NATURAL MODES OF VIBRATION OF BUILDING STRUCTURES CE 131 Matrix Structural Analysis Henri Gavin Fall, 2006

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1 NATURAL MODES OF VIBRATION OF BUILDING STRUCTURES CE 131 Matrix Structural Analysis Henri Gavin Fall, Mass and Stiffness Matrices Consider a building frame modeled by a set of rigid, massive flos suppted by flexible, massless columns This provides the simplest representation of a building f the purposes of investigating lateral dynamic responses, as produced by earthquakes strong winds The lateral position of mass i with respect to the ground will be given the variable r i (t), k i is the lateral stiffness of the columns in sty i, and the mass of mass i is m i F a three-sty building, this kind of representation is shown in Figure 1 m 3 r 3 k 3 k m m 1 r r 1 k 1 Figure 1 A simplified model of a building frame with massive rigid flos and light flexible columns Exercise 1: Show that the mass matrix and stiffness matrix f this threesty building can be written: m k 1 + k k 0 M = 0 m 0 and K = k k + k 3 k 3 (1) 0 0 m 3 0 k 3 k 3 F an n-sty building modeled in this way, the mass and stiffness matrices are m m M = m n ()

2 Natural Modes of Vibration and K = k 1 + k k 0 0 k k + k 3 k k 3 k 3 + k 4 k k 4 k n 1 0 k n 1 k n 1 + k n k n (3) k n k n Coupled Second Order Differential Equations The coupled n second der differential equations can be written in matrix fm as: M r(t) + Cṙ(t) + Kr(t) = f(t), r(0) = d o, ṙ(0) = v o, (4) where C is a symmetric non-negative definite damping matrix and f(t) is a vect of n external hizontal fces applied to the n masses Exercise : Write out the three dinary differential equations f n = 3 using the mass and stiffness matrices of equation 1 and a diagonal damping matrix Convince yourself that each of these three differential equation involves two me adjacent flo displacements and because of this, the three differential equations are inter-related coupled 3 Natural Modes F the time-being, assume that the structural system has almost no damping and no external fcing In this case M r(t) + Kr(t) = 0, r(0) = d o, ṙ(0) = v o, (5) and one may presume that the natural responses will be sinusoidal with frequency ω n rad/s and a vect of amplitudes r, r = [ r 1 r r 3 ] T Substituting the displacements and accelerations r(t) = r sin ω n t, r(t) = r ω n sin ω n t, into equation 5 and eliminating sin ω n t we obtain, K r ω nm r = 0, (6) which may be re-written as the generalized eigenvalue problem, [K ω nm] r = 0

3 Natural Modes of Vibration 3 The square of the natural frequencies are the eigenvalues and the amplitudes of natural vibration are the associated eigenvects As long as M and K are positive definite, the natural frequencies will be positive A planar building frame with n rigid flo masses will have n natural frequencies, ω ni, and n natural mode shapes, r i, i = 1,, n F a natural frequency ω ni and natural mode shape r i satisfying equation 6, this equation may be pre-multiplied by r T i to obtain which is called the Rayleigh quotient f mode i ω ni = rt i K r i r T i M r i, (7) The natural modes are mass-thogonal and stiffness-thogonal This means that { 0 i j r T i M r j = i = j m i and so that ω ni = k i /m i { 0 i j r T i K r j = i = j k i, The n natural mode vects r 1,, r n may be arranged column-wise into a modal matrix, R, R = [ r 1 r n ] Exercise 3: Use the WEAVE module entitled Building Vibrations - Natural Modes to investigate the effect of different mass and stiffness distributions on the natural mode shapes F the combinations of mass and stiffness shown in Table 1, use the module to determine natural frequencies and natural mode vects Write the three natural frequencies and sketch the three mode vects f the six cases shown in Table 1 Table 1 Six cases of mass and stiffness distribution case: units m ton m ton m ton k N/mm k N/mm k N/mm

4 Natural Modes of Vibration 4 4 Proptional Damping In general, mode vects that are mass-thogonal and stiffness-thogonal will not also be damping-thogonal In many lightly-damped structures, however, the damping may be approximately modeled by a matrix that is proptional to mass and stiffness, C = αm + βk (8) This representation of damping is called Rayleigh damping proptional damping Exercise 4: Show that if the units of all terms in C are N/mm/s, the units of M is tons and the units of K is N/mm, then the unit of α is (1/seconds) and the unit of β is seconds Exercise 5: Show that if the damping matrix is proptional to the mass and stiffness matrices, then { r T 0 i j i C r j = c i = αm i + βki i = j 5 Modal Codinates At any point in time, the lateral displacement of the flo masses is given by the vect r(t), r(t) = [r 1 (t), r (t),, r n (t)] T Because the set of natural mode vects fills the n-dimensional space of flo displacement vects, the flo displacement vects can be written as a weighted sum of the natural mode vects r 1 (t) r (t) r n (t) r(t) = r 1 q 1 (t) + r q (t) + + r n q n (t), = r 11 r 1 r n1 r 1 (t) r (t) r n (t) q 1(t) + = r 1 r r n r 11 r 1 r 1n r 1 r r n r n1 r n r nn q (t) + + q 1 (t) q (t) q n (t) r 1n r n r nn q n(t), r(t) = R q(t) (9) The vect q(t) is called the vect of modal codinates In a free vibration, q(t) are sinusoidal functions with a single frequency, q 1 (t) oscillates only at the first natural frequency, ω n1, q (t) oscillates only at the second natural frequency, ω n, and so on The free vibration of the masses, r(t), can involve all the modes of vibration, and can oscillate at all of the natural frequencies The elements of the modal codinate vect represent the amount of each mode present in the total response Exercise 6: Show that the n by n matrices R T M R and R T K R are diagonal matrices

5 Natural Modes of Vibration 5 6 Un-coupled Second Order Differential Equations Substituting equation 9 into equation 4 results in M R q(t) + C R q(t) + K Rq(t) = f(t), q(0) = R 1 d o, q(0) = R 1 v o Pre-multiplying both sides of this equation by the transpose of the modal matrix results in: R T M R q(t) + R T C R q(t) + R T K Rq(t) = R T f(t), q(0) = R 1 d o, q(0) = R 1 v o Because the modal matrix is mass-thogonal and stiffness-thogonal, and assuming the modal matrix is also damping-thogonal (eg, the damping is proptional), then the equation above may be written m m m n q 1 (t) q (t) q n(t) + c c c n r 11 r 1 r n1 r 1 r r n r 1n r n r nn q 1 (t) q (t) q n(t) k k kn f 1 (t) f (t) f n(t), q 1 (t) q (t) q n(t) m i q i (t) + c i q i (t) + k i q i (t) = r T i f(t) (10) f each mode i = 1,, n This represents n un-coupled second der differential equations in terms of the modal codinates q i (t) All of the solutions pertaining to a single degree of freedom oscillat are relevant to equation 10 Diving both sides of equation 10 by m i, q i (t) + c i m i q i (t) + k i q m i (t) = 1 r T i m i f(t), i q i (t) + ζ i ω ni q i (t) + ω niq i (t) = 1 m i r T i f(t), where ζ i is the damping ratio associated with mode i, and = ζ i = c i c ci = c i m i k i Exercise 7: Use the WEAVE module entitled Building Vibration - Natural Modes to determine values of α and β that will give approximately 5 percent damping in the first mode and approximately 1 percent damping in the third mode f cases, 4, and 6 shown in Table 1 Does increasing α increase the damping in the lower-frequency modes the higher-frequency modes? Does increasing β increase the damping in the lower-frequency modes the higher-frequency modes?

6 Natural Modes of Vibration 6 7 Initial Displacements and Free Response If the initial displacements, d o are proptional to the i-th natural mode vect, r i, then the free response ensuing from that initial displacement will consist entirely of the i-th mode, and will have no components from other modes Exercise 8: Use the WEAVE module entitled Building Vibration - Natural Modes to investigate this property of natural modes F a fixed distribution of mass and stiffness, set the initial displacement proptional to each of the three mode shape vects, and observe that the free response consists almost entirely of that mode Now select some other set of initial displacements and observe that the free response contains all three modes Print a few plots of these mode-shape and free response plots and discuss the results in a sht paragraph 8 Exple! Exercise 9: Use the WEAVE module entitled Building Vibration - Natural Modes to exple the effects of very large and very small values of mass, damping, and stiffness What happens if you increase α and/ β so that the damping is me than 100 percent? What happens if α is positive and β is slightly negative, and vice-versa? What happens if one of the stiffness coefficients is much much larger than the other coefficients? What happens if one of the stiffness coefficients is slightly negative? What happens if one of the mass coefficients is very negative?

Control of Earthquake Induced Vibrations in Asymmetric Buildings Using Passive Damping

Control of Earthquake Induced Vibrations in Asymmetric Buildings Using Passive Damping Rakesh K. Goel, California Polytechnic State University, San Luis Obispo Abstract This paper summarizes the results