Solution of nth-Order Linear Differential Equation with Constant Coefficients
The equations in Equation 9.33 are described as linear simultaneous differential equations with constant coefficients. The solution of these equations can be written as the sum of a transient solution and a steady-state solution. These are also referred to as homogeneous and particular solutions.
The steady state, or particular solution, is the part of the total solution that satisfies exactly the differential equation, including the forcing function. The transient solution satisfies the differential equation with the forcing function set equal to zero.
A linear differential equation of the nth order can be written in a general form as
To specify the problem completely, initial conditions equal in number to the order of the equation must be given. This usually means specifying x and its derivative up to order n – 1 at t = 0.
The solution to the homogeneous equation [/(t) = 0] is of the form
The nth derivative will be
This equation substituted into Equation 9.46 for f(t) equal to zero leads to an nth degree polynomial for a known as the characteristic equation.
Cn<rn + Cn-itrn 1 + • • • + Сгсг2 + Ci<r + Co= 0
There are n roots or values of a that will satisfy this polynomial. Thus, the general solution for x will be
x = Axe‘r, t + A2e<T2‘ + • ■ ■ + Anea(9.49)
where 2,… <rn are the roots of Equation 9.48. These roots may be positive or negative real numbers or complex numbers with positive or negative real parts. If a complex root exists, its conjugate must also exist. Complex roots always appear in pairs of the form
и = a ±ib
Au A2,…, A„ are real or complex constants to be determined from the initial conditions.
Let us examine the behavior of a term containing a particular type of root. Consider the following items.
If a and b are positive real quantities, it is obvious that item 1 will become infinitely large as f approaches This is an unstable transient motion. Conversely, item 2 is stable. Item 3 can be written as
e‘bt = cos bt + і sin bt
the complex root with a positive real part represents an oscillatory motion having an amplitude that increases without limit as t approaches 00. This situation is also unstable. Conversely, item 4 is a stable oscillatory root. Thus, to investigate the stability of the transient solution, one need only consider the roots of the polynomial given by Equation 9.57.
The particular solution is referred to as the steady-state solution since, for a stable system, it remains while the transient solution vanishes as t approaches °°. The particular solution is of the same form as fit) if /(f) can be expressed as a polynomial in t or in the form of
fit) = Aeы or
fit) = A cos a>t + В sin at For example, suppose /(f) were given by
/(f) = At3 + ВеШІ + C cos bit
Then the steady-state solution for x would take the form x = at3 + bt2 + ct + d + fеш‘ + g cos <ot
Notice that when f(t) is a polynomial, x must include all orders of t up to the highest in f(t), even though some of the lower-order terms may be missing in fit).
As a fairly simple (and well-used) example of the foregoing, consider the second-order damped system governed by the equation
x + 2 £wnx + w„2x = f(t) (9.50)
The characteristic equation becomes
<t2 + 2£ып(т + Ып = 0
The roots of this equation are
<гіл = Шп(-£± V<T2-1 (9.51)
C is referred to as the critical damping ratio for reasons that are now obvious. When £ is positive but less than unity, <r will be complex and x will be a damped oscillation. When £ is greater than unity, a will be a negative real number, and the motion will be aperiodic.
Suppose f(t) is given by
fit) = A + Веш (9.52)
The particular solution will be of this same form.
x = Хп+Х, еш
Substituting this into the original equation gives
Xi<i>n — o)2 + 2 £(ti(oni) є‘,>1 + o)nXо= A + Be*0*
In order for this equation to be satisfied for all t, the coefficients of like terms involving t must identically satisfy the equation. Thus,
(tiff — со2 + 2 £(tiO)ni
x – A л0 — 2 (tin
It is instructive to clear the denominator of X of the complex number, so
v _ о w<* ~ ы2 — 2g(ti(ti„i
‘ (шп2 – (ti2)2 + 4^i(tii(ti„2
If we further denote the frequency ratio а>Ішп by r, Xt can be written as
В 1 – r2 – 2£ri 1 w7(l – r2)2 + 4£2r2
Bla>n is simply the static value of X, for a> = 0. Thus the ratio of Xx to its static value can be written as
o>„2Xi _ 1 – r2 2£r
В “ (1 – r2)2 + 4£2r2 ‘ (l-r2)2 + 4£2r2
The complete solution to Equation 9.50, forced in the manner given by Equation 9.52, is thus
= 4- ———з—і
(X’e +Х2Є ) + W +4 r1 x, and x2 are constants to be determined by the values of x and x at t = 0.
The complex form for x and the forcing function may seem somewhat mysterious at first if one is not used to this type of mathematical treatment. Certainly there is nothing imaginary about the longitudinal motion of an airplane. The complex notation is simply a convenient mathematical device to show the amplitude and phase relationships between various terms in the system.
For example, consider Equation 9.53. This can be written in the form
so that the steady-state solution becomes
x = —2 + —2 [С,(г, О – iC2(r, £)]«" (9.55)
The complex quantity in the brackets can be further expressed as
[Ci(r, o-iC2(r, D] = Re~i*
b = Vc,2 + c22
ф = tan 1 ~r
Thus Equation 9.55 becomes
x = A + ^ (9.56)
If Веш is pictured as a force vector of magnitude В rotating around the origin in the complex plane, the time-dependent part of x, x – A/w„2, can be pictured as a displacement vector of magnitude BRIa>n2 rotating at the same
rate of o) but lagging behind the force vector by an angle of ф. This is illustrated in Figure 9.3.
іo„ is known as the undamped natural frequency, since this is the frequency at which the force-free system will oscillate when perturbed and with £ = 0. It is interesting to examine Equation 9.56 when the system is forced at its natural frequency. For this case r=1.0, so C,(r, () = 0 and Сг(г, О = (2£)_1 and ф therefore equals тг/2 or 90°. Thus the displacement of a second-order damped system, when forced at its natural frequency, will always lag the force by 90°, regardless of the damping.