Random Vibrations: Response of a linear system

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Random vibrations, stochastic vibration, dynamics,

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<ul><li><p>,H </p><p>x</p><p> y H xsystem </p><p>input output </p><p> x y </p><p>Response of a linear system </p><p> sinx x t</p><p> siny H x t </p></li><li><p> sinx x t </p><p> siny H x t </p><p> sinx x t </p><p> siny H x t </p><p> ( ) sin ( )x x t ( ) ( ) sin ( ) ( )y x H t </p><p>2</p><p>0</p><p>1 ( )( ) lim</p><p>2xx</p><p>xS</p><p>22</p><p>0</p><p> ( ) ( )1( ) lim</p><p>2yy</p><p>x HS</p><p> 2</p><p>( ) ( ) ( )yy xxS H S </p><p>Response spectrum of a linear system </p></li><li><p>( )F t</p><p>( )u t</p><p> k c</p><p>( )F t</p><p>( )u tm t</p><p>t</p><p>;2</p><p>e</p><p>k c</p><p>m k m </p><p> A single-mass-spring system under random loading </p></li><li><p> 1 2</p><p>2 22</p><p>1( )</p><p>1 2e e</p><p>H H</p><p>k</p><p> 2</p><p>2arctan with 0</p><p>1</p><p>e</p><p>e</p><p> (0.1) </p><p> frequency</p><p>; natural frequency</p><p> damping factor ; 2</p><p>e e k m</p><p>c k m</p><p>Dynamic single degree of freedom system </p><p> k c</p><p>( )F t</p><p>( )u tm</p></li><li><p>FFS</p><p>( )H </p><p>uuS</p><p>1</p><p>2k</p><p>0</p><p>0</p><p>2</p><p>u</p><p> Analysis of the single-mass-spring system </p></li><li><p>FFS</p><p>2H</p><p>uuS</p><p>2</p><p>1</p><p>2k</p><p>e</p><p>0</p><p>2area</p><p>4</p><p>e S</p><p>k</p><p>0S</p><p>21 k</p><p>2</p><p>0S k</p><p> Response spectrum for a single-mass-spring system under </p><p>stationary load with white-noise spectrum </p></li><li><p> wind </p><p>10 s </p></li><li><p> respons </p><p>10 s Te = 0.5 s </p></li><li><p>FFS</p><p>0S ( )FF eS </p><p>white-noise approximation </p><p>real spectrum </p><p> e </p><p> Approximation of the load spectrum by a white-noise </p><p> spectrum with intensity </p><p> 0 ( )FF eS S </p></li><li><p>S Du u u Fk </p><p> 2 2 22 2 2( )( )</p><p>( )21</p><p>FF eFFuu</p><p>ee</p><p>SSS</p><p>k k</p><p>Response to arbitrary load </p><p>Split the response into </p><p>a quasi static part and a white noise part: </p><p>22</p><p>2 2</p><p>( )1</p><p>4</p><p>F e FF eu</p><p>F</p><p>S</p><p>k</p></li><li><p>3 6 2</p><p>3 2</p><p>0</p><p>9 5</p><p>70 10 N ; 8 10 Nm ; 20,000 kg</p><p>50 10 N s ; 3 m ; 0.01</p><p>50 years 1.5 10 s ; 10 Nm</p><p>F</p><p>p</p><p>EI m</p><p>S l</p><p>T M</p><p>( )F t</p><p> T t</p><p>F</p><p> e </p><p>( )FFS </p><p>0S</p><p>( ) ( )F t m u t</p><p>l HE 200 A </p><p>Portal frame with data </p></li><li><p>Spring stiffness k : </p><p>6</p><p>3</p><p>247.1 10 N/m</p><p>EIk</p><p>l </p><p>Natural frequency: </p><p>3</p><p>2419 rad/se</p><p>k EI</p><p>m m l </p><p>Mean response </p><p>10.0098 m 9.8 mmu F</p><p>k </p><p>Variance of the response: </p><p>2 6 20</p><p>21.47 10 m</p><p>4</p><p>eu</p><p>S</p><p>k</p><p>Standard deviation: </p><p>0.0012 m 1.2 mm u </p><p>( ) ( )F t m u t</p><p>l HE 200 A </p></li><li><p>( )F t( )u t</p><p> = full-plastic hinge A</p><p>Considered limit state of the frame </p></li><li><p>1(0) </p><p> ( ) at </p><p>u</p><p>u t t t</p><p>( )u t</p><p>0 T t</p><p>( )u t</p><p>10 t T t</p><p> The event failure in [0,T] </p></li><li><p>( )u t</p><p>t</p><p>1a</p><p>2a 3a</p><p>ia</p><p>na</p><p>T</p><p> ( )aF </p><p>Peak values of the narrow-band process </p></li><li><p> Yield occurs if: 1</p><p>4A p pM F l M </p><p>Corresponding value of u: 4</p><p>0.0188 m 18.8 mmp p</p><p>p</p><p>F Mu</p><p>k kl </p><p>Reliability index: 18.8 9.8</p><p>7.51.2</p><p>p u</p><p>u</p><p>u </p><p> Exceedance probabtilities: </p><p>APT: </p><p>single peak: </p><p>50 year: </p><p>( )u t</p><p>t</p><p>1a</p><p>2a 3a</p><p>ia</p><p>na</p><p>T</p><p>example portal frame </p><p>14{ ( ) } 1 ( ) ( ) 3 10p N NP u t u </p><p>213{ } exp 6.1 10</p><p>2i pP a u</p><p>3{ ( ) in [0, ]} { } 2.7 10p i pP u t u T n P a u </p></li><li><p> Fatigue strength of steel </p><p>2 double amplitude [N/mm ]s </p><p>400 </p><p>100 </p><p>200 </p><p> 50 </p><p> 20 5 6 710 10 10 N</p><p>reinforced beam </p><p>welded beam </p><p>s</p><p>st</p><p>average </p><p>exceedance probability </p></li><li><p>Fatigue </p><p>105 106 Nf </p><p>S </p><p>bfS</p><p>CN</p><p>S = stress range </p><p>Nf = number of cucles till fracture </p><p>C, b = material constants </p><p>2</p></li><li><p>Fatigue Miner Rule </p><p>)2</p><p>b1(}22{</p><p>C</p><p>Tf)]T(D[E</p><p>ds2</p><p>sexp</p><p>s</p><p>C</p><p>)s2(Tf)]T(D[E</p><p>)s(N</p><p>ds)s(fn</p><p>)s(N</p><p>)s(n)]T(D[E</p><p>)1Diffailure(N</p><p>n...</p><p>N</p><p>n</p><p>N</p><p>nD</p><p>bs</p><p>e</p><p>2</p><p>2</p><p>2s</p><p>b</p><p>e</p><p>i</p><p>i</p><p>2</p><p>2</p><p>1</p><p>1</p></li><li><p>54</p><p>3</p><p>2</p><p>1</p><p>00 1 2 3 4 x</p><p>( )x</p><p> 0.50 1.772</p><p>0.75 1.225</p><p>1.00 1.000</p><p>1.25 0.906</p><p>1.50 0.886</p><p>1.75 0.919</p><p>2.00 1.000</p><p>x x</p><p> The Gamma function </p><p>for integers: (x) = (x-1)! </p></li><li><p>/ 2 / 2k c c k</p><p>v1</p><p>2</p><p>105 tons</p><p>300 tons</p><p>m</p><p>m</p><p> Schematisation of the coal-dust mill </p></li></ul>

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