Introduction to Random vibrations

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Random vibrations basics

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<ul><li><p>Random sine function </p><p> ( ) sinx t c t </p><p>stochastic phase angle, for example uniform on [0,2 ]</p><p>deterministic amplitude</p><p>deterministic circle frequency</p><p>c</p><p>f() </p><p> 2 </p></li><li><p>Random sine function </p><p>x and deterministic, uniform on (0,2) </p><p>2 2</p><p>0 0</p><p>1{ ( )} ( ) ( ) sin( ) 0</p><p>2x t x t f d x t d</p><p>22 2 2 2</p><p>2 2</p><p>0 0</p><p>1{ ( )} ( ) ( ) sin ( )</p><p>2 2</p><p>xx t x t f d x t d</p><p> ( ) sinx t x t </p></li><li><p>1.0 1.0 </p><p> Two important integrals </p><p>0.5 </p><p>( )f y</p><p>2y y</p><p>( )f y</p><p>2</p><p> 22 2</p><p>0 0</p><p>sin 0 sin 0.5 2y dy y dy</p></li><li><p>1( ) sin( ) ( )k k k k</p><p>N</p><p>k</p><p>x t x t x t </p><p>sum of sine functions with random phase angle: </p><p>{ ( )} { ( )} 0kx t x t 2 2 21 { ( )} { ( )}</p><p>2k kx t x t x </p></li><li><p>0</p><p>2</p><p>0</p><p>10</p><p>1</p><p>1,2,3, 10 ; 4 ; 2 ; 5</p><p> exp ; 2 ; 0.1</p><p>( ) sin ; 0 50 ; random phase angle</p><p>k k k</p><p>k k</p><p>k k k</p><p>k</p><p>k T k T</p><p>x a b a b</p><p>x t x t t</p><p>t</p><p>( )x t</p><p> 0 5 10 15 20 25 30 35 40 45 50 </p><p>10 </p><p>8 </p><p>6 </p><p>4 </p><p>2 </p><p>0 </p><p>2 </p><p>4 </p><p>6 </p><p>8 </p><p>10 </p><p>Process with 11 sine functions </p><p>k=1,11, xk = 1.0 </p><p> 1= 4.0, 2 = 4.2. 10 = 6.0 rad/s </p></li><li><p> 1</p><p> ( ) cos sin sin cosk k k k k k</p><p>N</p><p>k</p><p>x t x t x t </p><p>0</p><p>2( ) sink k</p><p>T</p><p>A x t t dtT</p><p> 0</p><p>2( ) cosk k</p><p>T</p><p>B x t t dtT</p><p>2 2 2k k kx A B </p><p> 2 2 21</p><p>1{ ( )}</p><p>2k k</p><p>N</p><p>k</p><p>x t A B</p></li><li><p> T T</p><p>x</p><p>t</p><p>2kx</p><p>k</p><p>2kx</p><p>k2</p><p>T</p><p> 4</p><p>T</p><p>base time T</p><p>2</p><p>T</p><p>base time 2T</p><p>k</p><p>2( ) 2k k kS x 2 2</p><p>lim kxxx</p><p>S</p><p>Definition of the variance spectrum </p></li><li><p> 1</p><p> ( ) cos sin sin cosk k k k k k</p><p>N</p><p>k</p><p>x t x t x t </p><p>0</p><p>2( ) sink k</p><p>T</p><p>A x t t dtT</p><p> 0</p><p>2( ) cosk k</p><p>T</p><p>B x t t dtT</p><p>2 2 2k k kx A B </p><p> 2 2 21</p><p>1{ ( )}</p><p>2k k</p><p>N</p><p>k</p><p>x t A B</p><p>2</p><p>10</p><p> 2 2( ) lim withkxx k k k</p><p>xS</p><p>T</p><p>2 2</p><p>0</p><p>1( ) ( ) ( )</p><p>2k xx k xxx x S S d </p></li><li><p>2 2</p><p>2 2</p><p>1 1( ) ( ) cos sin withi tx</p><p>T T</p><p>T T</p><p>S x t e dt x t t i t dt T </p><p>*( ) ( )xx x xS S ST</p></li><li><p>0</p><p>( )ii xxm S d </p><p>The standard deviation </p><p>0( )x m </p><p>central frequency: </p><p>0 1 0m m </p><p>2</p><p>0 2 0m m </p><p> width of the spectrum </p><p> 2</p><p>1</p><p>0 2</p><p>1m</p><p>qm m</p><p>moments of the spectrum </p></li><li><p> a) narrow-band process </p><p> b) wide-band process </p><p> c) process with two distinctive frequencies </p><p>0 02T </p><p>16%x</p><p>x</p><p>x</p><p>t</p><p>t</p><p>xxS</p><p>xxS</p><p>0</p><p>0</p><p>21</p><p>xxS</p><p>0</p><p>t</p><p>0.5q </p><p>0.6q </p><p>q </p></li><li><p>Fig. A.2: Stochastic process as a series of stochastic variables. </p><p>( )x t</p><p>t t</p><p>( )x t</p><p>( )x t t </p><p>( )x t n t </p><p> t t n t t</p><p>Alternative Approach (Annex) </p></li><li><p> random variable f(x) x x </p><p> two randomvariabels f(x,y) x x y y covxy </p><p> n random variables f(x) x x covxy </p><p> random process f(x(t)) x(t) x(t) covxy(t1,t2) </p><p> stationary random proc f(x) x x Rxx(t) </p></li><li><p>Description of continuous processes </p><p>Gaussian process: </p><p>Mean value for every point in time </p><p>Covariance for every two points in time </p><p>General process: </p><p>Multidimensional probability distribution for every </p><p>set X(t1), X(t2), X(t3), . </p><p>Stationary Gaussian process: </p><p>Mean value X </p><p>Autocovariance function RXX() </p></li><li><p>Spectrum </p><p>0</p><p>2( ) ( )cosxx xxS R d </p><p>0</p><p>( ) ( )cos( )xx xxR S d </p><p>2</p><p>0</p><p>(0) ( )x xx xxR S d </p></li><li><p>2 1 2 1</p><p>2 1 0 2 1 0 1 0</p><p>2 1 0 2 1 0 1 0</p><p>2 1 2 1</p><p>2 1 2 2 1 2</p><p>2 1 2</p><p>( ) ( ) ( ) ( )</p><p>1( ) ( ) cos ( ) ( ) ( )</p><p>21</p><p>( ) ( )sin ( ) ( ) ( )2</p><p>( ) ( ) ( ) ( )</p><p>( ) ( ) ( ) ( ) ( ) ( )</p><p>( ) ( ) ( )</p><p>nn</p><p>n</p><p>R a R S a S</p><p>R R S S S</p><p>R R S S Si</p><p>dR R S i S</p><p>d</p><p>R R R S S S</p><p>R R g t dt S</p><p> 1</p><p>2 1 2 1</p><p>( ) ( ) ( )</p><p>( ) ( ) ( ) ( ) ( ) ( )</p><p>Hereby is the Fourier transform of ( ) defined by:</p><p>( ) ( )</p><p>complex gonjugate of ( ) real function</p><p>i t</p><p>G S</p><p>R R g t dt S G S</p><p>G g t</p><p>G g t e dt</p><p>G Gg t</p><p>Table A.1: Properties of Fourier transforms. </p></li><li><p>( ) 1S </p><p>1 </p><p>( ) 2 ( )S </p><p>2area </p><p>0 0( ) ( ) ( )S </p><p>1area </p><p>0 0 </p><p>( )S </p><p>1area </p><p>1 </p><p>( )S </p><p>1area </p><p>0 1 2 </p><p>( )R </p><p>1area </p><p>( ) ( )R </p><p>( )R </p><p>( ) 1R </p><p>1</p><p>( )R </p><p>0( ) cosR </p><p>1</p><p>0</p><p>( )</p><p>sin cos</p><p>R </p><p>( )R </p><p>( )R </p><p>1</p><p>( )</p><p>sin1 </p><p>R </p><p>1( ) ( ) ( ) iR S R e dt </p><p>1 </p><p>2 </p><p>3 </p><p>4 </p><p>5 </p><p>( )R e</p><p> ( )R 2 2</p><p>1 2( )S</p><p>1</p><p>0</p><p>( )</p><p> cos</p><p>R</p><p>e</p><p>( )R </p><p>1</p><p> 2 22 2</p><p>0 0</p><p>1( )S</p><p>0 0 </p><p>00 0( ) cos sin4</p><p>R e </p><p>0 0 </p><p>4</p><p>0</p><p>22 2 2 2 2</p><p>0 0</p><p>( )4</p><p>S</p><p>2</p><p>0 1 </p><p>6 </p></li><li><p> ( ) sinx t c t </p><p>sine function </p><p> X= 0 </p><p> 2( ) ( )1</p><p>cov ( ) cos2</p><p>x t x t xxR c </p><p>2 21 1(0) ; 22 2</p><p>x xx xR c c </p><p>Fig. A.4: Auto-covariancefunction of with uniform on . </p><p>( )xxR </p><p>212c</p><p>c and deterministic, uniform on (0,2) </p><p>SXX() = c2 (-k) </p></li><li><p> ( ) sinx t c t </p><p>1 2 1 2( ) ( ) 1 ( ) 2 ( )cov {[ ( ) ] [ ( ) ]}x t x t x t x tE x t x t </p><p> 1 2</p><p>2</p><p>( ) ( ) 1 2 1 2cov { ( ) ( )} { sin sin }x t x t E x t x t E c t t </p><p> 1 2</p><p>2</p><p>( ) ( ) 1 2</p><p>2</p><p>0</p><p>cov sin sin2</p><p>x t x t</p><p>ct t d</p><p>1 2</p><p>2 2</p><p>( ) ( ) 1 2 1 2</p><p>2 2</p><p>1 2 1 2</p><p>2</p><p>1 2</p><p>2 2</p><p>0 0</p><p>2 2</p><p>0 0</p><p>1 1cov cos cos 2</p><p>2 2 2 2</p><p>cos sin 24 8</p><p>1cos</p><p>2</p><p>x t x t</p><p>c ct t d t t d</p><p>c ct t t t</p><p>c t t</p></li><li><p> 1</p><p>2 </p><p> sin </p><p>k</p><p>k</p><p>k k k k</p><p>N</p><p>k k k</p><p>k</p><p>S</p><p>a S</p><p>y a t</p><p>random </p><p>generator </p><p>Sk</p><p> Generation of a random process </p><p>To Remember (1): </p></li><li><p>Construction of the variance spectrum </p><p>x</p><p>t</p><p>Sx () = (1/) 0 T x(t) exp (it) dt </p><p>Sxx() = ( /T) Sx Sx* </p><p>2</p><p>0</p><p>(0) ( )x xx xxR S d </p><p>Sk</p><p>To Remember (2): </p></li></ul>