Transmission and Dissipation of Vibration in a Dynamic Vibration Absorber-Roller System Based on Particle Damping Technology

Abstract

The research of rolling mill vibration theory has always been a scientific problem in the field of rolling forming, which is very important to the quality of sheet metal and the stable operation of equipment. The essence of rolling mill vibration is the transfer of energy, which is generated from inside and outside. Based on particle damping technology, a dynamic vibration absorber (DVA) is proposed to control the vertical vibration of roll in the rolling process from the point of energy transfer and dissipation. A nonlinear vibration equation for the DVA-roller system is solved by the incremental harmonic balance method. Based on the obtained solutions, the effects of the basic parameters of the DVA on the properties of vibration transmission are investigated by using the power flow method, which provides theoretical guidance for the selection of the basic parameters of the DVA. Furthermore, the influence of the parameters of the particles on the overall dissipation of energy of the particle group is analyzed in a more systematic way, which provides a reference for the selection of the material and diameter and other parameters of the particles in the practical application of the DVA. The effect of particle parameters on roll amplitude inhibition is studied by experiments. The experimental results agree with the theoretical analysis, which proves the correctness of the theoretical analysis and the feasibility of the particle damping absorber. This research proposes a particle damping absorber to absorb and dissipate the energy transfer in rolling process, which provides a new idea for nonlinear dynamic analysis and stability control of rolling mills, and has important guiding significance for practical production.

1 Introduction

During the rolling process, the rolling mill undergoes various forms of vibrational behavior [1,2,3,4], which restrict the high-speed, continuous, and intelligent development of the rolling. Although a large number of new technologies as well as new processes have been adopted in recent years in strip production [5,6,7,8,9], the vibration problem still exists. Vibration in mills is very harmful for strip products, so vibration control of rolling mills is a critical issue that needs to be addressed urgently.

In the field of mill vibration control, many useful explorations have been made by international scholars. Belli et al. [10] designed a control system to deal with torsional vibrations of hot rolling mills based on a frequency domain method. Schlacher et al. [11] proposed two nonlinear control concepts on the basis of the factor of energy and differential theory in order to suppress the third octave chatter phenomenon in mills. Furumoto et al. [12] improved the damping effect in the mill stabilizing device by installing a chamber in the device to reduce the vibration of the mill when the thinner and harder strips are rolled. Lu et al. [13] proposed a data-driven mill vibration analysis method, meanwhile, it was shown that controlling the rolling speed, cumulative strip length, tension and roll radius helps to suppress mill vibration.

As a vibration control technology, DVA has been widely used in various fields due to its many advantages. Noori et al. [14] applied dynamic vibration absorbers to a double-deck railroad tunnel system to reduce railway induced ground-borne vibrations. Tian et al. [15] improved the ride comfort of in-wheel drive electric vehicles (IWD-EVs) by applying dynamic absorbers to the in-wheel powertrain system. Su et al. [16] suggested a dynamic vibration absorber with negative stiffness to control the longitudinal vibration of the propulsion shaft, and the results showed that the absorber has good properties of vibration-absorbing in a broad frequency scope. Mizuno et al. [17] designed an active DVA to suppress vibrations caused by the reciprocating motion of the displacer in a cryopump.

Conventional DVAs provide excellent control for vibrations that have determined frequency or vary within a very small scope. However, in engineering practice, conventional DVAs do not work when the vibrational frequency varies in a wide scope. In order to broaden the damping frequency band, several methods have been used, such as the distributed installation of multiple DVAs [18], the introduction of negative stiffness mechanisms in DVAs [19], the use of particle damping techniques [20], and the design of mass units in DVAs in an adjustable form [21]. Among them, particle damping technology dissipates the energy of the primary vibration system through particle-to-particle collision and friction, which has some advantages such as low noise, good damping effect and properties suitable for harsh environments [22, 23], and has been applied in many fields. However, there are few reports on the application of particle damping techniques in mill vibration control. In this paper, for the two-roll mill shown in Figure 1b, a DVA based on particle damping technology as shown in Figure 1a is designed to control the vertical vibration of the rolls by installing the DVA on the balance beam of the mill. In our previous work [24], we studied the vibration properties of the roll under this DVA control and optimized the mass ratio, frequency ratio and damping ratio of the absorber by adaptive genetic algorithm to offer the theoretical direction for the prototype manufacturing of the absorber. On this basis, the vibration transmission in the coupled roll-absorber system and the energy dissipation mechanism of the particle damping are investigated in depth in this paper.

Figure 1

a The dynamic vibration absorber, b two-roll mill

For all forms of mechanical vibration systems, the nature of vibration is the transfer of energy in the structure. Power flow of vibration is an important tool for describing energy transfer in vibrating structures, and it is used as a stand-alone parameter to depict the system response. Yang et al. [25] investigated the power flow properties for nonlinear vibration isolation systems with negative stiffness mechanisms, and the results showed that the power flow of nonlinear systems is sensitive to the initial conditions. Dai et al. [26] studied the power flow behavior of impact oscillators consisting of linear or quasi-zero stiffness (QZS) nonlinear constraints. Wu et al. [27] studied the vibration power flow of a beam with a crack and found that the presence of crack significantly changed the transmission properties of the power flow. For the vibration system composed of roll and DVA, the larger the power flow from the roll to the DVA, the better the vibration absorption effect of the DVA. By analyzing the effect of parameters in DVA on the power flow transfer characteristics and systematically analyzing the effect of particle parameters on the overall dissipation of energy, it has a theoretical direction for the parameter choice of DVA.

The paper is organized as follows. In Section 2.1, a three-degree-of-freedom model of the DVA-roller system is developed. The system is solved in Section 2.2 by using the IHBM. The influence of the basic parameters in the dynamic vibration absorber on the power flow transfer properties is analyzed in Section 2.3. Section 3 systematically analyzes the effect of parameters such as particle diameter, material, and fill ratio on the overall dissipation of energy. The amplitude squeezing rate of the roller by different particle parameters is experimentally investigated in Section 4. Conclusions are drawn in Section 5.

2 Analysis of Power Flow Characteristics

Power flow is usually used to study vibration systems. The effective tool uses the statistical analysis of energy transfer instead of the complicated equation solution to reveal the energy transfer and transmission of the system by considering the relationship between the forces and velocities in the system comprehensively [28]. DVA is a subassembly attached to a vibrating body. It achieves vibration control by absorbing and dissipating the energy of the vibrating body. Therefore, for an absorption vibration system, the greater the power flow from the source to the absorber under the same working conditions, the better the vibration absorption effect of the absorber. In this section, we will analyze the effect of the basic parameters of the DVA on the power flow transmission properties in the vibrating system.

2.1 Dynamical Equation of the System

Since particle damping has numerous influencing parameters, it is very difficult to model multiple particles accurately, and in this paper, multiple particles are equivalent to a single particle based on the principle of equivalent simplification [29] for the study. As shown in Figure 2, a three-degree-of-freedom model of the DVA-roller system is established.

Figure 2

Three-degree-of-freedom model of vibrating system

The dynamic equation of the system can be written as Eq. (1).

$$\left\{ \begin{gathered} m_{1} \ddot{x}_{1} + \left( {c_{0} + c_{1} } \right)\dot{x}_{1} + c_{2} \left( {\dot{x}_{1} - \dot{x}_{2} } \right) + c^{\prime}_{0} \dot{x}_{1}^{3} + \left( {k_{0} + k_{1} } \right)x_{1} + k_{2} \left( {x_{1} - x_{2} } \right) \hfill \\ { + }k^{\prime}_{0} x_{1}^{3} { = }F_{0} {\text{cos}}\omega t, \hfill \\ m_{2} \ddot{x}_{2} - c_{2} \left( {\dot{x}_{1} - \dot{x}_{2} } \right) + c\left( {\dot{x}_{2} - \dot{x}_{3} } \right) - k_{2} \left( {x_{1} - x_{2} } \right){ + }k\left( {x_{2} - x_{3} } \right){ = }0, \hfill \\ m_{b} \ddot{x}_{3} - c\left( {\dot{x}_{2} - \dot{x}_{3} } \right) - k\left( {x_{2} - x_{3} } \right){ = }0. \hfill \\ \end{gathered} \right.$$

(1)

Let $z_{1} = x_{1}$ , $z_{2} = x_{1} - x_{2}$ , $z_{3} = x_{2} - x_{3}$ , Eq. (1) can be written as

$$\left\{ \begin{gathered} m_{1} \ddot{z}_{1} + \left( {c_{0} + c_{1} } \right)\dot{z}_{1} + c_{2} \dot{z}_{2} + c^{\prime}_{0} \dot{z}_{1}^{3} + \left( {k_{0} + k_{1} } \right)z_{1} + k_{2} z_{2} \hfill \\ { + }k^{\prime}_{0} z_{1}^{3} { = }F_{0} {\text{cos}}\omega t, \hfill \\ m_{2} \ddot{x}_{2} - c_{2} \dot{z}_{2} + c\dot{z}_{3} - k_{2} z_{2} { + }kz_{3} { = }0, \hfill \\ m_{b} \ddot{x}_{3} - c\dot{z}_{3} - kz_{3} { = }0. \hfill \\ \end{gathered} \right.$$

(2)

By adding $- m_{2} \ddot{z}_{1}$ to both sides of the second row, $m_{b} \ddot{z}_{2}$ to both sides of the third row, we can simplify Eq. (2) to

$$\left\{ \begin{gathered} m_{1} \ddot{z}_{1} + \left( {c_{0} + c_{1} } \right)\dot{z}_{1} + c_{2} \dot{z}_{2} + c^{\prime}_{0} \dot{z}_{1}^{3} + \left( {k_{0} + k_{1} } \right)z_{1} + k_{2} z_{2} \hfill \\ { + }k^{\prime}_{0} z_{1}^{3} { = }F_{0} {\text{cos}}\omega t, \hfill \\ m_{2} \ddot{z}_{2} + c_{2} \dot{z}_{2} - c\dot{z}_{3} + k_{2} z_{2} - kz_{3} { = }m_{2} \ddot{z}_{1} , \hfill \\ m_{b} \ddot{z}_{3} + c\dot{z}_{3} + kz_{3} { = }m_{b} \left( {\ddot{z}_{1} - \ddot{z}_{2} } \right). \hfill \\ \end{gathered} \right.$$

(3)

2.2 The Solution of the System Solved by IHBM

IHBM proposed by Lau and Cheung [30] is an efficient method for the analysis of high-dimensional strongly nonlinear vibrating systems. It has been successfully used in various nonlinear systems.

$${\varvec{\overline{M}\ddot{Z}}} + \left( {{\overline{\varvec{C}}}^{\left( 3 \right)} + {\overline{\varvec{C}}}} \right){\dot{\varvec{Z}}} + \left( {{\overline{\varvec{K}}}^{{\left( {3} \right)}} + {\overline{\varvec{K}}}} \right){\varvec{Z}} = {\overline{\varvec{F}}},$$

(4)

where,

$${\varvec{Z}} = \left[ {\begin{array}{*{20}c} {z_{1} } \\ {z_{2} } \\ {z_{3} } \\ \end{array} } \right],\,{\overline{\varvec{F}}} = \left[ {\begin{array}{*{20}c} {F_{0} \cos \left( {\omega t} \right)} \\ 0 \\ 0 \\ \end{array} } \right],\,{\overline{\varvec{M}}} = \left[ {\begin{array}{*{20}c} {m_{1} } & 0 & 0 \\ { - m_{2} } & {m_{2} } & 0 \\ { - m_{b} } & {m_{b} } & {m_{b} } \\ \end{array} } \right],\,{\overline{\varvec{C}}} = \left[ {\begin{array}{*{20}c} {c_{0} + c_{1} } & {c_{2} } & 0 \\ 0 & {c_{2} } & { - c} \\ 0 & 0 & c \\ \end{array} } \right],\,{\overline{\varvec{C}}}^{{\left( {3} \right)}} = \left[ {\begin{array}{*{20}c} {c^{\prime}_{0} \dot{z}_{1}^{2} } & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right],\,{\overline{\varvec{K}}} = \left[ {\begin{array}{*{20}c} {k_{0} + k_{1} } & {k_{2} } & 0 \\ 0 & {k_{2} } & { - k} \\ 0 & 0 & k \\ \end{array} } \right],\,{\overline{\varvec{K}}}^{{\left( {3} \right)}} = \left[ {\begin{array}{*{20}c} {k^{\prime}_{0} z_{1}^{2} } & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right].$$

Let $\tau = \omega t$, frequency $\omega$ and the solution of Eq. (4) is expressed in incremental form as

$$\left\{ \begin{gathered} \omega = \omega_{0} + \Delta \omega , \hfill \\ {\varvec{Z}} = {\varvec{Z}}_{{0}} + \Delta {\varvec{Z}}. \hfill \\ \end{gathered} \right.$$

(5)

Substituting Eq. (5) into Eq. (4) and omitting the higher order terms, we obtain the incremental equations about $\Delta \omega$ and $\Delta {\varvec{Z}}$.

$$\begin{gathered} \omega_{0}^{2} {\overline{\varvec{M}}}{\Delta }{\varvec{Z^{\prime\prime}}} + \omega_{0} \left( {{\overline{\varvec{C}}}{ + 3}{\overline{\varvec{C}}}^{\left( 3 \right)} } \right){\Delta }{\varvec{Z^{\prime}}} + \left( {{\overline{\varvec{K}}} + {3}{\overline{\varvec{K}}}^{{\left( {3} \right)}} } \right){\Delta }{\varvec{Z}} \hfill \\ = {\overline{\varvec{R}}} - \left( {2\omega_{0} {\varvec{\overline{M}Z^{\prime\prime}}}_{0} + {\varvec{\overline{C}Z^{\prime}}}_{{0}} + 3{\overline{\varvec{C}}}^{\left( 3 \right)} {\varvec{Z^{\prime}}}_{{0}} } \right)\Delta \omega , \hfill \\ \end{gathered}$$

(6)

where, ${\varvec{\overline{R} = \overline{F}}} - \omega_{0}^{2} {\varvec{\overline{M}Z^{\prime\prime}}}_{{0}} - \omega_{0} {\varvec{\overline{C}Z^{\prime}}}_{{0}} - \omega_{0} {\overline{\varvec{C}}}^{\left( 3 \right)} {\varvec{Z^{\prime}}}_{0} - {\overline{\varvec{K}}\varvec{Z}}_{{0}} - {\overline{\varvec{K}}}^{{\left( {3} \right)}} {\varvec{Z}}_{{0}} .$

At this point, the first step of the IHBM is complete, and the second step is the harmonic balance process. The excitation, response and their increments are written as Fourier series.

$$z_{j0} \left( \tau \right) = \frac{{m_{j0} }}{2} + \sum\limits_{k = 1}^{n} {\left[ {m_{jk} \cos \left( {k\tau } \right) + n_{jk} \sin \left( {k\tau } \right)} \right]} = {\varvec{C}}{\text{s}}{\varvec{A}}_{j0} \, \left( {j = 1 \cdots 3} \right),$$

$$\Delta z_{j} \left( \tau \right) = \frac{{\Delta m_{j0} }}{2} + \sum\limits_{k = 1}^{n} {\left[ {\Delta m_{jk} \cos \left( {k\tau } \right) + \Delta n_{jk} \sin \left( {k\tau } \right)} \right]} = {\varvec{Cs}}\Delta {\varvec{A}}_{j} \, \left( {j = 1 \cdots 3} \right),$$

where,

$${\varvec{C}}{\text{s}} = \left( {\begin{array}{*{20}c} {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} & {{\text{cos}}\tau } & \cdots & {{\text{cos}}\left( {l\tau } \right)} & {{\text{sin}}\tau } & \cdots & {{\text{sin}}\left( {l\tau } \right)} \\ \end{array} } \right)$$

$$\left\{ \begin{gathered} {\varvec{A}}_{j0} = \left( {\begin{array}{*{20}c} {m_{j0} } & {m_{j1} } & \cdots & {m_{jn} } & {n_{j1} } & \cdots & {n_{jn} } \\ \end{array} } \right)^{{\text{T}}} \hfill \\ \Delta {\varvec{A}}_{j} = \left( {\begin{array}{*{20}c} {\Delta m_{j0} } & {\Delta m_{j1} } & \cdots & {\Delta m_{jn} } & {\Delta n_{j1} } & \cdots & {\Delta n_{jn} } \\ \end{array} } \right)^{{\text{T}}} \hfill \\ \end{gathered} \right. \, \left( {j = 1 \cdots 3} \right).$$

Write as matrix form

$${\varvec{Z}_{0}}{ = }{\varvec{S}}{\varvec{A}}_{0}, {\mkern 1mu}\Delta {\varvec{Z}} = {\varvec{S}}\Delta{\varvec{A}},$$

(7)

where,

$${\varvec{S}} = \left[ {\begin{array}{*{20}c} {{\varvec{C}}{\text{s}}} & {} & {} \\ {} & {{\varvec{C}}{\text{s}}} & {} \\ {} & {} & {{\varvec{C}}{\text{s}}} \\ \end{array} } \right],\,{\varvec{A}}_{{0}} = \left[ {\begin{array}{*{20}c} {{\varvec{A}}_{{{10}}} } \\ {{\varvec{A}}_{{{20}}} } \\ {{\varvec{A}}_{{{30}}} } \\ \end{array} } \right],\,{\Delta }{\varvec{A}} = \left[ {\begin{array}{*{20}c} {{\Delta }{\varvec{A}}_{{1}} } \\ {{\Delta }{\varvec{A}}_{2} } \\ {{\Delta }{\varvec{A}}_{3} } \\ \end{array} } \right].$$

Substituting Eq. (7) into Eq. (6) and executing the Galerkin process, we have

$$\begin{gathered} \int_{0}^{2\uppi } {\delta \left( {\Delta {\varvec{Z}}} \right)^{{\text{T}}} \left[ {\omega_{0}^{2} {\overline{\varvec{M}}}\Delta {\varvec{Z^{\prime\prime}}} + \omega_{0} \left( {{\overline{\varvec{C}}}{ + 3}{\overline{\varvec{C}}}^{\left( 3 \right)} } \right)\Delta {\varvec{Z^{\prime}}} + \left( {{\overline{\varvec{K}}} + {3}{\overline{\varvec{K}}}^{{\left( {3} \right)}} } \right)\Delta {\varvec{Z}}} \right]} {\text{d}}\tau \hfill \\ = \int_{0}^{2\uppi } {\delta \left( {\Delta {\mathbf{Z}}} \right)^{{\text{T}}} \left[ {{\overline{\varvec{R}}} - \left( {2\omega_{0} {\varvec{\overline{M}Z^{\prime\prime}}}_{0} + {\varvec{\overline{C}Z^{\prime}}}_{{0}} + 3{\overline{\varvec{C}}}^{\left( 3 \right)} {\varvec{Z^{\prime}}}_{0} } \right)\Delta \omega } \right]} {\text{d}}\tau . \hfill \\ \end{gathered}$$

(8)

Integrating Eq. (8), then we organize it into a linear equation on $\Delta \omega$ and $\Delta {\varvec{Z}}$ as

$${\varvec{K}}_{T} {\Delta }{\varvec{A}} = {\varvec{R}} - {\varvec{R}}_{m} \Delta \omega ,$$

(9)

where, ${\mathbf{R}}$ is the unbalanced force matrix,

$${\varvec{K}}_{T} = \omega_{0}^{2} {\varvec{M}} + \omega_{0} {\varvec{C}}{ + 3}\omega_{0} {\varvec{C}}^{\left( 3 \right)} + {\varvec{K}} + 3{\varvec{K}}^{\left( 3 \right)} ,$$

$${\varvec{R}} = {\varvec{F}} - \left( {\omega_{0}^{2} {\varvec{M}} + \omega_{0} {\varvec{C}} + \omega_{0} {\varvec{C}}^{\left( 3 \right)} + {\varvec{K}} + {\varvec{K}}^{\left( 3 \right)} } \right){\varvec{A}}_{0} ,$$

$${\varvec{R}}_{m} = \left( {2\omega_{0} {\varvec{M}} + {\varvec{C}} + {3}{\varvec{C}}^{\left( 3 \right)} } \right){\varvec{A}}_{0} ,$$

$${\varvec{M}} = \int_{0}^{2\uppi } {{\varvec{S}}^{{\text{T}}} {\overline{\varvec{M}}\varvec{S}}^{\prime \prime } } {\text{d}}\tau ,\,{\mathbf{C}} = \int_{0}^{2\uppi } {{\varvec{S}}^{{\text{T}}} {\overline{\varvec{C}}\varvec{S}}^{\prime } } {\text{d}}\tau ,$$

$${\varvec{C}}^{\left( 3 \right)} = \int_{0}^{2\uppi } {{\varvec{S}}^{{\text{T}}} {\overline{\varvec{C}}}^{\left( 3 \right)} {\varvec{S}}^{\prime } } {\text{d}}\tau ,\,{\varvec{K}} = \int_{0}^{2\uppi } {{\varvec{S}}^{{\text{T}}} {\overline{\varvec{K}}\varvec{S}}} {\text{d}}\tau ,$$

$${\varvec{K}}^{\left( 3 \right)} = \int_{0}^{2\uppi } {{\varvec{S}}^{{\text{T}}} {\overline{\varvec{K}}}^{\left( 3 \right)} {\varvec{S}}} {\text{d}}\tau ,\,{\varvec{F}} = \int_{0}^{2\uppi } {{\varvec{S}}^{{\text{T}}} {\overline{\varvec{F}}}} {\text{d}}\tau ,$$

One unknown needs to be chosen as the control parameter and solved iteratively by the Newton-Raphson method because there are $6n + 3$ equations and $6n + 4$ unknowns in Eq. (9).

The relevant parameters are chosen according to the two-roll mill shown in Figure 1b, and the basic parameters of the absorber in Figure 1a are initially chosen, as shown in Table 1. In order to validate the accuracy of IHBM, the time domain curves of the rolls with the amplitude of external excitation of 150 kN and vibrational frequency of 190 Hz are plotted. They are compared with the curves obtained by fourth-order Runge-Kutta methods as shown in Figure 3.

Table 1 Vibration system parameters

Figure 3

Displacement and velocity of each part of the vibration system

2.3 Vibration Power Flow

Suppose that

$$\omega_{n1} = \sqrt {\frac{{k_{0} + k_{1} }}{{m_{1} }}} ,\,\omega_{n2} = \sqrt {\frac{{k_{2} }}{{m_{2} }}} ,\,\omega_{n3} = \sqrt {\frac{k}{{m_{b} }}} ,\,\omega_{1} = \frac{\omega }{{\omega_{n1} }},\,\omega_{2} = \frac{\omega }{{\omega_{n2} }},\,\omega_{3} = \frac{\omega }{{\omega_{n3} }},\,\zeta_{2} = \frac{{c_{2} }}{{2m_{2} \omega_{n2} }},\,\zeta_{3} = \frac{c}{{2m_{b} \omega_{n3} }},\,\zeta_{1} = \frac{{c_{0} + c_{1} }}{{2m_{1} \omega_{n1} }},\,\delta_{1} = \frac{{k_{2} }}{{k_{0} + k_{1} }},\,\delta_{2} = \frac{k}{{k_{2} }},\,\gamma_{1} = \frac{{c_{2} }}{{m_{1} \omega_{n1} }},\,\gamma_{2} = \frac{c}{{m_{2} \omega_{n2} }},\,k^{\prime\prime}_{0} = \frac{{k^{\prime}_{0} }}{{k_{0} + k_{1} }},\,c^{\prime\prime}_{0} = \frac{{\left( {k_{0} + k_{1} } \right)^{2} c^{\prime}_{0} }}{{m_{1}^{3} \omega_{n1}^{3} }},\,F = \frac{{F_{0} }}{{k_{0} + k_{1} }},\,\tau = \omega t.$$

Then, Eq. (3) can be written as

$$\left\{ \begin{gathered} \omega_{1}^{2} \ddot{z}_{1} + 2\zeta_{1} \omega_{1} \dot{z}_{1} + \gamma_{1} \omega_{1} \dot{z}_{2} + c^{\prime\prime}_{0} \omega_{1}^{3} \dot{z}_{1}^{3} + z_{1} + \delta_{1} z_{2} + k^{\prime\prime}_{0} z_{1}^{3} \hfill \\ = F\cos \left( {\omega t} \right), \hfill \\ \omega_{2}^{2} \ddot{z}_{2} + 2\zeta_{2} \omega_{2} \dot{z}_{2} - \gamma_{2} \omega_{2} \dot{z}_{3} + z_{2} - \delta_{2} z_{3} = \omega_{2}^{2} \ddot{z}_{1} , \hfill \\ \omega_{3}^{2} \ddot{z}_{3} + 2\zeta_{3} \omega_{3} \dot{z}_{3} + z_{3} = \omega_{3}^{2} \left( {\ddot{z}_{1} - \ddot{z}_{2} } \right). \hfill \\ \end{gathered} \right.$$

(10)

The input power flow can be obtained as

$$P_{in} = \frac{1}{\Delta t}\int_{t}^{t + \Delta t} {F\cos \left( {\omega t} \right)\dot{z}_{1} {\text{d}}t} .$$

The power flow from the upper work roll to the fixed end is

$$P_{1} = \frac{1}{\Delta t}\int_{t}^{t + \Delta t} {\left( {2\zeta_{1} \omega_{1} \dot{z}_{1} + c_{0}^{\prime \prime } \omega_{1}^{3} z_{1}^{3} + z_{1} + k_{0}^{\prime \prime } z_{1}^{3} } \right)\dot{z}_{1} } {\text{d}}t.$$

The power flow from the upper work roll to the particle container is

$$P_{2} = \frac{1}{\Delta t}\int_{t}^{t + \Delta t} {\left( {2\zeta_{2} \omega_{2} \dot{z}_{2} + z_{2} } \right)\dot{z}_{2} } {\text{d}}t.$$

The power flow from the upper work roll through the particle container to the particle group is

$$P_{3} = \frac{1}{\Delta t}\int_{t}^{t + \Delta t} {\left( {2\zeta_{3} \omega_{3} \dot{z}_{3} + z_{3} } \right)\dot{z}_{3} } {\text{d}}t.$$

Among them, the power flow $P_{1}$ reflects the transmission of vibration from the roll to the strip and frame, which is closely correlated with the quality of the rolled product and the stability of the mill. The larger the value of $P_{1}$ , it means that the strip and frame to withstand more vibration, which is very unfavorable for rolling production. Therefore, when choosing the parameters of DVA, the value of $P_{1}$ should be as small as possible.$P_{2}$ and $P_{3}$ reflect the transmission of vibration from the roll to the DVA. The larger the $P_{2}$ and $P_{3}$, the better the vibration absorption effect of the DVA. However, when the influence of a parameter on $P_{1}$, $P_{2}$ and $P_{3}$ is obvious, we decide how to choose this parameter mainly based on $P_{1}$, which is directly related to the quality of the strip.

In the rolling production process, the vibrational frequency will vary in a wide scope with the changes of roll speed, reduction, strip thickness, etc. Therefore, we made the frequency scoped in the scope of 20–400 Hz and investigated the influence law of the basic parameters of DVA on each power flow, as shown in Figures 4, 5 and 6. Some of these parameters have a very minor effect on some of the power flows and are therefore not shown in the figure. For conveniently describing effects of parameters on power flow, the scope of 20–172 Hz is specified as the low frequency band (LFB), the scope of 172–324 Hz is specified as the main resonance frequency band (MFB), and the scope of 324–400 Hz is specified as the high frequency band (HFB), respectively.

It follows from Figure 4 that $k_{2}$ only has a more pronounced effect. When $k_{2} = 49.27{\mkern 1mu}\space {\text{N/m }}$, $P_{2}$ has a large value in the full scope of frequency bands. As $k_{2}$ increases, $P_{2}$ decreases gradually over the full frequency scope. Since $P_{2}$ is the power flow transferred from the roll to the particle container, the larger its value, the better the vibration absorption effect. Therefore, it is more advantageous to select a smaller stiffness coefficient of $k_{2}$.

Figure 4

Effect of $k_{2}$ on power flow

As can be seen from Figure 5, $c_{2}$ has a more pronounced effect on all power flows. When $c_{2}$ increases, $P_{in}$ increases in LFB as well as in MFB and decreases in the HFB. $P_{1}$ increases in LFB and decreases in MFB as well as in HFB; $P_{2}$ and $P_{3}$ both increase in the whole frequency band scope. As the external excitation is determined, $P_{1}$ directly reflects the transmission of vibration to the frame and strip, the smaller the power flow of this part, the more beneficial the rolling production process. Therefore, as the frequency of external excitation $\omega$ is in LFB, a smaller $c_{2}$ should be selected; while when $\omega$ is in MFB as well as HFB, selecting a larger $c_{2}$ not only enables $P_{1}$ to reach a lower level, but also increases the values of $P_{2}$ and $P_{3}$ .

Figure 5

Effect of damping coefficient $c_{2}$ on power flows

As shown in Figure 6, mass $m_{2}$ only has a more pronounced effect on power flow $P_{3}$. As $m_{2}$ increases, $P_{3}$ decreases over the full frequency scope. $P_{3}$ reflects the vibration transmission to the particle group, and since the particle group is the main energy dissipator of the DVA, it is advantageous to select a smaller $m_{2}$.

Figure 6

Effect of mass $m_{2}$ on power flow

In summary, under different rolling conditions, different basic parameters of the absorber are chosen to keep a high standard of power flow from the roll to the DVA.

3 Energy Dissipation of Particle Group

The particle group is the major energy dissipator in the absorber, and its energy consumption effect is affected by several factors, such as diameter, fill ratio and particle material. The mechanism of the influence of the variation of these parameters on the energy consumption of the particle group is not clear, so it is of great practical importance to study it systematically. In this paper, we will explore the mechanism of energy dissipation of particle damping by the software PFC3D.

PFC3D is a simulation software for analyzing the dynamic behaviors of an aggregate of spherical particles of any size. The FISH language embedded in this software allows the visualization of containers, particles and motion processes [31]. PFC3D can track the volume work, kinetic energy, bond energy, boundary work, damping work, friction work, and strain energy during particle collisions. However, the energy dissipation pathways are mainly damping work, friction work and strain energy at all collision contacts. Therefore, we focus on these three energies during the simulation.

The model domain is defined in PFC3D and a wall is generated based on the shape of the inner walls of the particle container. As shown in Figure 7, the wall is composed of several triangular walls and a specified number and size of particles are produced within the wall. The material of the particles is achieved by defining different material parameters.

Figure 7

a The inner wall of the particle container, b Wall generated in PFC3D, c Generation of particles of specified size and number (The figure shows 3mm particles with 50% fill rate)

Particle damping has numerous influencing factors, and in this section we analyze the effect of material, fill rate, particle diameter and mixed particles on the overall energy consumption. Furthermore, an energy wastage factor is utilized to assist in assessing the effect of energy consumption. The energy wastage factor is calculated as follows.

$$\eta = \frac{{H_{l} }}{{H_{s} }},$$

(11)

where $H_{l}$ is the wastage of total energy in one period of vibration; $H_{s}$ is the total energy in one period of vibration.

When the fill ratio is $\lambda$, assuming that the corresponding number of particles is $n$, the total volume is

$$V_{p} = \frac{{n\uppi d^{3} }}{6},$$

(12)

where, $d$ is the diameter of the particle.

According to the pore ratio [32], we can get

$$1.91V_{p} = \lambda V,$$

(13)

where, $V$ is the volume of the particle container and is set to 23859 mm³.

Then we can obtain that when the fill ratio is $\lambda$, the corresponding number of particles is

$$n = \frac{3.14 \cdot \lambda \cdot V}{{\uppi d^{3} }}.$$

(14)

3.1 Effect of Different Particle Diameter

The material of the particles is designated as steel in PFC3D and the fill ratio is set to 50%. The overall energy consumption is analyzed for particle diameters of 2 mm, 3 mm, 4 mm and 5 mm, respectively. The filling numbers for various particle diameters can be evaluated from Eq. (14), as shown in Table 2. To illustrate the accumulative processes of energy wastage of particle group, the energy consumption tracking curve is shown in Figure 8 for the vibration of 3 mm steel particles under the excitation of 20 and 300 Hz as an example.

Table 2 The number of particles for different diameters

Figure 8

a Energy consumption accumulation at 20 Hz, b Energy consumption accumulation at 300 Hz

From Figure 8, it can be seen that the strain energy stored by the particles and dissipated through conversion to mechanical energy has the largest share in the energy wastage of the steel, followed by the damping work and finally the friction work. In addition, the energy consumption increases in an irregular stairway and is more obvious at low frequencies. This is because the collision between the rising wall of the container and the falling particles leads to a sudden increase in the wastage of energy after the end of the collision, and the particles move upward movement while the container downward movement. This stage of energy consumption shows a slow growth plateau period, and so on, thus forming a stepped curve. When the vibrational frequency is 300 Hz, most of the particles are in the state of suspension due to the high-frequency vibration, and the contact between the particles is intensive and rapid, which results in the steps of the energy consumption curve exhibiting small and dense characteristics.

The frequency is made to vary in the scope of 20 to 300 Hz, and the effect of diameter variation on the energy consumption and energy wastage factor is analyzed, as shown in Figure 9.

Figure 9

a Energy consumption of different diameters, b Energy wastage factor of different diameters

It is seen from Figure 9a that the particles with a diameter of 2 mm consume the most energy in the scope of 20 Hz to 300 Hz, while the difference in energy consumption for particles with diameters of 3 mm, 4 mm and 5 mm is not significant. It is seen from Figure 9b that the energy wastage factor of the particle group of the same diameter is higher in the high-frequency vibration, which is due to the fact that when high-frequency vibration occurs, the vast majority of particles are in suspension and the accumulated kinetic energy of the particle group becomes less, leading to the increase of the wastage factor. Furthermore, at the same vibrational frequency, the energy wastage factor is higher for particles with larger diameters. This is due to the fact that under the same vibration conditions, the kinetic energy of particles with larger diameter is also smaller, causing an increase in the energy wastage factor. Since the smaller diameter particles consume more energy when the working conditions are the same, the smaller diameter particles should be selected in practical engineering applications.

3.2 Effect of Different Particle Material

The diameter of the particles is defined as 3 mm and the fill ratio is set to 50%. The overall energy consumption is analyzed when the particle materials are steel, glass, and POM plastic, respectively. The number of fillings corresponding to different particle materials is shown in Table 3.

Table 3 The number of particles for different materials

From Figure 10a, it can be seen that the steel particles consume the most energy in the scope of 20 to 300 Hz, followed by the glass particles and finally the POM plastic particles. It is seen from Figure 10b that the energy wastage factors of the particles of the three materials are very close as the low vibrational frequency occurs. With the increase of frequency, the loss factor of steel particles increases rapidly, while the energy wastage factor of glass particles and POM plastic particles grows slowly and has a large difference with that of steel particles. Thinking about the energy consumption and energy wastage factor, the choice of steel particles is more conducive to the control of vibration.

Figure 10

a Energy consumption of different materials, b Energy wastage factor of different materials

3.3 Effect of Different Fill Ratio

The particle diameter is set to 3 mm and the particle material is defined as steel. The overall energy consumption is analyzed for fill ratios of 10%, 30%, 50%, 70%, and 90%, respectively. The number of particles for different fill ratios is shown in Table 4.

Table 4 The number of particles for different fill ratios

From Figure 11a, it can be seen that the higher the fill ratio is in the scope of 20 to 300 Hz, the more energy is consumed by the particle group. It is seen from Figure 11b that the larger the fill ratio, the higher the energy wastage factor at the same vibrational frequency. When the fill ratio is the same, the higher the excitation frequency, the higher the energy wastage factor. Hence, the particle group in the absorber should be chosen to have as large a fill rate as possible.

Figure 11

a Energy consumption of different fill ratios, b Energy wastage factor of different fill ratios

3.4 Effect of Particle Mixing

In this section, we study the energy consumption of particle groups in two mixing methods, which are mixing of particles with different diameters and materials. The fill ratio is set to 50% in both mixing methods, and the mixing ratio is 1:1.

For different diameters of particle mixing methods, the material is designated as steel, the mixing mode is shown in Table 5.

Table 5 The number of particles for different diameter mixes

As is shown in Figure 12a, it is seen that when particles of different diameters are mixed, the difference in energy loss between the mixing mode is small and the difference in energy consumption between mixed filling and single particle size filling is also small. From Figure 12b, it can be found that all the mixing schemes have higher energy loss factors at high excitation frequencies, and when the vibration frequencies are the same, the difference in energy loss factors between the modes is smaller.

Figure 12

a Energy consumption for mixing of different particle size, b Energy wastage factor for mixing of different particle size

For the particle mixing method of different materials, the diameter is set to 3 mm and the mixing mode is shown in Table 6.

Table 6 The number of particles for different material mixes

As is shown in Figure 13a, the energy consumption when mixing steel particles with glass particles is the highest when the vibrational frequency is low and decreases rapidly with the increase of frequency. When the frequency reaches about 90 Hz, its energy consumption starts to lag behind that of pure steel particles, when the frequency reaches about 104 Hz, its energy consumption starts to lag behind that of steel particles mixed with POM plastic particles. When the frequency reaches 132 Hz, its energy consumption decreases at a slower rate and is on par with that of the mixed scheme of glass particles and POM plastic particles at 300 Hz. It is seen from Figure 13b that the energy wastage factor of the pure steel particles is always greater than the remaining three mixing schemes in the scope of 20 to 300 Hz. Therefore, taking into account the energy consumption and energy wastage factor of the particle mixing mode, it is more advantageous to fill the container with pure steel particles only.

Figure 13

a Energy consumption for mixing of different material, b Energy wastage factor for mixing of different material

In summary, the larger the fill ratio and the smaller the particle diameter, the better the energy consumption effect of the particle group. Among the three materials of steel, glass and POM plastic pellets, steel has the best energy dissipation effect. In addition, the difference in energy consumption between filling with a mixture of particles of different diameters and filling with particles of a single diameter is small. When mixed and filled with particles of different materials, the energy consumption effect is not as good as that of pure steel particles.

4 Experimental Research

4.1 Experimental Platform

Based on the analysis of the power flow, the parameters of the DVA prototype are determined, as shown in Table 7. The prototype is manufactured according to the parameters in Table 7, as shown in Figure 14a.

Table 7 Parameters of the DVA prototype

Figure 14

a DVA prototype, b Vibration test platform. 1-Three-way acceleration sensor. 2-Upper work roll. 3- DVA prototype. 4-Donghua DH5922D data acquisition instrument. 5-Donghua DHDAS Software Platform

Since the energy loss data during vibration could not be captured, the accuracy of the simulation is validated by qualitatively comparing the squeezing rate of the DVA on the acceleration amplitude of the roll with the energy consumption. The vibration experiment platform is composed of a three-way acceleration sensor, DH5922D data acquisition instrument, DHDAS software platform and two-roll mill, as shown in Figure 14b. Since vibration is transmitted from the roll to the bearing housing, the sensor is attached to the bearing housing to prevent the roll rotation from affecting the data sampling. The triaxial acceleration sensor is capable of collecting vibration data in the vertical, horizontal and axial directions simultaneously. Only vertical vibration data are used in this paper.

To ensure the reliability of the data, the reduction, the material and thickness of the composite sheet, roll speed and other conditions are always consistent. The operating conditions of the mill are shown in Table 8. In all experiments, the vibration signal of the roll without DVA is the same signal. In order to compare the experimental results with the particle energy simulation results, the experimental results are quantified as follows: the upper peak of the roll vibration signal before and after the installation of DVA is enveloped, and the average value of the envelope is obtained. The effect of different particle parameters on energy consumption is analyzed by comparing the squeezing rate of DVA with the amplitude of the roll.

Table 8 Rolling mill operating conditions

As an example, the acceleration simulation results of the rolls and the experimental results are compared and analyzed by filling the container with steel balls of 2 mm diameter and the fill ratio is 50%, as shown in Figure 15.

Figure 15

Comparison of simulation results and experimental results

Figure 15a shows the simulation and experimental comparison results. From Figure 15a, it is seen that the steady-state acceleration amplitude of the rolls calculated by IHBM is 0.15 m/s² after installing the absorber, and the maximum acceleration amplitude of the rolls collected experimentally is about 0.12 m/s². The reason for the error is that the modeling method of equating multiple particles to a single particle does not fully reflect the mechanism of particle damping, thus causing the simulation results to be slightly worse than the experimental results. Establishing a more accurate model of particle damping is also a difficulty as well as a direction to work on at present Figure 15b shows the experimental results before and after the filling of particles in the absorber. It is seen from Figure 15b that as the DVA is not filled with particles, its control effect on the roll vibration is worse than that of the DVA that has been filled with particles. The peak envelope of the three curves is obtained, as shown in Table 9. From Table 9, it can be obtained that the squeezing rate of DVA without particles on roll amplitude is 18.64%, and the squeezing rate of DVA with particles on roll amplitude is 34.77%, which indicates that the absorber in this paper has a good vibration control function.

Table 9 Average peak values of the envelope of the three curves in Figure 15b

4.2 Particle Size Experiment

The particles are filled and experiments are carried out according to the program in Section 3.1.

From Figure 16 and Table 10, it is seen that as filled with 2 mm particles, the amplitude squeezing of DVA on roll is 34.77%; as filled with 3 mm particles, the squeezing is 31.84%; as filled with 4 mm particles, the squeezing is 30.89%; and as filled with 5 mm particles, the squeezing is 31.32%. It can be found that particles with a diameter of 2 mm have the best effect on vibration control. Through the spectrum curve shown in Figure 17, we can see that under the current working condition, the main vibrational frequency is concentrated between 273.438 and 286.133 Hz, so the experimental results of particle size are basically consistent with the analysis of energy consumption.

Figure 16

Particle size experimental results

Table 10 Average peak values of particle size experimental signals

Figure 17

Frequency spectrum of vibration signal

4.3 Particle Material Experiment

Experiments are carried out according to the program in Section 3.2.

As can be seen from Figure 18 and Table 11, the amplitude squeezing of DVA on the roll is 31.84% when filled with steel particles; 28.56% when filled with glass particles; and 27.96% when filled with POM plastic particles. The experimental results are basically consistent with the energy dissipation analysis of the particle material.

Figure 18

Particle material experimental results

Table 11 Average peak values of particle material experimental signals

4.4 Particle Fill Ratio Experiment

The experiments are carried out according to the program in Section 3.3.

From Figure 19 and Table 12, it can be seen that the amplitude squeezing of DVA on the rolls is 22.26% when the fill ratio is 10%; 24.42% when the fill ratio is 30%; 31.84% when the fill ratio is 50%, 37.62% when the fill ratio is 70%, and 40.47% when the fill ratio is 90%. The experimental results are basically coincide with the energy consumption analysis of particle fill ratio.

Figure 19

Fill ratio experimental results

Table 12 Average peak values of fill ratio experimental signals

4.5 Particle Mixing Experiment

The experiments are carried out according to the program in Section 3.4.

From Figure 20 and Table 13, it is seen that the amplitude squeezing of DVA on roll was 34.77% when filling only 2 mm particles; 35.89% when mixing 2 mm and 3 mm particles, 34.94% when mixing 2 mm and 4 mm particles, and 35.55% when mixing 2 mm and 5 mm particles. It is found that there is very little difference between the schemes with different diameter mixes.

Figure 20

Mixing experiment results with different particle diameters

Table 13 Average peak values of experimental signals for particle diameter mixing

From Figure 21 and Table 14, it is seen that the amplitude squeezing of DVA on roll is 31.84% when filled with steel particles only; 29.77% when mixing steel particles with glass particles, 30.89% when mixing steel particles with POM plastic particles, and 29.68% when mixing glass particles with POM plastic particles. It is seen that the control effect of pure steel particles is better than the effect of mixing particles with the different materials.

Figure 21

Mixing experiment results with different particle materials

Table 14 Average peak values of experimental signals for particle material mixing

In summary, when the material is the same, the higher the fill ratio and the smaller the diameter, the better the control effect of DVA on the roll. For the various materials mentioned in the paper, the vibration control effect of steel particles is the best when the diameter of the particles and the fill ratio are the same. Furthermore, when the particles with different diameters are mixed, the difference between the schemes is small. When particles of different materials are mixed, the vibration control effect is not as good as that of pure steel particles.

5 Conclusions

In this paper, the transmission and dissipation properties of vibration energy in a system consisting of roll and absorber have been investigated theoretically and experimentally for a proposed dynamic vibration absorber (DVA) based on particle damping technology.

(1) The influence of the basic parameters of the DVA on the power flow transmission properties of the vibrating system has been investigated. The results show that a smaller stiffness factor, a larger damping factor and a smaller mass can keep the vibration energy transmitted from the roll to the DVA at a higher level, while a larger damping factor also contributes to the stability of the rolling process.

(2) The influence law of numerous parameters of the particles on the overall energy consumption has been investigated by the software of PFC3D. The results show that smaller particle diameters and larger fill ratios can increase energy consumption when the particle materials are the same. When the fill ratio and the particle diameter are the same, the steel particles have the most energy consumption compared to the glass particles and POM plastic particles. In addition, when 2 mm steel particles are mixed with steel particles of other diameters, the energy consumption differs less from that of filled with only 2 mm steel particles. When steel particles are mixed with particles of other materials, the energy consumption is not as effective as that of pure steel particles in the scope of 90 Hz to 300 Hz. In engineering practice, the vertical vibrational frequency of the mill is generally higher than 100 Hz. Therefore, in practical applications, small diameter steel particles and a larger fill ratio should be selected to achieve better vibration control with DVA.

(3) Particle experiments are conducted by using copper-aluminum composite plates at a spindle speed of 10 Hz. By comparing the amplitude squeezing rate of DVA on rolls caused by different particle parameters with the analysis of energy consumption. The experimental results are basically consistent with the theoretical analysis. The correctness of the theoretical analysis and the feasibility of the particle damping absorber are proved.