Free Vibrations of Beam System Structures with Elastic Boundary Conditions and an Internal Elastic Hinge

Abstract

The study of the dynamic properties of beam structures is extremely important for proper structural design. This present paper deals with the free in-plane vibrations of a system of two orthogonal beam members with an internal elastic hinge. The system is clamped at one end and is elastically connected at the other. Vibrations are analyzed for different boundary conditions at the elastically connected end, including classical conditions such as clamped, simply supported, and free. The beam system is assumed to behave according to the Bernoulli-Euler theory. The governing equations of motion of the structural system in free bending vibration are derived using Hamilton's principle. The exact expression for natural frequencies is obtained using the calculus of variations technique and the method of separation of variables. In the frequency analysis, special attention is paid to the influence of the flexibility and location of the elastic hinge. Results are very similar with those obtained using the finite element method, with values of particular cases of the model available in the literature, and with measurements in an experimental device.

1. Introduction

The study of the dynamic properties of beam systems is very important in structural design as they are the cornerstone for many resistant structures.

The issue is relevant virtually in all fields of engineering: structures composed of beams can resist by virtue of its geometry. Such structures can be found from large scale, such as bridges and buildings located in seismically active regions to microbeam systems used in modern electronic equipment which is subject to vibration environment.

As Laura and his coworkers pointed out [1], many excellent books and technical papers deal with vibrating beam systems, including [2–6].

Many researchers have analyzed the vibration of beam systems. Reference [1] dealt with the determination of the fundamental frequency of vibration of a frame elastically restrained against translation and rotation at the ends, carrying concentrated masses. Reference [7] proposed a hybrid analytical/numerical method to do dynamic analysis of planar serial-frame structures. Reference [8] presented an elastic- and rigid-combined beam element to determine the dynamic characteristics of a two-dimensional frame composed of any number of beam segments. In his paper, Mei [6] considered the vibration in multistory planar beam structures from the wave vibration standpoint. Reference [9] analyzed in-plane vibrations of portal frames with elastically restrained ends. An approximate solution is obtained by means of a variational method.

In the particular case of L-beam structures, early studies have been done by [10–12]. In 2003, [13] extended the previous papers [12] by relaxing the restrictions on the motion of the open frame. In 2005, [14] determined natural frequencies and mode shapes of elastically restrained L-beams. They applied the separation of variables method for the determination of the exact eigenfrequencies and mode shapes and calculated the eigenvalues numerically by applying the Newton method strategy to the corresponding frequency equation. Reference [15] used a formulation by the Rayleigh-Ritz method together with the introduction of artificial linear and torsional springs for computing the natural frequencies and modes for the in-plane vibrations of complex planar beam structures.

The presence of an internal hinge in beams has been treated in several papers including [16–22]. Here, we deal with the vibration of L-beams structures assuming an internal hinge in different positions of the structural system.

The two parts of the L-shaped geometry are joined at right angle, with the end of one of them clamped and the end of the other elastically restrained. Figure 1 depicts the structure under study.

Figure 1 

Beam system structure.

Classical structural models do not consider the properties of the connection stiffness, because they are based only on models with pinned or rigid joints. Many authors have studied structures with flexibly connected members as it is known that the behavior of the connection plays an important role in analysis and design. Reference [23] presented a computer-based method for geometrically nonlinear beam system with semirigid beam-to-column connections. Reference [24] studied wooden framed structures; they considered that mechanical connections are recognized as extremely important elements in the aspect of strength and structural safety. Reference [25] studied steel frames and observed that the interstory shears generally increase when the connections stiffness is taken into account. As known, ideal supports used in many structural models do not fit exactly real supports.

In the current presentation it is assumed that the beams are adequately modeled using Euler-Bernoulli theory, so that the effects of shear deformation and rotatory inertia are considered to be small, and they are neglected in the analysis. The cross sections of beam elements have double symmetry (the shear centre and centroid are coincident), so it can be assumed that there is no coupling between bending displacements and torsional rotations.

The beam system is modeled in Mathematica code [26] using the method of separation of variables to obtain the exact values of the natural frequency coefficients.

Finally, numerical results are shown for slender beam systems by considering the effects of different stiffness of the elastic connections. A comparison is made with results obtained by the authors with the finite element method, [27] and with values available in the literature. In addition, some particular cases are also compared with the experimental results of a specially constructed device.

2. Theory

A more realistic model of an L-geometry beam system is presented to analyze the natural vibration problem. The structure under study has elastic restrains at F and a clamped end H as it is shown in Figure 1, with (𝛼₁+𝛼₃=𝜋/2). It is assumed that the elastic internal hinge at P can be located in different positions. The structure has three beam members: FO beam, with length 𝑙₁; OP beam, with length 𝑙₂; and PH beam, with length 𝑙₃. Each of them has uniform properties throughout its length. The influence of the type of connection between beam-elements of the structural system and elastic conditions on the outer edge F is considered in the study. The external end H has a classical clamped condition, while the external end F is supported by two translational springs of stiffness 𝑡_𝑤 and 𝑡_𝑢 and a rotational spring of stiffness 𝑟_𝑧, Figure 2(a). At point P, there is an internal hinge, elastically restrained against rotation between beams 2, OP, and 3, PH; this semirigid connection is materialized by a rotational spring of stiffness 𝑟_𝑚, Figure 2(b).

(a)

(b)

Figure 2 

(a) Elastic boundary conditions at end F; (b) elastic internal hinge at joint P.

The flexural rigidity, the mass density, the length, and the area of the cross section of each beam are 𝐸_𝑖𝐼_𝑖, 𝜌_𝑖, 𝑙_𝑖, and 𝐴_𝑖, with 𝑖=1,2,3.

Three coordinate systems are located as shown in Figure 1. Respectively, each coordinate origin is at the point F, O, or P. At abscissa 𝑥_𝑖 (0≤𝑥_𝑖≤𝑙_𝑖) and at any time 𝑡: 𝑤_𝑖 is the flexural displacement in the transverse direction of the beam’s neutral axis, 𝜃_𝑖=𝜕𝑤_𝑖/𝜕𝑥_𝑖 is the section rotation, and 𝑢_𝑖 is the axial displacement. The deformation of a beam in the 𝑥 direction is not taken into account. The beams are considered infinitely rigid in the 𝑥 direction.

The sign convention used for the positive shear force spins an element clockwise (up on the left and down on the right). Likewise the normal convention for a positive bending moment elongates the reference fiber of the beam indicated by the dotted line. Figure 3 shows the sign convention to be employed.

Figure 3 

Sign convention for positive transverse displacement (𝑤), shear force (𝑄), and bending moment (𝑀).

For free vibration, the bending moment and the shear force expressions are 𝑄_𝑖𝑥_𝑖,𝑡=𝐸_𝑖𝐼_𝑖𝜕³𝑤_𝑖𝜕𝑥³_{𝑖(𝑥𝑖,𝑡)},𝑀_𝑖𝑥_𝑖,𝑡=𝐸_𝑖𝐼_𝑖𝜕²𝑤_𝑖𝜕𝑥²_{𝑖(𝑥𝑖,𝑡)}. (1) At time 𝑡, the kinetic energy of the beams is given by 𝑇^∗=³_𝑖=112^𝑙𝑖₀𝜌_𝑖𝐴_𝑖𝜕𝑤_𝑖𝜕𝑡_{(𝑥𝑖,𝑡)}²𝑑𝑥_𝑖+𝜌₁𝐴₁𝑙₁2𝜕𝑢₁𝜕𝑡_(𝑙1,𝑡)². (2) The last term of the kinetic energy expression is due to the rigid body translation of beam 1 of length 𝑙₁. Due to the assumption of infinity axial rigidity, 𝜕𝑢₁/𝜕𝑡 at 𝑥₁=𝑙₁ is equal to 𝜕𝑤₂/𝜕𝑡 at 𝑥₂=0; therefore in (2) the expression (𝜕𝑢₁/𝜕𝑡)²|_(𝑙1,𝑡) can also be replaced by (𝜕𝑤₂/𝜕𝑡)²|_(0,𝑡).

On the other hand, the potential energy of the mechanical system is given by 𝑈^∗=12³_𝑖=1^𝑙𝑖₀𝐸_𝑖𝐼_𝑖𝜕²𝑤_𝑖𝜕𝑥²_{𝑖(𝑥𝑖,𝑡)}²𝑑𝑥_𝑖+12𝑟_𝑚𝜕𝑤₂𝜕𝑥_{2(𝑙2,𝑡)}−𝜕𝑤₃𝜕𝑥_3(0,𝑡)²+12𝑟_𝑧𝜕𝑤₁𝜕𝑥_1(0,𝑡)²+12𝑡_𝑤𝑤₁(0,𝑡)²+12𝑡_𝑢𝑢₁(0,𝑡)², (3) which involves the work of the elastic constrains. And again due to the assumption of infinity axial stiffness, 𝑢₁(0,𝑡)=𝑢₁(𝑙₁,𝑡)=𝑤₂(0,𝑡); therefore in (3) the expression (𝑢₁(0,𝑡))² can also be replaced by (𝑤₂(0,𝑡))².

To express equations in dimensionless form, the nondimensional parameter is introduced: 𝑋_𝑖=𝑥_𝑖𝑙_𝑖,with𝑋_𝑖∈[0,1]∀𝑖=1,2,3. (4) The displacements 𝑢_𝑖 and 𝑤_𝑖, and 𝜃_𝑖 may be expressed in terms of the dimensionless coordinates as follows: 𝑊_𝑖𝑋_𝑖=𝑤,𝑡^𝑖𝑥_𝑖,𝑡𝑙_𝑖,𝜃_𝑖𝑋_𝑖,𝑡=𝜕𝑊_𝑖𝜕𝑋_{𝑖(𝑋𝑖,𝑡)}=𝜕𝑤_𝑖𝜕𝑥_{𝑖(𝑥𝑖,𝑡)},𝑈_𝑖𝑋_𝑖=𝑢,𝑡^𝑖𝑥_𝑖,𝑡𝑙_𝑖. (5) The characteristics of beam 1 are used as “reference”: 𝐸𝐼=𝐸₁𝐼₁,𝜌𝐴=𝜌₁𝐴₁,𝑙=𝑙₁, (6) to define the ratios: 𝜈_𝐸𝐼𝑖=𝐸_𝑖𝐼_𝑖𝐸𝐼,𝜈_𝜌𝐴𝑖=𝜌_𝑖𝐴_𝑖𝜌𝐴,𝜈_𝑙𝑖=𝑙_𝑖𝑙, (7) the dimensionless spring stiffness: 𝑇_𝑤=𝑡_𝑤𝑙³𝐸𝐼,𝑇_𝑢=𝑡_𝑢𝑙³,𝑅𝐸𝐼_𝑧=𝑟_𝑧𝑙𝐸𝐼,𝑅_𝑚=𝑟_𝑚𝑙,𝐸𝐼 (8) and the dimensionless frequency coefficient: 𝜆⁴=𝑙⁴𝜔²𝜌𝐴,𝐸𝐼 (9) where 𝜔 is the circular natural frequency of the vibrating system in radians per second.

The expression of the energy functional of the system of beams is 𝐽=𝑇^∗−𝑈^∗.

Hamilton’s principle requires that 𝛿∫^𝑡𝑏_𝑡𝑎𝐽𝑑𝑡 taken between arbitrary intervals of time (𝑡_𝑎,𝑡_𝑏) at which the positions of the mechanical system are known is equal to zero. That means that the system should execute a motion which makes the functional 𝐽 stationary on the space of admissible functions: 𝛿^𝑡𝑏_𝑡𝑎𝑇^∗−𝑈^∗𝛿𝑑𝑡=0,_{𝑡𝑏𝑡𝑎}𝑇^∗−𝑈^∗1𝑑𝑡=𝛿2𝜌𝐴𝑙³×^𝑡𝑏_𝑡𝑎³_𝑖=1¹₀𝑣_𝜌𝐴𝑖𝑣³_𝑙𝑖𝜕𝑊_𝑖𝜕𝑡_{(𝑋𝑖,𝑡)}²𝑑𝑋_𝑖+𝑣_𝜌𝐴1𝑣_𝑙1𝑣²_𝑙2𝜕𝑊₂𝜕𝑡_(0,𝑡)²−1𝑑𝑡2𝐸𝐼𝑙^𝑡𝑏_𝑡𝑎³_𝑖=1¹₀𝑣_𝐸𝐼𝑖𝑣_𝑙𝑖𝜕²𝑊_𝑖𝜕𝑋²_{𝑖(𝑋𝑖,𝑡)}²𝑑𝑋_𝑖+𝑑𝑡_{𝑡𝑏𝑡𝑎}𝑅_𝑚𝜕𝑊₂𝜕𝑋_2(1,𝑡)−𝜕𝑊₃𝜕𝑋_3(0,𝑡)²+𝑑𝑡_{𝑡𝑏𝑡𝑎}𝑅_𝑧𝜕𝑊₁𝜕𝑋_1(0,𝑡)²+𝑑𝑡_{𝑡𝑏𝑡𝑎}𝑇_𝑤𝑣_𝑙1²𝑊₁(0,𝑡)²+𝑑𝑡_{𝑡𝑏𝑡𝑎}𝑇_𝑢𝑣_𝑙2²𝑊₂(0,𝑡)²𝑑𝑡. (10)(11) Taking into account the boundary conditions at the ends, the compatibility, and equilibrium conditions at the joints between beam elements and applying the procedure of calculus of variations in (10), the following boundary and eigenvalue problem is obtained 𝜕⁴𝑊₁𝜕𝑋⁴_{1(𝑋1,𝑡)}+𝑘⁴₁𝜕²𝑊₁𝜕𝑡²_(𝑋1,𝑡)𝜕=0,₄𝑊₂𝜕𝑋⁴_{2(𝑋2,𝑡)}+𝑘⁴₂𝜕²𝑊₂𝜕𝑡²_(𝑋2,𝑡)𝜕=0,₄𝑊₃𝜕𝑋⁴_{3(𝑋3,𝑡)}+𝑘⁴₃𝜕²𝑊₃𝜕𝑡²_(𝑋3,𝑡)=0, (12) with 𝑘⁴_𝑖=(𝜌_𝑖𝐴_𝑖/𝐸_𝑖𝐼_𝑖)𝑙⁴_𝑖, 𝑖=1,2,3.

Consider 𝑣_𝑙1𝑊₁(1,𝑡)=0,𝑣_𝑙2𝑊₂(1,𝑡)=𝑣_𝑙3𝑊₃(0,𝑡),𝜕𝑊₁𝜕𝑋_1(1,𝑡)=𝜕𝑊₂𝜕𝑋_2(0,𝑡),𝑣_𝐸𝐼1𝑣_𝑙1𝜕²𝑊₁𝜕𝑋²_1(1,𝑡)=𝑣_𝐸𝐼2𝑣_𝑙2𝜕²𝑊₂𝜕𝑋²_2(0,𝑡),𝑣_𝐸𝐼2𝑣_𝑙2𝜕²𝑊₂𝜕𝑋²_2(1,𝑡)−𝑅_𝑚𝜕𝑊₃𝜕𝑋_3(0,𝑡)−𝜕𝑊₂𝜕𝑋_2(1,𝑡)𝑣=0,_𝐸𝐼3𝑣_𝑙3𝜕²𝑊₃𝜕𝑋²_3(0,𝑡)−𝑅_𝑚𝜕𝑊₃𝜕𝑋_3(0,𝑡)−𝜕𝑊₂𝜕𝑋_2(1,𝑡)𝑣=0,_𝐸𝐼1𝑣_𝑙1²𝜕³𝑊₁𝜕𝑋³_1(0,𝑡)+𝑇_𝑤𝑣_𝑙1𝑊₁()𝑣0,𝑡=0,_𝐸𝐼2𝑣²_𝑙2𝜕³𝑊₂𝜕𝑋³_2(1,𝑡)−𝑣_𝐸𝐼3𝑣²_𝑙3𝜕³𝑊₃𝜕𝑋³_3(0,𝑡)𝑣=0,_𝐸𝐼2𝑣_𝑙2²𝜕³𝑊₂𝜕𝑋³_2(0,𝑡)+𝑇_𝑢𝑣_𝑙2𝑊₂(0,𝑡)−𝑘⁴𝑣_𝑙1𝑣_𝑙2𝑊₂()𝑣0,𝑡=0,_𝐸𝐼1𝑣_𝑙1𝜕²𝑊₁𝜕𝑋²_1(0,𝑡)−𝑅_𝑧𝜕𝑊₁𝜕𝑋_1(0,𝑡)𝑣=0,_𝑙3𝑊₃(1,𝑡)=0,𝜕𝑊₃𝜕𝑋_3(1,𝑡)=0. (13) Using the well-known separation of variables method, solution of (12) is assumed to be of the form: 𝑊₁𝑋₁,𝑡=^∞_𝑛=1𝑊_1𝑛𝑋₁𝑇(𝑡),𝑊₂𝑋₂,𝑡=^∞_𝑛=1𝑊_2𝑛𝑋₂𝑇(𝑡),𝑊₃𝑋₃,𝑡=^∞_𝑛=1𝑊_3𝑛𝑋₃𝑇(𝑡). (14) The functions 𝑊_1𝑛, 𝑊_2𝑛, and 𝑊_3𝑛 represent the corresponding transverse modes of natural vibration of each beam member and are given by 𝑊_1𝑛𝑋₁=𝐶₁cosh𝜆_𝑛𝛼₁𝑋₁+𝐶₂senh𝜆_𝑛𝛼₁𝑋₁+𝐶₃cos𝜆_𝑛𝛼₁𝑋₁+𝐶₄sen𝜆_𝑛𝛼₁𝑋₁,𝑊_2𝑛𝑋₂=𝐶₅cosh𝜆_𝑛𝛼₂𝑋₂+𝐶₆senh𝜆_𝑛𝛼₂𝑋₂+𝐶₇cos𝜆_𝑛𝛼₂𝑋₂+𝐶₈sen𝜆_𝑛𝛼₂𝑋₂,𝑊_3𝑛𝑋₃=𝐶₉cosh𝜆_𝑛𝛼₃𝑋₃+𝐶₁₀senh𝜆_𝑛𝛼₃𝑋₃+𝐶₁₁cos𝜆_𝑛𝛼₃𝑋₃+𝐶₁₂sen𝜆_𝑛𝛼₃𝑋₃, (15) where 𝛼_𝑖=𝑣_𝑙𝑖⁴𝑣_𝜌𝐴𝑖/𝑣_𝐸𝐼𝑖 is a mechanical and geometrical parameter, with 𝑖=1,2,3, 𝜆_𝑛=⁴𝑙⁴𝜔²_𝑛𝜌𝐴/(𝐸𝐼) is the dimensionless frequency coefficient of mode of vibration 𝑛, and 𝐶₁,𝐶₂,…,𝐶₁₂ are arbitrary constants to be determined.

Replacing expressions (15) in (14) and these ones in (13), a linear system of equations in the unknown constants 𝐶₁,𝐶₂,…,𝐶₁₂ is obtained.

For a nontrivial solution to exist the determinant of the coefficient matrix in the linear system of equations should be equal to zero and the roots of the transcendental frequency equation are the dimensionless frequency coefficients of the mechanical system in Figure 1.

3. Finite Element Method

The authors solved some numerical examples using the finite element method, using the software ALGOR 23.1 [27]. The three members of the structure are divided into 100 beams elements, respectively, each beam element with three degrees of freedom.

The internal hinge elastically restrained was modeled by a very small beam element, 300 times smaller than the length of the beam. The moment of inertia of the section was varied in order to obtain stiffness values that are equivalent to the stiffness constants of the spring connecting the two sections at location P.

4. Experimental Model

An experimental model was built of steel to compare the results obtained by the analytical and finite element models.

The experimental beam system has two parts of equal length 𝑙 (see Figure 4). It was tested under two different boundary conditions (clamped and free at 𝑥₁=0) while the other end (𝑥₃=1) was clamped. The presence of an internal hinge was not considered. The geometry of the structural system is described by 𝑙=𝑙₁=𝑙₂+𝑙₃=0.50 m; 𝐴=𝐴_𝑖=4.064×10⁻⁵ m²; 𝐼=𝐼_𝑖=3.468×10⁻¹¹ m⁴; and the material properties are 𝝆=𝜌_𝑖=7870 kg/m³ and 𝐸=𝐸_𝑖=2.1×10⁶ kg/cm², with 𝑖=1,2,3.

Figure 4 

Experimental system device; C-C model.

In order to measure the natural frequencies, an optical proximity sensor was used as it is seen in Figure 4.

Figures 5 and 6 show the spectrum of the first ten natural frequencies of the free-clamped (Case a) and the clamped-clamped (Case b) beam systems, respectively.

Figure 5 

Spectrum of the first natural frequency; Case a: F-C model.

Figure 6 

Spectrum of the first natural frequency; Case b: C-C model.

5. Numerical Results

Frequency coefficients were obtained by the exact analytical solution, with the finite element method, FEM, [27] and with the experimental model.

Table 1 presents the first ten coefficients of natural frequency of vibration of a beam structure with free-clamped boundary conditions without internal hinge, Figure 5.

Table 1 

Comparison of the first ten frequency coefficients 𝜆_𝑛=⁴𝑙⁴𝜔²_𝑛(𝜌𝐴/𝐸𝐼) for L-beams: Case a: F-C.

For the analytical solution the parameters are taken as 𝑣_𝜌𝐴𝑖=1,𝑣_𝐸𝐼𝑖𝑣=1,∀𝑖=1,2,3,_𝑙1=2𝑣_𝑙2=2𝑣_𝑙3=1. (16) The analytical solution for Case a: F-C model, was obtained with 𝑇_𝑢=𝑇_𝑤=𝑅_𝑧=0; 𝑅_𝑚→∞. It is remarkable that for the first ten frequencies, the analytical results are very close to the experimental ones. The analytical results are also compared with previous published ones, [7]. As it can be seen in the table all of them are in excellent agreement.

The exact analytical solution for C-C structure was obtained with 𝑇_𝑢=𝑇_𝑤=𝑅_𝑧=𝑅_𝑚→∞.

Table 2 shows the first ten coefficients of natural frequency of vibration of the beam structure with clamped-clamped boundary conditions. Again the first ten frequency coefficients obtained by the analytical model are very similar to the experimental model. They also are in excellent agreement with the FEM solution and previous published results [14].

Table 2 

Comparison of the first ten frequency coefficients 𝜆_𝑛=⁴𝑙⁴𝜔²_𝑛(𝜌𝐴/𝐸𝐼) for L-beams: Case b: C-C.

Table 3 shows the frequency coefficients for pinned-clamped L-beams obtained by the analytical procedure and the finite element method. For the analytical solution the spring constants are assigned in two different ways to model the simply supported-clamped system:(1)𝑣_𝐸𝐼𝑖=1, 𝑣_𝜌𝐴𝑖=1, for all 𝑖=1,2,3, 𝑣_𝑙2=𝑣_𝑙3=1/2,  𝑇_𝑤→∞, 𝑇_𝑢→∞, 𝑅_𝑧=0, and 𝑅_𝑚→∞.(2)𝑣_𝐸𝐼𝑖=1, 𝑣_𝜌𝐴𝑖=1, for all 𝑖=1,2,3, 𝑣_𝑙2=0.99, 𝑣_𝑙3=0.01, 𝑇_𝑤→∞, 𝑇_𝑢→∞, 𝑅_𝑧→∞, and 𝑅_𝑚=0.The percentage differences between both sets of results are shown in file |Δ|%. Although both sets of results correspond to a simply supported-clamped system, the small percentage differences, less than 0.7%, are due to the different numerical computational aspects between the analytical models (1) and (2). The results were determined using the Mathematica software [26] with five significant figures.

Table 3 

Frequency coefficients 𝜆_𝑛=⁴𝑙⁴𝜔²_𝑛(𝜌𝐴/𝐸𝐼) for simply supported-clamped beam system; comparison of results.

Table 4 compares the first five natural coefficients of vibration for a free-clamped system with Morales published results [28]. The characteristics of Morales frame are 𝑙₁=2.215 m and 𝑙₂+𝑙₃=4.249 m, 𝐸₁𝐼₁=0.0147Nm², 𝐸₂𝐼₂=𝐸₃𝐼₃=0.0267Nm², 𝑚₁=6×10⁻³kg/m, and 𝑚₂=𝑚₃=4.5×10⁻⁵kg/m. For the analytical solution, the parameters are assumed as 𝑙₁=2.215m, 𝑙₂=2.249m, 𝑙₃=2.000m, 𝑣_𝑙1=1, 𝑣_𝐸𝐼1=1, 𝑣_𝜌𝐴1=1, 𝑣_𝑙2=1.0153, 𝑣_𝐸𝐼2=1.8163, 𝑣_𝜌𝐴2=0.0075, 𝑣_𝑙3=0.0903, 𝑣_𝐸𝐼3=1.8163, 𝑣_𝜌𝐴3=0.0075, 𝑇_𝑤=0, 𝑅_𝑧=0, 𝑇_𝑢=0, and 𝑅_𝑚→∞. It can be verified that our results are in excellent agreement whit those obtained by Morales.

Table 4 

Non dimensional frequency coefficients 𝜆_𝑛=⁴𝑙⁴𝜔²_𝑛(𝜌𝐴/𝐸𝐼) for free-clamped beam system; comparison of results.

Traditionally, the analysis and design of beam structures have been based on the assumption that the boundary conditions are rigid. The disadvantage of this model is that the flexibility of the real boundary conditions is neglected and therefore the real natural frequencies cannot be obtained.

Thus, we now analyze the case of beam system structure with elastic boundary conditions at edge F and clamped at the other end H, while the internal elastic hinge (at section P) is assumed to have 𝑅_𝑚→∞. Numerical simulations are carried out to investigate the effect of the rigidity of the boundary conditions on the natural frequencies of the system.

In Figure 7 the effect of rigidity 𝑇_𝑢 of the translational spring on the frequency coefficients of the structure is shown. It can be seen that the first frequency coefficient 𝜆₁ increases with increasing the value of 𝑇_𝑢, from 𝑇_𝑢=0, 𝜆₁=1.5208 to 𝑇_𝑢→200, 𝜆₁=3.3762. For values larger than 200 the fundamental coefficient stays practically the same: 𝑇_𝑢→∞, 𝜆₁=3.3920. Its increase is of 124%.

Figure 7 

Effect of 𝑇𝑢 on the first five natural frequency coefficients. 𝑙2=𝑙3, 𝑣𝑙1=𝑣𝑙2=0.5, 𝑣𝐸𝐼(2)=𝑣𝐸𝐼(3)=1, 𝑣𝜌𝐴(2)=𝑣𝜌𝐴(3)=1, 𝑅𝑚→∞, 𝑅𝑧=0, and 𝑇𝑤→∞.

The second frequency is 3.3947 for 𝑇_𝑢=0 and 4.4566 for 𝑇_𝑢→∞; its increase is of 31% and the most significant change occurs between 𝑇_𝑢=200 and 𝑇_𝑢=1000. The higher frequency coefficients exhibit a similar behavior.

From Figure 7 it could be concluded that the first and second frequencies interchange their modal shape for a value of 𝑇_𝑢 between 100 and 1000. A similar behaviour occurs in the case of third and fourth frequencies for a value of 𝑇_𝑢 between 1000 and 10000.

Figure 8 shows the effect of 𝑇_𝑤 on the first natural frequency coefficients. Again it is observed as a matter of course that the natural frequency coefficients increase with the spring constant parameter.

Figure 8 

Effect of 𝑇𝑤 on the first five natural frequency coefficients. 𝑙2=𝑙3, 𝑣𝑙1=𝑣𝑙2=0.5, 𝑣𝐸𝐼(2)=𝑣𝐸𝐼(3)=1, 𝑣𝜌𝐴(2)=𝑣𝜌𝐴(3)=1, 𝑅𝑚→∞, 𝑅𝑧=0, and 𝑇𝑢→∞.

Figure 9 shows that the effect of the spring 𝑅_𝑧 is small.

Figure 9 

Effect of 𝑅𝑧 on the first five natural frequency coefficients. 𝑙2=𝑙3, 𝑣𝑙1=𝑣𝑙2=0.5, 𝑣𝐸𝐼(2)=𝑣𝐸𝐼(3)=1, 𝑣𝜌𝐴(2)=𝑣𝜌𝐴(3)=1, 𝑅𝑚→∞, 𝑇𝑤→∞, and 𝑇𝑢→∞.

Table 5 presents the case of a clamped-clamped structure with an elastic joint at 𝑙₂=0.5𝑙₁, and 𝑅_𝑚 is the constant rigidity of the rotational spring at P that connects the beam members indicated as OP and PH. In this table 𝑅_𝑚 assumes different values from infinity to zero. It can be seen that all the first frequency coefficients decrease in value, except for 𝜆₄, which practically remains constant.

Table 5 

Effect of 𝑅_𝑚 on the first five non dimensional frequency coefficients of a C-C system, with an elastic joint at 𝑙₂ = 0.5𝑙₁, beam-beam connection.

Table 6 presents the case of a clamped-clamped structure with an elastic connection at P, 𝑙₂→0, 𝑙₃=𝑙₁. In this table, 𝑅_𝑚 assumes different values, from →∞ to 0. It can be seen that the first, the third, and the fifth frequency coefficients remain practically constant and the second coefficients decrease by 20% and the fourth decrease by 11%.

Table 6 

Effect of 𝑅_𝑚 on the first five non dimensional frequency coefficients of a C-C beam system, with an elastic joint at 𝑙₂→0, 𝑙₃=𝑙₁ (exact analytical solution).

Table 7 presents the frequency coefficients of a simply supported-clamped structure with an elastic joint at 𝑙₂=0.5𝑙₁.

Table 7 

Effect of 𝑅_𝑚 on the first five non dimensional frequency coefficients of a SS-C beam system, with an elastic joint at 𝑙₂ = 0.5𝑙₁, beam-beam connection (exact analytical solution).

In the present conditions, the first, the second, and the fifth frequency coefficients decrease in value and the third and fourth remain practically constant.

Table 8 contains the frequency coefficients of a simply supported-clamped beam system with an elastic connection at P, 𝑙₂→0, 𝑙₃=𝑙₁. It can be seen that all the frequency coefficients decrease in value between 8% (𝜆₁) and 3% (𝜆₅).

Table 8 

Effect of 𝑅_𝑚 on the first five non dimensional frequency coefficients of a SS-C beam system, with an elastic joint at 𝑙₂→0, 𝑙₃=𝑙₁ (exact analytical solution).

Figures 10 and 11 show graphically the variation of first three natural frequency coefficients for various locations of the elastic hinge: 𝑙₂/𝑙→0 to 𝑙₂/𝑙→1, with different combinations of 𝑅_𝑧, 𝑇_𝑤, 𝑇_𝑢 and 𝑅_𝑚. In both figures, it is clear that for the fundamental frequency coefficient the differences in the values of frequencies are not so significant.

Figure 10 

Variation of first three natural frequency coefficients with location of the elastic hinge, 𝑙1=𝑙2+𝑙3=𝑙, 𝑙2/𝑙∈(0,1), 𝑣𝐸𝐼(2)=𝑣𝐸𝐼(3)=𝑣𝜌𝐴(2)=𝑣𝜌𝐴(3)=1, 𝑅𝑧=3, 𝑇𝑤=10, 𝑇𝑢=5, and 𝑅𝑚=2.

Figure 11 

Variation of first three natural frequency coefficients with location of the elastic hinge, 𝑙1=𝑙2+𝑙3=𝑙, 𝑙2/𝑙∈(0,1), 𝑣𝐸𝐼(2)=𝑣𝐸𝐼(3)=𝑣𝜌𝐴(2)=𝑣𝜌𝐴(3)=1, 𝑅𝑧=50, 𝑇𝑤=100, 𝑇𝑢=100, and 𝑅𝑚=20.

6. Conclusions

The model presented in this paper allows studying the effect of the variation in the rigidity of the elastic supports and the elastic joint on the dynamical behavior of the beam system structure. The model enables an efficient, simple, and straightforward way of computing natural frequency coefficients for L-Shaped-Structures with many different boundary conditions at simply supported, clamped, free, and elastically restrained. It was observed that the loss of rigidity of the translational springs significantly changes the natural frequency coefficients, especially the fundamental frequency coefficient. The presence of an elastic joint, modeling a beam-beam connection, also implies changes in the natural frequencies when the rigidity is lost.

Although the model with elastic connection and supports does not represent the inherent complexities of real systems such as nonlinearity or damping, it provides conceptual insights regarding the fact that the loss of rigidity can cause significant changes in the dynamic behavior of the structure. In this sense, this model can be seen as a useful first approach to study the real system. The proposed analytical solution may be applied to include more complicating effects, such as elastic constraints at both ends, more than one internal elastic hinge, and another geometry with 𝛼₁+𝛼₃≠𝜋/2. Finally, it has been demonstrated that an elastic approach constitutes a reliable tool to deal with beam system structures.

Acknowledgments

The present work has been sponsored by Secretaría General de Ciencia y Tecnología of Universidad Nacional del Sur at the Department of Engineering and by Consejo Nacional de Investigaciones Científicas y Técnicas. The authors are indebted to the Chief of Mechanical Vibration Laboratory at the Department of Engineering, Ing. Santiago Maiz for his cooperation and useful suggestions, and to Mr. Gabriel Leguizamón for his careful preparation of the experimental device.