Generally, nonlinear effects became significant in all the continuous systems in this book when the vibratory displacements became large. Most of them were used as homework problems in the classes taught by the first author. Huang, The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions, J. Only one coordinate, measured along the string, is sufficient to define the classical problem of the transversely vibrating taut string, and the governing partial differential equation of motion is only of second order. This is done by making classical assumptions of structural mechanics about the material linearly elastic, isotropic, homogeneous. Inplane vibrations would be the same as for the membrane, and seldom of interest. If it is only approximate, then an approximate is determined.
Cowper, The shear coefficient in Timoshenkos beam theory, J. The frequencies do not depend on the actual cross-sectional size. As in the membrane vibration problem Chap. Challenging end-of-chapter problems reinforce the concepts presented in this detailed guide. For other cases, it is necessary to use an approximate method.
All work in this chapter, unlike others, involves only free vibra-tions with no damping. Like the string, the membrane is assumed to be perfectly flexible. Typical structures are still more complicated. Sonalla, Nonperiodic vibration of a cantilever beam subjected to various initial conditions, J. Thus, the solution given by 8. Part C: The fundamental frequency of a cantilever beam is 3 5160 4.
The examples given above string, bar, shaft, beam are all one-dimensional problems. That is, they are the proper values which, if chosen, permit us to obtain a nontrivial solution satisfying both the differential equation and boundary conditions, all of which are homogeneous. El-Azhari, Vibrations of free hollow circular cyl-inders, J. In this situation, the rod is frequently called a shaft, as used in some mechanical equipment. The middle surface of the shell may have an arbitrary curvature, and the wall thickness may vary arbitrarily.
This is called material damping also known in the technical literature as hysteretic or structural damping. Therefore, take 3 5160 2 1 3259. But this would typically be much more complicated than the general procedures laid out above. It does not enter the linearized analysis for the string. In addition, we will present solutions for the vibration problem of these components with various boundary conditions.
It therefore must remain straight, which is unrealistic and imposes a significant, frequency-raising con-straint along the free edge. The ill-conditioning problem can be effectively eliminated by using orthogonal polynomials Legendre polynomials, in this case in place of the ordinary polynomials in 6. It is the result of the first authors 50 years of research in the field of vibrations of continuous systems, and having taught a graduate-level course of the same title at Ohio State University for 35 years. Increasing mass density , weight density g , or axial acceleration g, which would be the sum of downward gravitational acceleration and upward base acceleration all cause proportional increases in. Like the finite element methods, they are based on energy principles instead of differential equations and, if used properly, will converge to exact frequencies and mode shapes if sufficient d. Huang, The effect of rotary inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions, J.
Frequencies for circular cross-sections, an obvious special case, were also reported. Then the abscissa of Fig. Or, alternatively, the rod could undergo transverse displacements w. The fourth-order frequency determinant is written and eigenvalues l are determined by plotting the determinant versus l and finding its zeros. As we have seen in previous chapters, this method involves the potential and kinetic energies of the vibrating body. Assume that the beam is made of titanium.
Static or dynamic displacements may Figure 1. It is the result of the first authors 50 years of research in the field of vibrations of continuous systems, and having taught a graduate-level course of the same title at Ohio State University for 35 years. Nevertheless, understanding of the behavior of the relatively simple continuum models can often help greatly in the understanding of the more complicated structure, either a single part of it, or the entire body. Most of them require significant thought and time spent more than one hour each. Students or readers should have beforehand at least a basic understanding of ordinary differential equations and, preferably, some background in the vibrations of discrete systems. Generalize the conclusion reached in Part B to a beam of variable cross-section having arbitrary boundary conditions, and carefully explain how you reach this conclusion. As will be seen, such curvature introduces significant complexity to the beam problem.