Investigating the Sommerfeld effect in the rotating shaft with internal and external damping using the Timoshenko beam theory

Abstract

The Sommerfeld effect is a destructive phenomenon in rotating systems with a non-ideal electrical source, which causes instability in the system by applying a dynamic jump around critical speeds. In this article, the Sommerfeld effect has been investigated for the first time using the Timoshenko beam theory for an eccentric continuous shaft with internal and external damping. After deriving the governing equations and finding displacement functions using the semi-analytical method, the Sommerfeld effect near the critical speeds is detected using the instantaneous power balance method. As confirmation of the correctness of the derived relations, it has been shown that for thin shafts, there is a good consistency between the results obtained from the Euler–Bernoulli and Timoshenko theories in the early modes. However, it was observed that at higher critical speeds, the jump amplitude decreases, and the unstable speed range increases significantly, so the probability of entering the vicinity of the instability range in the next mode is not unexpected. Since no effect has been ignored in this study, the dynamic analysis of the Sommerfeld jump in thick shafts is also possible. Despite the common belief that Timoshenko beam theory is only considered suitable for studying thick shafts, it has been shown that the effects of shear deformation are significant in high-speed systems, even for non-thick shafts, and regardless of them in higher modes, it causes a calculation error in determining the point of occurrence of the Sommerfeld phenomenon.

Keywords

Sommerfeld effect non-ideal source Timoshenko beam theory shear deformation rotor dynamics

1. Introduction

A non-ideal energy source is affected by the response of the vibration system, which will result in a limitation in the power supply. In contrast, an ideal energy source guarantees unlimited output power. Although the problems based on the non-ideality of the source are more realistic, it has been considered a serious challenge in theoretical and practical engineering studies for a short time. In the rotor-dynamics, when a vibrating system is driven by a non-ideal electric source such as a DC motor, in the pre-resonance region, an increase in the input voltage causes an increase in the rotational speed (Samantaray, 2022). In the vicinity of the natural frequency of the system, due to the increase in the oscillation amplitude and energy loss of the system, the power provided by the non-ideal source no longer leads to an increase in the rotational speed, and a significant part of the energy is spent on the lateral vibrations of the shaft. In this condition, reaching the natural frequency, the system suddenly jumps from the critical speed to a higher speed. This phenomenon, caused by the sudden transformation of energy used for non-useful oscillation into rotational kinetic energy, is called the Sommerfeld effect. If this harmful effect is ignored, the vibration system will be irreparably damaged (Cveticanin et al., 2018). The Sommerfeld effect was identified for the first time by Sommerfeld (1902) during the experimental investigation of the vibration of a cantilever beam excited by a non-ideal electric source. Sometime later, Kononenko (1969) explained this phenomenon and its destructive consequences in dynamic systems in his book. Recently it was introduced the effective factor in reducing or aggravating this non-linear effect as the system’s damping. Therefore, the effect of various types of dampings on the Sommerfeld phenomenon was investigated in a wide range of vibration systems with non-ideal electrical sources (Balthazar et al., 2003; El-Badawy, 2007). It was found that linear and non-linear electromechanical vibration absorbers can be suitable for reducing the Sommerfeld phenomenon’s destructive effects by reducing the jump’s amplitude (Felix and Balthazar, 2009). In addition to discretely modeled systems, it has been reported that continuous systems with internal damp and external viscous damp are also affected by the Sommerfeld effect (Jha and Dasgupta, 2020; Bharti et al., 2021; Sinha and Samantaray, 2023), and instability occurs around certain critical speeds (Dasgupta et al., 2010). Although passing through resonance in some modes is accompanied by a jump in non-ideal systems, it has been shown that proper dry damping—despite its disadvantages—can reduce this jump to a good extent (Chakraborty et al., 2019). Recently, several studies have been conducted on the dynamic behavior of disc and shaft vibration systems and the effect of disc eccentricity and various types of damping on the Sommerfeld effect has been reported around the system’s natural frequencies (Bharti et al., 2019; Dasgupta, 2022a, 2022b; Iskakov et al., 2022; Jha and Dasgupta, 2019, 2022; Karthikeyan et al., 2015; Sghaier et al., 2019).

What is evident in the mentioned studies is that passing resonance in all kinds of vibration systems with non-ideal sources and how to control the Sommerfeld effect has become a challenge for designers and researchers. The effective factors in strengthening this phenomenon are identified by finding the most suitable elements of the system (e.g., the use of various types of absorbents), and the harmful effects of this phenomenon are then reduced. Despite the investigations carried out on rotating shafts as a continuous medium using the Euler–Bernoulli beam theory and the study on the effects of shaft eccentricity and internal and external damping on the jump driven by non-ideal source (Dasgupta and Rajamohan, 2017; Dasgupta and Rajan, 2018), the investigation of this nonlinear dynamic effect by considering the effects of the shear deformation effect (the Timoshenko beam theory) has not been reported.

This article uses the Timoshenko beam theory to investigate the dynamic behavior of a rotating shaft with a non-ideal energy source, with external viscous and internal damping, for the first time. Obtaining the governing equations of motion, the displacement functions of the shaft are determined according to the boundary conditions. A ninth-order polynomial is obtained through instantaneous power balance during steady-state whirl. Afterward, a transfer function is calculated through this characteristic polynomial assuming supply voltage as a gain parameter, which determines the Sommerfeld jump with the aid of the root loci technique at the presence of multiple roots around critical speeds. Although the effect of shear deformation in thin shafts is negligible, in the absence of documentation investigating the Sommerfeld effect for thick shafts, to validate the obtained results, first, a thin shaft similar to the results of Dasgupta and Rajamohan (2017) has been explored. After validating the results and ensuring the correctness of the driven relations, in another case, this dynamic effect has been investigated at different critical speeds. Despite the popular opinion that Timoshenko beam theory is only suitable for investigating thick shafts, we have shown that at high speeds, the shear deformation effects are not unforgivable, even for non-thick shafts.

2. Mathematical modeling of the system

The schematic diagram of the flexible shaft driven through a non-ideal DC motor with external viscous and internal damping and constant eccentricity of the shaft is shown in Figure 1. The coupling between the shaft and the motor is considered transversely flexible and lightweight. In this study, the torsional vibration of the shaft is ignored. Therefore, the rotation speed does not change along the shaft length (z-axis).

2.1. Equations of motion of the system

The governing equations of motion of the flexible shaft according to Timoshenko beam theory, which is moved by a non-ideal DC motor, are obtained in Appendix A, as the following in the complex form

{\begin{cases} ρ A \frac{\partial^{2} U}{\partial t^{2}} + R_{a} \frac{\partial U}{\partial t} + K A G (\frac{\partial ψ}{\partial z} - \frac{\partial^{2} U}{\partial z^{2}}) = ρ A Ω^{2} a e^{i (Ω t + ϕ)} \\ E I \frac{\partial^{2} ψ}{\partial z^{2}} + K A G (\frac{\partial U}{\partial z} - ψ) + μ_{i} I \frac{\partial}{\partial t} (\frac{\partial^{2} ψ}{\partial z^{2}}) - i μ_{i} I Ω \frac{\partial^{2} ψ}{\partial z^{2}} = I_{d} (\frac{\partial^{2} ψ}{\partial t^{2}} - 2 i Ω \frac{\partial ψ}{\partial t}) \\ I_{r} \frac{\partial Ω}{\partial t} + R_{r} Ω = T_{m} - T_{L} \end{cases}

(1)

In this study, assuming the constant rotational speed, it can be written as $\partial Ω / \partial t = 0$ .

2.2. Boundary conditions

Short bearings on both sides support the shaft, and the bending moment and displacement in the supports are zero. The boundary conditions corresponding to the assumption of pin-pin support at the points z = 0 and z = L are as follows (Dasgupta et al., 2010; Sghaier et al., 2019)

\begin{array}{l} {U |}_{z = 0, L} = 0 \\ {[E I \frac{\partial ψ}{\partial z} + μ_{i} I \frac{\partial}{\partial t} (\frac{\partial ψ}{\partial z}) - i μ_{i} I Ω \frac{\partial ψ}{\partial z}]}_{z = 0, L} = 0 \end{array}

(2)

where the first term is related to the displacement, and the second is the bending moment in the supports.

2.3. Calculation of critical speeds

To calculate the critical speeds of the shaft, it is sufficient to solve the governing equations regardless of the shaft eccentricity as the external harmonic excitation factor. In this situation, displacement and slope functions are presented as follows according to the boundary conditions

\begin{array}{l} U (z, t) = \sum_{n = 0}^{\infty} U_{0 n} \sin (\frac{n π}{L}) z \cdot e^{i ω t} \\ ψ (z, t) = \sum_{n = 0}^{\infty} ψ_{0 n} \cos (\frac{n π}{L}) z \cdot e^{i ω t} \end{array}

(3)

By substituting equation (3) in equation (1), leads to the following matrix form equation

[\begin{array}{l} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}] (\begin{array}{l} U_{0 n} \\ ψ_{0 n} \end{array}) = (\begin{array}{l} 0 \\ 0 \end{array})

(4)

where

\begin{array}{l} c_{11} = - K A G {(\frac{n π}{L})}^{2} + ρ A ω^{2} - i R_{a} ω, c_{12} = c_{21} = K A G (\frac{n π}{L}) \\ c_{22} = - E I {(\frac{n π}{L})}^{2} - K A G + I_{d} (ω^{2} - 2 Ω ω) - i μ_{i} I {(\frac{n π}{L})}^{2} (ω - Ω) \end{array}

(5)

If the determinant of the coefficient matrix is placed equal to zero, the characteristic equation is obtained as follows

\begin{array}{l} \frac{I_{d} ρ A}{K A G} ω^{4} - {\frac{2 ρ A I_{d} Ω}{K A G} + i [\frac{μ_{i} I ρ A}{K A G} {(\frac{n π}{L})}^{2} + \frac{R_{a} I_{d}}{K A G}]} ω^{3} \\ - {(I_{d} + \frac{ρ A E I}{K A G} + \frac{μ_{i} I R_{a}}{K A G}) {(\frac{n π}{L})}^{2} + ρ A - i [\frac{μ_{i} I Ω ρ A}{K A G} {(\frac{n π}{L})}^{2} + 2 \frac{I_{d} Ω R_{a}}{K A G}]} ω^{2} \\ + {(2 I_{d} Ω + \frac{μ_{i} I Ω R_{a}}{K A G}) {(\frac{n π}{L})}^{2} + i [\frac{E I R_{a}}{K A G} {(\frac{n π}{L})}^{2} + R_{a} + μ_{i} I {(\frac{n π}{L})}^{4}]} ω + (E I - i μ_{i} I Ω) {(\frac{n π}{L})}^{4} = 0 \end{array}

(6)

that is a fourth-order polynomial in complex form, whereby separating real and imaginary values, it may be written as

\begin{array}{l} ω^{4} - 2 Ω ω^{3} - [(\frac{K A G}{ρ A} + \frac{E I}{I_{d}} + \frac{μ_{i} I R_{a}}{ρ A I_{d}}) {(\frac{n π}{L})}^{2} + \frac{K A G}{I_{d}}] ω^{2} \\ + (2 \frac{K A G}{ρ A} + \frac{μ_{i} I R_{a}}{ρ A I_{d}}) {(\frac{n π}{L})}^{2} Ω ω + \frac{E I K A G}{ρ A I_{d}} {(\frac{n π}{L})}^{4} = 0 \end{array}

(7)

\begin{array}{l} [1 + \frac{μ_{i} I ρ A}{I_{d} R_{a}} {(\frac{n π}{L})}^{2}] ω^{3} - [2 Ω + \frac{μ_{i} I Ω ρ A}{I_{d} R_{a}} {(\frac{n π}{L})}^{2}] ω^{2} \\ - [\frac{K A G}{I_{d}} + \frac{E I}{I_{d}} {(\frac{n π}{L})}^{2} + \frac{μ_{i} I K A G}{I_{d} R_{a}} {(\frac{n π}{L})}^{4}] ω + \frac{μ_{i} I K A G}{I_{d} R_{a}} Ω {(\frac{n π}{L})}^{4} = 0 \end{array}

(8)

Equations (7) and (8) must be established simultaneously. The relationship between critical rotational speed and natural frequency in resonance conditions using equation (8) can be calculated as follows

Ω_{n} = ω \cdot \frac{[1 + \frac{μ_{i} I ρ A}{I_{d} R_{a}} {(\frac{n π}{L})}^{2}] ω^{2} - [\frac{K A G}{I_{d}} + \frac{E I}{I_{d}} {(\frac{n π}{L})}^{2} + \frac{μ_{i} I K A G}{I_{d} R_{a}} {(\frac{n π}{L})}^{4}]}{[2 + \frac{μ_{i} I ρ A}{I_{d} R_{a}} {(\frac{n π}{L})}^{2}] ω^{2} - \frac{μ_{i} I K A G}{I_{d} R_{a}} {(\frac{n π}{L})}^{4}}

(9)

By substituting equation (9) in equation (7) and some mathematical simplification, the system characteristic equation will be obtained in terms of ω as

ω^{6} + M ω^{4} + N ω^{2} + P = 0

(10)

where

\begin{array}{l} M = - 2 \frac{K G}{ρ} {(\frac{n π}{L})}^{2} + {(\frac{R_{a}}{ρ A})}^{2} + \frac{1}{I_{d}} [E I {(\frac{n π}{L})}^{2} + K A G] \\ N = {(\frac{K G}{ρ})}^{2} {(\frac{n π}{L})}^{4} - [\frac{K G}{ρ I_{d}} {(\frac{n π}{L})}^{2} - \frac{{R_{a}}^{2}}{ρ^{2} A^{2} I_{d}} - \frac{2 R_{a} K G}{ρ^{2} A μ_{i} I}] [E I {(\frac{n π}{L})}^{2} + K A G] \\ P = {(\frac{K G}{ρ})}^{2} \frac{E I}{I_{d}} {(\frac{n π}{L})}^{6} \end{array}

(11)

The real roots of equation (10) are the natural frequencies of the vibrating system. The critical speeds of the shaft in different modes are calculated by putting the acceptable natural frequencies obtained from equation (10) in equation (9). Investigating the Sommerfeld effect, it is important to know the critical speeds with the highest vibration amplitude and, therefore, the highest power dissipate at these speeds. Consequently, the most likely occurrence of the Sommerfeld effect is in the neighborhood of calculated critical speeds.

2.4. Steady state response

To find the solution to the steady state of the problem, taking into account the eccentricity of the shaft in equation (1) and considering the boundary conditions mentioned in equation (2), the displacement and rotation functions of the shaft are considered as follows

\begin{array}{l} U (z, t) = \sum_{n = 1}^{\infty} U_{0 n} \sin (\frac{n π}{L}) z \cdot e^{i (Ω t + ϕ_{n})} \\ ψ (z, t) = \sum_{n = 1}^{\infty} ψ_{0 n} \cos (\frac{n π}{L}) z \cdot e^{i (Ω t + ϕ_{n})} \end{array}

(12)

Substituting equation (12) in equation (1) gives

\begin{array}{l} {[- K A G {(\frac{n π}{L})}^{2} - i R_{a} Ω + ρ A Ω^{2}] U_{0 n} + [K A G (\frac{n π}{L})] ψ_{0 n}} \sin \frac{n π}{L} z \cdot e^{i (Ω t + ϕ_{n})} = - ρ A Ω^{2} a e^{i (Ω t + ϕ)} \\ {[K A G (\frac{n π}{L})] U_{0 n} + [- E I {(\frac{n π}{L})}^{2} - K A G - I_{d} Ω^{2}] ψ_{0 n}} \cos \frac{n π}{L} z \cdot e^{i (Ω t + ϕ_{n})} = 0 \end{array}

(13)

Multiply the first relation of equation (13) by

\sin m π / L z

, integrating from 0 to L concerning z and noting the orthogonal property of harmonic functions as

\int_{0}^{L} \sin \frac{n π}{L} z \cdot \sin \frac{m π}{L} z d z = {\begin{cases} 0; n \neq m \\ \frac{L}{2}; n = m \end{cases}

(14)

and by simultaneously solving the second equation (13), yields the steady state unbalance response of the shaft is given as

U_{0 n} = \frac{2 ρ A Ω^{2} a \frac{(1 - \cos n π)}{n π} [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2} + \frac{I_{d}}{K A G} Ω^{2}]}{\sqrt{{E I {(\frac{n π}{L})}^{4} + [(I_{d} - \frac{ρ E I}{K G}) {(\frac{n π}{L})}^{2} - ρ A] Ω^{2} - \frac{ρ I_{d}}{K G} Ω^{4}}^{2} + {R_{a} Ω [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2} + \frac{I_{d}}{K A G} Ω^{2}]}^{2}}}

(15)

According to the expression $1 - \cos n π$ in equation (15), this relation calculates the amplitude of shaft oscillation in odd modes. The phase angle in each mode is

ϕ_{n} = ϕ - \tan^{- 1} (\frac{R_{a} Ω [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2} + \frac{I_{d}}{K A G} Ω^{2}]}{E I {(\frac{n π}{L})}^{4} + [(I_{d} - \frac{ρ E I}{K G}) {(\frac{n π}{L})}^{2} - ρ A] Ω^{2} - \frac{ρ I_{d}}{K G} Ω^{4}})

(16)

In equations (15) and (16), compared to the Euler–Bernoulli beam theory (Dasgupta and Rajamohan, 2017; Dasgupta and Rajan, 2018), due to the effect of shear deformation, terms containing the shear modulus G have been added to the equations. Using equation (15) with the separation of the displacement function into real and imaginary values, the displacement functions in x and y directions due to the shaft eccentricity a from the first relation of equation (12) can be written as

\begin{array}{l} U_{1} (z, t) = \sum_{n = 1}^{\infty} U_{0 n} \sin (\frac{n π}{L}) z \cdot \cos (Ω t + ϕ_{n}) \\ U_{2} (z, t) = \sum_{n = 1}^{\infty} U_{0 n} \sin (\frac{n π}{L}) z \cdot \sin (Ω t + ϕ_{n}) \end{array}

(17)

2.5. DC motor model

The torque applied to the shaft is supplied by the non-ideal brushed DC motor, which is usually modeled as follows

i_{m} = \frac{V_{s} - V_{e}}{R_{m}} = \frac{V_{s} - μ_{m} Ω}{R_{m}}

(18)

The torque applied by the motor can be written as

T_{m} = μ_{m} i_{m} = \frac{μ_{m}^{2}}{R_{m}} (\frac{V_{s}}{μ_{m}} - Ω)

(19)

where Vs is the constant electric potential applied to the motor terminals, μ_m is the constant value named the motor characteristic, i_m is the electric current drawn by the motor, and

V_{e} = μ_{m} Ω

is the back electromotive force developed in the motor coils.

2.6. Instantaneous power balance method

The electric power supplied by the DC motor in a steady state is obtained from the following

W_{m} = T_{m} Ω = (\frac{μ_{m} V_{s}}{R_{m}} - \frac{μ_{m}^{2}}{R_{m}} Ω) Ω

(20)

To calculate the mechanical power dissipated by the damping forces, first, it is necessary to decouple the system governing equations as follows

\begin{array}{l} (E I - i μ_{i} I Ω) \frac{\partial^{4} U}{\partial z^{4}} + μ_{i} I \frac{\partial^{5} U}{\partial z^{4} \partial t} - \frac{μ_{i} I ρ A}{K A G} \frac{\partial^{5} U}{\partial z^{2} \partial t^{3}} - (I_{d} + \frac{ρ A E I}{K A G} + \frac{μ_{i} I R_{a}}{K A G} - i \frac{μ_{i} I Ω ρ A}{K A G}) \frac{\partial^{4} U}{\partial z^{2} \partial t^{2}} + \\ (2 i I_{d} Ω - \frac{E I R_{a}}{K A G} + i \frac{μ_{i} I Ω R_{a}}{K A G}) \frac{\partial^{3} U}{\partial z^{2} \partial t} + R_{a} \frac{\partial U}{\partial t} + (ρ A - 2 i \frac{I_{d} Ω R_{a}}{K A G}) \frac{\partial^{2} U}{\partial t^{2}} + \frac{I_{d}}{K A G} (R_{a} - 2 i Ω ρ A) \frac{\partial^{3} U}{\partial t^{3}} \\ + \frac{I_{d} ρ A}{K A G} \frac{\partial^{4} U}{\partial t^{4}} = ρ A Ω^{2} a (1 + \frac{I_{d} Ω^{2}}{K A G}) e^{i (Ω t + ϕ)} \end{array}

(21)

\begin{array}{l} (E I - i μ_{i} I Ω) \frac{\partial^{4} ψ}{\partial z^{4}} + μ_{i} I \frac{\partial^{5} ψ}{\partial z^{4} \partial t} - \frac{μ_{i} I ρ A}{K A G} \frac{\partial^{5} ψ}{\partial z^{2} \partial t^{3}} - (I_{d} + \frac{ρ A E I}{K A G} + \frac{μ_{i} I R_{a}}{K A G} - i \frac{μ_{i} I Ω ρ A}{K A G}) \frac{\partial^{4} ψ}{\partial z^{2} \partial t^{2}} + \\ (2 i I_{d} Ω - \frac{E I R_{a}}{K A G} + i \frac{μ_{i} I Ω R_{a}}{K A G}) \frac{\partial^{3} ψ}{\partial z^{2} \partial t} + R_{a} \frac{\partial ψ}{\partial t} + (ρ A - 2 i \frac{I_{d} Ω R_{a}}{K A G}) \frac{\partial^{2} ψ}{\partial t^{2}} + \frac{I_{d}}{K A G} (R_{a} - 2 i Ω ρ A) \frac{\partial^{3} ψ}{\partial t^{3}} \\ + \frac{I_{d} ρ A}{K A G} \frac{\partial^{4} ψ}{\partial t^{4}} = 0 \end{array}

(22)

Similar to previous studies (Dasgupta et al., 2010; Samantaray et al., 2010), the power dissipated due to flexural vibrations through internal and external damping according to the decoupled equations (21) and (22) is

\begin{array}{l} W_{d} = \int_{0}^{L} R_{a} {{(\frac{\partial U_{1}}{\partial t})}^{2} + {(\frac{\partial U_{2}}{\partial t})}^{2} + \frac{I_{d}}{K A G} [\frac{\partial^{3} U_{1}}{\partial t^{3}} (\frac{\partial U_{1}}{\partial t}) + \frac{\partial^{3} U_{2}}{\partial t^{3}} (\frac{\partial U_{2}}{\partial t})] \\ - \frac{E I}{K A G} [\frac{\partial^{3} U_{1}}{\partial z^{2} \partial t} (\frac{\partial U_{1}}{\partial t}) + \frac{\partial^{3} U_{1}}{\partial z^{2} \partial t} (\frac{\partial U_{2}}{\partial t})]} d z \\ + \int_{0}^{L} μ_{i} I {\frac{\partial^{5} U_{1}}{\partial z^{4} \partial t} (\frac{\partial U_{1}}{\partial t}) + \frac{\partial^{5} U_{2}}{\partial z^{4} \partial t} (\frac{\partial U_{2}}{\partial t}) - \frac{ρ A}{K A G} [\frac{\partial^{5} U_{1}}{\partial z^{2} \partial t^{3}} (\frac{\partial U_{1}}{\partial t}) + \frac{\partial^{5} U_{2}}{\partial z^{2} \partial t^{3}} (\frac{\partial U_{2}}{\partial t})]} d z \\ + \int_{0}^{L} \frac{μ_{i} I R_{a}}{K A G} [\frac{\partial^{4} U_{1}}{\partial z^{2} \partial t^{2}} (\frac{\partial U_{1}}{\partial t}) + \frac{\partial^{4} U_{2}}{\partial z^{2} \partial t^{2}} (\frac{\partial U_{2}}{\partial t})] d z \end{array}

(23)

The non-ideal motor is loaded depending on the value of Ω, so the power transferred to flexural vibration is

\begin{array}{l} W_{r} = T_{L} Ω = \int_{0}^{L} μ_{i} I Ω {\frac{\partial^{4} U_{2}}{\partial z^{4}} (\frac{\partial U_{1}}{\partial t}) - \frac{\partial^{4} U_{1}}{\partial z^{4}} (\frac{\partial U_{2}}{\partial t}) + \frac{ρ A}{K A G} [\frac{\partial^{4} U_{2}}{\partial z^{2} \partial t^{2}} (\frac{\partial U_{1}}{\partial t}) - \frac{\partial^{4} U_{1}}{\partial z^{2} \partial t^{2}} (\frac{\partial U_{2}}{\partial t})]} d z \\ + \int_{0}^{L} 2 \frac{I_{d} Ω R_{a}}{K A G} [\frac{\partial^{2} U_{2}}{\partial t^{2}} (\frac{\partial U_{1}}{\partial t}) - \frac{\partial^{2} U_{1}}{\partial t^{2}} (\frac{\partial U_{2}}{\partial t})] d z \\ + \int_{0}^{L} \frac{μ_{i} I Ω R_{a}}{K A G} [- \frac{\partial^{3} U_{2}}{\partial z^{2} \partial t} (\frac{\partial U_{1}}{\partial t}) + \frac{\partial^{3} U_{1}}{\partial z^{2} \partial t} (\frac{\partial U_{2}}{\partial t})] d z \\ + \int_{0}^{L} 2 \frac{ρ A I_{d} Ω}{K A G} [\frac{\partial^{3} U_{2}}{\partial t^{3}} (\frac{\partial U_{1}}{\partial t}) - \frac{\partial^{3} U_{1}}{\partial t^{3}} (\frac{\partial U_{2}}{\partial t})] d z \\ + \int_{0}^{L} 2 I_{d} Ω [- \frac{\partial^{3} U_{2}}{\partial z^{2} \partial t} (\frac{\partial U_{1}}{\partial t}) + \frac{\partial^{3} U_{1}}{\partial z^{2} \partial t} (\frac{\partial U_{2}}{\partial t})] d z \\ - \int_{0}^{L} ρ A Ω^{2} a (1 + \frac{I_{d} Ω^{2}}{K A G}) [c o s (Ω t + ϕ) (\frac{\partial U_{1}}{\partial t}) + s i n (Ω t + ϕ) (\frac{\partial U_{2}}{\partial t})] d z \end{array}

(24)

Ignoring the torsional vibration of the shaft (assuming constant Ω along the length of the shaft), the load torque can be written as follows

\begin{array}{l} T_{L} = \frac{W_{r}}{Ω} = \int_{0}^{L} μ_{i} I {\frac{\partial^{4} U_{2}}{\partial z^{4}} (\frac{\partial U_{1}}{\partial t}) - \frac{\partial^{4} U_{1}}{\partial z^{4}} (\frac{\partial U_{2}}{\partial t}) + \frac{ρ A}{K A G} [\frac{\partial^{4} U_{2}}{\partial z^{2} \partial t^{2}} (\frac{\partial U_{1}}{\partial t}) - \frac{\partial^{4} U_{1}}{\partial z^{2} \partial t^{2}} (\frac{\partial U_{2}}{\partial t})]} d z \\ + \int_{0}^{L} 2 \frac{I_{d} R_{a}}{K A G} [\frac{\partial^{2} U_{2}}{\partial t^{2}} (\frac{\partial U_{1}}{\partial t}) - \frac{\partial^{2} U_{1}}{\partial t^{2}} (\frac{\partial U_{2}}{\partial t})] d z \\ + \int_{0}^{L} \frac{μ_{i} I R_{a}}{K A G} [- \frac{\partial^{3} U_{2}}{\partial z^{2} \partial t} (\frac{\partial U_{1}}{\partial t}) + \frac{\partial^{3} U_{1}}{\partial z^{2} \partial t} (\frac{\partial U_{2}}{\partial t})] d z \\ + \int_{0}^{L} 2 \frac{ρ A I_{d}}{K A G} [\frac{\partial^{3} U_{2}}{\partial t^{3}} (\frac{\partial U_{1}}{\partial t}) - \frac{\partial^{3} U_{1}}{\partial t^{3}} (\frac{\partial U_{2}}{\partial t})] d z \\ + \int_{0}^{L} 2 I_{d} [- \frac{\partial^{3} U_{2}}{\partial z^{2} \partial t} (\frac{\partial U_{1}}{\partial t}) + \frac{\partial^{3} U_{1}}{\partial z^{2} \partial t} (\frac{\partial U_{2}}{\partial t})] d z \\ - \int_{0}^{L} ρ A Ω a (1 + \frac{I_{d} Ω^{2}}{K A G}) [\cos (Ω t + ϕ) (\frac{\partial U_{1}}{\partial t}) + \sin (Ω t + ϕ) (\frac{\partial U_{2}}{\partial t})] d z \end{array}

(25)

Therefore, the net power that maintains the flexural vibrations is the difference between the power transferred to the flexural vibrations by the electric source and the power dissipated from the vibrations, that is

W_{f} = W_{r} - W_{d}

(26)

If W_f < 0, the flexural vibration is stable because the net energy of the system decreases in the flexural mode, and as a result, the amplitude of the flexural vibration decreases. Likewise, if W_f > 0, the flexural vibration becomes unstable; if W_f = 0, the flexural vibrations persist with a constant energy state. Therefore, the power transfer mechanism in a marginally stable state can be written as

W_{m} - R_{r} Ω^{2} = T_{L} Ω = W_{r} = W_{d}

(27)

R_{r} Ω^{2}

is the power dissipated due to the net rotational damping acting on the spinning shaft. If the shaft rotation speed is close to one of the critical speeds, the effects of other modes can be ignored. Considering the nth term near the corresponding critical speed and performing an analysis of the power balance between the supplied power W_m and the dissipated power

W_{d} + R_{r} Ω^{2}

, it may be written as

\begin{array}{l} (\frac{U_{s} μ_{m}}{R_{m}} - \frac{μ_{m}^{2}}{R_{m}} Ω) Ω = \frac{1}{2} L Ω^{2} {U_{0 n}}^{2} {R_{a} (1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2} - \frac{I_{d}}{K A G} Ω^{2}) \\ + μ_{i} I {(\frac{n π}{L})}^{2} [{(\frac{n π}{L})}^{2} - \frac{ρ A}{K A G} Ω^{2}]} + R_{r} Ω^{2} \end{array}

(28)

By substituting the vibration amplitude from equation (15), a ninth-order polynomial in terms of rotational speed is obtained as follows

a_{9} Ω^{9} + a_{8} Ω^{8} + a_{7} Ω^{7} + a_{6} Ω^{6} + a_{5} Ω^{5} + a_{4} Ω^{4} + a_{3} Ω^{3} + a_{2} Ω^{2} + a_{1} Ω + a_{0} = 0

(29)

All constants of equation (29) are according to the vibration system and electric source parameters, which are given in Appendix B. This relation is considered the system characteristic equation in investigating the Sommerfeld effect.

2.7. Stability criteria of the steady state response

The real positive roots of equation (29) represent the rotation speeds of the system that establish the electrical–mechanical power balance. Some of these balances are stable, and some are unstable. The stability condition of a vibrating system with a non-ideal DC motor (Samantaray et al., 2006) is

{\frac{d}{d Ω} (W_{m} - W_{d} - R_{r} Ω^{2}) |}_{Ω = \bar{Ω}} < 0

(30)

where

\bar{Ω}

is the acceptable roots of equation (29). Substituting equations (20) and (23) in equation (30) gives

\frac{d}{d Ω} {\begin{cases} (\frac{V_{s} μ_{m}}{R_{m}} - \frac{μ_{m}^{2}}{R_{m}} Ω) Ω - R_{r} Ω^{2} - \\ \frac{2 L ρ^{2} A^{2} Ω^{6} a^{2} \frac{{(1 - \cos n π)}^{2}}{n^{2} π^{2}} {R_{a} (1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2} - \frac{I_{d}}{K A G} Ω^{2}) + μ_{i} I {(\frac{n π}{L})}^{2} [{(\frac{n π}{L})}^{2} - \frac{ρ A}{K A G} Ω^{2}]} {[1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2} + \frac{I_{p}}{K A G} Ω^{2}]}^{2}}{{E I {(\frac{n π}{L})}^{4} + [(I_{p} - \frac{ρ E I}{K G}) {(\frac{n π}{L})}^{2} - ρ A] Ω^{2} - \frac{ρ I_{p}}{K G} Ω^{4}}^{2} + {R_{a} Ω [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2} + \frac{I_{p}}{K A G} Ω^{2}]}^{2}} \end{cases}}_{Ω = \bar{Ω}} < 0

(31)

By establishing equation (31) as a condition of stability, stable rotational speeds of the system in the vicinity of the critical speed are obtained.

3. Characteristics of the Sommerfeld effect in odd modes

Let’s rewrite equation (29) in transformation function form. According to Appendix B, the constants with even indices are defined as linear coefficients of the input voltage

G (s) = \frac{N (s)}{D (s)} = \frac{b_{8} s^{8} + b_{6} s^{6} + b_{4} s^{4} + b_{2} s^{2} + b_{0}}{a_{9} s^{9} + a_{7} s^{7} + a_{5} s^{5} + a_{3} s^{3} + a_{1} s}

(32)

where s is the complex form of the rotational speed of the shaft, N(s) and D(s) are polynomials in terms of s, and Vs is the gain parameter so that the characteristic

D (s) + V_{s} N (s) = 0

results in equation (29). The b_i constants are also listed in Appendix B. Now it is possible to study the properties and roots of the characteristic equation using standard control theoretic algorithms. The poles and zeros of the transfer function in equation (32) are symmetrical concerning the real and imaginary axes.

According to equation (32), while increasing the voltage from zero to infinity, Sommerfeld phenomenon occurs if more than one positive real root is obtained for each certain voltage. By drawing the root loci diagram of the transformation function and finding the poles and zeroes for each mode, the unstable speed intervals duo to the Sommerfeld effect are determined.

4. Results and discussion

Case a) To numerically study the Sommerfeld effect and ensure the obtained relationships, first, a non-ideal system is considered according to the example given in reference (Dasgupta and Rajamohan, 2017) with the following specifications

\begin{array}{l} E = 200 (GPa), G = 77 (GPa), ρ = 7800 (kg / m^{3}), K = 0.88, A = 9.81 \times 10^{- 5} (m^{2}), L = 1.5 (m) \\ a = 0.001 (m), R_{r} = 10^{- 5} (N . m . s / rad), R_{a} = 10^{- 13} \times E I (N . s / m^{2}), μ_{i} = 10^{- 6} \times E (N . s / m^{2}) \\ μ_{m} = 0.44 (N . m / A), R_{m} = 800 (Ω), I = 1.994 \times 10^{- 9} (m^{4}), I_{d} = ρ I . \end{array}

The Campbell diagram of the system for the first nine modes is shown in Figure 2. Also, by performing the calculations of equations (9) and (10), the forward critical speeds in the first to third modes are 100.14 (rad/s), 400.42 (rad/s), and 900.51 (rad/s), respectively.

Dimensionless quantities are used to study the steady state of the system. Therefore, dimensionless rotational speed ω^* = Ω/ω₁ and dimensionless whirling amplitude β_n = U_0n ω₁²/(π²g) are defined, which are briefly represented by β in each mode, and g is the acceleration of gravity. The root loci of the transfer function, considering the voltage parameter Vs as the gain parameter in a negative feedback system, is shown in Figure 3.

According to Figure 3, points a and b are the break-in and breakaway points of the root loci diagram, which are obtained for 46.3 (V) and 47.2 (V), respectively. Point b indicates the natural frequency. For Vs < 46.3 (V), there is only a single positive real root located on the left side of point b. For 46.3 (V)<V_s < 47.2 (V), three positive real roots are found, and for Vs > 47.2 (V), one positive real root can be obtained on the right side of point a. It is mentioned that the Sommerfeld effect occurred near the first natural frequency is equal to (ω* = 1) 100.14 (rad/s). It can also be seen from the values of the real axis in Figure 3 that the break-in point is at speed higher than the first natural frequency, and the breakaway point is sufficiently close to the first natural frequency.

Using equations (20) and (23), the power diagram with the angular speed of the shaft has been drawn (Figure 4). The intersection of the curves related to the dissipated power of the system (W_d+R_rΩ²) for different voltages and the mechanical power applied by the motor (W_m) shows the satisfying speed of the energy balance equation in equation (17). According to the previously mentioned results, in the range of 46.3 (V)<V_s < 47.2 (V), three crossing points are observed, which confirms the occurrence of the Sommerfeld effect in the vicinity of the first natural frequency.

Figure 5 shows the diagram of the dimensionless rotational speed with the supply voltage. If the voltage is assumed to increase slowly, the speed also increases. Near the first critical speed, there is not much change in the angular speed with increasing voltage (points a to b). In this situation, at the same time as the angular speed remains constant, the amplitude of the lateral vibration of the shaft increases, as shown in Figure 6. In continuing the increasing path of the voltage, a jump occurs from point b to the stable and equal voltage to point d. During this jump, while the angular speed of the shaft increases suddenly, the amplitude of the vibration also decreases suddenly. The reason for this can be found in the solution of equation (29). If three positive real roots of equation (29) are obtained for the rotational speed for each voltage supplied to the system, one of the roots does not satisfy the stability condition, which forms an unstable curve from point b to c in Figures 5 and 6. Therefore, the system cannot vibrate at the unstable interval, and to maintain the system’s stability, the vibrations continue at the third point, which is the same voltage as point b and satisfies the stability condition (point d).

Suppose the voltage slowly decreases from higher values, approaching point c, which is close to the first critical speed, while gradually reducing the speed, the vibration amplitude increases and again, as in the previous state. In that case, the system jumps from point c to a, which occurs as a sudden increase in the amplitude and then a gradual decrease, and in the speed, it occurs as a sudden decrease and then a gradual decrease.

Regarding the change process of the vibration amplitude and rotational speed of the non-ideal power supply voltage, the obtained results are very similar to the studies conducted with the Euler–Bernoulli theory for the continuous shaft (Dasgupta and Rajamohan, 2017) (Figures 5 and 6) and also the shaft-disc model (Samantaray et al., 2006). With the difference that the models considered in the previous studies were effective for thin shafts and due to ignoring the effects of shear deformation and surface inertia moment, it is not recommended to use them for thick shafts.

Case b) To study the Sommerfeld phenomenon and observe the effect of shear deformation on this phenomenon, a non-ideal system with the following characteristics is considered

\begin{array}{l} E = 200 (GPa), G = 77 (GPa), ρ = 7800 (kg / m^{3}), K = 0.88, A = 1.963 \times 10^{- 3} (m^{2}), L = 2 (m) \\ a = 10^{- 4} (m), R_{r} = 10^{- 5} (N . m . s / rad), R_{a} = 10^{- 13} E I (N . s / m^{2}), μ_{i} = 10^{- 6} E (N . s / m^{2}) \\ μ_{m} = 0.2 (N . m / A), R_{m} = 100 (Ω), I = 3.068 \times 10^{- 7} (m^{4}), I_{d} = ρ I . \end{array}

In this case, the critical speeds obtained from equation (10) are: ω₁ = 156.1 (rad/s), ω₃ = 1400.8 (rad/s), ω₅ = 3867.7 (rad/s), ω₇ = 7511.7 (rad/s) and ω₉ = 12266.0 (rad/s). The obtained results state that at high speeds, especially at more than the 7th natural frequency, considering the effects of shear deformation based on Timoshenko beam theory affects the value of the natural frequency and vibration amplitude (Figure 7).

Similar to previous case, taking points a, b, c, and d as the defining points of the Sommerfeld effect, Table 1, shows the occurrence of this phenomenon at different critical speeds using the Euler–Bernoulli and the Timoshenko beam theories.

As expected, the similarity of the results obtained in the two theories in the first and third critical speeds indicates that regardless of the effects of shear deformation at not-so-high speeds is recommended in thin shafts. But since in the governing equations of motion, the number of terms related to the shear deformation are multiplied by the rotational speed; it is not far from the expectation to see differences in the results of different theories at high speeds even though the shaft is not thick. Considering the dependence of the vibration amplitude and the dissipated energy of the system on the natural frequencies, the difference between the results obtained from the Timoshenko beam theory and the Euler–Bernoulli beam theory affects the occurrence point of the Sommerfeld effect so that the Euler–Bernoulli theory calculates the jump caused by this effect a little further. In other words, in the investigated shaft, according to Table 1, in the path of increasing the system speed, the ninth critical speed and, as a result, the corresponding jump will occur at the speed ω* = 78.5778 and with the voltage versus = 4372.7 (V). At the same time, calculations based on the Euler–Bernoulli theory report this point at a speed of ω* = 81.0404 and with a voltage of versus = 4402.3 (V).

Another important issue regarding the Sommerfeld effect at high frequencies is that after the jump that occurs, there is a possibility that the system will jump to near the next critical speed instead of being in a stable state. For example, in the seventh critical speed, the system jumps from ω* = 48.1211 to ω* = 71.8674, which is not far from the ninth critical speed of the system (ω* = 78.5778). The proximity of the peak points of the fifth critical speed and above is proof of this, which is shown in Figure 8.

The importance of the Sommerfeld effect at higher critical speeds is due to the fact that, according to Figure 8, in the above modes, despite the reduction of the jump amplitude caused by this phenomenon, the amount of sudden change in rotation speed increases significantly.

In other words, the speed instability interval around higher critical speeds is larger. For example, in the increasing process of speed around the first critical speed, the dimensionless rotational speed jumps from ω* = 1 to ω* = 1.15246 (15%), and this jump in the third critical speed jumps from ω* = 8.97373 to ω* = 11.5208 (28%) until the ninth critical speed of speed instability is calculated from ω* = 78.5778 to ω* = 136.183 (73%). A sudden change in speed at high frequencies leads to a sudden change in system power, manifesting as a mechanical shock to the system components.

Table 1.

Comparison of the voltage and dimensionless rotational speed of the points defining the Sommerfeld effect for case (b) in the Euler–Bernoulli and Timoshenko beam theories.

Mode	Theory	a		b		c		d
Mode	Theory	ω _a *	V _a	ω _b *	V _b	ω _c *	V _c	ω _d *	V _d
1	E-B	0.99116	32.98	1.00000	36.90	1.02056	32.98	1.15247	36.90
1	T	0.99097	32.97	1.00000	36.89	1.02024	32.97	1.15246	36.89
3	E-B	8.84625	307.23	9.00448	372.11	9.39333	307.23	11.6176	372.11
3	T	8.82767	306.77	8.97373	369.03	9.38181	306.77	11.5208	369.03
5	E-B	24.4657	879.37	25.0122	1589.2	26.6931	879.37	46.8924	1589.2
5	T	24.3363	875.69	24.7771	1485.0	26.6457	875.69	46.3696	1485.0
7	E-B	47.8700	1775.7	49.0243	2366.1	53.2313	1775.7	72.5766	2366.1
7	T	47.3555	1760.2	48.1211	2306.3	52.7937	1760.2	71.8674	2306.3
9	E-B	79.1659	3024.3	81.0404	4402.3	89.8264	3024.3	136.911	4402.3
9	T	77.7098	2981.6	78.5778	4372.7	88.5676	2981.6	136.183	4372.7

5. Conclusion

In this article, the Sommerfeld effect was investigated based on Timoshenko beam theory in the non-ideal motor-shaft system. First, the governing equations of motion were obtained by considering internal and external damping, and the characteristic equation and critical speeds were calculated. Then, for the first time, the power balancing method for the shaft was carried out based on Timoshenko beam theory, and the stability conditions of the system were obtained with the linear model of the non-ideal electric source. The limitations and restrictions of this study are as follows:

(1) This study concerns synchronous whirl of the rotor shaft. Therefore, instability of asynchronous whirl due to the presence of material damping has not been studied in this article. The results presented in this article are valid only when the asynchronous whirl is stable and then the steady state whirl is only due to the eccentricity induced synchronous whirl.

(2) In reality, the escape from resonance capture can occur much before the predicted critical DC motor voltage values. The presented results are valid when the DC motor supply voltage is increased at a very slow rate. If a high voltage is directly applied or the voltage is increased at a higher rate then inertial effects can aid in achieving escape through the resonances. These details can be obtained only from transient simulations which are not presented in this research.

(3) The inductance of the DC motor has been neglected in this study. In actual rotor coast up, the motor inductance may significantly affect the dynamics during the transit through resonance and speed jumps may occur elsewhere from those predicted through the presented steady state power balance analysis.

(4) This article does not consider the mechanism of excitation of different synchronous whirl modes. Whereas the present article considers synchronous whirl in all the modes, the even modes may not be excited at the corresponding critical speeds due to the nature of the eccentricity induced excitation. For example, second mode may not be excited in the absence of moment unbalance. These can be validated through transient simulations which are out of the scope of this research.

(5) The continuous cylindrical rotor shaft is assumed to have a parallel eccentricity, that is, the geometric center line is parallel to the mass center line. Thus the mass distribution over the circular shaft section is non-uniform. Such non-uniformity causes asymmetry in shaft bending stiffness which in turn may lead to parametric excitation and instabilities. In addition, the shaft supports have been idealized. Support stiffness and damping, and their asymmetry may also influence the results. The stretching of the shaft due to bending, or the stretching non-linearity, has been neglected. These aspects, which are known to significantly affect the rotor dynamics, have not been studied in this article.

So far, the use of Timoshenko theory has been introduced only for the analysis of thick shafts. Still, it was shown that in identifying and investigating the Sommerfeld effect, the effects of shear deformation and area moment for non-thick shafts also appeared at high critical speeds and, regardless of them, caused errors in calculating the speed and voltage of the Sommerfeld jump so that without using the Timoshenko beam theory, the jump caused by the Sommerfeld phenomenon is calculated and identified at a further point. It was also observed that at higher critical speeds, although the amount of amplitude jump is decreased, due to the Sommerfeld effect, the amount of speed jump and, as a result, the instability range of the system around critical speed significantly increases. Since significant and sudden changes in speed, especially at high speeds, cause mechanical impact and destruction of vibrating system components, to investigate the behavior of the vibrating system and find a solution to reduce the destructive effect of the Sommerfeld phenomenon, using the Timoshenko beam theory is more reliable than other theories. In this study, unlike other studies, considering all of the effects of the real mechanical system, such as the effects of shear deformation, area moment, internal damping, gyroscopic effects, etc., investigating the Sommerfeld effect in thick shafts is also made possible by using the provided equations.

Footnotes

Declaration of conflicting interests

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

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hossein Movahedi

Appendix A

The bending moments due to the internal forces in x-z and y-z planes are written as (Dasgupta et al., 2010; Sghaier et al., 2019) (A.1)

{\begin{cases} M_{1} = E I \frac{\partial ψ_{2}}{\partial z} + μ_{i} I \frac{\partial}{\partial t} (\frac{\partial ψ_{2}}{\partial z}) - μ_{i} I Ω \frac{\partial ψ_{1}}{\partial z} \\ M_{2} = - E I \frac{\partial ψ_{1}}{\partial z} - μ_{i} I \frac{\partial}{\partial t} (\frac{\partial ψ_{1}}{\partial z}) - μ_{i} I Ω \frac{\partial ψ_{2}}{\partial z} \end{cases}

where indexes 1, and 2 are assigned to the x and y axes, respectively. The assumptions related to the Timoshenko beam theory are established as follows (A.2)

{\begin{cases} β_{1} = \frac{\partial U_{1}}{\partial z} - ψ_{1}, V_{1} = K A G β_{1} \\ β_{2} = \frac{\partial U_{2}}{\partial z} - ψ_{2}, V_{2} = K A G β_{2} \end{cases}

According to Figure 9, the equilibrium relations of shear forces are: (A.3)

{\begin{cases} - V_{1} + (V_{1} + \frac{\partial V_{1}}{\partial z} d z) - R_{a} \frac{\partial U_{1}}{\partial t} d z = ρ A d z \frac{\partial^{2} U_{1}^{c}}{\partial t^{2}} \\ - V_{2} + (V_{2} + \frac{\partial V_{2}}{\partial z} d z) - R_{a} \frac{\partial U_{2}}{\partial t} d z = ρ A d z \frac{\partial^{2} U_{2}^{c}}{\partial t^{2}} \end{cases}

where (A.4)

\begin{array}{l} U_{1}^{c} = U_{1} + [a_{1} \cos (Ω t + ϕ) - a_{2} \sin (Ω t + ϕ)] \\ U_{2}^{c} = U_{2} + [a_{1} \sin (Ω t + ϕ) + a_{2} \cos (Ω t + ϕ)] \end{array}

a₁ and a₂ are the amounts of shaft eccentricity in directions 1 and 2. By substituting equation (A.2) in equation (A.3), it may be written (A.5)

{\begin{cases} K A G (\frac{\partial^{2} U_{1}}{\partial z^{2}} - \frac{\partial ψ_{1}}{\partial z}) - R_{a} \frac{\partial U_{1}}{\partial t} = ρ A \frac{\partial^{2} U_{1}^{c}}{\partial t^{2}} \\ K A G (\frac{\partial^{2} U_{2}}{\partial z^{2}} - \frac{\partial ψ_{2}}{\partial z}) - R_{a} \frac{\partial U_{2}}{\partial t} = ρ A \frac{\partial^{2} U_{2}^{c}}{\partial t^{2}} \end{cases}

Figure 9.

The balance of shear forces on the surface element

As well as, for torque balance, it can also be written (A.6)

{\begin{cases} E I \frac{\partial^{2} ψ_{1}}{\partial z^{2}} + K A G (\frac{\partial U_{1}}{\partial z} - ψ_{1}) + μ_{i} I \frac{\partial}{\partial t} (\frac{\partial^{2} ψ_{1}}{\partial z^{2}}) + μ_{i} I Ω \frac{\partial^{2} ψ_{2}}{\partial z^{2}} = I_{d} (\frac{\partial^{2} ψ_{1}}{\partial t^{2}} + 2 Ω \frac{\partial ψ_{2}}{\partial t}) \\ E I \frac{\partial^{2} ψ_{2}}{\partial z^{2}} + K A G (\frac{\partial U_{2}}{\partial z} - ψ_{2}) + μ_{i} I \frac{\partial}{\partial t} (\frac{\partial^{2} ψ_{2}}{\partial z^{2}}) - μ_{i} I Ω \frac{\partial^{2} ψ_{1}}{\partial z^{2}} = I_{d} (\frac{\partial^{2} ψ_{2}}{\partial t^{2}} - 2 Ω \frac{\partial ψ_{1}}{\partial t}) \end{cases}

Equation (A.5) and (A.6) are rewritten in the complex form to obtain the governing equations of motion as below (A.7)

{\begin{cases} K A G (\frac{\partial^{2} U}{\partial z^{2}} - \frac{\partial ψ}{\partial z}) - R_{a} \frac{\partial U}{\partial t} = ρ A \frac{\partial^{2} U}{\partial t^{2}} - ρ A Ω^{2} a e^{i (Ω t + ϕ)} \\ E I \frac{\partial^{2} ψ}{\partial z^{2}} + K A G (\frac{\partial U}{\partial z} - ψ) + μ_{i} I \frac{\partial}{\partial t} (\frac{\partial^{2} ψ}{\partial z^{2}}) - i μ_{i} I Ω \frac{\partial^{2} ψ}{\partial z^{2}} = I_{d} (\frac{\partial^{2} ψ}{\partial t^{2}} - 2 i Ω \frac{\partial ψ}{\partial t}) \end{cases}

where according to the definition (A.8)

ψ = ψ_{1} + i ψ_{2}, U = U_{1} + i U_{2}, a = a_{1} + i a_{2}

Appendix B

The constants related to equation (17), taking into account the effects of shear deformation and external viscous damping, are as follows

a_{0} = - E^{2} I^{2} {(\frac{n π}{L})}^{8} \frac{μ_{m}}{R_{m}} V_{s} a_{1} = E^{2} I^{2} {(\frac{n π}{L})}^{8} (\frac{μ_{m}^{2}}{R_{m}} + R_{r})

a_{2} = - {2 E I {(\frac{n π}{L})}^{4} [{(I_{d} - \frac{ρ E I}{K G})}^{2} {(\frac{n π}{L})}^{2} - ρ A] + R_{a}^{2} [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2}]} \frac{μ_{m}}{R_{m}} V_{s}

\begin{array}{l} a_{3} = {2 E I {(\frac{n π}{L})}^{4} [{(I_{d} - \frac{ρ E I}{K G})}^{2} {(\frac{n π}{L})}^{2} - ρ A] + R_{a}^{2} [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2}]} (\frac{μ_{m}^{2}}{R_{m}} + R_{r}) \\ + ρ A L a \frac{1 - \cos n π}{n π} [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2}] {R_{a} [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2}] + μ_{i} I {(\frac{n π}{L})}^{4}} \end{array}

(B.1)

a_{4} = - {- \frac{2 E I ρ I_{d}}{K G} {(\frac{n π}{L})}^{4} + {[{(I_{d} - \frac{ρ E I}{K G})}^{2} {(\frac{n π}{L})}^{2} - ρ A]}^{2} + \frac{2 R_{a}^{2} I_{d}}{K A G} [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2}]} \frac{μ_{m}}{R_{m}} V_{s}

\begin{array}{l} a_{5} = {- \frac{2 E I ρ I_{d}}{K G} {(\frac{n π}{L})}^{4} + {[{(I_{d} - \frac{ρ E I}{K G})}^{2} {(\frac{n π}{L})}^{2} - ρ A]}^{2} + \frac{2 R_{a}^{2} I_{d}}{K A G} [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2}]} (\frac{μ_{m}^{2}}{R_{m}} + R_{r}) \\ - ρ A L a \frac{1 - \cos n π}{n π} \frac{μ_{i} I}{K A G} {(\frac{n π}{L})}^{2} [(I_{d} - \frac{ρ E I}{K G}) {(\frac{n π}{L})}^{2} - ρ A] \end{array}

a_{6} = {\frac{2 ρ I_{d}}{K G} [{(I_{d} - \frac{ρ E I}{K G})}^{2} {(\frac{n π}{L})}^{2} - ρ A] - {(\frac{R_{a} I_{d}}{K A G})}^{2}} \frac{μ_{m}}{R_{m}} V_{s}

\begin{array}{l} a_{7} = {\frac{2 ρ I_{d}}{K G} [{(I_{d} - \frac{ρ E I}{K G})}^{2} {(\frac{n π}{L})}^{2} - ρ A] - {(\frac{R_{a} I_{d}}{K A G})}^{2}} (\frac{μ_{m}^{2}}{R_{m}} + R_{r}) \\ - ρ A L a \frac{1 - \cos n π}{n π} \frac{I_{d}}{{(K A G)}^{2}} [R_{a} I_{d} + ρ μ_{i} I {(\frac{n π}{L})}^{2}] \end{array}

a_{8} = - {(\frac{ρ I_{d}}{K G})}^{2} \frac{μ_{m}}{R_{m}} V_{s} a_{9} = {(\frac{ρ I_{d}}{K G})}^{2} (\frac{μ_{m}^{2}}{R_{m}} + R_{r})

The b_i constants in the equation (32) are (B.2)

\begin{array}{l} b_{0} = \frac{a_{0}}{V_{s}} = - E^{2} I^{2} {(\frac{n π}{L})}^{8} \frac{μ_{m}}{R_{m}} \\ b_{2} = - {2 E I {(\frac{n π}{L})}^{4} [{(I_{d} - \frac{ρ E I}{K G})}^{2} {(\frac{n π}{L})}^{2} - ρ A] + R_{a}^{2} [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2}]} \frac{μ_{m}}{R_{m}} \\ b_{4} = \frac{a_{4}}{V_{s}} = - {- \frac{2 E I ρ I_{d}}{K G} {(\frac{n π}{L})}^{4} + {[{(I_{d} - \frac{ρ E I}{K G})}^{2} {(\frac{n π}{L})}^{2} - ρ A]}^{2} + \frac{2 R_{a}^{2} I_{d}}{K A G} [1 + \frac{E I}{K A G} {(\frac{n π}{L})}^{2}]} \frac{μ_{m}}{R_{m}} \\ b_{6} = \frac{a_{6}}{V_{s}} = {\frac{2 ρ I_{d}}{K G} [{(I_{d} - \frac{ρ E I}{K G})}^{2} {(\frac{n π}{L})}^{2} - ρ A] - {(\frac{R_{a} I_{d}}{K A G})}^{2}} \frac{μ_{m}}{R_{m}} \\ b_{8} = \frac{a_{8}}{V_{s}} = - {(\frac{ρ I_{d}}{K G})}^{2} \frac{μ_{m}}{R_{m}} \end{array}

Appendix C

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