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Design of stop-band filter by use of curved pipe segments and shape optimization

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Abstract

It is often desirable to isolate some parts of a compound slender spatial structure from vibrations in connected substructures. An example of such a problem is found in domestic pipe systems where vibration from pumps and valves installed in an assembled pipe system should be suppressed before it reaches installations in dwellings. By use of Floquet theory and a shape optimization procedure a stop-band design is tuned into a specified frequency range. The structures with prescribed stop-band characteristics are composed of curved and straight pipe segments. For comparison vibro-acoustic energy transmission analysis is made on periodic piping systems where a finite series of the same substructures are implemented as a stop-band filter. The influence of production tolerances of such a stop-band filter is also assessed.

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Abbreviations

CRS:

Controlled random search;

IL:

Insertion loss;

MMA:

Method of moving asymptotes;

Obj :

Objective function

A i :

fluid area of the cross section

A p :

solid area of the cross section

c :

speed of the plane dilatation wave in the solid material

E :

Young’s modulus

\(\tilde{{E}}_{\rm ref} \) :

time average power flow through reference structure

\(\tilde{{E}}_{\rm system} \) :

time average power flow through analysed structure

G :

shear modulus

J :

axial moment of inertia

J p :

polar moment of inertia

K :

non-dimensional wave-number

K B :

propagation constant (Block parameter)

L :

total length of substructure

m :

number of frequencies

n :

number of design parameters

q :

external distributed load

R :

radius of pipe segment curvature

r g :

radius of gyration

s :

centre line length coordinate

u :

displacement along x-axis

U :

non-dimensional displacement along x-axis

v :

displacement along y-axis

V :

non-dimensional displacement along y-axis

w :

displacement along z-axis

W :

non-dimensional displacement along z-axis

x :

in-plane transversal cross section coordinate

y :

out-of-plane transversal cross section coordinate

z :

in-plane longitudinal cross section coordinate

α :

rotation around x-axis

β :

rotation around y-axis

γ :

rotation around z-axis

Γ:

non-dimensional rotation around z-axis

ε :

ratio between r g and R

θ :

angle of pipe segment curvature

λ :

periodic constant

ν :

Poisson’s ratio

ρ f :

fluid mass density

ρ s :

solid mass density

ψ :

objective function for bound formulation

ω :

circular frequency

Ω:

non-dimensional frequency

B :

Boundary integral equation and interfacial continuity matrix for a pipe string

B s :

Boundary integral equation matrix for a single pipe segment

f :

vector containing forces and moments

F :

free space Green’s matrix for forces and moments

G :

free space Green’s matrix for displacements and rotations

u :

vector containing displacements and rotations

v :

vector containing the design parameters R j , θ j and L

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Acknowledgments

Financial support from the Danish Agency for Research and Innovation is gratefully acknowledged. The author is indebted to Prof. S. V. Sorokin and Prof. N. Olhoff of Aalborg University for very useful discussions and supervision.

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Authors and Affiliations

  1. Department of Mechanical and Manufacturing Engineering, Aalborg University, Pontoppidanstraede 101, 9220, Aalborg East, Denmark

    Alf Søe-Knudsen

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Correspondence to Alf Søe-Knudsen.

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Søe-Knudsen, A. Design of stop-band filter by use of curved pipe segments and shape optimization. Struct Multidisc Optim 44, 863–874 (2011). https://doi.org/10.1007/s00158-011-0691-2

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