smart technologies for safety engineering

24 Smart Technologies for Safety Engineering
EA
4 4
t
F.(t)
0
4
t
1
4
EA
4 4
t
F.(t)
t
Figure 2.7 Impulse virtual distortion in an element, showing the process of matrix B ε , D ε composition
capturing the influence of inertia on structural response. Again, the response of the structure
to the impulse force distortion at nodes is obtained by the Newmark algorithm.
It will be demonstrated in subsequent sections that by establishing the influence matrices B ε ,
D ε , B f , D f , the remodeling of stiffness and mass in the structure becomes feasible. A nonlinear
constitutive relation can also be accounted for. However, linear geometric relations (small
strains) are assumed. Examples of VDM applications in dynamics are demonstrated in Chapters
3 and 6.
2.5.2 Stiffness Remodeling in Dynamics
In signal processing performed in many fields of engineering, the output response of a system
is expressed as an integral of the product of the input excitation and transfer function (i.e. the
system’s response to an impulse function like Dirac’s delta) over some period of time.
For a simple harmonic oscillator of mass m and natural frequency ω, the convolution of the
two functions determines the displacement u(t) due to a series of impulses f (τ)dτ over the
1
3
4
1
3
5
4
2
1
2
t
F.(t)
0
p 4x=1
t
Figure 2.8 Impulse virtual distortion at a node, showing the process of matrix B f , D f composition
1
1
3
3
4
4
1
1
3
3
5
5
0
p 4y=1
4
2
4
2
2
2
1
t
F.(t)
t