Calculation of the natural frequencies and mode shapes of a Euler–Bernoulli beam which has any combination of linear boundary conditions

Abstract

There are well-known expressions for natural frequencies and mode shapes of a Euler-Bernoulli beam which has classical boundary conditions, such as free, fixed, and pinned. There are also expressions for particular boundary conditions, such as attached springs and masses. Surprisingly, however, there is not a method to calculate the natural frequencies and mode shapes for a Euler–Bernoulli beam which has any combination of linear boundary conditions. This paper describes a new method to achieve this, by writing the boundary conditions in terms of dynamic stiffness of attached elements. The method is valid for any boundaries provided they are linear, including dissipative boundaries. Ways to overcome numerical issues that can occur when computing higher natural frequencies and mode shapes are also discussed. Some examples are given to illustrate the applicability of the proposed method.

Keywords

Dynamic stiffness general boundary condition mode shape natural frequency numerical stable equations

1. Introduction

The study of vibration in one-dimensional systems, such as beams, is of great importance in mechanical and structural engineering. This has also helped researchers to understand the behavior of more complex systems. It is well-known that a beam can be described by Euler–Bernoulli beam theory under certain conditions (Bishop and Johnson, 1960). This can be extended for thin plates (Warburton, 1954). Fundamental properties of beams are their natural frequencies and mode shapes, which are a function of the beam geometry and material, and the boundary conditions at each end of the beam.

Previous work has involved the determination of the natural frequencies and mode shapes for different boundary conditions. Much of this occurred many years ago. In a classic textbook (Blevins, 2001), Blevins gives tables for the natural frequencies and mode shape of beam with different boundary configurations, including masses and springs. However, these results are limited to the fixed values of mass and springs provided. Sample graphical solutions for this type of problem were also given by Kinsler et al. (1999). Amabili and Garziera (1999) have proposed a simple method of choice for admissible functions in the Rayleigh–Ritz method. They have considered less-constrained problems of a beam, where the rigid constraints are replaced by elastic ones. They have demonstrated the methodology through examples.

Some researchers have developed expressions for beams subject to elastic boundary conditions. Rao and Mirza (1989) gave exact expressions for natural frequencies and mode shapes for boundaries involving translational and rotational springs. Li (2000) has also discussed a similar problem concerning beams with elastic supports. Instead of using traditional trigonometric and hyperbolic functions, he expressed the displacement functions as a linear combination of a Fourier series and an auxiliary polynomial function. More recently, Melanson and Zu (1998) have determined the natural frequencies and mode shapes for rotating Timoshenko beams for classical boundary conditions. Further, in Oliveto et al. (1997) a complex mode superposition method for a beam with viscous-end dampers is investigated in detail up to the third mode, and in Svedholm et al. (2016) the authors apply the viscous damping model to moving loads over bridges in both time and frequency domains. Aristizabal-Ochoa (2007) considers springs and masses attached to the ends of uniform shear beam columns, observing the influence on the natural frequencies and the buckling modes.

In a paper closely related to the work proposed in this paper, Zarek and Gibbs (1981) described a procedure to calculate expressions for the complex eigenvalues and eigenvectors of a Euler–Bernoulli beam for any combination of linear boundary conditions. In their paper they described the boundaries in terms of mechanical impedances.

This paper develops the work of Zarek and Gibbs (1981). Two important advances are made. The first involves the reformulation of their approach to facilitate the derivation of compact analytical expressions for natural frequencies and mode shapes. These expressions are general and are applicable for any linear combination of any boundaries that can be described in terms of dynamic stiffnesses, including boundaries that dissipate energy. Moreover, the expressions, together with the accompanying matrices, are written in such a way that natural frequencies and mode shapes can be easily determined using a computer. The second advance involves the inclusion of some recent work that takes into account the numerical issues that can occur when computing these quantities at higher frequencies as discussed in Gonçalves et al. (2007, 2018).

The paper is organized as follows: Sections 2 and 3 describe the procedures to calculate natural frequencies and mode shapes respectively; some examples are given in Section 4, including some animations to help visualize the behavior as the boundary conditions change; and the paper is closed in Section 5 with a summary.

2. Calculation of natural frequencies

Figure 1 shows a uniform Euler–Bernoulli beam with general linear boundary conditions which are defined in terms of the dynamic stiffnesses of the elements attached to each end of the beam. Dynamic stiffness is a complex function of frequency, and for the elements considered here are given by $D^{s} = F / W$ and $D^{r} = M / Θ$ , where the superscripts $(\cdot)^{s}$ and $(\cdot)^{r}$ represent shear and rotary, respectively; F and M are the complex force and moment amplitudes applied to the respective elements, and W and Θ are the resulting complex linear and rotary displacements, respectively. Note that all the variables are frequency domain quantities. The subscripts $(\cdot)_{0}$ and $(\cdot)_{l}$ are related to the ends of the beam at $x = 0$ and $x = l$ , respectively.

The equation of motion for free vibration of a uniform and homogenous Euler–Bernoulli beam is given by (Bishop and Johnson, 1960)

EI \frac{\partial 2 w}{\partial x^{2}} + ρ A_{0} \frac{\partial 2 w}{\partial t^{2}} = 0

(1)

where E is the Young’s modulus, I is the second moment of area, A₀ is the cross-sectional area and ρ is the density. Assuming a harmonic solution of the form

w (x, t) = W (x) e^{j ω t}

, where ω is angular frequency and t is time, the solution for the displacement as a function of x can be written as

W (x) = A cos kx + B sin kx + C \cosh kx + D \sinh kx

(2)

where

W (x)

is the transverse displacement,

k = ω^{1 / 2} (ρ A / EI) 1 / 4

is the bending wavenumber, and

A, B, C, D

are constants to be determined depending on the boundary conditions. The forces and moments acting on the beam at each end, due to the motion of the beam and the dynamic stiffnesses of the elements forming the boundaries, are given by (Fahy and Walker, 2005)

\begin{matrix} F (0) = - D_{0}^{s} W (0), M (0) = - D_{0}^{r} Θ (0), \\ F (l) = D_{l}^{s} W (l), M (l) = D_{l}^{r} Θ (l) \end{matrix}

(3a−d)

Noting that the bending moment and shear force are related to the displacement by $M = EI \partial 2 W / \partial x^{2}$ $F = EI \partial 3 W / \partial x^{3}$ (Fahy and Walker, 2005), equations (2) and (3) can be combined to give

\begin{matrix} [\begin{matrix} α_{1} & - 1 & α_{1} & 1 \\ 1 & α_{2} & - 1 & α_{2} \\ sin kl - α_{3} cos kl & - (sin kl + α_{3} cos kl) & \sinh kl - α_{3} \cosh kl & \cosh kl - α_{3} \sinh kl \\ cos kl + α_{4} sin kl & sin kl - α_{4} cos kl & - (\cosh kl + α_{4} \sinh kl) & - (\sinh kl + α_{4} \cosh kl) \end{matrix}] \\ {\begin{matrix} A \\ B \\ C \\ D \end{matrix}} = {\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}} \end{matrix}

(4)

where $α_{1} = \frac{D_{0}^{s}}{{EIk}^{3}}, α_{2} = \frac{D_{0}^{r}}{EIk}, α_{3} = \frac{D_{l}^{s}}{{EIk}^{3}}, α_{4} = \frac{D_{l}^{r}}{EIk}$ . The nontrivial solution of equation (4) requires that the determinant of the matrix must be zero. This results in the transcendental frequency equation, given by

\begin{matrix} [β_{1} sin kl + β_{2} cos kl] \sinh kl \\ + [β_{3} sin kl + β_{4} cos kl] \cosh kl + β_{5} = 0 \end{matrix}

(5)

where

\begin{matrix} {\begin{matrix} β_{1} \\ β_{2} \\ β_{3} \\ β_{4} \\ β_{5} \end{matrix}} = [\begin{matrix} 0 & 0 & 2 α_{1} & - 2 α_{2} & 0 \\ 1 + α_{2} α_{4} & - (1 + α_{1} α_{3}) & 1 + α_{2} α_{4} & - (1 + α_{1} α_{3}) & 0 \\ - (1 - α_{2} α_{4}) & - (1 - α_{1} α_{3}) & - (1 - α_{2} α_{4}) & - (1 - α_{1} α_{3}) & 0 \\ α_{2} + 2 α_{4} & (2 - α_{1} α_{4}) α_{3} & α_{4} & 0 & - 1 \\ α_{2} & α_{1} α_{3} α_{4} & α_{4} & 0 & 1 \end{matrix}] {\begin{matrix} α_{1} \\ α_{2} \\ α_{3} \\ α_{4} \\ 1 \end{matrix}} \end{matrix}

The roots of equation (5) give the natural frequencies for any combination of linear boundary conditions, described in terms of their dynamic stiffness.

There are various ways to obtain the roots of the transcendental equation in equation (5) to calculate the natural frequencies, some of which are better than others. It can be seen that equation (5) contains the hyperbolic functions $\sinh kl$ and $\cosh kl$ to represent the evanescent waves on the beam at the boundaries. These grow exponentially as kl increases, and may lead to numerical issues when trying to determine

\begin{matrix} {\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \end{matrix}} = [\begin{matrix} - (1 + α_{1} α_{2}) & - (1 + α_{1} α_{2}) α_{3} & 1 - α_{1} α_{2} - 2 α_{2} α_{3} & 2 α_{2} - (1 - α_{1} α_{2}) α_{3} \\ (1 + α_{1} α_{2}) α_{3} & - (1 + α_{1} α_{2}) & 2 α_{1} + (1 - α_{1} α_{2}) α_{3} & - 1 + α_{1} α_{2} - 2 α_{1} α_{3} \\ - 1 + α_{1} α_{2} + 2 α_{2} α_{3} & - 2 α_{2} - (1 - α_{1} α_{2}) α_{3} & 1 + α_{1} α_{2} & - (1 + α_{1} α_{2}) α_{3} \\ 2 α_{1} + (1 - α_{1} α_{2}) α_{3} & - 1 + α_{1} α_{2} + 2 α_{1} α_{3} & (1 + α_{1} α_{2}) α_{3} & - (1 + α_{1} α_{2}) \end{matrix}] {\begin{matrix} cos k_{i} l \\ sin k_{i} l \\ \cosh k_{i} l \\ \sinh k_{i} l \end{matrix}} \end{matrix}

a high natural frequency (Gonçalves et al., 2014, 2018). For a common double-precision binary floating-point computer, the limit corresponds to the value that can be stored, which is determined by the maximum significant digit available. This is

2^{53}

. For many boundary conditions, this problem occurs above about the 12th natural frequency. To avoid this, equation (5) may be written in a numerically stable form by noting that

\sinh kl = (e^{kl} - e^{- kl}) / 2

and

\cosh kl = (e^{kl} + e^{- kl}) / 2

. The modified form of equation (5) is

\begin{matrix} [β_{1} sin kl + β_{2} cos kl] \frac{1 - e^{- 2 kl}}{1 + e^{- 2 kl}} + β_{3} sin kl \\ + β_{4} cos kl + β_{5} \frac{2 e^{- kl}}{1 + e^{- 2 kl}} = 0 \end{matrix}

(6a)

At high frequencies, $e^{- kl} \to 0$ and equation (6a) can then be written in an approximate form as

tan kl \approx - \frac{β_{2} + β_{4}}{β_{1} + β_{3}}

(6b)

3. Calculation of mode shapes

Examining equation (2), it can be seen that five variables are needed to plot the displacement as a function of x. These are the constants A, B, C, and D, related to the amplitudes of the four waves, and the wavenumber k. If the displacement is normalized by one of the constants, then apart from the wavenumber, which has already been determined from equations (5) and (6), which involves all four boundary conditions, then only three constants need to be calculated. This requires the use of three out of the four equations in equation (4), and hence only three boundary conditions are involved in this calculation. As an example, the expression for the mode shape of the i-th natural frequency is calculated using the first three equations to give

Ψ_{i} (x) = \frac{a_{1} cos k_{i} x + a_{2} sin k_{i} x + a_{3} \cosh k_{i} x + a_{4} \sinh k_{i} x}{b_{1} cos k_{i} l + b_{2} sin k_{i} l + b_{3} \cosh k_{i} l + b_{4} \sinh k_{i} l}

(7)

where

and

\begin{matrix} {\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \end{matrix}} = [\begin{matrix} - α_{1} & 0 & - 1 \\ 0 & - (1 + α_{1} α_{2}) & 0 \\ - (α_{1} + 2 α_{3}) & 0 & 1 \\ 2 & - (1 + α_{1} α_{2}) & 0 \end{matrix}] {\begin{matrix} α_{2} \\ α_{3} \\ 1 \end{matrix}} \end{matrix}

Note that the formulation for the mode shape is not unique, being dependent upon which constant is used as the normalizing factor. In the case above, because the normalizing constant is A, the normalized dynamic stiffness $α_{4}$ does not appear in equation (7). Similar to calculation for the natural frequencies, the hyperbolic terms can lead to numerical issues for large values of $k_{i} l$ . This problem can be overcome by multiplying both the numerator and denominator of equation (7) by $e^{- k_{i} l}$ . This produces a numerically stable expression for the calculation of mode shapes for high values of $k_{i} l$ , given by

Ψ_{i} (x) = \frac{[c_{1} cos k_{i} x + c_{2} sin k_{i} x] e^{- k_{i} l} + \frac{1}{2} [c_{3} e^{- k_{i} (l - x)} + c_{4} e^{- k_{i} (l + x)} + c_{5} e^{- k_{i} (2 l - x)} + c_{6} e^{- k_{i} x}]}{[b_{1} cos k_{i} l + b_{2} sin k_{i} l] e^{- k_{i} l} + \frac{1}{2} [b_{3} (1 + e^{- 2 k_{i} l}) + b_{4} (1 - e^{- 2 k_{i} l})]}

(8)

where

\begin{matrix} {\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \\ c_{4} \\ c_{5} \\ c_{6} \end{matrix}} = [\begin{matrix} - (1 + α_{1} α_{2}) & - (1 + α_{1} α_{2}) α_{3} & 1 - α_{1} α_{2} - 2 α_{2} α_{3} & 2 α_{2} - (1 - α_{1} α_{2}) α_{3} & 0 \\ (1 + α_{1} α_{2}) α_{3} & - (1 + α_{1} α_{2}) & 2 α_{1} + (1 - α_{1} α_{2}) α_{3} & - 1 + α_{1} α_{2} - 2 α_{1} α_{3} & 0 \\ - 1 + 2 α_{1} + α_{1} α_{2} + (1 + (2 - α_{1}) α_{2}) α_{3} & - 1 - 2 α_{1} + α_{1} α_{2} - (1 - (2 + α_{1}) α_{2}) α_{3} & 0 & 0 & 0 \\ - 1 - (2 - α_{1}) α_{2} - (1 - 2 α_{1} - α_{1} α_{2}) α_{3} + & - 1 - 2 α_{1} + α_{1} α_{2} - (1 - (2 + α_{1}) α_{2}) α_{3} & 0 & 0 & 0 \\ α_{2} & 0 & 1 + α_{1} α_{2} & 0 & 1 \\ α_{2} & 0 & - (1 + α_{1} α_{2}) & 0 & 1 \end{matrix}] {\begin{matrix} cos (k_{i} l) \\ sin (k_{i} l) \\ (\frac{1 + e^{- 2 k_{i} l}}{2}) \\ (\frac{1 - e^{- 2 k_{i} l}}{2}) \\ 1 \end{matrix}} \end{matrix}

k_il, given by

4. Examples

The procedure described in the previous section to determine the natural frequencies and mode shapes for a beam, works for any combination of boundaries described in terms of their dynamic stiffness. Because the method is general and can cope with all combinations of boundary conditions, the frequency and mode shape equations cannot be written down in a simple form. However, if at least two of the boundary conditions are classical such that the dynamic stiffness is either zero or infinity, then the frequency equation simplifies considerably. In this section, two examples are given to illustrate this. The first is for a beam in which the boundaries include two classical boundary conditions and the other two have real types of dynamic stiffnesses of different signs, that is, a mass where the dynamic stiffness is frequency dependent and a spring with the dynamic stiffness being frequency independent. The second is where there are three classical boundary conditions and the beam is connected to a viscous damper, which has a positive imaginary dynamic stiffness.

4.1. Mass and stiffness boundaries

A beam with stiffness and mass boundary conditions on the left- and right-hand sides, respectively, is shown in Figure 2. The left-hand boundary has a pinned condition, which corresponds to an infinite translational stiffness, and has a torsional stiffness s_t. On the right-hand side, there is a concentrated point mass m and there is no rotational constraint. The boundary conditions are therefore, $α_{1} = \infty, α_{2} = \frac{s_{t}}{kEI}, α_{3} = \frac{- ω^{2} m}{k^{3} EI}, α_{4} = 0$ .

The resulting frequency equation is determined using equation (6a) to give

\begin{matrix} (- 2 μ kl sin kl + (1 + μ σ_{t}) cos kl) \frac{1 - e^{- 2 kl}}{1 + e^{- 2 kl}} - (1 + μ σ_{t}) sin kl \\ + \frac{σ_{t}}{kl} (cos kl + \frac{2 e^{- kl}}{1 + e^{- 2 kl}}) = 0 \end{matrix}

(9)

where

σ_{t} = s_{t} l / EI

and

μ = m / ρ A_{0} l

are nondimensional stiffness and nondimensional mass, respectively. If the boundary conditions are

α_{2} = α_{3} = 0

α_{2} = α_{3} = \infty

then equation (9) reduces to

tan kl = \frac{1 - e^{- 2 kl}}{1 + e^{- 2 kl}}

, if they are

α_{2} = \infty, α_{3} = 0

, then it reduces to

cos kl (1 + e^{- 2 kl}) + 2 e^{- kl} = 0

, and if

α_{2} = 0, α_{3} = \infty

then it reduces to

sin kl = 0

. To illustrate the behavior of the first natural frequency as

σ_{t}

changes, the first natural frequency, normalized by

ω_{0} = \sqrt{\frac{3 EI}{l^{3} (m + 0.23 m_{b})}},

which is the approximate fundamental natural frequency of a cantilever beam with a tip mass [3], is plotted in Figure 3 for a value of

μ = 1

and a wide range of values of

σ_{t}

Examining Figure 3, it can be seen that for $σ_{t} << 1$ the natural frequency tends to zero, which corresponds to the fundamental natural frequency of a pinned-free beam with a tip mass. For $σ_{t} >> 1$ the natural frequency tends to the fundamental natural frequency of a cantilever beam with a tip mass. (To illustrate this effect, an animation is presented as supplementary material). This example also shows that it is possible to calculate the natural frequencies and mode shapes of a beam with infinite dynamic stiffness, for example the pinned condition, using very high values of stiffness.

4.2. Viscous damper boundary

The second example is shown in Figure 4. It is a cantilever beam with a viscous damper attached to the right end. The nondimensional boundary conditions are, therefore, $α_{1} = \infty$ , $α_{2} = \infty$ , $α_{3} = \frac{j ω c_{v}}{{EIk}^{3}},$ and $α_{4} = 0$ .

The frequency equation is determined from equation (5) to give

j \frac{ξ}{kl} (sin kl - \frac{1 - e^{- 2 kl}}{1 + e^{- 2 kl}} cos kl) + cos kl + \frac{2 e^{- kl}}{1 + e^{- 2 kl}} = 0

(10)

where

ξ = c_{v} l / \sqrt{ρ A_{0} EI}

is a nondimensional damping parameter. If

α_{3} = 0

, corresponds to a damping coefficient of zero, then equation (10) reduces to

(1 + e^{- 2 kl}) cos kl + 2 e^{- kl} = 0

, and if

α_{3} = \infty

, corresponds to an infinite damping coefficient, then it reduces to

tan kl = \frac{1 - e^{- 2 kl}}{1 + e^{- 2 kl}}

. In general equation (10) is complex, so its roots are complex. There are various methods to solve nonlinear transcendental equations with complex roots, but here Muller’s method (Muller, 1956), which is a variation of the secant method, is used. The i-th complex root of equation (10) is given by

k_{i} l = Re {k_{i} l} + j Im {k_{i} l}

, which is related to the complex i-th natural frequency by

{\hat{ω}}_{i} = ω_{i} / (EI / ρ A_{0} l^{4})^{1 / 2} = k_{i}^{2} l^{2}

. Setting

{\hat{s}}_{i} = j {\hat{ω}}_{i}

, the roots of the equation (10) can be written as

\begin{matrix} Re {{\hat{s}}_{i}} = \frac{- Im {ω_{i}}}{(\frac{EI}{ρ A_{0} l^{4}})^{1 / 2}} = - 2 Re {k_{i} l} Im {k_{i} l}, \\ Im {{\hat{s}}_{i}} = \frac{j Re {ω_{i}}}{(\frac{EI}{ρ A_{0} l^{4}})^{1 / 2}} = Re {k_{i} l} 2 - Im {k_{i} l} 2 \end{matrix}

(11a, b)

These quantities have clear physical significance. The imaginary part is the damped natural frequency, the real part is $ξ_{i} | {\hat{s}}_{i} |$ , where $| {\hat{s}}_{i} |$ is the undamped natural frequency, and the damping ratio is $ξ_{i} = - Re {{\hat{s}}_{i}} / | {\hat{s}}_{i} |$ . The complex roots for the first two modes are plotted in Figure 5 as ξ is increased from $10^{- 2}$ to $10^{3}$ . It can be seen that when damping is negligibly small, the roots tend to those of a cantilever beam (clamped-free), having only an imaginary part. For high values of damping, the first root becomes purely real (natural frequency approaches zero); after a value correspondent to critical damping for that mode, the system becomes overdamped. The second root becomes purely imaginary again, and its value tends to the root of a beam with clamped-pinned boundary conditions. The comparison of the second mode shape for small and large values of damping is shown in Figure 6.

The mode shapes are also complex functions, which can be plotted considering the magnitude and phase. These two mode shapes can also be observed in the supplementary material provided with this article (animation videos).

5. Summary

This paper has described a basic methodology to calculate the natural frequencies and mode-shapes of Euler–Bernoulli beams for any combination of linear boundary conditions. The boundary conditions are described in terms of dynamics stiffnesses in a nondimensional form. Some examples have been given (including some animations), which involve boundaries described in terms of real dynamic stiffness such as mass and stiffness, and an example involving a viscous damper, which has an imaginary dynamic stiffness. Alternative expressions are also given for natural frequencies and mode shapes, such that they can be computationally evaluated at high frequencies to avoid issues with hyperbolic functions.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Supplemental Material

Supplemental material for this article is available online.

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Supplementary Material

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