Periodic and non-periodic combination resonance in kinematically excited system of rods

Abstract

A theoretical investigation has been made of periodic and non-periodic combination resonances of the system of rods under vertical and horizontal kinematic excitations. The elements of the system are connected with articulated points. The coupling of the elements of the system through internal longitudinal forces, which are transverse forces at the ends of neighboring rods, are taken into account. The equations of motion are obtained from the Lagrange equations. The mathematical analysis of the equations of motion is accomplished by using Tondl’s technique. The steady state solutions and their stability are determined. Resonance curves for the stationary states have been determined numerically.

Introduction

The non-linear responses of elementary structural components, such as beams and plates, have been studies extensively. This is due to the fact that the governing equations of motion of motion are readily derivable and, in some cases, they are solvable numerically or in an approximate closed from by using perturbation techniques. Thus, experiments can be verified by the theory and vice versa. More complex systems, such as systems of beams, have not been investigated as thoroughly because the equations of motion are more difficult to derive and solve. Despite this drawback, the motivation for experimentally advancing the state of art still exist because the results can, in the short run, qualitatively guide the design and, in the long run, inspire, guide, and verify the analysis.

There has been extensive research performed on the non-linear behavior of beams. In particular, the cases of subharmonic, superharmonic, combinations, and autoparametric resonances have been well documented for beams [1], [2], [3]. Some of the more recent discoveries include descriptions of the chaotic behavior and interaction between widely spaced modes. Haddow and Hasan [4] experimentally investigated the response of a parametrically excited flexible cantilever beam with a rectangular cross section. They reported various periodic and chaotic behaviors in addition to what they described as external low subharmonic response. Burton and Kolowith [5] performed analytical and experimental investigations on a system similar to that of Haddow and Hasan. The results of Burton and Kolowith are similar to those of Haddow and Hasan. Lau et al. [6] studied the response of a uniform hinged–clamed beam to a primary resonance of the first mode and a combination subharmonic resonance of the first two modes, whereas Chen et al. [7] studied the response of a uniform beam to primary resonance of either the first or the second mode. Nayfeh et al. [8] studied the resonances of non-uniform hinged–clamped beams to primary resonances of either the first or the second mode. Lau et al. [9] used the method of incremental harmonic balance with multiple time scales to study almost periodic vibrations of a uniform hinged–clamped beam. Yamamoto et al. [10], [11], and Ibrahim et al. [12] investigated combination internal resonances in extensional beams. Elangar and El-bassiouny [13] used the method of multiple scales to studied harmonic, subharmonic, superharmonic, simultaneous sub/superharmonic and combination resonance resonances of self excited two coupled second order system to multifrequency excitations. Elangar and El-bassiouny [14] investigated the response of three-degree-of-freedom systems to multifrequency excitations. El-Bassiouny [15] studied primary resonance of three-degree-of-freedom system with cubic non-linearities in which the third mode subjected to harmonic excitation. El-Bassiouny and Abdelhafez [16] analyzed the predication of bifurcations for external and parametric excited one-degree-of-freedom system with quadratic, cubic, and quartic non-linearities. Sridhar et al. [17], [18] investigated the combination internal resonance $\text{Ω≅2ω}{}_2\text{+ω}{}_1$ in clamped circular plates. They found that multimode vibrations occur only when $\text{Ω≅ω}{}_3$. Sridhar et al. [17], [18] and Hadian and Nayfeh [19] found that the equilibrium solutions of the modulation equations undergo a Hopf bifurcation, which results in periodic, quasiperiodic, and chaotic motions. In these systems, the non-linearity is cubic. On the other hand, Bux and Roberts [20], Cartmell and Roberts [21], and Nayfeh et al. [22] investigated combination internal resonances with quadratic non-linearities, namely two-beam structures. Dugundji and Mukhopadhyay [23] experimentally and theoretically investigated the response of a thin cantilever beam with an external base excitation at a frequency close to the sum of the natural frequencies of the first bending and first torsional mode. Forys and Niziol [24] pursued analytical investigations of a system of three rods jointed to make a portal frame. The upper corners of the rectangular frame had masses that translated but did not rotate. Forys and Niziol limited their analytical investigation to autoparametric resonances of transverse vibrations. They presented results for cases in which the vertical beams were longer than the horizontal beam. The results showed how the amplitudes of vibration varied with the excitation frequency for various values of damping and how the corner masses affected the response amplitudes. Forys [25] considered a structure similar to the one of Forys and Niziol [24] although without the corner masses. Forys limited his investigation to combination resonances of symmetric modes and plotted the response amplitude of each beam against the frequency of excitation. Forys [26] studied the harmonic resonance of the system of three rods under vertical kinematic excitation. Elnaggar and El-Bassiouny [27] investigated harmonic resonance of the same system of three rods under horizontal kinematic excitation.

In this paper we confine ourselves to the combinations resonances of the above system of three rods which subjected to vertical and horizontal kinematic excitations. Such system is an essential element of many buildings and structures e.g. engine rooms. Internal couplings between rods are important and are taken into account.