A4.1 Introduction
The equations which define the longitudinal short-period motion and the directional oscillation. Eqns (4.84) and are of the
where the undamped natural frequency of the motion is:
and the damping ratio is:
represents an angular control input such as or -
The damping coefficient. R, the damped natural frequency, J. and the forcing function. are defined at the relevant equations. The solution of Eqn. has two parts, namely the Complementary Function and the Particular Integral.
The form of the equation is typical of systems which may be represented by a damped spring-mass mechanism. In the simple case of a spring supporting a the variable, X, is the deflection. The coefficient of the term is effectively the mass it has been normalized in Eqn. The coefficient of the X term is the spring stiffness, say If there is no damping the
x
term is zero and the natural frequency of vibration is then simply = The damping term is out of phase relative to the stiffness term.A4.2 The Complementary Function
The Complementary Function part of the solution of the equation represents the transient motion of the system. It is described by placing the right-hand side of
Rigid airframe dynamics
Eqn. (A4.1) to zero,
and applying the differential operator:
+
Providing that they are not equal the general roots of this equation are:
X
-
The physical meaning depends upon the value of the damping ratio,
If the roots are
-
and the ComplementaryFunction can be written as:
where A and are coefficients determined by system boundary conditions.
When is negative this represents an oscillation of increasing magnitude. but if is positive the oscillation decays.
For = 0, the solution becomes:
When = the roots are equal and in this special case the solution is:
and the system is said to be critically damped.
When
>
the system is over-damped:The system is convergent in this case, while if - the form of the equation is similar but the system is divergent.
loading a n d structural layout
An alternative of expressing the Complementary Function for I is:
where and are coefficients determined by the boundary conditions, and are equivalent to A and B of Eqns to which are and
respectively. Although mathematically identical to (A4.8) is more convenient to use in some circumstances and of value in understanding the physical interpretation of the transient motion.
A4.3 The Particular Integral
The Particular Integral represents the steady state of the system and is defined as a given solution to the and right-hand sides of Eqn. It is thus dependent upon the expression of the forcing function
A4.3.1
Exponential forcing function the equation of motion in the form:where
gives the maximum value of the forcing function
K the measure of rate of application of the control For this case the Particular Integral is of the form:
- +
k')]This may be verified differentiation and substitution into Eqn. The complete solution then becomes:
The and B areevaluated by assuming initial rest conditions. At time = 0, X = 0 and
x
= and then:and:
airframe dynamics
Note the in brackets tends to unity as k tends to infinity.
Using this and differentiating Eqn. (A4.10):
and for
x
= at = 0:where A is given by:
Note that k tends to infinity the term in the brackets tends to Finally:
A4.3.2
Step functionThis is a special case of the exponential forcing function where the rate of application, k, tends to infinity so that tends to zero. In this case Eqn. becomes:
is directly proportional to it is convenient to examine the implications of (A4.12) in of a unit step input, that is by assuming = 1. See Fig. A4.1.
loading and layout
Fig. A4.1 Response of a second-order linear a step
For 1 , the special critically damped case:
X 1 -
+
At
X
= and A = 1Rigid airframe dynamics
wand so:
For - 1:
A4.3.3 Sinusoidal forcing function In this case the basic equation is of the
where is the forcing frequency.
One form of the Particular Integral is:
where and the magnitude of the forcing function.
The corresponding complete solution is:
If the forcing function is assumed to be applied to a static condition at then X is zero at this time, from where:
The specification of a second boundary condition is less since
x
is not necessarily zero at time However,x
will be zero when X is a maximum but due to the interaction between the two frequencies, wand the time at which X i s a maximum is not directly apparent.A special case of particular importance is when the forcing frequency coincides with the undamped natural frequency, o. In this case is zero and using
reduces to:
where A
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In this case the second derivative of X with respect to is zero at time = and this leads to
A
= The full of the equation then becomes:An alternative form of Eqn. is:
where use has been made of Eqn. (A4.8) and
-
The boundary condition of X at enables either or to be determined
= -
+
= 2
from where
For the special case where so that = is infinity and 90":
Again equating the second derivative of X with respect to to zero gives tan =
and when this is substituted into Eqn. (A4.20) it yields the same solution as that given by Eqn.
A further special case is when the forcing frequency, is equal to the natural damped frequency, Here pow equal to It is convenient to express the Particular Integral in the form:
Rigid dynamics
Following the same procedure as used above for the case when 0:
=
and
A =
+
Substitution of these expressions for A and B and the Particular Integral from ( A 4 . 2 1 ) into Eqn. gives the full solution for this case as: