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NOTE: FiS)

NOTE: FiS)

n DENOTES A POSITIVE INTEGER; a AND b DENOTE POSITIVE REAL NUMBERS. •»„ IS THE DIRAC DELTA FUNCTION.

tu IS HEAVISIDE'S UNIT STEP FUNCTION WHICH IS DEFINED BY u = 0 FOR t < a. u • 1 FOR t > a.

Solution of Linear Differential Equations. Linear differential equations with constant coefficients may be solved by taking the Laplace transform of each term of the differential equation, thereby reducing a differential equation in t to an algebraic equation in s. The solution may then be transformed back to the time domain by taking the inverse Laplace transform. This procedure simplifies the analysis of the response of complex physical systems to frequency-dependent stimuli, such as the response of an onboard control system to periodic disturbance torques.

The solution to the linear differential equation

with an = 1 and forcing function x(t) is given by

where X(s)= £(*(/)), f(*)= £ a,*' is the characteristic polynomial of Eq. (F-17) and '=°

are the initial conditions.

Any physically reasonable forcing function, including impulses, steps, and ramps, may be conveniently transformed (see Table F-l). The analysis of the algebraic transformed equation is generally much easier than the original differential equation. For example, the steady-state solution, f(oo), of a differential equation is obtained from the Laplace transform by using the final value theorem, Eq. (F-6).

The first term on the right-hand side of Eq. (F-18) is the forced response of the system due to the forcing function and the second term is the free response of the system due to the initial conditions. The forced response, £~'(X(s)/£(s)), consists of two parts: transient and steady state.

Solving the differential equation (Eq. (F-17)) is equivalent to finding the inverse Laplace transform of the algebraic functions of s in Eq. (F-18). One technique involves expressing rational functions of the form

as a sum of partial fractions (n > m) using the fundamental theorem of algebra. The characteristic polynomial, £(s), may be factored as

where —p, is the ith zero of with multiplicity m, and

The partial fraction expansion is

where

__1 dm>~k and b„=0 unless m = /i. The coefficients Cn are the residues of ^(5) at the poles —pt. If no roots are repeated, Eq. (F-22) may be rewritten as

where

The zeros of f(j) may be determined using various numerical methods [DiStefano, et al., 1967].

The inverse Laplace transform of expressions in the form of Eq. (F-23) may be obtained directly from Table F-l. Other techniques for computing inverse Laplace transforms include series expansions and differential equations [Spiegel, 1965].

Example: Forced Harmonic Oscillator. The equation describing a 1-degree-of-freedom gyroscope (Sections 6.5 and 7.8) is d*9 D d$ , K8 L

where Ic is the moment of inertia of the gyroscope about the output axis, D is the viscous damping coefficient about the output axis, K is the restoring spring constant about the output axis, L is the angular momentum of the rotor, and u(t) is the angular velocity about the input axis which is to be measured (see Fig. 6-45).

We assume that the input angular velocity is sinusoidal* with amplitude, A, and frequency, ue; i.e^

•This is not as severe a restriction as it might seem because any physically reasonable a(t) may be expanded in a Fourier series. The result For a general u(0 is then obtained by linear superposition.

772 where

S(s) = S2+DS/Ig+K/1c X(s) = t"(ALcosw„l/IG ) = ALs[+ <o2)/c]

The characteristic polynomial. £(s), may be factored as

where

ALs/Ic

(s2 + u2)(s+Pl)(i+P2) The second term on the right-hand side of Eq. (F-29) is given in Table F-l as D80/lc + s80+80

= (-jp^ ) {(D8J1C + 0o)l>xp(-/>,/)-exp(-p2l)]

The first term on the right-hand-side of Eq. (F-29) may be expanded in partial fractions as

2(Pi ~ iue)(Pi~ ><•>,) 2(p, + iwe)(p2 + iue) 1 f Pi^P(-Pi') Pi^M-Pi')

Equation (F-30) and the last term on the right-hand-side of Eq. (F-31) are the transient response of the system to the initial conditions and the forcing function. The transient response decays with a time constant and frequency given by

0 0

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