3. Estructura general del SGSSI en Colombia
3.1. Subsistemas del SGSSI
Ritz vector analysis is an approach, which finds natural frequencies and mode shapes representing the dynamic properties of a structure. The use of Ritz vectors is known to be more efficient than using Eigen vector analysis for calculating such dynamic properties. This method is an extension of the Rayleigh-Ritz approach, which finds a natural frequency by assuming a mode shape of a multi-degree of freedom structure and converting it into a single degree of freedom system.
We now assume that the displacement vector in the equation of motion for a structure of n – degrees of freedom can be expressed by combining p number of
Ritz vectors. Here, p is smaller than or equal to n.
Mu( t )+ Cu( t )+ Ku( t )= p( t ) (1)
p i i i 1 u( t ) ψ z ( t ) Ψz( t ) = =
∑
= (2) where,M : Mass matrix of the structure C : Damping matrix of the structure K : Stiffness matrix of the structure
( )
u t : Displacement vector of the structure with n – degrees of freedom
( )
z t : Generalized coordinate vector ( )
p t : Dynamic load vector
i ψ : i - th Ritz vector ( ) i z t : i - th Generalized coordinate T 1 i p
Ψ= ψ ψ ψ : Ritz vector matrix
From the above assumption, the equation of motion of n – degrees of freedom
can be reduced to the equation of motion of p – degrees of freedom.
( ) ( ) ( ) ( )
Mz t + Cz t + Kz t = p t (3)
where,
T
M Ψ MΨ= : Mass matrix of the reduced equation of motion
T
C Ψ CΨ= : Damping matrix of the reduced equation of motion
T
K Ψ KΨ= : Stiffness matrix of the reduced equation of motion
( ) T ( )
p t =Ψ p t : Dynamic load vector of the reduced equation of motion
The following eigenvalue is formulated and analyzed for the reduced equation of motion:
2
i i i
Kφ=ω Mφ (4)
where,
i
φ : Mode shape of the reduced equation of motion
ωi : Natural frequency of the reduced equation of motion
Using the above eigenvalue solution and assuming the classical damping matrix, the reduced equation of motion can be decomposed into the equation of motion for a single degree of freedom for each mode as follows:
2 ( ) ( ) 2 ( ) ( ) T i i i i i i T Ψ p t q t + q t + q t = Ψ MΨ ξ ω ω (5) 1 ( ) ( ) p i i i z t φ q t = =
∑
(6) where ( ) i q t : i - th mode coordinateξi : i - th mode damping ratio
The eigenvalue solution of the reduced equation of motion, ω , represents an i
approximate solution for the natural frequency of the original equation of motion.
ω ωi= i (7)
where,
ωi : Approximate solution for i- th mode shape
A mode shape of a structure is a vector, which defines the mapping relationship between the displacement vector of the equation of motion and the mode coordinate. The approximate mode shape obtained by Ritz vector analysis is thus defined by the relationship between the displacement vector of the original equation of motion, ( )u t , and the mode coordinate, ( )q ti , as noted below.
[ ]
1 ( ) ( ) ( ) p i i i u t Ψz t Ψφ q t = = =∑
(8)Accordingly, the approximate solution for the i – th mode shape is defined as
i i
φ =Ψφ (9)
where,
i
The approximate mode shape vector in Ritz vector analysis retains orthogonality for the original mass and stiffness matrices similar to that for eigenvalue analysis.
The approximate solution for natural frequencies and mode shapes in Ritz vector analysis is used for calculating modal participation factors and effective modal masses similar to a conventional eigenvalue analysis.
When a time history analysis is carried out by modal superposition on the basis of the results of Ritz vector analysis, the above equation of motion (5) is used. The Ritz vector, which assumes the deformed shape of a structure, is generally created by repeatedly calculating the displacement due to loads applied to the structure. The user first specifies the initial load vector. The basic assumption here is that the dynamic loading changes with time, but the spatial distribution for each degree of freedom follows the initial load vector specified by the user. Next, the first Ritz vector is obtained by performing the first static analysis for the specified initial load vector.
(1) (1)
Kψ = r
(1) 1 (1)
ψ = K r−
where,
K : Stiffness matrix of the structure
(1)
ψ : First Ritz vector
(1)
r : User specified initial load vector
The first Ritz vector thus obtained is assumed as the structural displacement. However, the above static analysis ignores the effect of the inertia force developed by the dynamic response of the structure. Accordingly, the displacement due to the inertia force is calculated through additional repeated calculations. The distribution of acceleration for the structure is assumed to follow the displacement vector calculated before, which is the first Ritz vector. The inertia force generated by the acceleration is calculated by multiplying the mass vector. The inertia force is then assumed to act as a loading, which induces additional displacement in the structure, and static analysis is carried out again.
(2) (1)
Kψ = Mψ
(2) 1 (1)
ψ = K Mψ−
where,
M : Mass matrix of the structure
(2)
ψ : Second Ritz vector
The second Ritz vector thus obtained in the above equation also reflects a static equilibrium only. Assuming the above equation is expressed without considering
the acceleration distribution, the above process is repeated in order to calculate the number of Ritz vectors specified by the user.
The user may specify a multiple number of initial load vectors. The number of Ritz vectors to be generated can be individually specified for each initial load vector. However, the total number of Ritz vectors to be generated can not exceed that of real modes, which exist in the equation of motion. Also, those Ritz vectors already generated in the repetitive process are deleted once linearly dependent Ritz vectors are calculated. For this reason, the generation cycle ends if linearly independent Ritz vectors can not be calculated any longer. This means that the initial load vectors specified by the user alone can not find the specified number of modes.
The initial load vectors that can be specified in the MIDAS programs are an inertia force due to ground acceleration in the global X, Y or Z direction, a user- defined static load case and a nonlinear link force vector. The inertia force due to ground acceleration in the global X, Y or Z direction is mainly used to find the Ritz vector related to the displacement resulting from the ground acceleration in the corresponding direction.
The user-defined static load case is used to find the Ritz vectors for a dynamic load with specific distribution. A common static load case (dead load, live load, wind load, etc.) may be used, or an artificially created static load case may be used to generate Ritz vectors.
The member force vectors of nonlinear link elements are used to generate Ritz vectors. The member forces generated in each nonlinear link element are applied to the structure as a load vector. For only the degrees of freedom checked by the user among the 6 degrees of freedom in an element, initial load vectors having unit forces individually are composed and used for generating the Ritz vectors. However, the member force vectors of link elements do not have to be used in the analysis of a structure, which contains nonlinear link elements. The user specifies initial load vectors at his/her discretion, which should adequately reflect the structural deformed shape under the given analysis condition.
When compared with eigenvalue analysis, Ritz vector analysis has the following advantages:
Ritz vectors are founded on static analysis solutions for real loads. Even if a smaller number of modes are calculated in Ritz vector analysis, the effects of higher modes are automatically reflected. For example, the first mode shape in a Ritz vector analysis can be different from that in an eigenvalue analysis, which is attributed to representing the effects of higher modes. Also, Ritz vector analysis finds only the mode shapes pertaining to the loads acting on the structure, thereby eliminating the calculations for unnecessary modes. Ritz vector analysis thus reduces the number of modes for finding accurate results. Ritz vector
analysis requires a less number of modes to attain sufficient modal mass participation compared to eigenvalue analysis.