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Systems with Multiple Degrees of Freedom

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In this chapter we consider structures with multiple degrees of freedom. Since we often label the number of degrees of freedom N, we call them NDOF systems. In this and the next several chapters we will analyze idealized discrete structures, wherein the parts associated with mass are considered rigid and the parts associated with restoring forces have no mass. This simplification will result in equations of motion that are coupled ordinary differential equations. While we will relax the assumptions that decouple mass and stiffness in the later chapters of the book, this idealization gives a great deal of insight into the dynamic response of structural systems.

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  1. 1.

    Note that the concept of degree of freedom , which we often abbreviate DOF, is associated with the independent motion variables. The number of independent motion variables required to describe the motion is the number of degrees of freedom of the system.

  2. 2.

    The reason we assume that the columns are massless and the masses are rigid is that it allows us to treat the system as discrete rather than continuous. This assumption avoids some complex phenomena that we will look at later in the book.

  3. 3.

    To be consistent with what we did with the SDOF system, we should assume u = φ e λt. If we do so, then the eigenvalue equation that results is K φ = −λ 2 M φ. Since both K and M are positive definite, we can premultiply that equation with φ T to show that − λ 2 must be a positive real number. Therefore, λ must be purely imaginary. That is tantamount to assuming that the solution is u = φ e iωt, which is the same as assuming the sinusoidal form we have selected here.

  4. 4.

    John William Strutt, Lord Rayleigh, 1842–1919, made numerous contributions to dynamics, including his book The Theory of Sound (1877).

  5. 5.

    The acceleration a n is not to be confused with the use of a(t) for the vector of modal components in modal analysis. If there is a subscript, then it is the discrete acceleration in the numerical integration scheme. If it is a function of time, then it is a vector of modal components.

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Hjelmstad, K.D. (2022). Systems with Multiple Degrees of Freedom. In: Fundamentals of Structural Dynamics. Springer, Cham.

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-89943-1

  • Online ISBN: 978-3-030-89944-8

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