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yC 1 2

S S 1

y t y t ...

y t

=

( ) ( )

( )

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

×

(3.175)

In many cases, only the first modal response is used, which is called the fundamental modal response to represent the displacement, that is,

x t

( )

^≈u1 1y t

( )

(3.176)

and,

x t

( )

^{≈u1 1}y t

( )

(3.177)

3.3.2  m^odaL contriBUtion i^ndicator

In order to determine if a specific mode should be used, criteria that are generally referred to as modal participation indicators or modal contribution indicators are needed. That is, these indica-tors will be used to establish if a specific peak response of a structure is sufficiently accurate when truncated modal superposition is used. The number of truncated modes is determined by the values of the indicators. For this purpose, Wilson (2004) suggested the modal mass ratio, which is helpful for proportionally damped systems. Chopra (2006) suggested another indicator called the modal contribution factor. In the following discussion, practical approaches that will be valid for both proportionally and nonproportionally damped systems are explored.

Before the discussion on the selection of various parameters that can be used as the modal contribution indicators, the criterion of selection is first described. There are two basic criteria for selection of the indicators.

3.3.2.1  Theory of the Indicator

The existence and application of an indicator should be scientifically sound, which means that the possible candidates should be mathematically consistent and legitimate.

Chopra (2006) states that the modal contribution indicators should have the following three properties:

i. The indicators should be dimensionless.

ii. The indicators should be independent of how the mode shapes are normalized.

iii. The sum of the modal contribution indicators over all modes should be unity.

As a matter of fact, these three properties have the same necessary essence. Namely, the indi-cators should be referenced by a given standard and the most convenient standard is unity. For example, the quantity of the modal participation factor cannot be directly used as the indicator. This is because it cannot satisfy the above conditions. Thus, it is necessary to find alternative quantities to determine the number of modes for the modal truncations.

However, the quantity used as an indicator of a specific mode should also be an amount of the percentage quantitatively describing the magnitude of the contribution. That is, if a specific mode has a larger contribution, the indicator should be proportionally larger. It is understood that the structural responses under earthquake excitations are dynamic quantities, which are the result of the convolutions of the ground excitations and the imposed response functions of structures.

Practically speaking, several additional modes are often acquired to improve the accuracy of the response computation, namely,

S S S= P+ f<<n (3.178)

where S is the number of truncated modes in the response computation, S_p is the number calcu-lated from various modal participation indicators, and S_f is the number of a few additional modes.

Usually,

Sf≥1 (3.179)

The greater the irregularity of a structure, the larger the S_f that should be considered. The con-cept of structural irregularity will be discussed later.

Using Equation 3.179, the burden of considering the influence of the excitations is removed.

Thus, a single equation can be used to cover the essence of conditions (i) to (iii) listed above. Note that is, if the i^th generic modal participation indicator is denoted as γi, then

γi i 1

n

1

∑

= ^{= (3.180)}

Using the quantity γi, the idea that the larger the value of γi, the greater the contribution should be, is explored, and practically, individual modes may not be counted; instead, the summation of the first several indicators can be compared and a preset value G can be used as the criterion, that is, if

γ_i

i 1 Sp

∑

= ^{≥G (3.181)}

then the number Sp is specified.

Yet, the equations or the above-mentioned three requirements are not sufficient. In addition, it is best that the indicators are all nonnegative numbers. That is,

γi≥0, i=1 2, , , S (3.182)

First, this is because the modal participations or contributions are counted by using the concept that the larger the value of γi, the greater the contribution. This concept therefore implies the use of absolute values. Secondly, if some of the indicators become negative, then the summation of the first

S_p modal participation indicators will not monotonically increase, and will not be convenient to use.

That is, if there are two numbers for the required modes, namely, S₂ and S₁, it may be required that

The second aspect for the selection of indicators is related to using the indicator in practical design.

First, it is noticed that to compute different types of responses, different numbers of modes may be needed to guarantee the accuracy of modal truncation.

Secondly, it is understood that computing responses at different locations may require different numbers of modes to guarantee the accuracy of modal truncation.

It is also noted that the consideration of accuracy of modal truncation is related to the dynamic behavior of a structure. The response due to the Duhamel convolution contains two factors, namely, the structure itself and the external excitation. In this case, any indicator that does not involve the factors of the ground excitation cannot be absolutely precise. Instead of seeking more accurate modal contributions by including earthquake, as well as other excitations, Equation 3.178 is used and several additional modes are included.

In addition, to calculate the modal participation indicator, the fewer pieces of information needed, the easier it will be to obtain the quantity.

3.3.2.2.1  Modal Mass Ratio

Having discussed the criteria for selecting the modal contribution factors, one of the oldest param-eters, the concept of modal mass ratio (Wilson 2004), is considered. It is defined as

is the total mass of the structure. Here, it is noted that the mass matrix is taken to be diagonal (see Equation 3.187). Let

which is called the effective mass of the i^th mode. Note that if the generic mode shape u_i is used to replace ui, Equation 3.186 is still valid.

It can be seen that the modal mass ratio will satisfy all four conditions described in Equations 3.180 through 3.183. Plus, to obtain the modal mass ratio, only the mass matrix and the mode shapes

of the first few modes are needed. Thus, the modal mass ratio can be a good index to indicate the pro-portion of the contribution of the corresponding mode. It is also easy to use for practical engineers.

In later chapters on practical damper designs, a preset value to compare the first S summations of the modal mass ratios is provided. The idea is that if the modal mass ratio is large enough, which implies that this particular mode will contribute significantly to the total responses, the mode should be considered. Otherwise, it can be dismissed.

In more general cases, there can also be complex-valued mode shapes or complex modes, as well as overdamped pseudo modes. The corresponding modal participation factor and modal mass ratio will be more complicated. However, Equation 3.180 will still hold, although conditions described in Equations 3.181 through 3.183 may be violated.

Note that in practice, it is often assumed that the mass matrix is diagonal, that is,

M =^{diag m}

( )

j j 1, 2, , n^{= …} (3.187)

Equation 3.187 is used in NEHRP 2000 (BSSC 2000) and many other codes. Mathematically, however, Equation 3.187 is not a necessary condition to use in defining the modal participation fac-tor and modal mass ratio. Thus, in the following derivations, a diagonal mass matrix is not required, unless specifically stated.

3.3.2.2.2  Other Indicators

In addition to the modal mass ratio, several other kinds of indicators exist. For example, the modal contribution factor γCi is given by

γCi nst

rst

= r (3.188)

where r_n^st and r^st are respectively the static response of the i^th mode and of the total external force, (Chopra, 2006).

As a brief summary, modal truncation can save significant computational time and provide suf-ficiently accurate response estimations, with the proper number of selected modes, which can be determined by modal contribution indicators. The modal mass ratio is a comparatively better and simpler criterion; this indicator is used in the practical damper designs discussed in Chapters 7 and 8.

3.3.3  r^EsPonsE c^{omPUtation of} t^rUncatEd m^odaL sUPErPosition

In this section, examples are used to demonstrate the procedure of modal truncation as well as to compare the modal contribution indicators described above.

3.3.3.1  Computation Procedure

Suppose the number S is obtained from Equation 3.178 through a certain quantity of the modal participation indicators. It is now possible to compute the structural responses by the truncated S modes. In the following discussion, a method to carry out the truncated modal superposition is explained in detail. The response computation is also used to compare the above-mentioned modal participation indicators to examine their accuracy.

Note that the modal truncation discussed here is for proportionally damped systems only. In other words, all the modes of concern are normal modes. The system with complex modes and/or overdamped subsystems will be discussed in Chapter 4.

The procedure of modal decoupling for proportionally damped systems was theoretically explained in the previous section. It was shown that, generally, the decoupled systems, namely, the individual modes, can be used to compute the modal responses, such as described in Equation

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