The small-amplitude tip displacement of a turbine blade

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Ordinary differential equations

Mathematical Methods for Physics, Ch Higher-order linear equations, the Wronskian and variation of parameters

Problem

The small-amplitude tip displacement

y(t)

of a turbine blade, whose effective stiffness changes as the blade heats up after start-up, obeys

\tau t\,y'' + \frac{3t^2}{\tau}\,y' + y = 0, \qquad t \gt 0\,\mathrm{s},

where

y

is measured in millimetres,

t

in seconds after start-up, and

\tau = 1.0\,\mathrm{s}

is the blade's heating time. Two independent solutions

y_1(t)

and

y_2(t)

of this equation have Wronskian

W(y_1,y_2)(1.0\,\mathrm{s}) = 4.0\,\mathrm{mm^2\,s^{-1}}

. ((a)) Write the equation in the standard form

y'' + a_1(t)y' + a_0(t)y = 0

and identify

a_1(t)

. ((b)) Use Abel's identity to write

W(y_1,y_2)(t)

in terms of

W(y_1,y_2)(1.0\,\mathrm{s})

, without solving the equation for

y_1

y_2

. ((c)) Evaluate

W(y_1,y_2)(2.0\,\mathrm{s})

, to two significant figures.

Related theory

Ordinary differential equations

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