Vibration of a two-member open frame

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Abstract

The dynamics of a two-member open frame structure undergoing both in- and out-of-plane motion is examined. The frames are modelled using the Euler–Bernoulli beam theory and are further generalized by permitting an arbitrary angle between the beams and the attachment of a payload at the end of the second beam. The equations of motion are derived using Hamilton's principle and the orthogonality conditions are presented. It is shown that the in- and out-of-plane motions can be decoupled by including the axial deformation components in the assumed displacement fields. The natural frequencies of the system and the contribution of each member into the system potential energy are examined via numerical examples.

Introduction

Frame structures can be categorized as closed or open [1]. Closed frames are frames formed by chains of beams in which both ends are fixed (e.g., [2], [3], [4], [5], [6], [7], [8], [9], [10]). The simplest example of a closed frame is a three-beam member portal frame, (e.g., [5], [6], [7], [8], [9], [10]). Open frames are chains of beams that have one end fixed and the other end free (e.g., [1], [11], [12], [13], [14]). This paper is concerned with open frame structures. While the most common focus of frame analysis is civil engineering structures, the kind of structure being considered here has broader instances of application such as space-based antenna structures and electrical or electronic components operating in severe dynamic environments such as that might be found in a rocket launch.

The dynamics of two-member open frame structure which comprises a cantilever beam with a second beam attached to its free end has been considered by Bang [13], Oguamanam et al. [1] and Gürgöze [14]. Bang [13] and Gürgöze [14] examine L-shaped configurations without a tip mass while Oguamanam et al. [1] investigate configurations with arbitrary inclination angles with or without a tip mass. Oguamanam et al. [1] define an angle of inclination measured from the undeformed axis of the fixed beam (the first beam) to the undeformed axis of the beam with the free end (the second or distal beam). They provide analytical expressions for the frequency (or characteristic) equation, the mode shapes, and the orthogonality conditions for the case of planar motion of the frame. In addition to the study being limited to in-plane motion, axial deformation was ignored.

The advantages and disadvantages of analytical solutions for L-shaped structures versus finite-dimensional approximations solutions have been presented by Bang [13]. Oguamanam et al. [1] further highlight the advantages of the analytical solution over the finite element method (FEM) using specific examples. In particular, a 60-degrees-of-freedom FEM model is used to obtain accurate results for the first five natural frequencies. The disadvantage of using this model in a control system is stressed.

This study extends the work by Oguamanam et al. [1] by relaxing the restrictions on the motion of the structure. The formulation technique used in that work is cumbersome in light of the relaxation introduced here. Hence a substructure approach is adopted. The in- and out-of-plane motions uncouple and the system reduces to solving two sets of governing equations of motion. Neither the analytical expression for the frequency equation nor the analytical expressions for the mode shapes are included in the paper because of their extreme length and complexity. Nevertheless, the components of the matrices that lead to the frequency equations and expressions for the mode shapes, for both the in- and out-of-plane motions, are presented. The orthogonality conditions are also included.

Numerical simulations are performed to examine the effects of: the ratio of the lengths of the beams, the ratio of the mass at the tip to the mass of the first beam, and the orientation angle on the natural frequencies. The contribution of each member to the system potential energy is also investigated.

Section snippets

Equations of motion

A global reference frame is attached at the base of the structure and a non-inertial frame is attached at the joint as depicted in Fig. 1. The unit vectors along the x1-, y1-, and z1-axis of the fixed inertial frame Fa are, respectively, defined as a1, a2 and a3. Similarly, the unit vectors of the body fixed frame Fb are b1, b2 and b3 and they correspond to the x2-, y2-, and z2-axis of the non-inertial frame Fb which has its origin at the junction of the two beam segments.

The assumed

Frequency equation

To reduce the number of defining parameters the following non-dimensional variables are introduced:ξi=xiLi,ρ=ρ2A2ρ1A1,Mt=mtρ1A1L1,L=L2L1ui2=ρiLi2ω2Ei,λvi4=ρiAiLi4ω2EiIzz(i),λwi4=ρiAiLi4ω2EiIyy(i),λψi2=ρiLi2ω2Gi,σ=E2A2E1A1,η=G2J2G1J1,νvi=Izz(i)AiLi2,νwi=Iyy(i)AiLi2,χi=EiIzz(i)GiJi.By assuming a separable solution in the form (i=1,2)ui(xi,t)=LiUii)ejωt,vi(xi,t)=LiVii)ejωt,wi(xi,t)=LiWii)ejωt,ψi(xi,t)=Ψii)ejωt,the equations of motion can be written as (i=1,2)Ui″+λui2Ui=0,Vi″″−λvi4Vi=0,Wi

Orthogonality conditions

The orthogonality conditions are derived using the equations of motion (13), the boundary conditions , , , , and the compatibility conditions , , , , , , , , . The resulting expression for the in-plane motion can be expressed asmtL22(U2i(1)U2j(1)+W2i(1)W2j(1))+k=12ρkAkLk301(UkiUkj+WkiWkj)dxk=0and that for the out-of-plane motion may be written asmtL22V2i(1)V2j(1)+k=12ρkLk01(AkLk2VkiVkj+JkΨkiΨkj)dxk=0.

Numerical examples

The following examples have been chosen with several objectives in mind. The first is simply to confirm the formulation by making comparison to previous results. The second is to illustrate the effect that changing the relative lengths of the beams, while maintaining a constant system mass, has on the frequency behavior of the system. The third is to explore the effect that changing the size of the tip mass has on the frequency behavior of the system for a fixed value of L.

Summary

It has been shown that inclusion of the axial deformation component in the assumed displacement field for the two beams ultimately allows the in- and out-of-plane frequency equations to be uncoupled and solved independently. The first five frequencies, in the examples presented, support the expectation that they are dominated by either bending or torsion and there is no significant axial component in them. Therefore, while its inclusion in the formulation has provided analytical benefits the

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