Mech. Sci.

, 8, 165–178, 2017
https://doi.org/10.5194/ms-8-165-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.

Study on the dynamic performance of concrete

mixer’s mixing drum
Jiapeng Yang1 , Hua Zeng2 , Tongqing Zhu2 , and Qi An1
1 School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai,
200237, People’s Republic of China
2 Shanghai Electric Hydraulics & Pneumatics Co., Ltd., Shanghai, 201100, People’s Republic of China

Correspondence to: Qi An (anqi@ecust.edu.cn)

Received: 2 December 2016 – Revised: 11 March 2017 – Accepted: 17 May 2017 – Published: 16 June 2017

Abstract. When working, the geometric distribution shape of concrete in concrete mixing truck’s rotary drum
changes continuously, which cause a great difficulty for studying the dynamic performance of the mixing drum.
In this paper, the mixing system of a certain type of concrete mixing truck is studied. A mathematical formulation
has been derived through the force analysis to calculate the supporting force. The calculation method of the
concrete distribution shape in the rotary drum is developed. A new transfer matrix is built with considering the
concrete geometric distribution shape. 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. Results show that 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.

1 Introduction ing 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 prin-
The rotary drum of Concrete mixing truck is a special hol- ciple of the mixing drum. The dynamic performance of the
low rotor, the section geometry of which is complicated. The vehicle is studied under three conditions of the linear brak-
calculation of supporting force for the drum is difficult due to ing, the curve driving and the curve braking.
the inclined installation and the changing concrete state. With The rotary drum of concrete mixing truck is a kind of typi-
the rotation of the drum, the shape of concrete will change, cal tilt rotor, the dynamic characteristics of which is different
which lead to the complex vibration performance. from the rotors working in horizontal state. Research on the
At present, the research on the mechanical and vibration dynamic properties of concrete mixer’s rotating drum is not
performance of concrete mixing drum is still under devel- enough in the world. The existing literatures on the dynamic
opment. Li et al. (2013) studied the changing loads at both performance of the bearing rotor system are mainly aimed at
supporting points with different conditions including uphill, horizontally placed rotors. Matthew and Sergei (2015) and
downhill, uniformly driving, full load, no-load, emergency Shi et al. (2013) studied the dynamic characteristics of hor-
brake, etc. The test method is used to verify whether the static izontal and vertical rotor system supported by journal bear-
mechanical performance of the reducer meets the require- ing, and analyzed the effects of bearing elasticity on the orbit
ments of drum’s application. Y. D. Gao et al. (2013) analyzed of bearing center. Y. Gao et al. (2013) and Harsha (2005)
the mechanical properties of the mixing drum’s front sup- made some researches on rotor’s vibration properties sup-
porting by using software ANSYS. The effect of different ex- ported by rolling element bearings with taking bearing struc-
citing forces on drum’s natural frequency and vibration mode ture, installation errors into consideration. Liu et al. (2012)
is studied in detail. An improved scheme of the front support-

Published by Copernicus Publications.

166 J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum

Figure 1. Structure and mechanics analysis of the concrete mixer system. 1: gear reducer, 2: front bracket, 3: flange, 4: mixing drum,
5: subframe, 6: raceway, 7: rolling wheel 8: rear bracket.

and Qian et al. (2011) built the dynamic equations of the in- is supported by a pair of tapered roller bearings. The centers
clined rotor supported by journal bearing by the use of finite of the two wheels and the drum center form an angle 2γ . The
element method, and the equations are solved by Newmark whole rotating drum is inclined, and the angle of the instal-
method (Dukkipati et al., 2000). The influences of tilting an- lation is β which can be adjusted by a machine frame.
gle with different loads on the vibration performance of the The mixing drum is subjected to gravity G, centrifugal
system are analyzed in detail. Misaligned contact occurs even force Fc and the supporting forces of the roller wheels and
for taper roller bearings because of different contact condi- the bearing in gear reducer. Different from the horizontal ro-
tions between rollers and inner and outer raceways as well tor, the gravity and centrifugal force of the drum are not in
as with the retaining rib. Therefore, all rotors run with some the same plane when the drum rotates. As shown in Fig. 1,
degree of misalignment (Rahnejat and Gohar, 1979). Kabus the forces on the left are radial force Fr1 and axial force Fa1 .
et al. (2014) dealt with the same issue as Rahnejat and Go- The roller wheels do not bear the axial force, so the gravity
har (1979), where numerical analysis of non-Hertzian tilted component of the internal drum and its concrete along the ax-
rollers to races contacts are required for a misaligned rotor- ial direction is borne entirely by the bearing in gear reducer.
bearing system. Aleyaasin et al. (2000) and Tsai and Huang And the force on the right includes Fr2 by roller wheels (Fr2
(2013) used transfer matrix method to establish the dynamic is the resultant force of Fw1 and Fw2 ).
model for rotor-bearing system with multiple degrees of free- Before the mechanics analysis of the mixing drum, the fol-
dom. The critical speed, unbalanced response and vibration lowing assumptions are introduced: (a) the inner ring bearing
mode are studied. and the flange are tight fitted (it is true in actual installation),
It can be found through the analysis of literatures that al- and relative sliding does not occur between flange and inner
though some researches have studied bearing-rotor system, ring. (b) Machining and installing error of bearing housing
the studies on the dynamic performance of concrete mixing are neglected. (c) There is no bearing clearance error, and
truck’s rotary drum are few. In this paper, a type of con- roller-raceway contact is in the range of elastic deformation
crete truck’s mixing drum is taken as the research object. and under elastohydrodynamic conditions. (d) The rotating
The calculation method of supporting force is established on drum can be regarded as a hollow shaft, and the shape of the
the base of mechanics theory. The computational model of section and the geometric size of the cross section are not
the concrete shape in the rotary drum is implemented. A new change when the elastic deformation occurs. (e) The axial
transfer matrix is built to calculate the dynamic performance. deformation and the geometric error of the drum are ignored.
With a specific example, the dynamic response of the mixing As Fig. 1 shows, according to the force balance condition,
drum is studied numerically. the force balance equations of the mixing drum can be ob-
tained:
2 Mechanics analysis of mixing drum  q
2 2 2

1/2

 F r1 + F r2 − G cos β + F c + 2e1 GF c cos β/ e12 + e22 =0
The mixing system structure and force analysis of concrete 
 q
2 2
 a0 Fr1 + e + e G sin β − Fr2 (L6 − L5 − a0 ) = 0

mixing truck are shown in Fig. 1. The mixing drum is con- 1 2
,
nected to the gear reducer by flange, and the gear reducer is 
 F w1 cos γ + F w2 cos γ − Fr2 sin (α2 + α1 ) = 0
 Fw1 sin γ − Fw2 sin γ − Fr2 cos (α2 + α1 ) = 0

installed on the front bracket. The right side of the drum is 

supported by two rolling wheels that are mounted on the rear G sin β − Fa1 = 0
bracket. The wheels are installed symmetrically. Each wheel (1)

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J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum 167

y/m 0.5

−0.5

−1
−1 −0.5 0 0.5 1
x/m

Figure 2. Shape of a concrete cross section at a certain time (β =

12◦ , n = 3 r min−1 ).
Figure 3. Schematic diagram of circumferential motion in concrete
mixing process.
where α2 = arccos q e2 ,
e12 +e22
q
Fc +e1 G cos β/ e12 +e22
When φmaxω−φ0 < t ≤ 2π ω , the concrete makes an oblique pro-
α1 = arccos q 1/2 , e1 and e2 jectile motion, the dynamic mass center is:
G2 cos2 β+Fc2 +2e1 Fc G cos β/ e12 +e22
 R r2 R φ  
are the vertical distance from gravity center to x and y axis.  r r φm r cos φmax − r (φ − φmax ) sin φmax dφdr
1
x2 =

By solving Eq. (1), we can get the supporting forces Fa1 , Fr1 ,



 R r 2 R φm S
Fr2 , Fw1 , and Fw2 .

r cos φdφdr


 r 0
Since the concrete in the drum is a kind of slurry fluid, the + 1



 S
distribution shape of the concrete changes with the rotation

of the rotating drum. The shape of the concrete on a cross 
 y2 =
section should be as that shown in Fig. 2. Rotating speed
 R r2 R φ 
 −2 2
r1 r φm r sin φmax − r (φ − φmax ) cos φmax − 0.5ω g(φ − φmax ) dφdr


can not be too high, otherwise, the concrete will form a ring



 R r 2 R φm S
distribution in the rotary drum, and can not form an effective r 0 sin φdφdr



 +
 r 1
mixing. However, the speed can not be too low, otherwise, S
most concrete will always be in a horizontal state and this can (3)
not achieve the mixing effect. Therefore, the speed should
fall in a reasonable range. where φm = φmax − φ0 , S is area of the particle concrete el-
Taking one particle concrete element as the object, the mo- ement. According to the Eqs. (2) and (3), the shape of con-
tion analysis is shown in Fig. 3. In the mixing process, the crete at any time any cross section can be obtained. On this
motion state of the concrete can be divided into two parts. basis, the mass center of each cross section can be calculated,
One part of the concrete makes no movement in a circular and the eccentricities e1 and e2 can be confirmed. Finally, the
area, the radius of which is r1 . Another part of the concrete centrifugal force Fc at different time will be obtained.
undergoes a circular motion from point A to point B, and Based on the above theory, the distribution law of concrete
then from point B along BC moving to C due to the effect of in the mixing drum can be calculated numerically. Figure 4
gravity, centrifugal force and friction force. And then repeat shows the three dimensional distribution shape of concrete,
this process all the time (Li, 2013). where the inclination angle is β = 12◦ , and rotating speed is
The equation of dynamic mass center in the mixing pro- n = 3 r min−1 , 8 r min−1 , 15 r min−1 . It can be found by com-
cess is deduced by literature (Li, 2013): paring Fig. 4a, b, c that, for the same tilt angle, the concrete
When 0 ≤ t ≤ φmaxω−φ0 , the concrete moves in a circle, the shapes are different under different rotating speeds. It can be
dynamic mass center is: seen from Fig. 4d that with the increase of the rotating speed,
the highest level of concrete is increasing, the lowest liquid
Zr2


 1 level is gradually reduced.

 x1 = cos φdφdr Figure 5 shows the influences of inclined angle on the
S


concrete surface shape when β = 10◦ , 12◦ , 14◦ , 16◦ , n =


 r1
(2) 8 r min−1 . When the speed is fixed, with the increase of the

 Zr2 tilt angle, the concrete liquid level is gradually increasing,

 1
y = sin φdφdr but the maximum value of liquid level is almost unchanged.

 1 S



r1 Different cross sections have different concrete shapes. The

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168 J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum

Figure 4. Shapes of the concrete with different speeds. (a) β = 12◦ , n = 3 r min−1 . (b) β = 12◦ , n = 8 r min−1 . (c) β = 12◦ , n = 15 r min−1 .
(d) Concrete section shape (z = 4647 mm).

Figure 5. Shapes of the concrete with different tilt angles. (a) z = 3074 mm. (b) z = 4647 mm.

closer to the z axis origin, the higher the concrete liquid level, 3 Vibration model of mixing drum
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 According to previous research, it can be known that the
concrete liquid level on concrete distribution shape when distribution shape of concrete in the drum is continuously
β = 12◦ , n = 8 r min−1 , h = 0 m, −0.10 m, −0.15 m, and changing when working, which will affect the vibration
−0.20 m (at full load, the static height of the liquid level is performance of the drum. This paper used transfer matrix
as the benchmark 0 m, lower than this is a negative value). method to analyze the dynamic performance of the mixing
It can be seen that with the decrease of static concrete liq- drum with considering the concrete shape changing.
uid level height, the concrete surface level decreases at each The concrete mixing drum is lumped in a new system
cross section, however the maximum value of surface level shown in Fig. 7. The system is a multiple degrees of freedom
is almost unchanged. Different cross sections have different system consisting of n rigid lumped-mass discs and n − 1
concrete shapes. The position nearer to the z axis origin will mass-less elastic shaft segment. The supporting at both ends
has smaller concrete liquid level value. of the drum are simplified as spring-damping system. Fe j
is the centrifugal force caused by the concrete. Through the
mechanical analysis of the disc and shaft segment, the trans-
fer matrix of each node is established at both ends of the

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J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum 169

Figure 6. Shapes of the concrete with different concrete levels. (a) z = 3074 mm. (b) z = 4647 mm.

Figure 7. Lumped mass model of the concrete mixer rotating drum.

state vector. The equations consisted of transfer matrix can Figure 8. Mechanics analysis of shaft segment.
be solved by using the boundary conditions of the system,
and the natural frequencies and the vibration modes of the
Equation (4) can be written in matrix form:
system will be obtained.
Figure 8 is the dynamics analysis of shaft segment. The  R    L
y 1 l l 2 /2EI l 3 /6EI y
state vectors of both sides of the shaft segment can be ex- θ  0 1 l/EI l 2 /2EI  θ 
pressed as   =
M 
   . (5)
0 0 1 l  M 

Zj = yθ MQ j ,
T Q j 0 0 0 1 j
Q j

where y, θ, M, and Q denote the transverse displacement, Reduce it to a simplified form:

rotation angle, bending moment and shearing force on both ZjR = Bj ZjL , (6)
sections. According to the mechanics theory (shearing de-
formation and higher modal responses are neglected here). where EI is the bending stiffness of the shaft segment. l is
A more representative approach has been stated in reference the length of the shaft segment. The subscript j indicates the
Matsubara et al. (1988), we have: shaft segment number. The superscript L and R represent the
 R left-hand and right-hand respectively.
Qj = QLj
Figure 9 is the schematic diagram of mechanics analysis




of tilt disk with concrete loading. On the basis of the force


 R L L
 Mj = Mj + Qj lj

analysis, the relationship between the two sides of the cross



section can be written as:



Zlj

MjL lj QLj lj2 . (4)

 R    L  L
R L 1 R L y 1 0 0 0 y 0
 θ j = θj + Mj dz = θ j + + θ   0 1 0 0  θ   0 


 (EI)j (EI)j 2(EI)j M  =  0 (Jd − Jp )ω2 1 0 M  + −Ma  , (7)
 0 Q j mω2 0 0 1 j Q j −Fe j





lj M Ll2 QL l 3


 y R = y L + R θ R dz = y L + θ L l + j j + j j

 where Ma is the torque induced by the concrete, Fe is the
 j j j j j j

0 2(EI)j 6(EI)j centrifugal force induced in mixing process. Jd and Jp are the

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170 J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum

Figure 10 shows the mechanics analysis of tilt disk at sup-

porting point. The relationship between the two sides of the
cross section can be expressed by matrix:
 R    L
y 1 0 0 0 y
θ  0 1 0 0  θ 
=

M   0 (Jd − Jp )ω2 1 0  M 
Q j mω2 − K − iCω 0 0 1 j
Q j
 L
0
 0 
+−M  (8)
a
−Fe j

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:
Figure 9. Mechanics analysis of tilt disk with concrete loading. As for the left supporting point, the transfer matrix be-
tween the two ends of the state vector can be calculated by
using the Eq. (8) with Ma = 0 and Fe = 0.

ZjR = Dj ZjL + Fj (9)

Through the Eqs. (6), (9a), the cross section state vector of
the system can be expressed as:
 L L
 Z1R = Z1 L

 Z1L = D1 ZR1 + F1



 Z2 = B1 Z1


Z2R = D2 Z2L + F2 . (10)
 ...



 ZnL = Bn−1 ZnR



 R
Zn = Dn ZnL + Fn

Though the recursive relation of Eq. (10), we can establish

the relation between ZnR and Z1L :
 R    L  
y a11 a12 a13 a14 y b1
θ  a21 a22 a23 a24   θ  b2 
M  = a a32 a33 a34  M  + b3  (11)
31
Figure 10. Mechanics analysis of tilt disk at supporting point. Q n a41 a42 a43 a44 n Q 1 b4 n−1

Bring the boundary conditions M1L = 0, QL1 = 0, MnR = 0,

diameter and polar moment of inertia of the disc respectively. QRn = 0 into transfer matrix (11), we can get:
Jp = mj (R1j 2 + R 2 )/2, J = J /2. R is the inner diameter
2j d p 1  R    L    
of the tilt disk. R2 is the outer diameter of the tilt disk. M a a32 y b3 0
= 31 + = (12)
Q n a41 a42 θ 1 b4 n−1 0

By solving Eq. (12), the unbalanced response of the mixing

drum can be obtained.

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J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum 171

roller-raceway in the first and second row of the bearing is

(Houpert, 2001):
V Q1j
Ksz_1j = lim
V Q1j → 0 V δ1j
V δ1j → 0
1/3
0.8752E 0 2/3 Q1j
= −0.0590 −0.0590 −0.2743 −0.2743
(15)
(Rxi + Rxo )(Ryi + Ryo )
V Q2j
Ksz_2j = lim
V Q1j → 0 V δ2j
V δ1j → 0
1/3
Figure 11. Stiffness and damping model at left supporting point. 0.8752E 0 2/3 Q2j
=   (16)
−0.0590 −0.0590 −0.2743 −0.2743
Rxi + Rxo Ryi + Ryo

1


 Rxi =
ρI + ρIi




 1
 Ryi =


ρII + ρIIi (17)
1
 Rxo =



 ρ I + ρIo
1



 Ryo =

ρII + ρIIo
Figure 12. Mechanics analysis of spherical roller bearing.
where Q1j and Q2j are normal loads acting on the two rows
of rollers, and can be obtained according to the method of
literature (Ma et al., 2016). E 0 (N m−2 ) is the equivalent
4 Stiffness and damping
elastic modulus. Rx is the equivalent radius of the contact
4.1 Stiffness and damping at left supporting point point along rolling direction of the roller. Ry is the equiva-
lent radius of the contact point perpendicular to the rolling
The stiffness and damping at left supporting point is com- direction of the roller. The principal radii of spherical roller
posed of the stiffness and damping of the gear reducer’s main of contact in xy and xz planes are ρI = D2 , ρII = R1 . The
bearing and the front bracket. The stiffness and damping principal radii of the inner race of contact in xy and xz
model is shown in Fig. 11. The mass of the bearing and the 2γ 1
planes are ρIi = D(1−γ ) , ρIIi = − ri . The principal radii of the
damping of the front bracket are very small, and can be ne- 2γ
outer race of contact in xy and xz planes are ρIo = − D(1+γ ),
glected. Thus, the spherical roller bearing (SRB) in the gear
reducer is equivalent to a spring-damping system, and the ρIIo = − r1o . D is the roller diameter. R is the roller contour
front bracket is equivalent to a spring-mass system. Both sim- radius. ri is the inner contour radius. ro is the outer contour
plified models are in series. According to the series-parallel radius. γ = D · cos αc /dm . αc is the contact angle. dm is the
calculation formula of stiffness and damping, the total stiff- pitch circle diameter.
ness and damping can be expressed as (Wen et al. 1999): The oil film stiffness of the first and second column of
roller j is (Yang et al., 2016):
Kb1 Ks1 − ms1 ω2 /cos2 β


Kleft =
, (13) 1Q1j Q1.073
1j
Kb1 + Ks1 − ms1 ω2 /cos2 β Koil_1j = lim = (18)
1Q1j → 0 1hmin 0.073λ
Cleft = Cb1 , (14)
1hmin → 0
where Kb1 and Cb1 are the stiffness and damping of SRB, 1Q2j Q1.073
2j
Ks1 and ms1 are the vertical stiffness and mass of front Koil_2j = lim = (19)
1Q2j → 0 1h min 0.073λ
bracket.
The stiffness and damping calculation method of dou- 1hmin → 0
h
ble row spherical roller bearing (as shown in Fig. 12) has λ = 3.63E 0
−0.117 0.49
α (η0 vm )0.68 1 − e−0.68ki Rxi

0.466
not been studied in detail at present. However, based on
0.466
the definition of stiffness, the contact stiffness of the j th + 1 − e−0.68ko Rxo ] (20)

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172 J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum

The comprehensive radial stiffness of SRB is:

jX
=Z  −1
−1 −1
Kb1 = Ksz_1j + Koil_1j cos ϕ1j cos α1j
j =1
 −1 
−1 −1
+ Ksz_2j + Koil_2j cos ϕ2j cos α2j , (21)

where α (m2 N−1 ) is the pressure-viscosity coefficient of

the lubricant. η0 (Pa s) is the dynamic viscosity of lubri-
cant under the pressure 0.1 M Pa. vm (m s−1 ) is the average
speed of rolling bearings. e is the base of natural logarithms
(e = 2.71828). k is the ellipticity ratio. ϕ1j and ϕ2j are the
azimuth of the rollers.α1j and α2j are the actual contact an-
gles of roller-raceway contact. Z is the number of single row
rollers for SRB. Subscript i and o denote the inner and outer
rings of the bearing. Figure 13. Stiffness and damping model of a single rolling wheel.
1Q
According to the definition of damping C = lim 1u z
, the
1→0
damping in the first row of the bearing’s roller j between Table 1. Structural parameters of concrete mixer rotating drum.
inner raceway and outer raceway can be obtained (Yang et
al., 2016; Hamrock and Dowson, 1981): Parameters Value
 0.831 Q0.1095 Length of front cone L1 (mm) 1550
 2.16aRxi 1j Length of cylindrical section L2 (mm) 1400

 C 1j _i =
1.5
−0.1755 Length of middle cone L3 (mm) 1050



 E 0 α 0.735 0.02 1.02
η0 vm 1 − e−0.68ki
. (22) Length of posterior cone L4 (mm) 2200

 0.831 Q0.1095
2.16aRxo Length from wheel to bottom L5 (mm) 890

 1j
 C1j _o = Total length of the drum L (mm) 6490


1.5
E 0 −0.1755 α 0.735 η00.02 vm
1.02 1 − e−0.68ko
Diameter of drum head D1 (mm) 1933
Diameter of cylindrical drum D2 (mm) 2480
According to damping series relation of bearing, the compre- Diameter of middle drum D3 (mm) 2110
hensive damping in first row of the bearing at roller j is: Diameter of raceway D4 (mm) 1642

−1 Diameter of posterior drum D5 (mm) 1132
−1 −1
C1j = C1j _i + C1j _o cos ϕ1j cos α1j . (23) Diameter of drum entrance D6 (mm) 560

Also, the comprehensive damping in the second row of the

bearing at roller j is:
Crightt = 2 (Cb21 + Cb22 ) cos γ , (27)
−1 −1
−1
C2j = C2j _i + C2j _o cos ϕ2j cos α2j . (24)
where h −1  −1 i−1
The comprehensive damping of SRB is: Kbw = 2K1b21 + K1w1 + 2K1b22 + K1w2 cos2 γ ,
jX
=Z
Kb21 , Cb21 and Kb22 , Cb22 are the stiffness and damping of
Cb1 =


C1j + C2j . (25) the bearings in first roller wheel and the second roller wheel.
j =1 Literature (Li et al., 2016) gives the calculation method of
radial stiffness Kw1 , Kw2 between roller wheel and rotary
4.2 Stiffness and damping at right supporting point
drum. Literature (Wu, 2011) lists the parameters Ks2 , ms2 .

The stiffness and damping at right supporting point consists

of the stiffness and damping of the wheel-drum, bearing in 5 Specific example analysis
the wheel and the rear bracket. Fig.13 is the stiffness and
damping model of a single rolling wheel. According to the The dynamic performance of a concrete mixing truck’s mix-
series-parallel calculation formula of stiffness and damping, ing drum produced by a company is studied in this paper.
the total stiffness and damping can be expressed as (Wen, et The structure of the mixing drum is shown in Fig. 1, and
al., 1999): the parameters of the drum are listed in Table 1. The main
parameters of the gear reducer’s main bearing are shown in
Kbw Ks2 /cos2 β − ms2 ω2 cos2 γ


Table 2. The type of the tapered roller bearing supporting the
Kright = , (26) roller wheel is 32311.
Kbw + Ks2 /cos2 β − ms2 ω2

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J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum 173

Start

Input bearing parameters, drum parameters

and n, h, 

Calculate parameters of Generate transfer Calculate supporting

shaft segment and tilt disk matrix stiffness and damping

Calculate unbalanced response

Change
n,h, 
Save calculation results

N Finish
computation?

Output results

Draw curves

End

Figure 14. Calculation flow chart.

Figure 15. Residual volume 1(ω2 ) curve. (a) Residual volume curve n-1(ω2 ). (b) Local enlarged drawing of (a).

By using the calculation model established in this paper, a is at the intersection point of the curve and y = 0. It can be
series of curves are obtained by Matlab programming. Fig- seen from Fig. 15b, first-order critical speed of the drum is
ure 14 is the calculation flow chart of the programming. 390 r min−1 . In the actual working process, the rotating speed
Figure 15a is the residual volume curve when calculating of the mixing drum can not reach the critical speed. There-
the critical speed of the mixing drum. Figure 15b is local en- fore, this paper only study the dynamic performance of the
larged drawing of Fig. 15a. The critical speed of the drum drum within n = 3 ∼ 15 r min−1 .

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174 J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum

Figure 16. Vibration amplitude of the concrete mixing drum. (a) Vibration mode. (b) Response curve.

Table 2. Main parameters of the reducer bearing. 20

15 = 12
Parameters Value
= 13
10 = 14
Inner contour radius (mm) 98.85

Amplitude / m
Outer contour radius (mm) 98.85 = 15
5
Roller diameter (mm) 24
Roller contour radius (mm) 97.71 0
Number of single row roller 19
−5
Bearing inside diameter (mm) 120
Bearing outside diameter (mm) 215 −10
Bearing width (mm) 80
Radial clearance (mm) 0.083 −15
Nominal contact angle (◦ ) 10
−20
Elastic modulus (N m−2 ) 2.06 × 1011 0 1 2 3 4 5 6 7
Length / m
Poisson’s ratio 0.3
Dynamic viscosity of lubricant (Pa s) 0.1362
Figure 18. Effect of the drum’s tilt angle on vibration amplitude.
Pressure-viscosity coefficient (m2 N−1 ) 2.03 × 10−8

20
wheel supporting point. This vibration mode is related to the
15 n = 3 r min−1
distribution of the concrete in the drum.
n = 8 r min−1
10 Figure 17 depicts the influences of rotating speed on the
n = 15 r min−1
dynamic properties of the mixing drum when β = 12◦ and
Amplitude / m

5
h = 0 m. The vibration amplitude of the mixing drum de-
0 creases with the increase of the rotating speed. This is due
to the wider distribution of the concrete along the rotating
−5
direction and the more obviously symmetrical distribution of
−10 concrete which result in the great decrease of concrete’s ec-
centricity. Although the rotating speed increases, the exciting
−15
force becomes smaller due to the great decrease of concrete’s
−20 eccentricity.
0 1 2 3 4 5 6 7
Length / m Figure 18 shows the effects of drum’s inclination angle on
the vibration properties when n = 3 r min−1 and h = 0. The
Figure 17. Effect of rotation speed on drum’s vibration amplitude. peak amplitude gradually moves to the right with the incli-
nation angle increasing. The vibration amplitude value of the
peak’s left side decreases when tilt angle increases, while the
Figure 16a and b show vibration curve of the mixing drum right side increases. The peak amplitude is almost unchanged
with n = 3 r min−1 , β = 16◦ , h = 0 m. As shown in Fig. 16b, and gradually moves to the right with the inclination angle
the response curve distributes asymmetrically. The vibration increasing, which is caused by the right movement along the
amplitude first increases to the maximum value and then axial direction of the maximum eccentric position of the con-
gradually decrease (L = 5.6 m). The vibration amplitude of crete in the mixing drum with the increase of the inclination
the drum is almost not changed at the right side of the roller angle.

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J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum 175

25 6 Conclusions
20 h=0m
A mathematical formulation has been derived through the
h = −0.15 m
15 force analysis for calculating the supporting force. The calcu-
h = −0.20 m
10 h = −0.25 m lation method of the concrete distribution shape in the rotary
Amplitude / m

drum is developed based on some others’ previous studies.

5
The effects of rotating speed, tilting angle and concrete liq-
0 uid level on concrete shape are analyzed. A new transfer ma-
−5 trix is built with considering the concrete geometric distribu-
tion shape. The stiffness and damping calculation models of
−10
spherical roller bearings and rolling wheels are established.
−15 On the basis of these analysis, a method for calculating the
−20
dynamic performance of the mixing drum is established.
0 1 2 3 4 5 6 7 The effects of rotating speed, inclination angle and con-
Length / m
crete liquid level on the vibration performance of the mixing
Figure 19. Effect of concrete level on the rotary drum’s vibration drum are studied with a specific example. Speed-amplitude
amplitude. curves, inclination angle-amplitude curves and concrete liq-
uid level height-amplitude curves are obtained. It is found
that the vibration mode curves of the drum distribute asym-
Figure 19 presents the influences of concrete’s liquid level metrically. And with the increase of rotating speed, the vi-
height when n = 3 r min−1 and β = 12◦ . The maximum un- bration amplitude of the mixing drum decreases, the peak
balanced response amplitude of the drum increases with the amplitude gradually moves to the right with the inclination
decrease of concrete liquid level height h, which is because angle increasing, the amplitude value of the peak’s left side
that when hdecreases, the eccentric distance of the concrete decreases when tilt angle increases, while the right side in-
in the mixing drum is gradually increased which result in the creases. The maximum unbalanced response amplitude of
increase of concrete’s maximum centrifugal force. It can also the drum increases with the decrease of concrete liquid level
be seen that the vibration peak amplitude gradually moves to height, and the vibration peak moves to the left.
the left with the decrease of h.

Data availability. All the data used in this manuscript can be ob-
tained by requesting from the corresponding author.

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176 J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum

Appendix A: Notations

a0 distance between mass center and left supporting (mm)

C damping (Ns m−1 )
e eccentricity (mm)
E Young modulus (M Pa)
E0 equivalent Young’s modulus (M Pa)
Fa axial force (N)
Fc centrifugal force (N)
Fr radial force (N)
Fw roller wheel supporting force (N)
G gravity (N)
Jd diameter moment of inertia of the disc
Jp polar moment of inertia of the disc
k ratio of equivalent radii of roller-raceway contact
K stiffness (N m−1 )
L drum length (mm)
m disc mass (kg) Figure B1. Force analysis of the mixing drum.
M bending moment (N m)
Ma additional bending moment by gravity (N m)
n rotating speed (r min−1 ) Appendix B: Force analysis of the mixing drum
Q shearing force (N)
Rx equivalent radius of roller-inner raceway The force analysis of the mixing is shown in Figs. 1 and B1.
contact in rolling direction (mm)
According to the force balance principle, the forces of the
Ry equivalent radius of roller-inner raceway contact
perpendicular to the rolling direction (mm)
mixing drum in radial direction are balanced:
νm average speed of bearing (m s−1 )
Z roller’s number in each row of SRB
Fr1 + Fr2 = Fr , (B1)
α pressure-viscosity coefficient of lubricant (m2 N−1 )
β inclination angle (◦ )
Fr is the resultant force of Fc and G0 , the value of which is:
γ angle between roller wheel center q
1
2
and drum center (◦ ) Fr = G0 + Fc2 + 2e1 G0 Fc / e12 + e22 2 , (B2)
δ normal elastic deformation of roller-raceway contact (mm)
η0 dynamic viscosity of lubricant (Pa s) where G0 is a component of G, and it’s direction is perpen-
θ rotation angle of the section (rad) dicular to the axis:
ϕ azimuth angle of roller (rad)
ω angular velocity (rads−1 ) G0 = G cos β. (B3)
Superscript
L left-hand Combine Eqs. (A1), (A2) and (A3) into an equation, we can
R right-hand get:
Subscript
a axial direction q
1/2
Fr1 + Fr2 − G2 cos2 β + Fc2 + 2e1 G Fc cos β/ e12 + e22 = 0. (B4)
i inner raceway
j serial number of lumped disc
1j serial number of roller in first row of SRB The forces of the mixing drum in axial direction are bal-
2j serial number of roller in second row of SRB anced:
left left supporting point
o outer raceway Fa1 − G sin β = 0. (B5)
r radial direction
right right supporting point According to the moment balance principle, the moment at
x along the coordinate directionx the mass center is balanced:
y along the coordinate direction y q
z along the coordinate direction z a0 Fr1 + G sin β e12 + e22 − Fr2 (L6 − L5 − a0 ) = 0. (B6)

Fr2 and Fr have the opposite direction, and Fr2 is the resultant
force of Fw1 and Fw2 , which means:

Fr2 = Fw1 + Fw2 . (B7)

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J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum 177

Equation (A4) can be written as: Appendix C: Stiffness and damping model at left
supporting point
Fw1 cos γ + Fw2 cos γ = Fr2 sin (α2 + α1 ) (B8)
Fw1 sin γ − Fw2 sin γ = Fr2 cos (α2 + α1 ) , (B9) As shown in Fig. 11, Ks1 and ms1 are the vertical stiffness
and mass of front bracket. Kb1 and Cb1 are the radial stiffness
where α1 is the angle between Fc and Fr , α2 is the angle and damping of the spherical roller bearing.
between Fc and x axis as shown in Fig. 19. When considering the mass ms1 , the composite stiffness of
the front bracket is (Wen et al., 1999):
e2
α2 = arccos q (B10)
e12 + e22 Ks1 0 = Ks1 − ms1 ω2 (C1)
c , the di-
q
Fc + e1 G cos β/ e12 + e22 The vertical stiffness Ks1 0 should be changed into Ks1
α1 = arccos q
1/2 . (B11) rection of which is perpendicular to the mixing drum’s axial
G2 cos2 β + Fc2 + 2e1 Fc G cos β/ e12 + e22 direction:
Combine Eqs. (A4), (A5), (A6), (A8), (A9) into a set of equa- c
Ks1 = Ks1 0 /cos2 β. (C2)
tions, and let the right side of the equations equal 0, we can
get: The composite stiffness Ks1 c of the front bracket and the ra-

 q
1/2 dial stiffness Kb1 of the bearing are in series:


 Fr1 + Fr2 − G2 cos2 β + Fc2 + 2e1 G Fc cos β/ e12 + e22 =0
  −1

 1 1
Kleft = + c . (C3)

 q
e12 + e22 G sin β − Fr2 (L6 − L5 − a0 ) = 0




 a 0 F r1 + Kb1 Ks1


Fw1 cos γ + Fw2 cos γ − Fr2 sin (α2 + α1 ) = 0 . (B12) Bring Eqs. (A13) and (A14) into Eq. (A15), we can get:


Kb1 Ks1 − ms1 ω2 /cos2 β




Fw1 sin γ − Fw2 sin γ − Fr2 cos (α2 + α1 ) = 0


 Kleft =
. (C4)
Kb1 + Ks1 − ms1 ω2 /cos2 β





G sin β − Fa1 = 0


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178 J. Yang et al.: Study on the dynamic performance of concrete mixer’s mixing drum

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