STRUCTURAL DYNAMICS, Civil Engineering, Rice generalized modal evaluat - PDF document

STRUCTURAL DYNAMICS, Civil Engineering, Rice generalized modal evaluating the dynamic response damped linear Graduate Student. when used with the response has had for the the specification considerable uncertainty, practical applications. There are instances, however, more refined analysis that the generalized procedure be improved. objectives are: implementation of the modal method the dynamic response of non-classically the elements of this the attractiveness the method use by, then applied force-excited systems. Following the natural alternative forms, the more with the for single-degree-of- systems," is obtain the solution are: the physical of the pairs of characteristic vectors, the building blocks the definition of the free vibrational responses of some useful relationships vectors in the the analysis and the others that the transient multi-degree-of-freedom system the true series of similarly excited single-degree-of-freedom systems, the analysis be implemented with only effort beyond that required of the same comprehensive numerical solutions are solutions also are compared an approximate modal superposition classical modes of vibration," the two use is the derivation of the presented herein, final expressions of real-valued this paper linear cantilever system be excited the acceleration of which is the column of the moving base; differentiation with respect a column vector of ones; ones; [c] and [k] are the mass matrix, stiffness matrix of the symmetric; additionally, additionally, and [k] are positive semi-definite, semi-definite, is positive definite. vibration, the right-hand admits a of the a characteristic the homogeneous one obtains the characteristic characteristic + r[cl+ Ckl) {$I = (01 in which {0} is the null vector. Rather than through the solution system of desired values and the first reducing suggested in References characteristic value problem of the the and [ B] are symmetric, real matrices of vector of elements, of the lower elements represent elements represent may be Provided the of damping pairs with zero real parts. system with freedom, there of characteristic pair there corresponds a conjugate pair a pair of characteristic values defined the associated pair these expressions, real positive real-valued vectors of elements each. characteristic values the following values of associated vectors, the corresponding are the the characteristic of a system with undamped circular Furthermore, equation the same relating the frequencies of such a second section In the following developments, be referred undamped circular frequency of be referred to as the corresponding damped modal damping a function the corresponding frequency of Where confusion the symbol undamped and classically damped valued. Accordingly, an undamped the characteristic purely imaginary system satisfies the the [MI - ' PI = Ckl Cml - ' CCI (11) the natural modes are real-valued and equal to those of the associated undamped system. The exponent in this expression denotes the inverse of the modulus of each pair, frequency of of damping," damping," is proportional to either [m] or [k] or is a linear combination of the two, special case of modes, satisfies characteristic vectors distinct characteristic values, including these expressions is reviewed the third section of Appendix substituting this equation that the the resulting expression must be system for which which {$k} (14) {$j {$k} = 0 for rk # rj (15) {$j )'[kl {$k} = for rk # 'j (16) Furthermore, on substituting the latter equation into equation be recalled that the quantities expressionsare the those for undamped system. sensitivity of free vibrational characteristics the magnitude the damping for the three-storey building be of shear-beam type with uniform storey indicated. System the form several different these systems not satisfy characteristic values these systems of a standard computer reviewed in first section Systems considered different values are the characteristic vectors such that the the corresponding The components characteristic vectors system with the damper the bottom are also plotted Note that Free vibrational characteristics Quantity Floor First (a) For (b) For 1.9968 -0.22331 system considered values, particularly for the particularly sensitive Figure 3 a function from equation the damped than the necessarily lower frequencies of those corresponding mode, they may well be increases with increasing and that high values of the corresponding values of it is the damped, that are modal damping factors for each of these affected differently a change that an modal damping in Figure values of the assumption that the system also diagonalizes system in this undamped natural modes systems considered Damper in Top damping on triple matricial matricial {4j}, the terms denotes a transposed This approach, the subject is generally significant differences larger values of Selected values damping factors system in frequency values, modal solution a solution represented a linear combination of pair of associated vectors. modal solution given by constant and damping factors 1.4138 1.4138 00236 00236 1.4127 1.4142 1.3425 1.3744 02647 02357 the right-hand member of is the complex sum of the imaginary expression vanishes (18) reduces reduces {t,bj}erj'] (19) in which the quantity that a phase (19) entirely alternative forms of the identity between trigonometric functions, may be be {qjj } cos (pj t + Bj ) - {xi } sin (pi t + Oj)] (21) 2Cj{t,bj} = {Bj}+i{yj) (22) Alternatively, if one first evaluates the real-valued vectors, then substitutes into equation the identity trigonometric functions, e-~JpJr[{/3j}cospjt -{yj}sinpjt] (23) represent the exponentially decaying harmonic motions circular frequency, a damping factor, one-quarter the different configurations. result, each of the harmonic motion, not remain continuously, repeating the system. be seen when sin as the reference configurations, this fact a subsequent development. undamped systems, (21) reduces Such systems can execute simple harmonic motions in fixed, systems satisfying (21) reduces same expression as that Such systems can invariant configurations, with exponentially decaying the system initial excitation is given the modal solutions preceding section. In particular, the be expressed either the form form {bj} cos(pjt + oj) - {xi} sin(pjt + oj)] (27) j= 1 or n {x} = C e-[ipjr [{~j}cospjt-{~j}sinpj~] j= 1 The complex-valued participation factors, prescribed vector initial displacements vector of initial derivation of this fifth heading values of the constants and the following generalized version of is valid (see and on on (~(0)) (30) in which well-known expressions:" should be interpreted real-valued mode of the material convenient of the initial displacements velocities of sixth section values of kth, vanish, and that is given given { $k} erk'] 2 Ck = b,, + id, (36) (37) To clarify the meaning this expression equations (36)and one obtains It should now that any initial displacement excite only the kth vibration of the initial velocities needed excite only the corresponding initial displacements from equations result is the proportionality factors from equation and the displacements of time may may be expressed directly in in ({ 4k } -dk { xk} Ices Pkt - fdk {+k) analysis of of the system, it is a uniform of initial displacement changes, for a unitary set of initial value is displacements of system may from the following expression deduced deduced j= 1 If the product now expressed form analogous real-valued vectors with units of time per radian, equation last two expressions crucial final now be taken. impulse response as the response unit initial system with this function and its first derivative on making use functions multiplying now replaced corresponding expressions in which the course of this study, this transformation has also been it differs from the latter cosine functions, it the functions have clear physical meanings; and (b) reference configurations the configuration the instant for transient response The response of the system from the vibration presented preceding section acceleration of change of time interval moving base is differential displacements velocitychangesare then obtained from equation equation (-z)+ {&}hj(c -z)]Xg(z)dz j= 1 The displacements prescribed base The quantity represents the instantaneous pseudovelocity of circular frequency damping factor prescribed excitation; the corresponding deformation relative velocity of the system, respectively. multi-degree-of-freedom system be expressed pairs of first member of such pair variation of which is whereas the second member represents the temporal which is the same that of functions of the and are stepwise numerical evaluation of of a relative velocity, is normally associated pseudovelocity value, multi-degree-of-freedom system have analysis of such a system may implemented with only effort beyond in terms of and true relative velocity dimensionless vectors defined the relationship may then the vectors the pseudo-acceleration defined by by )()+ {/Ij”)pjDj(t)I (60) j= 1 Eflect of non-zero initial conditions Implicit in the foregoing development the assumption system with non-zero initial conditions, equation equivalent versions defined the addition of the free vibrational solution classically damped damped J( 1 - (3 )], in which For classically damped systems, for which the and that the well-known expression on the that the latter deformation function, participation factors steps involved the analysis of the transient may be Evaluate the and from determine the damped of the the participation factors, Evaluate the complex-valued application of equation compute the the vectors response of single-degree-of-freedom systems the pseudovelocity the true relative velocities, from equation and the corresponding storey deformations from the subscript displacements may from equation response vectors (60) satisfy in which represents the displacements of the structure forces associated with of the solution. from equations noting that, for from the first derivative of the fact high-frequency limiting maximum values zero, respectively; Illustrative example response of the system shown this section for two different excitations displacement pulse shown which the acceleration trace consists of sequence of the same peak values of the record of peak values of the acceleration, displacement of the latter motion assigned the values of values of these systems associated values of Simple base participation factors with the associated vectors These results, characteristics of for the various quantities the extreme right-hand the table. the information system may from any may then histories of storey deformation for systems subjected displacement pulse the solid lines in different values of the the fundamental undamped frequency of the system in per unit of time, systems with Centro earthquake each case, the response of the system considered a base excitation mode Common 1.2163; 0.1295 -0.27371 0.0290 -01093 +02485i 2.4378 0.0531 -01544 00645 01077 -00192 00401 -00127 -00256 -0.0075 -0.0272 -00243 -00491 -00384 00800 -00466 0629Oi 0.1751 -0.17961 0.0071 -00704 -00199i 1.1835 -0.3645 04391 03502 -00273 00143 25523 00851 -03278 -01772 -04564 -02467 -00248 01738 -00145 00076 0-0823 0-0096 -01273 -00070 Interfloor deformations a period exceeds the duration the excitation the fundamental undamped are the the natural undamped system. I I Interfloor deformations for system Figure I(a); system 2 subjected simple base Centro earthquake from Figures the damping generally larger floor displacements, particularly the structure higher modes of vibration important than for the lower parts. These trends be seen response spectra the absolute maximum floor displacements, and the corresponding These results, for systems with respect maximum value systems with real-valued characteristic values The information presented which all characteristic values (or roots) and associated characteristic vectors this case modal solution exponentially decaying general, there real-valued negative each associated with real-valued characteristic vector. of this section explain how be handled in forced vibration pair of such quantities that Response spectra for maximum displacements and storey deformations for system shown system with adding equations it is following expression may be equation (68b) from The modal solution of real-valued is given the sum exponentially decaying forced vibration, the functions the deformation relative velocity the prescribed system with the frequency following modified latter expression and the resulting motion from equation real-valued characteristic the associated real-valued characteristic given in section of Appendix Response spectra for maximum displacements and storey deformations The method systems presented preceding sections can, with lateral forces, following development, these forces in which vector with dimensionless time function. forces considered system is changes induced infinitesimal time dz. These be determined impulse-momentum relationship relationship ()= [m]-'{P}g(?)dz (77) The complex-valued participation factor for the resulting free vibration [equation (26)] steps taken corresponding solution the following expression for the in which The quantity displacement, whereas the function The latter function represents normalized displacement the natural and damping are the prescribed multi-degree-of-freedom system, is excited force of the same the static the system induced peak value of applied force. in which The vectors the dimensionless amplification function normalized pseudo-acceleration steady-state response of non-classically harmonic forces motion. However, modal superposition this paper systems having large number its application this section. exciting forces circular frequency first derivative derivative )=Ajsin(wt -Oj) and In the latter expressions, expressions, )= ~A~COS(CU~ -Oj) and 0, = tan-’ (3) in which 0 Oj II. Let xi([) be the displacement of the ith floor of the system and a$ and B$ be the corresponding substituting equations Further, on introducing the quantities quantities ()+ @jPt)’] Eij = tan-’ (%) equation (87) may lie in the range maximum value be determined steady-state response system shown in harmonic force applied to the The natural modes of damping factors part (d) for the forcing function upper part latter quantities force-excited systems Floor First mode Third mode Values of 2 -02527 -03123 04688 -02527 -00014 01607 -00470 -03040 00156 00658 00527 Values of Values of 05574 08580 -07631 06463 09338 01824 06082 07938 -09016 -04325 variation of exciting force that the exciting frequency the frequency ratios, phase angles defined displacement amplitudes phase angles defined values of sin The maximum displacements from equation in which peak value of applied force. maximum interfloor maximum displacements exciting frequencies, where they the undamped natural modes of Note that two sets of for maximum displacements and storey deformations for system in Figure harmonic force at the response of damped linear system minor computational response of the deformations and true relative velocities of series of similarly Comprehensive numerical have been maximum response system parameters, use of classical on the system itself, the approximate solution This study research project structures sponsored project has been under the administrative Damodaran Nair, is acknowledged with thanks. following first first (2) + [BI (2) = { yo) 1 ('41) in which [A] and [I?] are matrices of size 2n by 2n given by and {z) and { Y(t)) are vectors of of -Cml [O] [k] loll The solution of the homogeneous form of equation (Al) may be taken as where r is a characteristic value and { Z} is the associated characteristic vector of desired modal displacements, the upper elements represent the homogeneous form of obtains the characteristic value {$j}erJr into the homogeneous homogeneous ($i 1 = {O} and premultiplication E. VENTURA a transposed vector. Each the three represents a a {$j } (AW (AW (AW equation (A7) can be written as mfrf+cfrj+kf =O which is recognized to be the characteristic or frequency equation for a single-degree-of-freedom system with usual manner the roots any pair distinct characteristic and rk the orthogonality relations relations and [B] in equation (4) are real symmetric matrices, characteristics vectors vectors �iZk = 0 (A 14) and These relations for a conjugate pair making use for classically damped systems damped systems, these facts, the transpose of the symmetry the participation from the follows. Let the vector of the of this this and making use of the orthogonality condition all terms resulting expression, use of (A2), (A5) conditions that When expressed the vectors be written written = +Ckili(Zk} (A241 The participation then be determined from equation orthogonality condition this expression vanishes, except when which case Systems with associated characteristic motion represented two real-valued characteristic vectors, the quantities constants that can be use of relationship between exponential hyperbolic functions, be rewritten rewritten { Bj } cosh pjt - {yj } sinh pjt] where 'A28) (A291 For a system subjected to a set of unit initial the vectors be replaced then reduce the impulse function for an overdamped single-degree-of-freedom system that equation pseudo-acceleration of a participation factor base-excited system factor for force-excited system matrix of system a base-excited impulse response it indicates mode under consideration stiffness coefficient; stiffness matrix of system mass matrix number of degrees a system circular frequency of mode, respectively of associated characteristic value single-degree-of-freedom system period of mode of associated of excitation vector of interfloor maximum value of vector of displacements relative moving base for a displacements for force-excited system initial displacements initial velocities system, respectively of ith maximum value of of the characteristic vector a force-excited system LINEAR SYSTEMS modal damping factor configuration defined modal configurations modal configurations modal configurations natural mode, characteristic vector natural mode H. 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