NATURAL MODES OF VIBRATION OF BUILDING STRUCTURES CE 131 — Matrix Structural Analysis Henri Gavin Fall, 2006 1
Mass Mass and and Stiff Stiffne ness ss Matr Matrice icess
Consider a building frame modeled by a set of rigid, massive floors supported by flexible, flexible, massless columns. columns. This provides provides the simplest represen representatio tation n of a building building for the purposes of investigating lateral dynamic responses, as produced by earthquakes or strong winds. The lateral position of mass i with respect to the ground will be given the variable ri (t), ki is the lateral stiffness of the columns in story i, and the mass of mass i is mi . For a three-story building, this kind of representation is shown in Figure 1. m k
2
r
1
r
2
2
m k
3
3
m k
r
3
1
1
Figure 1. A simplified model of a building frame with massive rigid floors and light flexible columns.
Exercise 1: Show that the mass matrix and stiffness matrix for this three-
story building can be written:
m = 0 0
1
M
0 0 m2 0 0 m3
k +k = −k 0 1
and
K
2
2
−k 0 k + k −k −k k 2
2
3
3
3
3
.
(1)
For an n-story building modeled in this way, the mass and stiffness matrices are
m 0 = . ..
1
M
0
0 m2
··· ···
··· .. . ... 0
0 .. . 0 mn
(2)
2
Natural Modes of Vibration
and
k +k −k 0. .. = ... 0... 1
2
2
K
2
−k
0 k3
2
k2 + k3
−k
3
0 .. . .. . 0
···
··· ···
0
−
k3 + k4
...
−k
4
...
...
...
...
...
.. . 0
...
−k
4
0 .. .
...
0
−k
n−1
−k
kn
n−1
1
−
0
···
0 0
··· ··· ··· + kn kn
−k
n
kn
−
.
(3)
Coupled Second Order Differential Equations The coupled n second order differential equations can be written in matrix form as: M¨ r(t) + Cr˙ (t) + Kr(t) = f (t) ,
r˙ (0) = v ,
r(0) = d , o
o
(4)
where C is a symmetric non-negative definite damping matrix and f (t) is a vector of n external horizontal forces applied to the n masses. Exercise 2: Write out the three ordinary differential equations for 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 or more adjacent floor displacements and because of this, the three differential equations are inter-related or coupled .
3
Natural Modes
For the time-being, assume that the structural system has almost no damping and no external forcing. In this case M¨ r(t) + Kr(t) = 0 ,
r(0) = d , o
r˙ (0) = v , o
(5)
and one may presume that the natural responses will be sinusoidal with frequency ωn rad/s and a vector of amplitudes ¯r, ¯r = [¯r1 r¯2 r¯3 ]T . Substituting the displacements r(t) = ¯ r sin ωn t ,
and accelerations ¨r(t) =
2 n
−¯r ω
sin ωn t ,
into equation 5 and eliminating sin ωn t we obtain, K¯ r
2 n
− ω M¯r = 0 ,
which may be re-written as the generalized eigenvalue problem, [K
2 n
− ω M]¯r = 0 .
(6)
3
Natural Modes of Vibration
The square of the natural frequencies are the eigenvalues and the amplitudes of natural vibration are the associated eigenvectors. As long as M and K are positive definite, the natural frequencies will be positive. A planar building frame with n rigid floor masses will have n natural frequencies , ωni , and n natural mode shapes , ¯ri , i = 1, . . . , n. For a natural frequency ωni and natural mode shape ¯ri satisfying equation 6, this equation may be pre-multiplied by ¯rT i to obtain ωn2i
¯rT K¯ ri = T i , ¯ri M¯ri
(7)
which is called the Rayleigh quotient for mode i. The natural modes are mass-orthogonal and stiffness-orthogonal. This means that ¯rT r j = i M¯ and ¯rT r j = i K¯
0 i=j mi i = j ∗
0
ki
∗
i=j i=j
,
so that ωn2i = ki /mi . ∗
∗
The n natural mode vectors ¯r1 , . . . , ¯rn may be arranged column-wise into a modal ¯, matrix , R ¯ = [¯r1 . . . ¯rn ] . R 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. For the combinations of mass and stiffness shown in Table 1, use the module to determine natural frequencies and natural mode vectors. Write the three natural frequencies and sketch the three mode vectors for the six cases shown in Table 1.
Table 1. Six cases of mass and stiffness distribution. 1 2 3 4 5 6 case:
m1 m2 m3 k1 k2 k3
units
10 10 10 100 10 10 ton 10 10 10 10 100 10 ton 10 10 10 10 10 100 ton 100 1000 1000 1000 1000 1000 N/mm 1000 100 1000 1000 1000 1000 N/mm 1000 1000 100 1000 1000 1000 N/mm
4
Natural Modes of Vibration
4
Proportional Damping
In general, mode vectors that are mass-orthogonal and stiffness-orthogonal will not also be damping-orthogonal. In many lightly-damped structures, however, the damping may be approximately modeled by a matrix that is proportional to mass and stiffness, C = αM + β K .
(8)
This representation of damping is called Rayleigh damping or proportional 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 proportional to the mass and
stiffness matrices, then ¯rT r j = i C¯
5
0 ci = αmi + βk i ∗
∗
∗
i=j i=j
.
Modal Coordinates
At any point in time, the lateral displacement of the floor masses is given by the vector r(t), r(t) = [r1 (t), r2 (t), , rn (t)]T . Because the set of natural mode vectors fills the n-dimensional space of floor displacement vectors, the floor displacement vectors can be written as a weighted sum of the natural mode vectors
···
r(t) = ¯ r1 q 1 (t) + ¯ r2 q 2 (t) +
or
r (t) r¯ r¯ r (t) = r¯ q (t) + r¯ r ...(t) r¯... r¯... r (t) r¯ r¯ r (t) = r¯ r¯ r ...(t) r¯... r¯... 1
11
2
21
12 22
1
n1
n
or
or
n2
1
11
12
2
21
22
n
n1
n2
· · · + ¯r
n
q n (t) ,
r¯ q (t) + · · · + r¯ q (t) , r¯... · · · r¯ q (t) · · · r¯ q (t) .. .. . . . .. · · · r¯ q (t) 1n 2n
2
n
nn
1n
1
2n
2
nn
n
¯ q(t) r(t) = R
(9) The vector q(t) is called the vector of modal coordinates . In a free vibration, q(t) are sinusoidal functions with a single frequency, q 1 (t) oscillates only at the first natural frequency, ωn1 , q 2 (t) oscillates only at the second natural frequency, ωn2 , 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 coordinate vector represent the amount of each mode present in the total response. ¯ T MR ¯ and R ¯ T KR ¯ are diagonal Exercise 6: Show that the n by n matrices R matrices.
5
Natural Modes of Vibration
6
Un-coupled Second Order Differential Equations Substituting equation 9 into equation 4 results in ¯q ¯ q˙ (t) + KRq ¯ (t) = f (t) , ¨ (t) + CR MR
¯ q(0) = R
1
−
¯ q˙ (0) = R
d ,
1
−
o
v . o
Pre-multiplying both sides of this equation by the transpose of the modal matrix results in: ¯ T MR ¯q ¯ T CR ¯ q˙ (t) + R ¯ T KRq ¯ (t) = R ¯ T f (t) , ¨ (t) + R R
¯ q(0) = R
1
−
¯ q˙ (0) = R
d ,
1
−
o
v . o
Because the modal matrix is mass-orthogonal and stiffness-orthogonal, and assuming the modal matrix is also damping-orthogonal (e.g., the damping is proportional), then the equation above may be written
m1
0
0 . .. 0
m2
∗
∗
··· ···
or
···
.. . .. 0
.
0 .. . 0 m
∗
n
q¨1 (t) q¨2 (t)
.. . q¨ (t)
+
n
c1
0
0 . .. 0
c2
∗
.. .
∗
..
. 0
··· ···
r¯11 r¯12
r¯21 r¯22
.. .
.. .
r¯1
r¯2
n
0 .. .
···
q˙1 (t) q˙2 (t)
.. . q˙ (t)
0
n
c
∗
n
··· ···
r¯ r¯
n1 n2
.. .
··· n
···
r¯
nn
f 1 (t) f 2 (t)
.. . f (t) n
+
k1
0
0 . .. 0
k2
∗
∗
··· ···
···
.. . ..
. 0
0 .. . 0 k
∗
n
q1 (t) q2 (t)
.. . q (t) n
,
mi q¨ i(t) + ci q ˙i (t) + ki q i(t) = ¯rT i f (t) ∗
∗
=
∗
(10)
for each mode i = 1, . . . , n. This represents n un-coupled second order differential equations in terms of the modal coordinates q i(t). All of the solutions pertaining to a single degree of freedom oscillator are relevant to equation 10. Diving both sides of equation 10 by mi , 1 T c k ¯r f (t) , q¨i (t) + i q ˙i (t) + i q i (t) = mi mi mi i or 1 T ¯r f (t) , q¨i (t) + 2ζ i ωni q ˙i (t) + ωn2i q i (t) = mi i ∗
∗
∗
∗
∗
∗
∗
where ζ i is the damping ratio associated with mode i, and c ci ζ i = i = . 2 mi ki cci ∗
∗
∗
√
∗
∗
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 for cases 2, 4, and 6 shown in Table 1. Does increasing α increase the damping in the lower-frequency modes or the higher-frequency modes? Does increasing β increase the damping in the lower-frequency modes or the higher-frequency modes?
6
Natural Modes of Vibration
7
Initial Displacements and Free Response
If the initial displacements, d are proportional to the i-th natural mode vector, ¯ri , 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. o
Exercise 8: Use the WEAVE module entitled “Building Vibration - Natural
Modes” to investigate this property of natural modes. For a fixed distribution of mass and stiffness, set the initial displacement proportional to each of the three mode shape vectors, 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 short paragraph.
8
Explore! Exercise 9: Use the WEAVE module entitled “Building Vibration - Natural
Modes” to explore the effects of very large and very small values of mass, damping, and stiffness. What happens if you increase α and/or β so that the damping is more 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 ?