If the periodic loading is expressed in terms of individual harmonics of the exponential form of Eq. However, most of the problems in the text, which are intended to teach the fundamentals of dynamics, are quite simple in form allowing their solutions to be obtained using a hand calculator. Its range of applicability for this purpose will be broadened by reducing the natural frequency, i. As may be seen in Fig.
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Consider the periodic loading shown in Fig. The solution of Eq. One would have extreme difficulty in treating such cases by the time-domain approach described in the first part of this chapter. Suppose that the test of Prob. While in most cases, the mass and stiffness can be evaluated rather easily using simple physical considerations or generalized expressions as discussed in Chapter 8, it is usually not feasible to determine the damping coefficient by similar means because the basic energy-loss mechanisms in most practical systems are seldom fully understood.
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Full text of "Dynamics Of Structures Clough & Penzien"
Dynamic linearly elastic systems having continuously distributed properties are considered in Part Three. A simplified model of the earthquake-excitation problem is shown in Fig. The subject matter of this text can serve as the basis of a series of graduate-level courses.
On the other hand, if these masses are not fully concentrated so that they possess finite rotational inertia, the rotational displacements of the three points will also have to be considered, in which case the system has 6 DOF. If the bars could deform in flexure, the system stductures have an infinite number of degrees of freedom.
If the weight of the console is lb, determine the stiffness of the vibration isolation system required to reduce the vertical-motion amplitude of the console to 0. The treatment of random vibrations in Part IV is however stochastic random in form since the structurea considered can be characterized only in a statistical manner. Using this explicit formulation, the displacement at the end of the time step, Vi, can be calculated directly by solving Eq.
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Deflections sometimes develop in concrete bridge girders due to creep, and if the bridge consists of a long series of identical spans, these deformations will cause a harmonic excitation in a vehicle traveling over the bridge at constant speed.
However, a single zero-displacement crossing would occur if the signs of the initial velocity and displacement were different from each other. Thus for an impulse of very short- duration, a large part of the applied load is resisted by the inertia of the structure, and the stresses produced are much smaller than those produced by loadings of longer duration.
A straightforward evaluation of this summation for all values of n requires N 2 complex multiplications. The full extent of this coverage would depend, of course, upon whether the course is of quarter or semester duration.
AmazonGlobal Ship Orders Internationally. A stductures harmonic-loading machine provides an effec- tive means for evaluating the dynamic properties of structures in the field.
They are extremely efficient due to the fact that each summation is used immediately in the next summa- tion. Substituting this value into Eq. The advantages of this dynamixs procedure are as follows: On this basis, the approximate relationship may be used: The advantage of this approach is that a good approximation to the actual beam shape can be achieved by a truncated series of sine-wave components; thus a 3 DOF approximation would contain only three terms in the series, etc.
The force p i may be considered to include many types of force acting on the mass: For most problems in structural dynamics it may be assumed that mass does not vary with time, in which case Eq. Nonperiodic loadings may be either short-duration impulsive structurres or long- duration general forms of loads.
Since the peak of the frequency-response curve for a typical low damped structure is quite narrow, it is usually necessary to shorten the intervals of the discrete frequencies 0 Pj 1 P 2 2 Frequency ratio, P FIGURE Frequency-response curve for moderately damped system.
For the system starting from rest, i. E7- 1 a; three straight line approximations of the one and one-half cycle loading as sketched in Fig. To take account of viscous damping in evaluating the steady-state response of a SDOF system to periodic loading, it is necessary to substitute the damped-harmonic- response expressions of the form of Eq.
For this elastoplastic system, the response behavior changes drastically as the yielding starts and stops, and to clouvh best accuracy it would be desirable to divide each time increment involving such a change of phase into two subincre- ments. If a maximum value occurs in Phase I, the value of a at which it occurs can be determined by differentiating Eq.
The essential concept is represented by the following equations: Note that this free response of a critically-damped system does not include oscillation about the zero-deflection position; instead it simply returns colugh zero asymptotically in accordance with the exponential term of Eq.