The rotary drum of Concrete mixing truck is a special hollow rotor, the section geometry of which is complicated. The calculation of supporting force for the drum is difficult due to the inclined installation and the changing concrete state. With the rotation of the drum, the shape of concrete will change, which lead to the complex vibration performance.

At present, the research on the mechanical and vibration performance of concrete mixing drum is still under development. Li et al. (2013) studied the changing loads at both supporting points with different conditions including uphill, downhill, uniformly driving, full load, no-load, emergency brake, etc. The test method is used to verify whether the static mechanical performance of the reducer meets the requirements of drum's application. Y. D. Gao et al. (2013) analyzed the mechanical properties of the mixing drum's front supporting by using software ANSYS. The effect of different exciting forces on drum's natural frequency and vibration mode is studied in detail. An improved scheme of the front supporting structure with good performance is presented. Yan (2013) established the virtual simulation model of the mixing truck by the ADAMS software, and investigated the working principle of the mixing drum. The dynamic performance of the vehicle is studied under three conditions of the linear braking, the curve driving and the curve braking.

The rotary drum of concrete mixing truck is a kind of typical tilt rotor, the dynamic characteristics of which is different from the rotors working in horizontal state. Research on the dynamic properties of concrete mixer's rotating drum is not enough in the world. The existing literatures on the dynamic performance of the bearing rotor system are mainly aimed at horizontally placed rotors. Matthew and Sergei (2015) and Shi et al. (2013) studied the dynamic characteristics of horizontal and vertical rotor system supported by journal bearing, and analyzed the effects of bearing elasticity on the orbit of bearing center. Y. Gao et al. (2013) and Harsha (2005) made some researches on rotor's vibration properties supported by rolling element bearings with taking bearing structure, installation errors into consideration. Liu et al. (2012) and Qian et al. (2011) built the dynamic equations of the inclined rotor supported by journal bearing by the use of finite element method, and the equations are solved by Newmark method (Dukkipati et al., 2000). The influences of tilting angle with different loads on the vibration performance of the system are analyzed in detail. Misaligned contact occurs even for taper roller bearings because of different contact conditions between rollers and inner and outer raceways as well as with the retaining rib. Therefore, all rotors run with some degree of misalignment (Rahnejat and Gohar, 1979). Kabus et al. (2014) dealt with the same issue as Rahnejat and Gohar (1979), where numerical analysis of non-Hertzian tilted rollers to races contacts are required for a misaligned rotor-bearing system. Aleyaasin et al. (2000) and Tsai and Huang (2013) used transfer matrix method to establish the dynamic model for rotor-bearing system with multiple degrees of freedom. The critical speed, unbalanced response and vibration mode are studied.

It can be found through the analysis of literatures that although some researches have studied bearing-rotor system, the studies on the dynamic performance of concrete mixing truck's rotary drum are few. In this paper, a type of concrete truck's mixing drum is taken as the research object. The calculation method of supporting force is established on the base of mechanics theory. The computational model of the concrete shape in the rotary drum is implemented. A new transfer matrix is built to calculate the dynamic performance. With a specific example, the dynamic response of the mixing drum is studied numerically.

The mixing system structure and force analysis of concrete mixing truck are shown in Fig. 1. The mixing drum is connected to the gear reducer by flange, and the gear reducer is installed on the front bracket. The right side of the drum is supported by two rolling wheels that are mounted on the rear bracket. The wheels are installed symmetrically. Each wheel is supported by a pair of tapered roller bearings. The centers of the two wheels and the drum center form an angle 2γ. The whole rotating drum is inclined, and the angle of the installation is β which can be adjusted by a machine frame.

The mixing drum is subjected to gravity G, centrifugal force Fc and the supporting forces of the roller wheels and the bearing in gear reducer. Different from the horizontal rotor, the gravity and centrifugal force of the drum are not in the same plane when the drum rotates. As shown in Fig. 1, the forces on the left are radial force Fr1 and axial force Fa1. The roller wheels do not bear the axial force, so the gravity component of the internal drum and its concrete along the axial direction is borne entirely by the bearing in gear reducer. And the force on the right includes Fr2 by roller wheels (Fr2 is the resultant force of Fw1 and Fw2).

Before the mechanics analysis of the mixing drum, the following assumptions are introduced: (a) the inner ring bearing and the flange are tight fitted (it is true in actual installation), and relative sliding does not occur between flange and inner ring. (b) Machining and installing error of bearing housing are neglected. (c) There is no bearing clearance error, and roller-raceway contact is in the range of elastic deformation and under elastohydrodynamic conditions. (d) The rotating drum can be regarded as a hollow shaft, and the shape of the section and the geometric size of the cross section are not change when the elastic deformation occurs. (e) The axial deformation and the geometric error of the drum are ignored.

As Fig. 1 shows, according to the force balance condition, the force balance equations of the mixing drum can be obtained: Fr1+Fr2-(G2cos⁡2β+Fc2+2e1GFccos⁡β/e12+e22)1/2=0a0Fr1+e12+e22Gsin⁡β-Fr2L6-L5-a0=0Fw1cos⁡γ+Fw2cos⁡γ-Fr2sin⁡α2+α1=0Fw1sin⁡γ-Fw2sin⁡γ-Fr2cos⁡α2+α1=0Gsin⁡β-Fa1=0, where α2=arccos⁡e2e12+e22, α1=arccos⁡Fc+e1Gcos⁡β/e12+e22(G2cos⁡2β+Fc2+2e1FcGcos⁡β/e12+e22)1/2, e1 and e2 are the vertical distance from gravity center to x and y axis. By solving Eq. (1), we can get the supporting forces Fa1, Fr1, Fr2, Fw1, and Fw2.

Since the concrete in the drum is a kind of slurry fluid, the distribution shape of the concrete changes with the rotation of the rotating drum. The shape of the concrete on a cross section should be as that shown in Fig. 2. Rotating speed can not be too high, otherwise, the concrete will form a ring distribution in the rotary drum, and can not form an effective mixing. However, the speed can not be too low, otherwise, most concrete will always be in a horizontal state and this can not achieve the mixing effect. Therefore, the speed should fall in a reasonable range.

Taking one particle concrete element as the object, the motion analysis is shown in Fig. 3. In the mixing process, the motion state of the concrete can be divided into two parts. One part of the concrete makes no movement in a circular area, the radius of which is r1. Another part of the concrete undergoes a circular motion from point A to point B, and then from point B along BC moving to C due to the effect of gravity, centrifugal force and friction force. And then repeat this process all the time (Li, 2013).

The equation of dynamic mass center in the mixing process is deduced by literature (Li, 2013):

When 0≤t≤ϕmax⁡-ϕ0ω, the concrete moves in a circle, the dynamic mass center is: x1=1S∫r1r2cos⁡ϕdϕdry1=1S∫r1r2sin⁡ϕdϕdr When ϕmax⁡-ϕ0ω<t≤2πω, the concrete makes an oblique projectile motion, the dynamic mass center is: x2=∫r1r2r∫ϕmϕ[rcos⁡ϕmax⁡-rϕ-ϕmax⁡sin⁡ϕmax⁡]dϕdrS+∫r1r2r∫0ϕmcos⁡ϕdϕdrSy2=∫r1r2r∫ϕmϕ[rsin⁡ϕmax⁡-rϕ-ϕmax⁡cos⁡ϕmax⁡-0.5ω-2gϕ-ϕmax⁡2]dϕdrS+∫r1r2r∫0ϕmsin⁡ϕdϕdrS where ϕm=ϕmax-ϕ0, S is area of the particle concrete element. According to the Eqs. (2) and (3), the shape of concrete at any time any cross section can be obtained. On this basis, the mass center of each cross section can be calculated, and the eccentricities e1 and e2 can be confirmed. Finally, the centrifugal force Fc at different time will be obtained.

Based on the above theory, the distribution law of concrete in the mixing drum can be calculated numerically. Figure 4 shows the three dimensional distribution shape of concrete, where the inclination angle is β=12∘, and rotating speed is n=3 rmin-1, 8 rmin-1, 15 rmin-1. It can be found by comparing Fig. 4a, b, c that, for the same tilt angle, the concrete shapes are different under different rotating speeds. It can be seen from Fig. 4d that with the increase of the rotating speed, the highest level of concrete is increasing, the lowest liquid level is gradually reduced.

Figure 5 shows the influences of inclined angle on the concrete surface shape when β=10∘, 12∘, 14∘, 16∘, n=8 rmin-1. When the speed is fixed, with the increase of the tilt angle, the concrete liquid level is gradually increasing, but the maximum value of liquid level is almost unchanged. Different cross sections have different concrete shapes. The closer to the z axis origin, the higher the concrete liquid level, and the more area of concrete is not moving with the mixing drum.

Figure 6a and b show the effect of the height of static concrete liquid level on concrete distribution shape when β=12∘, n=8 rmin-1, h=0 m, -0.10 m, -0.15 m, and -0.20 m (at full load, the static height of the liquid level is as the benchmark 0 m, lower than this is a negative value). It can be seen that with the decrease of static concrete liquid level height, the concrete surface level decreases at each cross section, however the maximum value of surface level is almost unchanged. Different cross sections have different concrete shapes. The position nearer to the z axis origin will has smaller concrete liquid level value.

According to previous research, it can be known that the distribution shape of concrete in the drum is continuously changing when working, which will affect the vibration performance of the drum. This paper used transfer matrix method to analyze the dynamic performance of the mixing drum with considering the concrete shape changing.

The concrete mixing drum is lumped in a new system shown in Fig. 7. The system is a multiple degrees of freedom system consisting of n rigid lumped-mass discs and n-1 mass-less elastic shaft segment. The supporting at both ends of the drum are simplified as spring-damping system. Fej is the centrifugal force caused by the concrete. Through the mechanical analysis of the disc and shaft segment, the transfer matrix of each node is established at both ends of the state vector. The equations consisted of transfer matrix can be solved by using the boundary conditions of the system, and the natural frequencies and the vibration modes of the system will be obtained.

Figure 8 is the dynamics analysis of shaft segment. The state vectors of both sides of the shaft segment can be expressed as Zj=yθMQjT, where y, θ, M, and Q denote the transverse displacement, rotation angle, bending moment and shearing force on both sections. According to the mechanics theory (shearing deformation and higher modal responses are neglected here). A more representative approach has been stated in reference Matsubara et al. (1988), we have: QjR=QjLMjR=MjL+QjLljθjR=θjL+1(EI)j∫0ljMjRdz=θjL+MjLlj(EI)j+QjLlj22(EI)jyjR=yjL+∫0ljθjRdz=yjL+θjLlj+MjLlj22(EI)j+QjLlj36(EI)j. Equation (4) can be written in matrix form: yθMQjR=1ll2/2EIl3/6EI01l/EIl2/2EI001l0001jyθMQjL. Reduce it to a simplified form: ZjR=BjZjL, where EI is the bending stiffness of the shaft segment. l is the length of the shaft segment. The subscript j indicates the shaft segment number. The superscript L and R represent the left-hand and right-hand respectively.

Figure 9 is the schematic diagram of mechanics analysis of tilt disk with concrete loading. On the basis of the force analysis, the relationship between the two sides of the cross section can be written as: yθMQjR=100001000(Jd-Jp)ω210mω2001jyθMQjL+00-Ma-FejL, where Ma is the torque induced by the concrete, Fe is the centrifugal force induced in mixing process. Jd and Jp are the diameter and polar moment of inertia of the disc respectively. Jp=mj(R1j2+R2j2)/2, Jd=Jp/2. R1 is the inner diameter of the tilt disk. R2 is the outer diameter of the tilt disk.

Figure 10 shows the mechanics analysis of tilt disk at supporting point. The relationship between the two sides of the cross section can be expressed by matrix: yθMQjR=100001000(Jd-Jp)ω210mω2-K-iCω001jyθMQjL+00-Ma-FejL where Jd and Jp are the diameter and polar moment of inertia of the disc respectively. ω is the angular velocity, m is the mass of the disk. K and C are the stiffness and damping at the supporting point.

Equations (7) and (8) can be simplified expression as:

As for the left supporting point, the transfer matrix between the two ends of the state vector can be calculated by using the Eq. (8) with Ma=0 and Fe=0. ZjR=DjZjL+Fj Through the Eqs. (6), (9a), the cross section state vector of the system can be expressed as: Z1L=Z1LZ1R=D1Z1L+F1Z2L=B1Z1RZ2R=D2Z2L+F2…ZnL=Bn-1ZnRZnR=DnZnL+Fn. Though the recursive relation of Eq. (10), we can establish the relation between ZnR and Z1L: yθMQnR=a11a12a13a14a21a22a23a24a31a32a33a34a41a42a43a44nyθMQ1L+b1b2b3b4n-1 Bring the boundary conditions M1L=0, Q1L=0, MnR=0, QnR=0 into transfer matrix (11), we can get: MQnR=a31a32a41a42yθ1L+b3b4n-1=00 By solving Eq. (12), the unbalanced response of the mixing drum can be obtained.

The dynamic performance of a concrete mixing truck's mixing drum produced by a company is studied in this paper. The structure of the mixing drum is shown in Fig. 1, and the parameters of the drum are listed in Table 1. The main parameters of the gear reducer's main bearing are shown in Table 2. The type of the tapered roller bearing supporting the roller wheel is 32311.

By using the calculation model established in this paper, a series of curves are obtained by Matlab programming. Figure 14 is the calculation flow chart of the programming.

Figure 15a is the residual volume curve when calculating the critical speed of the mixing drum. Figure 15b is local enlarged drawing of Fig. 15a. The critical speed of the drum is at the intersection point of the curve and y=0. It can be seen from Fig. 15b, first-order critical speed of the drum is 390 rmin-1. In the actual working process, the rotating speed of the mixing drum can not reach the critical speed. Therefore, this paper only study the dynamic performance of the drum within n=3∼15 rmin-1.

Figure 16a and b show vibration curve of the mixing drum with n=3 rmin-1, β=16∘, h=0 m. As shown in Fig. 16b, the response curve distributes asymmetrically. The vibration amplitude first increases to the maximum value and then gradually decrease (L=5.6 m). The vibration amplitude of the drum is almost not changed at the right side of the roller wheel supporting point. This vibration mode is related to the distribution of the concrete in the drum.

Figure 17 depicts the influences of rotating speed on the dynamic properties of the mixing drum when β=12∘ and h=0 m. The vibration amplitude of the mixing drum decreases with the increase of the rotating speed. This is due to the wider distribution of the concrete along the rotating direction and the more obviously symmetrical distribution of concrete which result in the great decrease of concrete's eccentricity. Although the rotating speed increases, the exciting force becomes smaller due to the great decrease of concrete's eccentricity.

Figure 18 shows the effects of drum's inclination angle on the vibration properties when n=3 rmin-1 and h=0. The peak amplitude gradually moves to the right with the inclination angle increasing. The vibration amplitude value of the peak's left side decreases when tilt angle increases, while the right side increases. The peak amplitude is almost unchanged and gradually moves to the right with the inclination angle increasing, which is caused by the right movement along the axial direction of the maximum eccentric position of the concrete in the mixing drum with the increase of the inclination angle.

Figure 19 presents the influences of concrete's liquid level height when n=3 rmin-1 and β=12∘. The maximum unbalanced response amplitude of the drum increases with the decrease of concrete liquid level height h, which is because that when hdecreases, the eccentric distance of the concrete in the mixing drum is gradually increased which result in the increase of concrete's maximum centrifugal force. It can also be seen that the vibration peak amplitude gradually moves to the left with the decrease of h.

A mathematical formulation has been derived through the force analysis for calculating the supporting force. The calculation method of the concrete distribution shape in the rotary drum is developed based on some others' previous studies. The effects of rotating speed, tilting angle and concrete liquid level on concrete shape are analyzed. A new transfer matrix is built with considering the concrete geometric distribution shape. The stiffness and damping calculation models of spherical roller bearings and rolling wheels are established. On the basis of these analysis, a method for calculating the dynamic performance of the mixing drum is established.

The effects of rotating speed, inclination angle and concrete liquid level on the vibration performance of the mixing drum are studied with a specific example. Speed-amplitude curves, inclination angle-amplitude curves and concrete liquid level height-amplitude curves are obtained. It is found that the vibration mode curves of the drum distribute asymmetrically. And with the increase of rotating speed, the vibration amplitude of the mixing drum decreases, the peak amplitude gradually moves to the right with the inclination angle increasing, the amplitude value of the peak's left side decreases when tilt angle increases, while the right side increases. The maximum unbalanced response amplitude of the drum increases with the decrease of concrete liquid level height, and the vibration peak moves to the left.