116
5 Continuous Systems
Fig. 5.1 Flexural beam and sign convention
of separation of variables is used for solution. This method views v(x, t) as the
product of two functions, ϕ(x) and q(t), which depend on only x and only t and
satisfy ordinary differential equations. The non-trivial solutions of the homogeneous
differential equation D[ϕ(x)] = 0 for ϕ(x), i.e., the eigenfunctions of the
differential operator D, are used to represent v(x, t). The representation is similar
to that for the solution of MDOF systems, which is based on eigenvectors (modal
shapes).
For simplicity, we assume that the stiffness and the mass of the beams are spaceinvariant and that the systems are undamped. Also, the presentation is limited to time
domain analysis. Our considerations on the frequency domain analysis for MDOF
systems extend directly to the continuous systems considered in this chapter.
5.1 Flexural Beams
Consider the beam in Fig. 5.1 with mass per unit length m(x) and stiffness EI (x)
under the action of a spatially distributed, time-dependent force f (x, t). Denote the
beam displacement at location x and time t by v(x, t). The figure also shows the
positive sign convention for the shear force Q(x, t) and bending moment M(x, t).
5.1.1 Physical System and Equations of Motion
This section lists without proof the differential relationships for flexural beams.
Their derivation can be found in, e.g., [3]. These relationships also give the
equations of motion for this continuous system.
5 Continuous Systems
Fig. 5.1 Flexural beam and sign convention
of separation of variables is used for solution. This method views v(x, t) as the
product of two functions, ϕ(x) and q(t), which depend on only x and only t and
satisfy ordinary differential equations. The non-trivial solutions of the homogeneous
differential equation D[ϕ(x)] = 0 for ϕ(x), i.e., the eigenfunctions of the
differential operator D, are used to represent v(x, t). The representation is similar
to that for the solution of MDOF systems, which is based on eigenvectors (modal
shapes).
For simplicity, we assume that the stiffness and the mass of the beams are spaceinvariant and that the systems are undamped. Also, the presentation is limited to time
domain analysis. Our considerations on the frequency domain analysis for MDOF
systems extend directly to the continuous systems considered in this chapter.
5.1 Flexural Beams
Consider the beam in Fig. 5.1 with mass per unit length m(x) and stiffness EI (x)
under the action of a spatially distributed, time-dependent force f (x, t). Denote the
beam displacement at location x and time t by v(x, t). The figure also shows the
positive sign convention for the shear force Q(x, t) and bending moment M(x, t).
5.1.1 Physical System and Equations of Motion
This section lists without proof the differential relationships for flexural beams.
Their derivation can be found in, e.g., [3]. These relationships also give the
equations of motion for this continuous system.