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Vibration control of a flexible structure with electromagnetic actuators

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Abstract

This work presents the model of a shear-frame-type structure composed of six flexible beams and three rigid masses. Fixed on the ground, outside the structure, two voltage-controlled electromagnetic actuators are used for vibration control. To model the flexible beams, unidimensional finite elements were used. Nonlinear equations for the actuator electromagnetic force, noise in the position sensor, time delays for the control signal update and voltage saturation were also considered in the model. For controlling purposes, a discrete linear quadratic regulator combined with a predictive full-order discrete linear observer was employed. Results of numerical simulations, where the structure is submitted to an impulsive disturbance force and to a harmonic force, show that the oscillations can be significantly reduced with the use of the electromagnetic actuators.

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Abbreviations

A :

State matrix of the continuous state-space equations

a :

Air gap cross section

B :

Input matrix of the continuous state-space equations

C :

Output matrix of the state-space equations

c :

Control vector

f :

Force vector

G :

State matrix of the discrete state-space equations

H :

Input matrix of the discrete state-space equations

I :

Identity matrix

i :

Total electric current in one actuator

\(i_{{a_{1} }}\) :

Total electric current in actuator 1

\(i_{{a_{2} }}\) :

Total electric current in actuator 2

K n :

Beam element stiffness matrix

l a :

Length of the flexible beams between the ground and m 1

l b :

Length of the flexible beams between m 1 and m 2

l c :

Length of the flexible beams between m 2 and m 3

l n :

Length of a beam element

M n :

Beam element mass matrix

N :

Number of turns of the electromagnetic actuator coil

P :

Solution matrix of the algebraic Riccati equation

q :

Vector of state variables

\({\hat{\mathbf{q}}}\) :

Vector of estimated state variables

r :

Actuator electric resistance

s :

Distance between the actuator surface and the ferromagnetic mass surface

\(s_{{a_{1} }}\) :

Distance between actuator 1 surface and m 1 left surface

\(s_{{a_{2} }}\) :

Distance between actuator 2 surface and m 1 right surface

t :

Continuous time

\(U_{{a_{1} }}\) :

Total voltage in actuator 1

\(U_{{a_{2} }}\) :

Total voltage in actuator 2

u G :

Structure displacement vector

v G :

Structure velocity vector

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Author information

Authors and Affiliations

  1. Department of Mechanical and Materials Engineering, Military Institute of Engineering (IME), Praça General Tibúrcio 80, Urca, Rio de Janeiro, RJ, CEP 22290-270, Brazil

    Maurício Gruzman

  2. Department of Mechanical Engineering, Technical University of Denmark (DTU), 2800, Kongens Lyngby, Denmark

    Ilmar Ferreira Santos

Authors
  1. Maurício Gruzman
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  2. Ilmar Ferreira Santos
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Corresponding author

Correspondence to Maurício Gruzman.

Additional information

Technical Editor: Kátia Lucchesi Cavalca Dedini.

Appendix: matrices elements

Appendix: matrices elements

Below, all the terms that are different to zero from the global stiffness matrix K G , the global mass matrix M G (including m 1, m 2 and m 3) and the state matrix A are given.

$$\begin{aligned} k_{1,1} & = k_{3,3} = 192EI/l_{a}^{3} ;\,k_{1,5} = k_{5,1} = k_{3,5} = k_{5,3} = - 96EI/l_{a}^{3} ;\,k_{2,2} = k_{4,4} = 16EI/l_{a} \\ k_{2,5} & = k_{5,2} = k_{4,5} = k_{5,4} = 24EI/l_{a}^{2} ;\,k_{5,5} = (192EI/l_{a}^{3} ) + (192EI/l_{b}^{3} ); \\ k_{5,6} & = k_{5,8} = k_{6,5} = k_{6,10} = k_{8,5} = k_{8,10} = k_{10,6} = k_{10,8} = - 96EI/l_{b}^{3} ; \\ k_{5,7} & = k_{5,9} = k_{7,5} = k_{9,5} = - 24EI/l_{b}^{2} ;\,k_{6,6} = k_{8,8} = 192EI/l_{b}^{3} ;\,k_{7,7} = k_{9,9} = 16EI/l_{b} ; \\ k_{7,10} & = k_{10,7} = k_{9,10} = k_{10,9} = 24EI/l_{b}^{2} ;\,k_{10,10} = (192EI/l_{b}^{3} ) + (192EI/l_{c}^{3} ); \\ k_{10,11} & = k_{10,13} = k_{11,10} = k_{11,15} = k_{13,10} = k_{13,15} = k_{15,11} = k_{15,13} = - 96EI/l_{c}^{3} ; \\ k_{10,12} & = k_{10,14} = k_{12,10} = k_{14,10} = - 24EI/l_{c}^{2} ;\,k_{11,11} = k_{13,13} = k_{15,15} = 192EI/l_{c}^{3} ; \\ k_{12,12} & = k_{14,14} = 16EI/l_{c} ;\,k_{12,15} = k_{14,15} = k_{15,12} = k_{15,14} = 24EI/l_{c}^{2} . \\ \end{aligned}$$
$$\begin{aligned} m_{1,1} & = m_{3,3} = 156\rho Al{}_{a}/420;\,m_{1,5} = m_{3,5} = m_{5,1} = m_{5,3} = 27\rho Al{}_{a}/420 \\ m_{2,2} & = m_{4,4} = \rho Al_{a}^{3} /420;\,m_{2,5} = m_{4,5} = m_{5,2} = m_{5,4} = - 3.25\rho Al_{a}^{3} /420; \\ m_{5,5} & = (156\rho Al{}_{a}/420) + (156\rho Al{}_{b}/420) + m_{1} ; \\ m_{5,6} & = m_{5,8} = m_{6,5} = m_{6,10} = m_{8,5} = m_{8,10} = m_{10,6} = m_{10,8} = 27\rho Al{}_{b}/420; \\ m_{5,7} & = m_{5,9} = m_{7,5} = m_{9,5} = 3.25\rho Al_{b}^{2} /420;\,m_{6,6} = m_{8,8} = 156\rho Al{}_{b}/420; \\ m_{7,7} & = m_{9,9} = \rho Al_{b}^{3} /420;\,m_{7,10} = m_{9,10} = m_{10,7} = m_{10,9} = - 3.25\rho Al_{b}^{2} /420; \\ m_{10,10} & = (156\rho Al{}_{b}/420) + (156\rho Al{}_{c}/420) + m_{2} ; \\ m_{10,11} & = m_{10,13} = m_{11,10} = m_{11,15} = m_{13,10} = m_{13,15} = m_{15,11} = m_{15,13} = 27\rho Al{}_{c}/420; \\ m_{10,12} & = m_{10,14} = m_{12,10} = m_{14,10} = 3.25\rho Al_{c}^{2} /420;\,m_{11,11} = m_{13,13} = 156\rho Al{}_{c}/420; \\ m_{12,12} & = m_{14,14} = \rho Al_{c}^{3} /420;\,m_{12,15} = m_{14,15} = m_{15,12} = m_{15,14} = - 3.25\rho Al_{c}^{3} /420; \\ m_{15,15} & = (156\rho Al{}_{c}/420) + m_{3} . \\ \end{aligned}$$
$$\begin{aligned} A_{1,4} & = A_{2,5} = A_{3,6} = 1;\,A_{4,1} = (k_{s} - k_{a} - k_{b} )/m_{1} ;\,A_{4,2} = k_{b} /m_{1} ; \\ A_{4,4} & = - [\alpha m_{1} + \beta (k_{a} + k_{b} )]/m_{1} ;\,A_{4,5} = \beta k_{b} /m_{1} ;\,A_{4,7} = k_{i} /m_{1} ;\,A_{5,1} = k_{b} /m_{2} ; \\ A_{5,2} & = ( - k_{b} - k_{c} )/m_{2} ;\,A_{5,3} = k_{c} /m_{2} ;\,A_{5,4} = \beta k_{b} /m_{2} ;\,A_{5,5} = - [\alpha m_{2} + \beta (k_{b} + k_{c} )]/m_{2} ; \\ A_{5,6} & = \beta k_{c} /m_{2} ;\,A_{6,2} = k_{c} /m_{3} ;\,A_{6,3} = - k_{c} /m_{3} ;\,A_{6,5} = \beta k_{c} /m_{3} ; \\ A_{6,6} & = - [\alpha m_{3} + \beta k_{c} ]/m_{3} ;\,A_{7,4} = - k_{u} /L_{\text{avg}} ;\,A_{7,7} = - r/L_{\text{avg}} . \\ \end{aligned}$$

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Gruzman, M., Santos, I.F. Vibration control of a flexible structure with electromagnetic actuators. J Braz. Soc. Mech. Sci. Eng. 38, 1131–1142 (2016). https://doi.org/10.1007/s40430-015-0438-x

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