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Axial Wave Propagation

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Up to this point we have considered only discrete dynamical systems. The primary assumption in a discrete system is that elements providing restoring forces have no associated mass and that the masses are rigid. Once we remove these restrictions, we are in the realm of continuous systems. In a continuous system, both the mass and the elasticity are distributed throughout the body. All materials have this feature, so it is a more realistic model than the discrete system idealization. The price of this additional fidelity is more mathematical complication. The remainder of the book concerns continuous systems. In this chapter we will explore the simplest version of a continuous system in solid mechanics—the axial bar problem.

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Change history

  • 01 January 2022

    This book was inadvertently published without updating the following corrections:


  1. 1.

    The formulation in this chapter is linear. We use engineering strain for simplicity.

  2. 2.

    The value of the applied load can be zero, of course. To find the appropriate boundary condition at a free end take a tiny free body diagram of a piece at that point to expose the internal force. Equilibrium then determines the boundary condition.

  3. 3.

    The term general solution means a function that satisfies the domain equation but does not yet consider the boundary or initial conditions.

  4. 4.

    A dot over a function indicates partial derivative with respect to time.

  5. 5.

    It is easy to prove that these boundary terms vanish for any boundary conditions because either the displacement is prescribed or the axial force is prescribed at either end. Since the term that results from integration by parts has the product of an eigenfunction and the derivative of an eigenfunction, one of them must be zero at each end.

  6. 6.

    Note that A is the area of the cross section of the bar and should not be confused with An, which is the nth coefficient of the modal expansion.

  7. 7.

    The symbol u has two distinct uses here. The symbol without subscript refers to u(x, t), whereas the symbol with subscript is a component of the Ritz expansion. The context should make the distinction clear.

  8. 8.

    When we invoke the Ritz approximation, we go from a continuous problem to a discrete one. The condition “for all \(\bar {u}(x)\)” then reduces to “for all \(\bar {\mathbf u}\)”.

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Hjelmstad, K.D. (2022). Axial Wave Propagation. 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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