MOTION INDUCED INSTABILITIES - Theory of Bridge Aerodynamics - page 194

Chapter 8

MOTION INDUCED INSTABILITIES

8.1 Introduction

Static as well as dynamic structural response will in general increase with increasing
mean wind velocity. In some cases the response may develop towards what is perceived
as unstable behaviour, i.e. the response is rapidly increasing for even a small increase of
the mean wind velocity, as indicated in Fig. 6.3. It is seen from Eqs. 6.48 and 6.49 (see
also Eqs. 4.69 and 4.82) that in the limit the structural displacement response will
become infinitely large if the absolute value of the determinant to the non-dimensional

N by N impedance matrix

mod mod

2

⎧ ⎫

⎛ ⎡⎤ ⎞ ⎡⎤

1 1

⎪ ⎪

ˆ

( ) ( )

, V diag 2 i diag E ω =− I κ − ⋅ ω + ω ⋅ ⋅ ζζ − (8.1)

⎨ ⎜ ⎟ ⎬

⎢⎥ ⎢⎥ η ae

⎜ ⎟

⎣⎦

ae

ω ω

⎣⎦

⎪ ⎝ ⎠ ⎪

i i

⎩ ⎭

is zero. Thus, any stability limit may be revealed by studying the properties of the
impedance matrix. Obviously, unstable behaviour is caused by the effects of and

a ζ . The effects of a ζ is to change the damping properties of the combined structure
and flow system, while the effects of

κ

ae

is to change the stiffness properties. While we
in the entire chapter 6 ignored any motion induced changes to resonance frequencies
(defined as the frequency positions of the apexes of the modal frequency response
function) this can not be accepted in the search for any relevant instability limit. The
reason is explained in chapter 5.2, and as shown in Eq. 5.24, it involves taking into
account that the aerodynamic derivatives are modal quantities that have been normalised
by and are functions of the mean wind velocity dependent resonance frequencies. Thus
(see Eqs. 5.24, 6.51 and 6.52) the content of

κ

ae

⎡ ⎤ ⎡ ⎤

%$ %$

⎢ ⎥ ⎢ ⎥

ζ

κ = κ and = ζ (8.2)

⎢ ⎥ ⎢ ⎥

ae ae ij ae ae ij

⎢ ⎥ ⎢ ⎥

$% $%

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦