Multi-degree-of-freedom Systems

Consider a linear system with Nd degrees of freedom (DOFs) and its equation of motion of 

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

The diagonal matrix A contains the eigenvalues. The mth column vector 0 in the modal matrix associated with the mth lowest eigenvalue is called the mode shape of the mth mode of the dynamical system. Here, they are normalized such that  

the uncoupled modal equations of motion are obtained  

For a given set of model parameters, the response x is a zero-mean Gaussian process and the (/, / ) element of its power spectral density matrix function S is given by [249]  

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General transformation factor equations for distributed mass systems and multi-degree of freedom systems are given by Higgs 1964 (Chapter 5), and Clough 1993 (Chapter 2). These general methods can be used in determining transformation factors for nonprismatic members or members which have nonuniform mass distributions. 

This result makes clear that considerably greater insight is necessary before a set of adequate diagnostics are available to define a multi-degree-of-freedom system as sufficiently chaotic to apply statistical models of bound relaxation. 

H. WaaUcens, S. Wiggins, Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed, J. Phys. A Math. Gen. 37 (2004) L435-L445. 

This section considers the response measurement of a single-degree-of-freedom system or a single-channel measurement of a multi-degree-of-freedom system. Discrete data is sampled with a time step Ar and y denotes the measured response at time t = nAt. The measurement is different from the model response ... 

In this chapter, the Bayesian time-domain approach was introduced for identification of the model parameters and stochastic excitation parameters of linear multi-degree-of-freedom systems using noisy stationary or nonstationary response measurements. The direct exact formulation was presented but it turned out to be computationally prohibited for a large number of data points. Then, an approximated likelihood function expansion was proposed to resolve this obstacle. For a globally identifiable case with a large number of data points, the updated PDF... 

Each tie rod was considered to be a series of six lumped masses connected by springs. The tension in the rod was included and the ends of the rods were considered to be clamped. This multi degree of freedom system was excited by alternating forces imposed on two of the lumped masses to simulate the assumed loading shown in Figure 12-2. 

Manson, G. February 2003. Sharper eigenproblem estimates for uncertain multi-degree of freedom systems, Proceedings of IMAC XXI CDrom, Orlando, Florida, USA, pp. 8. 

Irschik, H. Nonstationary Random Vibrations of Yielding Multi-Degree-of-Freedom Systems Method of Effective Envelope Functions. Acta Mechanica 60 (1986), p. 265-280. 

For a multi-degree-of-freedom system, the evolutionary covariance submatrix of the state vectors, x.x., defining the modal responses, E (t) is given by Gasparlni (1979) as follows ... 

Suppose the equation of motion of a multi-degree-of-freedom system can be decoupled and, along the direction of the generalized coordinate X, the system is governed by... 

Vakakis A, Paipetis S (1986) The effect of a viscously damped dynamic absorber on a linear multi-degree-of-freedom system. J Sound Vib 105(l) 49-60 Val DV, Segal F (2005) Effect of damping model on pre-yielding earthquake response of structures. Eng Struct 27(14) 1968-1980... 

The differential equation that governs the motion of multi-degree-of-freedom systems, with one TMD located on the top floor (Fig. 1) and subjected... 

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