Modal coordinate

Using modal analysis, the linear transformation between the generalized coordinates and the modal coordinates is given by ... 

The zero value of hS T) for negative x is due to the causality of physical systems. The underlying meaning is that any input should not generate response in the past. Furthermore, the correlation matrix function Rg for the modal coordinates q has (m, m ) element between... 

Figure 5. Weighed sum of amplitude response function (expressed in generalized modal coordinates) for the base isolated building with and without TLCGD installed...

Equation (46) takes on a hyper matrix form for a multiple-degree-of-freedom main system (preferably described in modal coordinates) with several VTLCGDs attached at properly selected positions, and possibly converted into smaller units in parallel action at one and the same location. In such a case, fine-tuning in state space is recommended. 

We propose a simple method for the linearization of the equations, which are established in our case, based on the virtual work principle. The kinematic relations between the interconnected bodies are represented by the recursive equations. Under the small deformation assumption, the system generalized variables used in the equations are the relative joint coordinates at the connections and the deformation modal coordinates of the flexible bodies. In the linearization process, the differentiation of the kinematic terms with respect to the generalized variables must be performed. In our method, these partial derivatives are attained through the first and second order time differentiations of the body absolute angular velocities and through the first, second, and third order time differentiations of the mass center coordinates. This is the essential idea behind our method. The partial differentiation of the mechanical terms, for example, of the inertial tensors will also be presented. We have developed specific operators to perform the time differentiations. This method makes both the theoretical formulation and the programming task relatively simple, and allows fast computation. 

Vectors X,Y and the relative displacement vector U are observed in the body fixed frame Rd, while r and a are given in the inertial frame Rq, thus Y is constant in Rd. U(M) is the position vector of point M with respect to Mq, and U(G) is the position vector of point Gd with respect to Go. n is the number of selected modes for the deformed body. The set of vectors Oa defines a reduced basis of deformation of body k, which can be obtained by the finite-element method. For instance, we choose a set of vibration modes, and the are the corresponding modal coordinates. 

Unfortuately, proceeding in this manner yields a number of difficulties. The equations of motion which are produced as per Eqs.(2) are incorrectly linearized in the modal coordinates and there time derivatives derivatives. This can lead to significant error where motion induced stiffness is involved. In addition, the integration over the volume of each body and the indicated summations over all bodies of the system at each time step can be computationally expensive. Furthermore, the resulting equations of motions are highly coupled in the state derivatives, thus requiring computationally expensive decomposition of the mass matrix before the state derivatives can be solved for and the equations of motion can be temporally integrated. Each of these difficulties will be addressed subsequently. 

Linear strain energy theory assumes that the deformation components are independent. But in systems involving high rotation rates, high radial forces can occur and the coupling between radial and transverse deflections becomes significant. Unfortimately, the use of modal coordinates in modeling flexibility results in... 

The system discussed here consists of a cantilever beam attached to a base with prescribed angular velocity - an example used as a benchmark test by a number of authors. Refs. [14,17-19]. The generalized coordinates used in this problem are the four modal coordinates and the angular velocity of the base is given by... 

The principle of the two techniques is easiest illustrated by recalling that any response vector y t) can be expressed in modal coordinates q t) as... 

The damping ratios can be identified through the refinement of the EDD technique, namely, the enhanced frequency domain decomposition (EEDD, Brincker et al. 2001). The EEDD technique is based on the fact that the first singular value in the neighborhood of a resonant peak is the ASD of a modal coordinate. Hence, moving the partially identified ASD of the modal coordinate back in the time... 

The BSS problem has a similar pursuit with the output-only modal identification issue. The close similarity is implied between the modal expansion Eq. 11 and the BSS model Eq. 1. If the system responses are fed as mixtures into the BSS model, then the target of output-only identifying 4> and q(0 in Eq. 11 can be solved by blind recovery of A and s(0 using those BSS techniques dependent on the independence assumption since q(0 typically with incommensurable fi-equencies are independent on the modal coordinates. 

The aforementioned CP framework can then be cast into the modal identification whose physical interpretation is based on a new concept of independent physical system on modal coordinates (Yang and Nagarajaiah 2013b). The CP method is successful as shown in Figs. 6,7,8, and 9 and Tables 3 and 4 in identification of closely spaced couples with high-damped modes and in the presence of nonstationary seismic excitation. 

For nonproportional damping, the weU-known modal coordinate transformation, which diagonalizes the system mass and stiffness matrices, will not diagonalize the system damping matrix. Therefore, when a system with nonproportional damping exists, the equations of motion are coupled when formulated in n dimension physical space. Fortunately, the equations of motion can be uncoupled when formulated in 2n dimension state space, by means of complex analysis. 

Substituting Eq. 6 in Eq. 1 and left multiplying by we obtain n uncoupled differential equation of motion in modal coordinates ... 

As for the free vibrations, the motion of an n-dof classically damped structure can be thought as a linear combination of normal modes cj)j, each of them vibrating with the associated circular frequency coj and damping Indeed, using the modal coordinates q(t) as defined in Eq. 6 and Eq. 11 and left multiplying by 4>, we obtain n uncoupled differential equation of motion in modal coordinates ... 

It is worth mentioning that given a structure, knowing of it (1) the prevalent modal shape (O), (2) the prevalent period (TJ, and (3) the modal participation factor (F) if a seismic action is considered, the absolute structural acceleration (iij) associated to the single modal coordinate (Oj) can be evaluated according to Eq. 3 where Rs(Ts) is the value of the normalized acceleration spectrum, for a given value of the structural damping ( s) at the prevalent period of the structure. 

Assuming a given anal3flical function (Rfrs), being it dependent on the period (Tns) and the damping ( ns) of the nonstructural element, the required FRS, associable to the i modal coordinate, is equal to... 

The general scheme of spatial filtering can be used for different purposes. A first idea is to condense the information into the modal coordinates of the undamaged structure, which allows to reduce the daX yiJif) from a very large network of n sensors to a limited set of a,(t) time series from N modal sensors. This idea is motivated by the fact that the vibration of structures typically involves only a few mode shapes which are excited in a given frequency band of interest. 

The variational equations of motion are developed for flexible serial manipulators with rotary joints which account for full coupling between the rigid body motion and link deformation. Velocity and acceleration transformation equations are developed to conveniently transform Cartesian space equations into Joint space. Small deformation is assxamed such that vibration modal coordinates and mode shapes can represent the elastic motion of the flexible links. Flexibility and mass properties of the links are obtained by finite element method. A case study of an industrial robot is presented to show the effect of bending and torsional vibrations on end-effector motion. 

Such relations have been obtained by Jerkovsky [4] and Kim [5] for rigid mechanical systems. Recently Chang [6] has also determined somewhat similar relations as those of Eq. 11 for flexible mechanical systems. Collecting the joint and modal coordinates as presented in Eq. 8 and using the above notation, Eqs. (9) and (10) may be written in matrix form as ... 

The mode shapes, as presented previously for describing the deformations of the links, can be obtained from a finite element analysis of each of the bodies of the structure. The Lagrangian formalism enables a straightforward derivation of the equations of motion. The Lagrangian can be expressed in terms of nodal displacements and their time derivatives. Here it is formulated as a function of the joint angles 0, the joint deformation variables p and the link modal coordinates g, and their time derivatives, as follows L(0yp q p q) = ifJ5(, p,g,, p,g) -PE fP q), For a serial link manipulator with n joints and n links, the equations of motion take the form ... 

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