Dynamics of Aerospace Structures
Dynamics of Aerospace Structures ASEN 5022
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13 Transient Analysis of Structural Systems Chapter 13 TRANSIENT ANALYSIS OF STRUCTURAL SYSTEMS 13 2 131 MODAL APPROACH TO TRANSIENT ANALYSIS Consider the following largeorder nite element model equations of motion for linear structures Miil Dul Kul fl D 05M 8K 0 8 are constant 131 where the size of the displacement vector n ranges from several thousands to several millions Now suppose we would like to obtain the displacement response ul for expected applied force fl There are two approaches direct time integration and modal superposition We will rst discuss modal superposition techniques followed by direct time integration techniques for computing transient responses of structural systems To this end we decompose the displacement vector ul in terms of its modal components by um0an 032 where 11 is the mode shapes of the freevibration modes and ql is the generalized modal displace ment The mode shape matrix 11 has the property of simultaneously diagonalizing both the mass and stiffness matrices That is it is obtained from the following eigenvalue problem A diagaf i where a is the j th undamped frequency component of 131 In structural dynamics one often employs the following special form of mode shapes eigenvector wTszx WTszlzmg L J Um 13 2 13 3 131 MODAL APPROACH TO TRANSIENT ANALYSIS Substituting 132 into 131 and premultiplying the resulting equation by IJT results in the fol lowing uncoupled modal equation c39jzt a new gym w3q1r 1z 139 1 2 3 1r W 2gtT fa The above equation can be cast into a canonical form 135 1w A xiltzgt b 1r x qzm 411 1T 0 1 o 136 Al w2 oz8ai2 b 1 1 whose solution is given by t xileAquottx0 eA tTbptdt 137 0 It should be noted that the above solution provides only for one of the nvector generalized modal coordinates qt n x 1 Carrying out for the entire nmodal vector the physical displacement ut n x 1 can be obtained from 132 by 111 1 1 1111 1 11 1 1141 q qilQ2l quotqulT 138 While the solution method described in 132 138 appears to be straightfowrad its practical implementation needs to overcome several computational challenges which include 13 3 Chapter 13 TRANSIENT ANALYSIS OF STRUCTURAL SYSTEMS a b 0 13 4 When the size of discrete nite element model increases the task for obtaining a large number of modes m if not all m lt lt n becomes computationally expensive In practice it is customary to truncate only part of the modes and obtain an approximate solution of the form ul IJl n l m ql l m l q q1l q2l qml T m ltlt n 139 It is not uncommon to have mn lt 1100 11000 A typical vehicle consists of many substructures whose structural characteristics are distinctly different from one to another For example a fuselage has different structural characteristics from wing structures Likewise engine blocks are considerably stiffer than the car frame structure The impact of stiffness differences on the computed modes and mode shapes can lead to accuracy loss and frequently to an unacceptable level In modern manufacturing arrangements rarely an aerospace company or automobile company designs manufactures assembles and tests the entire vehicle system This means except for the nal performance evaluation each substructure can be modeled analyzed and tested before it can be assembled as a separate and independent structure An alternative approach is to numerically integrate the equations of motion 131 We will discuss computational procedures of two direction algorithms in the next section Their algorithmic properties will be examined later in the course 13 4 13 5 132 SOLUTION BY DIRECT INTEGRATION METHODS 132 SOLUTION BY DIRECT INTEGRATION METHODS There are two distinct direct time integration methods explicit and implicit integration formulas We summarize their computational sequences below 1321 Central Difference Method for Undamped Case D 0 First we eXpress the acceleration vector ii froml3 l as iit M1 f Kut 1310 Hence it is clear that if the mass matriX is diagonal the computation for obtaining the acceleration vector would be greatly simpli ed We now describe direction time integration by the central difference method Initial step Given the initial conditions 110 u0 ft obtain the velocity at the half step t h h At by 110 M1 f0 Ku0 ugh 2 110 h 110 1311 uhgt u0gt h new Subsequent steps 13 5 Chapter 13 TRANSIENT ANALYSIS OF STRUCTURAL SYSTEMS 13 6 Troml Z hnmax for n l nmax 1114 M 1 W Kun 1104 1104 h iin un 1 2 un h um 1312 end 1322 The Trapezoidal Rule for Undamped Case D 0 This method is also referred to Newmark s implicit rule with its free parameter chosen to be or 8 14 Among several ways of implementing the trapezoidal integration rule we will employ a summed form or halfinterval rule given as follows um um a 101 un un h um 1313 am 1 21104 um un 1 2 2un un In using the preceding formula one multiply the rst of l3 13 by M to yield Mun MunhMun i 1314 13 6 13 7 132 SOLUTION BY DIRECT INTEGRATION METHODS The term M iin I in the above equation is obtained from 131 as Miin fn I Dun KunI 1315 which when substituted into 1314 results in Mun Mun 91 fn Dun Kun U 1316 M hD um a M um a fn a Kun 5 Now multiply the second of 1313 by M I hD to obtain M 911 un 5 M 911 un h M hD um a 1317 Third substitute the second term in the righthand side of 1317 by 1316 one obtains M 91D W a M am mm h M um h fn a mm a u M hD h2K un a M un a 1104 a Dun 102 fn 1318 13 7 Chapter 13 TRANSIENT ANALYSIS OF STRUCTURAL SYSTEMS 13 8 Implicit integration steps Assemble A M hD h2K Factor A LU for n 0 nmax bn M un y 1104 h Dun 902 fn un A 1bn where A 1 U 1 L 1 um un unh 1319 am 1 2un un un 1 2un un end 133 DISCRETE APPROXIMATION OF MODAL SOLUTION The modalform solution of the equations of motion for linear structures given by 137 and 138 involves the convolution integral of the applied force For general applied forces an exact evaluation of the convolution integral can involve a considerable effort To this end an approximate solution is utilized in practice To this end 137 is expressed in discrete form at time t nh nh xinh eAi quot 1 x 0 eA quoth T bpit d 1320 0 13 8 13 9 134 ILLUSTRATIVE PROBLEMS Likewise at time I nh h x nh h is given by nhh xnh h eA quotW1 x0 eAquot quotWquot bin0dr 0 nh eA heA quothx0 eAiltquoth Tgtbpiltrgtdr 0321 0 nhh eAi nhh r h pi t dt nh The bracketed term in the above equation is xnh in View of 1320 and the second term is approximated as nhh A h h nhh A h h e quotquotTb itdt e quot7 611 b nh fnh p nh p 1322 ArllteAih I bpiltnhgt Substiutting this together the bracketed term by x nh into 1321 x nh h is approximated as IXznh hgt eAih xltnhgt A 1lteAquoth I b mam x qzrnh 11 c1 ltnh M 1323 Once xinh h is computed the physical displacement unh h is obtained Via the modal summation expression 138 13 9 13 11 134 ILLUSTRATIVE PROBLEMS It should be noted that the end condition w0 t wL t 0 is equivalent to kwl gt oo sz gt 00 However in computerimplementation itisimpractical to use kwl gt oo sz gt 00 due to limited oating point precision The applied force chosen are Step load fL2 t 100 0 5t 13 26 Sinusoidal load fL2 t sin2nff l i where the forcing frequency is set to ff 15w1 2 7139 with col being the fundamental frequency of the model problems Figures 1323 illustrate time responses of a beam with boundary springs subject to unit midspan step load The responses by solid red lines are those obtained by using the central difference method the ones with are by the trapezoidal rule and the blue lines by the canonical formulal323 The step increments used for the three methods are 15861E 5 for the canonical formula 1327 18928E 7 for the central difference method h 163445E 5 for the trapezoidal rule It should be noted that the stepsie for the central difference method is dictated by the computational stability whereas that of the canonical formula and the trapezoidal rule by accurcy considerations We hope to reVisit this issue later in the course 13 11