Identification of the dynamic characteristics of nonlinear structures

 
case, 
 
 
is 
 
 
located 
error shifted
to coordinates 
and 
instead of 
and 
as shown in Fig.5.12, although the
location task is effectively successfully completed.
Stiffness Error 
Fig.511 Stiffness 

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5
Location of Structural 
Nonlinearity
155
Fig.5.12 Stiffness Error Due to Nonlinearity 
 
5.4 EXPERIMENTAL INVESTIGATION
5.4.1 SIMULATION OF STRUCTURAL SYSTEM
WITH LOCALISED NONLINEARITY
To demonstrate the practical applicability of this proposed nonlinearity location method,
an experimental investigation was carried out. The experimental system is an essentially
linear frame 
made of mild steel coupled to an SDOF system with nonlinear
stiffness. This nonlinear SDOF system is simulated using an electro-dynamic shaker by
feeding the displacement signal of its moving table through a nonlinear analogue circuit
and then back to the shaker to produce a force which satisfies a prescribed nonlinear
function 
illustrates the setup of the simulated nonlinear structure.

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5
Location of 
Nonlinearity
156
Fig.5 13 Simulated Nonlinear Structure with 
Stiffness Nonlinearity
FRAME GEOMETRY
The frame structure is made of mild steel with Young’s modulus assumed to be
and density 
The geometry of the frame is illustrated in
813
Fig.5.14 Geometry of the Frame Structure

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5
Location of Structural Nonlinearity
157
SHAKER PROPERTIES
A Ling Dynamics shaker (type 
was used to simulate the nonlinear SDOF system
attached to the frame. The effective mass of the moving system of the shaker is 
and the spring/mass resonance is 
The effective stiffness is therefore
m,). The SDOF system
Fig.5.15.
model of the shaker is shown in
Fig.5.15 Shaker SDOF Model
According to electro-magnetic dynamics, the force F produced on the coil (figure 5.15) is
proportional to the current I in the coil: F=kI. However, due to the motion of the coil in
the magnetic field, a back emf is produced which is proportional to the velocity of the
coil: 
As a result, the relationship between the force F produced on the coil and the
voltage V applied on the coil becomes F= k(V- 
(R, L and are coil
resistance, coil inductance and excitation frequency, respectively). Clearly, if 
(a
prescribed nonlinear function) is to be satisfied, the effect of this back emf must be
compensated so that the force produced is proportional to the voltage applied to the power
amplifier. This compensation can be achieved by using a current power amplifier which
produces a current output (and so the force F) proportional to voltage input regardless of
the loading impedance (which changes dramatically around structural resonances). The
current amplifier and shaker can be looked on as a single unit, as shown in Fig.5.16. The
output/input (I/V) characteristic of this unit when the frame is attached is measured as
shown in Fig.5.17. From figure 5.17, it can be seen that although the effective
impedance of the shaker changes dramatically around the resonance, due to the back emf
effect, the force produced on the coil is always proportional to the voltage applied to the
current power amplifier.

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 Location of Structural Nonlinearity