Keywords

1 Introduction

A granular material damper is a damping element that consists of a container filled with a granular material placed on the mass part of the primary vibration system and which uses the motion of the granular material to produce a damping effect. The damping characteristics of a granular material damper are thus governed by the motion of the granular material, so we can understand those characteristics by analyzing the motion of the granular material.

Araki and Yokomichi et alia have applied a powder impact damper to vibration systems of a single degree of freedom and of multiple degrees of freedom and considered the damping effect due to the powder, which is understood as a single mass whose coefficient of restitution is zero [1]. Saeki et alia used the discrete element method to calculate the motion of individual granules to study the damping characteristics of granular materials [2]. Even though, in that research, the granular material is treated as a single mass the granules, in the other hand, are treated as individual masses and different analytical methods are applied, the movement of the granular material is large compared to the motion of the primary system. We can thus take the impact force of the granular material on the primary system as the basis of the damping mechanism. In addition to the work just cited, the effects of the size, quantity of the granules and the shape and number of the granular material containers on the damping characteristics have also been studied [3–5].

In general, previous researches have mostly assumed a large movement of the granular material. There is thus already considerable knowledge concerning the damping characteristics for the case in which granular material dampers are applied to structures that have large displacement amplitude.

For cases in which the displacement amplitude of the primary system is small as the movement of the granular material is, it may be difficult to attribute the basic damping effect due to the collision of the granular material on the primary system. That is to say, it is not sufficiently clear what factors we need to focus on to realize a high degree of damping from a granular material damper in a structure for which the natural frequency is relatively high and the displacement amplitude is small.

In the work reported here, we investigated the damping mechanism of a granular material damper for a structure that has a small vibration displacement.

2 Structure of a Granular Material Damper

In our granular material damper model (Fig. 1), the primary vibration system (primary system) has 1° of freedom and consists of a spring constant, a damping coefficient, and a mass (container). In this model, the primary system can move only in the x direction. Granular material is arranged in the container. Furthermore, to simplify the movement of the granular material, we take the granule shape to be cylindrical.

Fig. 1
figure 1

Model of damper with granular materials

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3 Numerical Simulation of a Granular Material Damper

3.1 Computational Model of the Primary System and the Granular Material

In the computation model of the granular material damper (Fig. 2), we denote the mass of the primary system as M, the spring constant of the spring that supports the primary mass as K, and the damping coefficient as C. Given a forced displacement of the base (x b ) as input to model an external vibration, we obtain the horizontal displacement of the container (x m ), the horizontal and vertical displacement of the granular material (\( z_{h} \left( {i,j} \right) \) and \( z_{v} \left( {i,j} \right) \)), and the angle of rotation, (\( z_{\theta } \left( {i,j} \right) \)). The subscripts i and j in the terms respectively indicate the row and column positions of an individual granule.

Fig. 2
figure 2

Computation model for a granular material damper

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The granules all have the same mass (m p ) and radius (r). The contact between two granules and between a granule and the inner wall of the container are modeled as springs and dampers as shown in Fig. 3.

Fig. 3
figure 3

Contact model for the granular material

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3.2 Equations of Motion for the Primary System and Granular Material

The model described in the previous section enables the equation of motion of the primary system to be represented as follows.

$$ \begin{aligned} M{\ddot{\it x}}_{m} & = - K\left( {x_{m} - x_{b} } \right) - C\left( {\dot{x}_{m} - \dot{x}_{b} } \right) \\ &\quad - \mathop \sum \limits_{i = 1}^{p} N_{{L\left( {i, 1} \right)}} \left[ {k_{a} \left\{ {r - \left( {z_{{h\left( {i, 1} \right)}} - x_{m} } \right)} \right\}^{\frac{3}{2}} + c_{a} \left( { - \dot{z}_{{h\left( {i, 1} \right)}} + \dot{x}_{m} } \right)} \right] \\ &\quad + \mathop \sum \limits_{i = 1}^{p} N_{{R\left( {i, q} \right)}} \left[ {k_{a} \left\{ {r - \left( {x_{m} + l - z_{{h\left( {i, q} \right)}} } \right)} \right\}^{\frac{3}{2}} + c_{a} \left( { - \dot{x}_{m} + \dot{z}_{{h\left( {i, q} \right)}} } \right)} \right] \\ &\quad + \mathop \sum \limits_{j = 1}^{q} N_{{B\left( {1, j} \right)}} sgn\left( {\dot{z}_{{h\left( {1, j} \right)}} - \dot{x}_{m} - r\dot{z}_{{\theta \left( {1, j} \right)}} } \right)\mu_{a} \left\{ {k_{a} \left( {r - z_{{v\left( {1, j} \right)}} } \right)^{\frac{3}{2}} + c_{a} \left( { - \dot{z}_{{v\left( {1, j} \right)}} } \right)} \right\} \\ \end{aligned} $$
(1)

In the equation above, \( N_{{L\left( {i, 1} \right)}} \) is the constant for determining if there is a contact between the left wall and the cylinder in the ith row from the bottom of the first column; it has the value 1 if there is contact and 0 otherwise. \( N_{{R\left( {i, q} \right)}} \) is the constant for determining if there is a contact between the right wall and the cylinder in the ith row from the bottom of column q. \( N_{{B\left( {1, j} \right)}} \) is a constant for determining if there is a contact between the bottom row of granules with the bottom surface of the container (the primary system). The function sgn in the equation determines the direction of the force of friction; it takes the value 1 when the value within the parentheses is positive, −1 when the value is negative, and 0 when the value is zero. The term μ a is the coefficient of friction between the granular material and the bottom surface of the primary system.

Because the contact of the granular material with the container walls or bottom is modeled as shown in Fig. 3, the spring constant k a , k b and the damping coefficient c a , c b are given by the equations according to the Hertz theory of contact.

The spring constant and damping coefficient are used to formulate equations of motion for multiple granules to obtain the horizontal, vertical and rotational displacements of individual granules. We used Intel Visual FORTRAN® to perform the computation on the basis of the equation of motion previously introduced. We used the Runge-Kutta method to solve the equation of motion.

4 Deriving the Equivalent Added Mass, Relative Motion Mass of the Granular Material and Damping Ratio

4.1 Equivalent Added Mass and Relative Motion Mass

The damping effect of the granular material damper is obtained by the movement of the granular material. Therefore, as a step towards a deeper understanding of the mechanism of damping by granular material, we consider the motion of the granular material as broadly either ‘moving’ or ‘not moving’ relative to mass M.

We modeled a single degree of freedom vibration system that involves granular material as shown in Fig. 4. Granular material of total mass m is placed in a single degree of freedom vibration system (the primary system) that is excited in forced vibration by foundation motion. One part of the granular material moves together with the container of mass M and the other part moves relative to the container. In Fig. 4, the mass that moves together with the container is represented as a rectangular block that is implicitly attached to mass M and we refer to that mass here as the equivalent added mass m eq . The remaining granular material mass (m − m eq ) is drawn as circles to represent individual granules that can move freely. We refer to that mass as the relative motion mass.

Fig. 4
figure 4

Vibration model

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In Fig. 4, the displacement of the primary system excited in forced vibration by foundation motion is denoted as x m , the displacement of the base is denoted as \( x_{b} \), and the relative displacement of the two is denoted as x. Given

$$ x = X{ \sin }(\omega t - \phi ) $$
(2)
$$ x = x_{m} - x_{b} $$
(3)

we obtain the internal force \( f_{i} \) from the relative displacement x as follows.

$$ f_{i} = c\dot{x} + kx $$
(4)

Because the equivalent added mass m eq is attached to the mass of the primary system M, the internal force \( f_{i} \) is then expressed by the following equation.

$$ f_{i} = - \left( {M + m_{eq} } \right){\ddot{\it x}}_{m} $$
(5)

Expressing the acceleration of the primary system \( {\ddot{\it x}}_{m} \) as

$$ {\ddot{\it x}}_{m} = A_{m} { \sin }(\omega t - \phi - \theta) $$
(6)
$$ \theta = { \cos }^{ - 1} \left( {\frac{kX}{{\left( {M + m_{eq} } \right)A_{m} }}} \right) $$
(7)

\( f_{i} \) is given by Eq. 8.

$$ f_{i} = - \left( {M + m_{eq} } \right)A_{m} { \sin }\left( {\omega t - \phi - { \cos }^{ - 1} \left( {\frac{kX}{{\left( {M + m_{eq} } \right)A_{m} }}} \right)} \right) $$
(8)

Expressing the relation of the internal force \( f_{i} \) and the relative displacement x as

$$ \frac{{\left( {f_{i} - kx} \right)^{2} }}{{\left( {c\omega X} \right)^{2} }} + \left( \frac{x}{X} \right)^{2} = 1 $$
(9)

the Lissajous pattern for \( \left( {f_{i} - kx} \right) \) and x is an ellipse that has no inclination. Using that property, we define the equivalent added mass m eq as the mass for which the inclination of the Lissajous pattern is zero.

4.2 Derivation of the Damping Ratio Using the Equivalent Added Mass

The work over one period of vibration (the energy loss within the system) is expressed as the area of the Lissajous pattern for the internal force and the displacement \( W_{f} \).

$$ W_{f} = \pi c\omega X^{2} $$
(10)

In practice, the shape of the Lissajous pattern is not necessarily elliptical, but by regarding the area of the pattern to be equal to, the apparent damping coefficient \( C_{eq} \) and \( \xi \) can be obtained with Eqs. 11 and 12.

$$ C_{eq} = \frac{{W_{f} }}{{\pi \omega X^{2} }} $$
(11)
$$ \xi = \frac{{C_{eq} }}{{2\sqrt {\left( {M + m_{eq} } \right)k} }} $$
(12)

We obtain the area of the Lissajous pattern W f from the computational results of Sect. 2, and obtain the damping ratio by the method described above. With those results, we discuss the forced vibration characteristics due to the granular material and the relation of the equivalent added mass to the relative motion mass.

5 Damping Characteristics of a Granular Material Damper for Various Total Masses of Material

We varied the total mass of the granular material in the container over the range from 1 to 7 kg and calculated the displacement and acceleration of the granular material and container for an acceleration of 1.0 m/s2 applied to the base. We used the calculated results and the method described in Sect. 3 to obtain the equivalent added mass, the relative motion mass, and the damping ratio. The damping ratio and relative motion mass for various total masses are presented in Fig. 5.

Fig. 5
figure 5

Damping ratios and relative motion masses for various total granular material masses

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In Fig. 5, up to a total mass of 5 kg, both the relative motion mass and the damping ratio increase. In that range, the relative motion mass is considered to increase the damping ratio by moving differently from the primary system. However, when the total mass of the granular material is 6 kg, the damping ratio increases further, even though the relative motion mass had been decreasing up to that point. Furthermore, when the total mass of the granular material is 7 kg, the relative motion mass again increases and the damping ratio decreases. We consider these results in light of the following additional inference.

The total mass of the granular material is plotted in relation to the relative motion mass and equivalent added mass in Fig. 6. The relative motion mass increase monotonically up to 5 kg, decrease somewhat in the range from 5 to 6 kg, and then increases again. The equivalent added mass, on the other hand, being the result of subtracting the relative motion mass from the total mass, is larger at the total masses of 6 and 7 kg total mass than at 5 kg.

Fig. 6
figure 6

Relative motion mass and equivalent added mass

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Comparing Figs. 6 and 5, we consider the relation of the relative motion mass and the equivalent added mass to the damping ratio. Because the relative motion mass increases up to the total mass of 5 kg, the damping characteristic is considered to improve. On the basis of that consideration, the behavior at the total mass of 6 kg cannot be explained. Specifically, even though the contribution of the relative motion mass to the damping ratio is decreasing at the total mass of 6 kg, the damping ratio is increasing.

We make the following inference concerning the behavior at the total mass of 6 kg. The relative motion mass is the mass of the granular material that is displaced relative to the primary system (“translation motion”). However, the motion of the granular material is not simply translation, but also includes rotation. The granular material follows the movement of the primary system (there is no relative motion mass), and even if there is rotational motion at that time, the rotation of the granules dissipates overall system energy and thus contributes to damping. That is to say, we can infer that parts of the equivalent added mass rotate when the total mass is 6 kg so that the damping ratio is larger than when the total mass is 5 kg.

6 Conclusions

We introduced the equivalent added mass and relative motion mass of granular materials as physical quantities to explain the damping mechanism of a granular material damper for cases in which the vibration amplitude is relatively small. We formulated a computational model for a granular material damper and calculate the motions of each part by numerical simulation. We confirmed the possibility of using the concepts of equivalent added mass and relative motion mass to distinguish damping mechanisms in a granular material damper.