability derivatives are involved in terms of the stability of the spiral mode?
Large C_l_beta –> Stable
Large C_n_beta –> Unstable
Yeckout Ch 7, Pg 373
True or False
Spiral stability is usually compromised for good Dutch roll characteristics that are typically achieved with relatively high directional stability and relatively low lateral stability.
True.
Yeckout Ch 7, Pg 373
The roll mode will have a _______ time constant that the sprial mode.
Smaller
Yeckout Ch 7, Pg 374
Consider the following roots and assume they are taken from a lateral directional transfer function. Which roots are associated with which lateral mode? What are the time constants for the roll and spiral mode.
s1,2 = -0.1135 +/- 2.4056i
s3 = -1.2700 + 0.0000i
s4 = -0.0037 + 0.0000i
s_12= -0.1135 +/- 2.4056i --> Dutch Roll s3 = -1.2700 + 0.0000i --> Roll mode s4 = -0.0037 + 0.0000i --> Spiral Mode
tau_ roll = 1/1.27 ~ 0.7874 seconds
tau_spiral = 1/0.0037 ~ 270.2703 seconds
Yeckout Ch 7, Pg 374
Consider the following roots. Is the spiral mode stable? Is the dutch roll mode stable?
s1,2 = 0.1135 +/- 2.4056i
s3 = -1.2700 + 0.0000i
s4 = -0.0037 + 0.0000i
Yes. The root for the spiral mode is s3 = -1.2700 + 0.0000i. Since the real part of the root is negative, that mode is stable.
No. The roots for the dutch roll mode are s1 and s2.
s1,2 = 0.1135 +/- 2.4056i
Since the real part of the root is positive, the dutch roll mode is unstable.
Yeckout Ch 7, Pg 375
Consider the following roots. Can the time constant for the dutch roll mode be determined directly from the roots? What about the time constants for the spiral and roll modes?
s1,2 = -0.1135 +/- 2.4056i
s3 = -1.2700 + 0.0000i
s4 = -0.0037 + 0.0000i
No. The equation for the time constant for the dutch roll mode involves complex roots s1 and s2. The inverse of the natural frequency, multiplied by the inverse of the damping factor gives the time constant for the dutch roll mode.
Yes. The inverse of the absolute value of the roots gives the time constants for the spiral and roll modes.
Yeckout Ch 7, Pg 376
What is the simplest of the five dynamic aircraft modes?
The roll mode.
Yeckout Ch 7, Pg 375
True or False
The higher the roll damping, the smaller the roll time constant.
True.
Yeckout Ch 7, Pg 376
The spiral mode will have a ______ real root, while the roll mode will have a ______ real root.
small
larger (bigger than the spiral root)
Yeckout Ch 7, Pg 382
