Vibration Analysis of Viscoelastic Timoshenko Cracked Beams with Massless Viscoelastic Rotational Spring Models

. Based on the equivalent bending sti ﬀ ness of the viscoelastic cracked beam with open cracks, the corresponding complex frequency characteristic equations of a Timoshenko viscoelastic cracked beam are obtained by using the method of separation of variables and the Laplace transform. The vibration characteristics of a viscoelastic Timoshenko cracked beams with the standard linear solid model and Kelvin-Voigt model are investigated. By numerical examples, the e ﬀ ects of the crack location, crack number, crack depth, and slenderness ratio on the vibration characteristics of the viscoelastic cracked beams are revealed.


Introduction
Viscoelastic materials [1,2] are widely used in civil, mechanical, and aerospace engineering, etc. In order to investigate the vibration fatigue characteristics of the viscoelastic structures, the modal analysis method can be used to derive the analytical solutions. By using the complex modal method and Laplace transform, Huang and Huang [3] studied the free vibration of Timoshenko viscoelastic beams satisfying the standard linear solid constitutive equations. Considering the axial forces, Chen et al. [4] studied the vibration characteristics of clamped-clamped Timoshenko viscoelastic beams constituted by the Kelvin-Voigt model. Peng [5] applied the complex modal method and differential quadrature method to analyze the transverse vibration characteristics of the elastic Euler-Bernoulli and Timoshenko beams resting on the viscoelastic foundation.
Additionally, the correspondence principle, integral transformation, finite element method (FEM), differential quadrature method [6,7], and other numerical methods [8,9] were applied to analyze the static and dynamic properties of the viscoelastic beam structures. This paper is organized as follows. Firstly, the equivalent flexural stiffness of the viscoelastic cracked beam established in reference [10] is used to present the motion equation of a Timoshenko viscoelastic beam with open cracks. Then, the general explicit analytical expressions for solving the complex frequency of Timoshenko viscoelastic cracked beams are derived by using the separation of variables method and Laplace transform. Finally, by numerical examples, the effects of the crack location, crack number, crack depth, and slenderness ratio on the vibration characteristics of the viscoelastic cracked beams are investigated.

Formulation of the Problem
According to the constitutive equation of the standard linear solid model [1,2], E 1 and E 2 are the elastic modulus of elastic elements, η is the viscous coefficient of a viscous element, and the relaxation modulus YðtÞ and shear modulus GðtÞ are defined as follows: where the Poisson's ratio ν is a constant, and Suppose that the superscript denotes the Laplace transform of the function with respect to the time t, and the Laplace transform of relaxation modulus and shear modulus are given as Y s ð Þ = q 0 + sq 1 where the parameter is λ 1 = 0:5ð1 + νÞ −1 , and s is the Laplace transform parameter. The physical model of the viscoelastic beam is given as shown in Figure 1. Let us consider a viscoelastic rectangular beam with length L (x -axis), width b (y -axis), and height h (z -axis). Here, wðx, tÞ and ϕðx, tÞ denote the transverse deflection of the axial line and rotation angle of the beam cross section subjected to the distributed transverse load qðx, tÞ, respectively. Assuming that the crack at the location x = x j ðj = 1, 2, ⋯, NÞ is always open, and it can be equivalent as a massless viscoelastic rotational spring. Then, the equivalent flexural stiffness of viscoelastic cracked beams in time domain and the Laplace domain established in reference [10] are given as follows: where ðEIÞ e ðx, tÞ is the equivalent bending stiffness of a viscoelastic beam with open cracks. Here, ðEI · Þ e ðx, tÞ is the first derivative of ðEIÞ e ðx, tÞ with respect to the time t, and the asterisk * denotes the convolution, i.e., f ðtÞ * gðtÞ And by using the expression for the rectangular cross section beams in references [11,12], the equivalent stiffness of crack at the location x = x j with the crack depth d j in time domain and Laplace domain are given as, respectively, where the moment of inertia of the neutral axis is given as I = ∬ Ω y 2 dydz:. The bending moment and shearing force of a Timoshenko viscoelastic cracked beam are written as follow, respectively, where κ is the shear correction factor of a Timoshenko beam, and the cross-section is given as Substituting equations (3), (5), and (6) into equations (7) and utilizing the inverse Laplace transform, one obtain The free vibration equations of the Timoshenko viscoelastic beam [4] are

Solutions
Introduce the following dimensionless variables and parameters The dimensionless forms of equations (8) and (9) are given 2 Journal of Applied Mathematics as follows Based on the separation of variables method [13], the vibration solutions [14] can be assumed as Here in i = ffiffiffiffiffi ffi −1 p , ω is the complex eigenfrequency, and the real part and imaginary part of ω are the natural frequency and decrement coefficient [15,16], respectively. W * ðξÞ, Φ * ðξÞ, M * ðξÞ, and F * s ðξÞ are the dimensionless mode functions of the transverse displacement, rotation angle, bending moment and shearing force for the cracked beam.
If there exists a nonzero solution of fCg, the determinant of the coefficients vector is zero, i.e., By utilizing MATLAB programs, the complex eigenfrequency ω can be obtained with the different boundary conditions.

Numerical Results and Discussion
To verify the correctness and applicability of the present exact analytical method (EAM), the numerical examples for comparisons have been provided. Let d 1 ⟶ 0, the present model is degenerated into the intact model of the standard linear solid mechanism. Huang and Huang [3] analyzed the vibration properties of the clamped-clamped beam. Based on the common physical and geometric parameters with reference [3], the analytical results of the first eigenfrequencies are shown in Table 1. It is noticed that the errors of the natural frequency and decrement coefficient are less than 2.5% and 6%, respectively.
For the sake of simplicity, the k-th eigenfrequency of the viscoelastic Timoshenko and Euler-Bernoulli beams are defined by ω TB and ω EB , respectively. Additionally, the real part (natural frequency) and imaginary part (decrement coefficient) of the k-th eigenfrequency ω k are defined by Re ðω k Þ and Im ðω k Þ. To consider the effect of viscous coefficient η, let slenderness ratio L/h = 50 and crack location x 1 = 0:4L. The first two eigenfrequencies of the simplysupported standard linear solid viscoelastic Timoshenko and Euler-Bernoulli beams when the crack depth d 1 /h = 0:0001 and d 1 /h = 0:4 are analyzed, respectively. In Tables 3 and 4, it is shown that when the slenderness ratio is high, the decrement coefficient Im ðω k Þ and natural frequency Re ðω k Þ of the first two eigenfrequencies of the Timoshenko model are close to those of the Euler-Bernoulli model, which means that the transverse shear deformation and moment of inertia has less influence on dynamic characteristics of beam when the slenderness ratio is high.
Then, let L/h = 10, the effects of the viscous coefficient η on the first three frequencies of the simply-supported viscoelastic intact beams with standard linear solid model (SLS) and Kelvin-Voigt model (KV) are analyzed in Tables 5 and 6, respectively. It is seen that, with the viscous coefficient increasing, the decrement coefficient Im ðω k Þ of the first three frequencies first increase and then decrease. When η ∈ 6:9 × ½10 4 , 10 8 GPa ⋅ h, the decrement coefficient Im ðω k Þ increases with the order of mode function increasing. While η ∈ 6:9 × ½10 9 , 10 12 GPa ⋅ h, the decrement coefficient tends to be a constant. There is a similar conclusion presented by Peng [5] based on the vibration properties of the Timoshenko elastic beam resting on the viscoelastic foundation.
Additionally, with the viscous coefficient η and order of mode function increasing, the natural frequency Re ðω k Þ of the first three frequencies with the SLS model increases, and then it almost remains a constant when η ≥ 6:9 × 10 8 GPa ⋅ h. However, the natural frequency Re ðω k Þ of the first three frequencies with KV model decreases first, and when η = 6:9 × 10 7 GPa ⋅ h, it reduces to zero [16].
Let κ ⟶ ∞, equations (30)-(36) are degenerated into the analytical expressions of the viscoelastic Euler-Bernoulli    Tables 7-10, respectively. It is found that, when η ∈ 6:9 × ½   Journal of Applied Mathematics 10 4 , 10 7 GPa ⋅ h, the coefficient E Im,1 of the SLS viscoelastic cracked beam decreases with the crack location ξ 1 decreasing when ξ 1 ≤ 0:5, and the crack number and viscous coeffi-cient increasing. While η ∈ 6:9 × ½10 8 , 10 12 GPa ⋅ h, the coefficient E Im,1 tends to be zero. It is revealed that the viscous coefficient has little effect on the variations of the first    To sum up, for a higher value of η, the crack depth, crack number, and order of mode function has very less effect on the decrement coefficient Im ðω k Þ of the viscoelastic beam. Therefore, the following analyses are mainly focused on the effects of crack depth, crack number, order of mode function, and slenderness ratio on the natural frequency Re ðω k Þ of the viscoelastic beams.
Next, to consider the effect of a crack, it is supposed that ω 0k and ω k are the k-th eigenfrequency of the viscoelastic intact and cracked beam, respectively, then the corresponding k-th natural frequency ratio is defined as λ k = Re ðω k Þ/

Journal of Applied Mathematics
Re ðω 0k Þ. In the case of a simple-supported Timoshenko viscoelastic beam with a single crack, the variations of the first three frequency ratio versus crack location ξ 1 of the cracked beam with different crack depth d 1 /h = 0:2, 0.4, and 0.6 are presented in Figure 2. It is noticed that the crack location has a significant effect on the natural frequency ratio. When ξ 1 = 0:5 because of the midspan moment of the 2nd modal functions is null in Figure 2(b), the 2nd natural frequency ratio is λ 2 = 1. Similarly, when ξ 1 = 1/3 or 2/3, the crack depth has no effect on the 3rd natural frequency ratio in Figure 2(c). It is concluded that, the k-th natural frequency ratio is independent with the crack depth when the crack is located at some critical position. Then, to consider the effect of crack depth d/h, the variations of the first two frequency ratio of the simplysupported viscoelastic beam with the symmetrically distributed cracks N are presented in Figure 3. It is found that the first two natural frequencies decrease with the crack number N and crack depth d j /h increasing. When N = 1,    Journal of Applied Mathematics the 2nd natural frequency ratio is independent with the crack depth in Figure 3(b).
Next, to consider the effect of slenderness ratio, the variations of the first two frequency ratio versus crack depth d 1 /h of the simply-supported beam with a single crack (ξ 1 = 0:5) are analyzed in Figure 4. It is noticed that the 1st frequency ratio increases with the slenderness ratio increasing, while the 2nd frequency ratio is independent with the slenderness ratio.
Finally, the variations of the first five circular frequencies of the simply-supported Timoshenko viscoelastic beam with a single crack for different slenderness ratio L/h are presented in Figure 5. It is found that the variations of the first five circular frequencies between the Timoshenko and Euler cracked beam models increase with the order of mode function increasing and slenderness ratio decreasing. When L/h = 50, the variations of the 1st circular frequency is E Re,1 = 0:1% and that of the 5th circular frequency is E Re,5 = 1:5%. While L/h = 20, 10, and 5, the corresponding variations of the 1st circular frequency are 0.2%, 1.2%, and 4.1% and those of the 5th circular frequency are 7.4%, 22.3%, and 45.5%. It is seen that the effects of the transverse shear deformation and moment of inertia on the variations of the first five circular frequencies is dependent with the slenderness ratio.

Conclusions
In this paper, the dynamic properties of a viscoelastic Timoshenko cracked beams based on the standard linear solid model and Kelvin-Voigt model are investigated. Based on the equivalent bending stiffness of the viscoelastic cracked beam with open cracks, the corresponding complex frequency characteristic equations of a Timoshenko viscoelastic cracked beam are obtained by using the method of separation of variables and the Laplace transform. Some conclusions arising from the numerical results can be summarized as follows: (1) For a higher value of η, the crack depth, crack number, and order of mode function has very less effect on the decrement coefficient Im ðω k Þ of the viscoelastic beam (2) The transverse shear deformation and moment of inertia has less influence on dynamic characteristics of beam when the slenderness ratio is high, while the variations of the first five circular frequencies between the Timoshenko and Euler cracked beam models increase with the order of mode function increasing and slenderness ratio decreasing (3) The first three natural frequencies of the viscoelastic cracked beam with the SLS model decrease with the crack number and crack depth increasing. And when the crack is located at some critical position, the k-th natural frequency is independent with the crack number, crack depth, and slenderness ratio

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The author declares that there are no conflicts of interest.