A more accurate dynamic model for dual-side excitation large vibrating screens

3.1. Physical parameters

Generally, with a three-dimension geometrical model built and the corresponding materials defined in a three-dimensional modeling software, the center of mass for DELVS can be located accurately, as Fig. 3 shows. Then the physical parameters of DELVS for $\mathbf{M}$, $\mathbf{F}$, $\mathbf{K}$ can be gained and reported in Table 1. They are consistent with the parameters of the experimental apparatus which will be presented later.

Fig. 3Physical parameters of DELVS

Furthermore, the damping matrix $\mathbf{C}$ should be determined as well. For a practical system, due to the various phenomena of energy consumption, it is difficult to construct an accurate damping matrix $\mathbf{C}$. Hence, a commonly-used proportional damping method is adopted in this study, which means that the damping matrix is regarded as the linear combination of mass matrix and stiffness matrix, namely [11, 12]:

As the damping ratio needs to be obtained by experiment and has a little influence on the movement under the forced vibration, it is feasible to take the constants $\alpha$ and $\beta$ far less than 1, thus, the authors let $\alpha =\beta =$ 0.01 [11].

3.2. Transverse stiffness of a MCCS

Then the stiffness of a MCCS should be determined. It is known that during the working process of DELVS, the oscillated screen structure makes a synchronous dynamic deformation of a MCCS along the transverse and longitudinal direction. Such deformations are both quantifiable on the condition that the transverse stiffness and longitudinal stiffness are available. As the MCCS compression deformation is within 3-7 mm, namely, far less than free length of MCCS and in its linearly-deformed range, the transverse stiffness and longitudinal stiffness are supposed to be a certain constant value [3, 5, 8, 9]. Since the dynamic equation governing DELVS system in Fig. 2 has been illustrated in section 2 and the longitudinal stiffness of a MCCS is well-determined and easily obtained, the key to conducting a numerical simulation of DELVS is to calculate the transverse stiffness $k_x$. Up to now, there is no uniform method for calculating the transverse stiffness of a MCCS. In this paper, several methods will be compared in the following section.

3.2.2. Indirect methods

As we know, the longitudinal stiffness of a MCCS can be exactly calculated. Therefore, some methods have been developed to obtain the transverse stiffness through the longitudinal one, such as the empirical method as well as the newly-developed elastic compression bar model method (ECBM).

(1) Empirical method. Maybe the empirical method is the most common-used formula for calculating the longitudinal stiffness of a MCCS when designing a vibrating screening. One can find in [6, 12] that the transverse stiffness is regarded as one third of the longitudinal stiffness, i.e.:

11

$k_{xi}^{emp}=\frac{k_{yi}^{emp}}{3},$

where $k_{yi}^{emp}$ is longitudinal stiffness of a MCCS and can be formulated as [14]:

12

$k_{yi}^{emp}=\frac{G{d}_i^4}{8n_i{D}_i^3},\begin{array}{l}\end{array}\left(i=1,2\right),$

where $G$ is the transversal elastic modulus of the spring. Similarly, the transverse stiffness of each group of compression damping spring system by the empirical method is $k_x^{emp}=(k_{x1}^{emp}+k_{x2}^{emp})\times 6$. Here the superscript “emp” means “Empirical method”.

(2) Elastic compressed bar model method (ECBM). From above we can conclude that the two direct methods are more complex than the empirical method. However, the empirical method was obtained by practical experience and a lack of accurate derivation or sufficient explanation. Here, we will present a simple method.

Fig. 4Sketch of the longitudinal deformation: a) and the transverse deformation, b) considered in ECBM

a)

b)

As Yang introduced in [15], a MCCS can be modeled as a compressed straight bar with constant cross-section, whose longitudinal and transverse elastic modulus are the same as those of the MCCS, respectively. The longitudinal deformation of MCCS is equivalent to the compression deformation of modeled bar while the transverse deformation of MCCS is equivalent to the shear deformation of the bar, as depicted in Fig. 4. The static compressed position is where the MCCS remains with the DELVS in a stationary state. The transverse stiffness formula by ECBM will be derived as follows. Note that the derivation is considered in the liner deformation range, which agrees well with the property of practical MCCS of the DELVS.

For a compressed straight bar with constant cross-section, the following relation holds [16]:

13

$\epsilon =\frac{F_y}{EA},$

where $\epsilon$ is the longitudinal strain induced by the imposed compress force $F_y$, and $A$ is the cross-section. According to mechanics of materials, if a bar with an initial deformed length (denoted by $y_0$) experiences an additional compression deformation (denoted by $\mathrm{\Delta }y$) in Fig. 4(a), then we may have:

14

$\epsilon =\frac{\mathrm{\Delta }y}{y_0},$

and furthermore:

15

$F_y=k_y\cdot \mathrm{\Delta }y,$

where $k_y$ is the longitudinal stiffness.

Also, when a transverse force $F_x$ is applied on the compressed bar, an offset (denoted by $\mathrm{\Delta }x$) of the top section occurs, as shown in Fig. 5(b). Assuming that the angle between the initial position and the rotated position of the center line of the compressed bar is $\xi$ in radian, then the shear strain is expressed as $\gamma =\mathrm{t}\mathrm{a}\mathrm{n}\xi$. As the shear deformation is so tiny that the angle $\xi$ might represent tan $\xi$. Thus, we have $\gamma =\xi$. For the compressed bar, the shear strain can be expressed as [17]:

16

$\gamma =\frac{F_x}{GA},$

and according to the geometrical relationship in Fig. 5(b) and adopt the $\gamma =\xi$, we hence obtain:

17

$\gamma =\frac{\mathrm{\Delta }x}{y_0},$

as well as:

18

$F_x=k_x\cdot \mathrm{\Delta }x,$

where $k_x$ is the transverse stiffness.

From Eqs. (13) to (18) the transverse stiffness formula by ECBM can be derived to be:

19

$k_{xi}^{ECBM}=\frac{G}{E}{k}_{yi}^{ECBM},$

where $k_{yi}^{ECBM}$ is longitudinal stiffness, as Eq. (12) demonstrates. Here the superscript “ECBM” means “Elastic compressed bar model method”. Similarly, the transverse stiffness of the damping spring system by ECBM is $k_x^{ECBM}=(k_{x1}^{ECBM}+k_{x2}^{ECBM})\times 6$.

Table 2Physical parameters of MCCS of DELVS

Symbols (Unit)	$G$ (GPa)	$E$ (GPa)	$d$ (m)	$N$	$D$ (m)	$H$ (m)	$\mathrm{}\mathrm{Q}\mathrm{U}\mathrm{O}\mathrm{T}\mathrm{E}\mathrm{}\mathrm{}\upsilon$	$h$ (m)
Outer spring	79	206	0.03	6	0.193	0.2909	0.3	0.05
Inner spring	79	206	0.02	7	0.115	0.2009	0.3	0.03

Based on the basic parameters of the experimental MCCS in Table 2, the transverse stiffness and longitudinal stiffness of the inner and outer MCCS are calculated according to the four stiffness calculation methods, as shown and compared in Table 3. The transverse stiffness values obtained by the four methods are entirely different. Among them, the value by the proposed ECBM method is the largest for both the inner spring and outer spring. Currently, we are hard to distinguish which method matches better well with the actual situation and which method is more fit for DELVS.

Table 3Transverse and longitudinal stiffness value of the MCCS in the experimental apparatus

Symbols (Unit)	Method	Outer spring ($i=$ 1)	Inner spring ($i=$ 2)
$k_{yi}$ (N/m)	NONE	1.8544×105	1.4841×105
$k_{xi}$ (N/m)	Unit force method	6.0701×104	3.7364×104
	Energy method	5.7506×104	3.4476×104
	Empirical method	6.1813×104	4.9470×104
	ECBM method	7.1115×104	5.6915×104

4.1. Numerical solution

The complete numerical simulation procedure is illustrated in the flowchart in Fig. 5. For a DELVS, vibration response of the screen structure can be obtained by the numerical solution of Eq. (7), which can be solved by a numerical direct time-integration method such as the Newmark-$\beta$ method [18, 19]. It is widely used in the numerical evaluation of the dynamic response of structures. Here Matlab was used to conduct the numerical simulation with the Newmark-$\beta$ method. In the simulation, the zero initial displacement, the zero initial velocity and the zero acceleration of DELVS are considered. Each transverse stiffness calculation method for a MCCS in section 2.2 will be adopted in order to demonstrate the theoretical vibration response. Note that the numerical stability of Newmark-$\beta$ scheme can be guaranteed when $\delta \ge$ 0.5, $\kappa =$ 0.25 [18]. In this paper, we set $\delta =$ 0.5 and $\kappa =$ 0.25. In addition, as the sampling frequency of the experimental test is 10.24 kHz, we make the time interval $\mathrm{\Delta }t=$1/10240 s. Other parameter is simulation time $T=$ 60 s.

Fig. 6 indicates the numerical solution result of the DELVS. The transverse time responses ($x$-displacement) with the Empirical method, the ECBM method, the Unit-force method and the Energy method are compared. From the whole time response, it is seen that the transverse motion all trend to stability at an amplitude of nearly 1.0 mm. Even in the local magnification curve of the response between 38.5 s and 39.0 s, there is no any significant difference. Thus, we can conclude that the transverse stiffness is not easy to affect the transverse time response.

Fig. 7 shows and compares the Lissajous curves indicating the relationship between the numerical transverse displacement and longitudinal displacement. These curves are, in fact, the numerical vibration trajectories of the DELVS by various transverse stiffness calculation methods. All the trajectories are adjusted according to the same scale for an intuitional demonstration. With the influence of rotation around $z$-direction in Fig. 2, the trajectories are more like slender ellipses than straight lines. Additionally, for the ECBM method, the ellipse has a larger minor axis than the others, which implies that a large stiffness may affect the trajectory of the DELVS.

Fig. 5Flowchart of the numerical solution for DELVS

Fig. 6Numerical transverse time response of the DELVS by various transverse stiffness calculation methods

4.2. Experimental test

The experimental system and equipment are shown in Fig. 8. Two same types of acceleration sensors (measurement range is 500 m/s², resolution ratio is 0.002 m/s², frequency range is 0.5-8 KHz) are magnetically attached on the vibration motor support of DELVS for obtaining the transverse and longitudinal vibration signal, respectively. These sensors are connected with a data acquisition instrument, and then signals can be collected and handled through a data acquisition software. During the experimental test, the vibration signal of each measuring point was acquired simultaneously, and only the steady-state signal was used. After obtaining the data, the software DASP (China Orient Institute of Noise & Vibration, China) was utilized for data processing.

Fig. 7Numerical vibration trajectories of DELVS by various transverse stiffness calculation methods

a) Uni-force method

b) Energy method

c) Empirical method

d) ECBM method

Fig. 8Experimental apparatus: a) experimental DELVS; b) displacement sensors; c) data acquisition instrument; d) data acquisition software

The measured acceleration signals were digitally filtered with a cutoff of 100 Hz and analyzed by numerical integration so that the experimental transverse displacement and longitudinal displacement were extracted, as viewed in Fig. 9. Note that the corresponding Lissajous curve was generated by the data acquisition software automatically. The magnitude of longitudinal displacement is larger than that of transverse displacement, for the reason that the vertical component force is larger than the horizontal component force. Besides, displacements in the two directions share the same frequency and phase. Rotation exists in the experimental DELVS as the vibration trajectory of DELVS is a slender ellipse, and the experimental ellipse is more like the numerical ellipse by ECBM.

Fig. 9Experimental transverse displacement: a) longitudinal displacement, b) the corresponding Lissajous curve

a)

b)

4.3. Numerical solution vs. experimental test

In view of quantification, the kinematic characteristics of DELVS, i.e., the transverse amplitude (denoted by ${\lambda }_x$), the longitudinal amplitude (denoted by ${\lambda }_y$), the compound amplitude (denoted by $\lambda$) and the vibration angle (denoted by $\tau$) are listed in Table 4, among which according to the test methods of vibrating screens, we have $\lambda =\mathrm{s}\mathrm{q}\mathrm{r}\mathrm{t}({\lambda }_x^2+{\lambda }_y^2)$ and $\tau =\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}({\lambda }_y/{\lambda }_x)$ [20].

Here the values of transverse amplitude and longitudinal amplitude were acquired from time history in the stable operation. We can find that although the transverse stiffness values by the corresponding calculation method are different, the numerical vibration characteristics are almost the same as those of the experiment, which verifies the 3-DOF theoretical model of DELVS and confirms the derivation in Section 2. However, it is still hard to distinguish which transverse stiffness calculation method is the best in view of transverse stiffness value itself. Fortunately, since the errors of the numerical simulation data by ECBM are 0.41 % for ${\lambda }_x$, –1.77 % for $\lambda$, and –0.56 % for $\tau$, respectively, and are more closely related to the experimental data, ECBM is verified to improve the accuracy of DELVS dynamic model successfully. It should be noted that as a vibrating screen works in the super-resonant mode, the stiffness of MCCS has little influence on the movement characteristics of the screen structure. However, as the two vibration motors with eccentric blocks inside are mounted on the left and right lateral plate for a DELVS, respectively, the screen structure is easier to experience a lateral oscillation than a traditional vibrating screen. In order to further study such lateral oscillation, a more accurate calculation method for the transverse and longitudinal stiffness of a MCCS, such as the proposed ECBM method, is necessary.