Abstract
The frequency-response curve is an important information for the structural design, but the conventional time-history method for obtaining the frequency-response curve of a multi-degree-of-freedom (MDOF) system is time-consuming. Thus, this paper presents an efficient technique to determine the forced vibration response amplitudes of a multi-span beam carrying arbitrary concentrated elements. To this end, the "steady" response amplitudes|Y(x)|sof the above-mentioned MDOF system due to harmonic excitations (with the specified frequencieswe) are determined by using the numerical assembly method (NAM). Next, the corresponding "total" response amplitudes|Y(x)|tof the same vibrating system are calculated by using a relationship between|Y(x)|tand|Y(x)|sobtained from the single-degree-of-freedom (SDOF) vibrating system. It is noted that, near resonance (i.e.,we/w≈1.0), the entire MDOF system (with natural frequencyw) will vibrate synchronously in a certain mode and can be modeled by a SDOF system. Finally, the conventional finite element method (FEM) incorporated with the Newmark's direct integration method is also used to determine the "total" response amplitudes|Y(x)|tof the same forced vibrating system from the time histories of dynamic responses at each specified exciting frequencywe. It has been found that the numerical results of the presented approach are in good agreement with those of FEM, this confirms the reliability of the presented theory. Because the CPU time required by the presented approach is less than 1% of that required by the conventional FEM, the presented approach should be an efficient technique for the title problem.
Highlights
In order to confirm the reliability of the presented approach, the complete solution for the forced vibration responses of the multi-span beam carrying various concentrated elements and subjected to external harmonic excitations is determined by using the conventional finite element method (FEM) and the step-by-step numerical integration method [24], and, corresponding to each specified exciting frequency ωe, the “total” vibration response amplitude for each point of the beam is obtained from the maximum absolute response |y(x, t)|max of its time history of transverse displacements, y(x, t)
The method based on Eqs (14) and (43) (or Eq (A.6) in Appendix A) is called NAM, because the “steady” response amplitudes |Y (x)|s are obtained from the numerical assembly method and the other method is called FEM, because the “total” response amplitudes |y(x, t)|max are obtained from time histories for transverse displacements of the vibrating system by using finite element method
Based on the relationship |Y (x)|t = [1 +] · |Y (x)|s, with |Y (x)|t and |Y (x)|s respectively denoting the total and steady response amplitudes at position x and ωe denoting the exciting frequency, one may determine the forced vibration response amplitudes of a multi-span beam carrying arbitrary concentrated elements by using NAM with the CPU time to be much less than that required by using FEM
Summary
Because the mathematical model of some vibrating systems can be established by using a uniform or non-uniform beam carrying various concentrated elements (such as lumped masses with rotary inertias, linear springs and/or rotational springs) with various boundary (supporting) conditions, the literature concerned is plenty [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. In order to confirm the reliability of the presented approach, the complete solution for the forced (transient) vibration responses of the multi-span beam carrying various concentrated elements and subjected to external harmonic excitations (as shown in Fig. 6) is determined by using the conventional FEM and the step-by-step numerical integration method [24], and, corresponding to each specified exciting frequency ωe, the “total” vibration response amplitude for each point of the beam is obtained from the maximum absolute response |y(x, t)|max of its time history of transverse displacements, y(x, t). Because the formulation of this manuscript is based on the “continuous” model, for the uniform beam shown in Fig. 1 carrying s sets of “intermediate concentrated elements” (ICE’s) with each set of ICE’s consisting of a linear spring kti, a rotational spring kθi and a lumped mass mi (with rotary inertia Ji), one can model the entire beam by using only s + 1 beam segments to achieve the “exact” solutions. This is the reason why the input data required by the developed computer program based on the formulation of NAM is simpler than those required by the conventional FEM, as a result, the storage memory and computing time required by NAM are much smaller than those required by FEM
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