自動(dòng)控制液壓板料剪切生產(chǎn)線-成品接料架設(shè)計(jì)【液壓剪板機(jī)】
自動(dòng)控制液壓板料剪切生產(chǎn)線-成品接料架設(shè)計(jì)【液壓剪板機(jī)】,液壓剪板機(jī),自動(dòng)控制液壓板料剪切生產(chǎn)線-成品接料架設(shè)計(jì)【液壓剪板機(jī)】,自動(dòng)控制,液壓,板料,剪切,生產(chǎn)線,成品,架設(shè),板機(jī)
Journal of Sound and Vibration (1999) 223(4), 645±657
Article No. jsvi.1999.2154, available online at http://www.idealibrary.com on
TIPPING LOADS OF MOBILE CRANES WITH
FLEXIBLE BOOMS
S. KILIC?ASLAN, T. BALKAN AND S. K. IDER
Department of Mechanical Engineering, Middle East Technical University,
06531 Ankara, Turkey
(Received 23 July 1997, and in ?nal form 4 January 1999)
In this study the characteristics of a mobile crane are obtained by using a
ˉexible multibody dynamics approach, for the determination of safe loads to
prevent tipping of a mobile crane. Only the boom of the crane is assumed to be
ˉexible since it is the only element that has considerable deˉections in
applications. The coupled rigid and elastic motions of the crane are formulated
and software is developed in order to carry out the dynamic analysis. The
variation of piston force with respect to boom angular position for dierent
boom motion times are simulated, load curves are generated and the results are
compared with the experimental results obtained from a 10 t mobile crane.
# 1999 Academic Press
1. INTRODUCTION
Cranes as mechanical systems are in general closed-chain mechanisms with
ˉexible members. In the problem of determination of safe loads which as a
function of the boom angular position, the solution of the dynamic equations is
necessary.
There are few studies related to the dynamics and control of mobile cranes for
various applications. In almost all of these studies the body ˉexibility is not
taken into consideration. A dynamic model for the control of a ˉexible rotary
crane which carries out three kinds of motion (rotation, load hoisting and boom
hoisting) simultaneously is derived by Sato and Sakawa [1]. Only the joint
between the boom and the jib is assumed to be ˉexible. The goal is to transfer a
load to a desired place in such a way that at the end of the transfer the swing of
the load decays as quickly as possible. The application of a hook load and safe
load indicator and limiter for mobile cranes is presented by Balkan where the
microprocessor-based control system for the determination of current hook load
is based on oil pressure and boom angle [2].
In this paper, mobile crane characteristics are determined by using ˉexible
multibody analysis. Kinematics and equations of motion of the ˉexible
multibody system are derived. Software has been developed to carry out
dynamic analysis of the crane. In the ˉexible dynamic analysis, the coupled rigid
and elastic motion of the system is formulated by using absolute co-ordinates
0022±460X/99/240645 13 $30.00/0
# 1999 Academic Press
646
S. KILIC?ASLAN ET AL.
and modal variables [3, 4]. Then, joint connections and prescribed motions are
imposed as constraint equations. The ˉexible body is modelled by the ?nite
element method and modal variables are used as the elastic variables by utilizing
modal transformation.
The variations of the piston force with respect to the boom angular positions
are analyzed for different boom motion times to illustrate the effect of ˉexibility
by using the developed software. Load curves are generated for various boom
motion times and compared to those of the manufacturer.
2. MODELLING OF THE CRANE
Since there is experimental work on the tipping load control of a COLES
Mobile 930 crane, for the application of the developed software, the structure of
the above mentioned crane and its parameters are used [5]. However, the method
of analysis can easily be applied to similar types of cranes with simple
modi?cations.
In general, mobile cranes are operated under blocked conditions by the
vertical jacks. The load is attached to the hook and the boom is hoisted. Since
the excessive raising of the load is dangerous, the height of the load is controlled
by lengthening the rope. During hoisting, lowering and transportation of the
load, the crane is not rotated, due to some restrictions such as very huge and/or
heavy loads, space problems, etc.
In every angular position of the boom, there is a maximum load above which
tipping might probably occur. Since the angular position of the boom changes
only during the up and down motion of the load, which is actually a planar
motion, modelling and analysis are carried out in two dimensions.
Figure 1. Schematic representation of the test crane.
MOBILE CRANES
(5)
C
n(5)
647
n(5)
1
n(2)
n2
G
A
n1
Body 1
D
Body 3
Body 5
1
2
n(1)
Body 2
n(1)
1
2
n(2)
n(3)
(3)
1
(1) (2)
n(3)
1
B
Body 4
a
A0
2
n(4)
(4)
O
n(4)
1
2
B0
Figure 2. Kinematic model of the test crane. Dimensions in mm: A0G 2000; GC 17 500; A0A
5823; A0D 5850; AD 565; BD 3455; OB02350; OA0805.
Schematic representation of the test crane is shown in Figure 1 and the
kinematic model of the test crane which can be represented by ?ve bodies is
shown in Figure 2.
Cross-section, material properties and dimensions of each body are obtained
from the technical data sheet and measured directly from the test crane. The
cross-section of Body 1 is a hollow polygon of thickness, t, as shown in Figure 3.
The cross-section dimensions increase from A0to G and decrease from G to C
linearly, and the dimensions at sections A0, G and C are shown in Figure 3.
Body 2 is a cylindrical rod 25 mm in diameter and Body 3 is a piston 180 mm in
e0
c0
A0G C
f0
t=20
e0453
c0200
f0120
160 160
440 207
120 120
Figure 3. Cross-section of Body 1. Dimensions in mm.
648
S. KILIC?ASLAN ET AL.
diameter with spool thickness of 20 mm. Body 4 is a cylinder with inner
diameter of 230 mm, outer diameter of 246 mm and length of 3440 mm.
Additionally, modulus of elasticity and mass density of the Body 1 are taken as
200 GPa and 5750 kg/m3, respectively. Mass density of other bodies are taken as
7850 kg/m3.
When dimensions (lengths and cross-sections) and elastic properties of the
bodies of the crane are considered, it is suf?cient to take only the Body 1 (the
boom) as ˉexible. In this case, other bodies are assumed to be rigid.
The following assumptions are considered in the analysis of the crane.
1. The mass of the hydraulic oil is included in the mass of the cylinder
(Body 4). Varying mass of the cylinder due to varying amounts of hydraulic oil
inside it is taken into consideration.
2. Hydraulic oil is assumed incompressible.
3. The hook load is considered as a point mass and connected to the end of
the boom with a rope which is taken as a rigid rod. This rope is free for planar
rotation about point C. This assumption is valid as long as the oscillations of the
rod about the vertical position are small and the rod remains in tension. These
conditions are satis?ed for normal operation speeds and hook loads.
4. The structural damping of the boom is taken into account by assuming
Rayleigh damping.
5. The distance between the load and the base is assumed to be kept constant
by varying the length of the rope during the up and down motion of the crane.
3. DYNAMIC EQUATIONS
Let nk represent a body reference frame relative to which the deformation of
Body k is de?ned and n represent a ?xed frame. Let xk represent the position of
the origin Q of nk in n, and ok be the angular velocity of Body k.
Using the ?nite element method, the deformation displacement vector uki of an
arbitrary point P in element i of Body k is
where fki is the element shape function matrix transformed to nk, Bki is the
element connectivity Boolean matrix and ak is the vector of body nodal
variables.
The velocity of P is written as
where qki is the position vector from Q to P in nk including deformation, ~qki is
the skew symmetric matrix of qki, Tk is the co-ordinate transformation from nk
to n, o" k TkiTok, wk is modal transformation used to reduce the elastic degrees
of freedom, and Zk is the vector of body modal variables. Equation (2) can be
expressed as
MOBILE CRANES
649
The joint connections and prescribed motions in the system of N
interconnected bodies are represented by kinematic constraint equations
expressed at velocity level as
where y is the system generalized speed vector given by
C is the constraint Jacobian matrix which can be formed by the velocity
inˉuence coef?cient matrices and g indicates the prescribed velocities.
Kane's equations are used to determine the equations of motion of the system
as
where l is the vector of constraint forces, M is the generalized mass matrix, Q,
Fs, Fd and F are vectors of Coriolis forces, elastic forces, damping forces and
applied forces, respectively and
650
v0
v(t)
0
S. KILIC?ASLAN ET AL.
t1t2
t3
Figure 4. Velocity pro?le with cycloidal acceleration and deceleration.
where Kk is the structural stiffness matrix and Dk is the structural Rayleigh
damping matrix of Body k. In the simulations, the weights of the structural
mass and stiffness matrices used in forming Dk correspond to a 2% damping
ratio.
When the space dependent terms in equations (8) and (9) are separated, a set
of time invariant matrices are obtained [3, 4]. These mass properties are
evaluated once in advance. Equation (6) and the derivative of equation (4)
represent linear equations for the accelerations y and the constraint forces l. The
accelerations obtained from these equations are numerically integrated by using
a variable step, variable order predictor±corrector algorithm to obtain the time
history of the generalized speeds and generalized co-ordinates.
The boundary conditions used for the description of the deformation of Body
1 are that for axial deformation A0is ?xed, and for bending A0is hinged and A
is ?xed. The ?rst axial mode, the ?rst bending mode of part A0A and the ?rst
two bending modes of part AC of Body 1 are taken as the modal co-ordinates
since the higher modes are observed negligible. Therefore the generalized speed
vector of the system is
MOBILE CRANES
651
The boom is driven by a hydraulic actuator which is controlled by the
operator. In general, throughout the motion, the hydraulic actuator is driven
with constant velocity v0so that the boom and piston oscillations are kept to a
minimum level. Moreover, to avoid impact loading, the actuator velocity is
increased from zero to v0at the beginning of the motion and decreased from v0
to zero at the end of the motion which can be assumed cycloidal in time. This
desired velocity pro?le is shown in Figure 4 and can be expressed as follows.
If the pivots of Bodies 1 and 2 were at different points, the system would be a
structure. The system is moveable owing to the special dimension obtained due
to the concurrency of the pivots. Thus, the constraint equations written for Body
1 and Body 2 are linearly dependent. For this reason, one of the constraint
equations is dropped to remove the linear dependency.
652
600
400
200
0
20
S. KILIC?ASLAN ET AL.
Experimental
Simulation
40
60
80
Boom angular position (degree)
Figure 5. Piston force with respect to boom angular position (32á4 kN hook load and 30 s
boom upward motion).
4. COMPUTER SIMULATION OF THE CRANE CHARACTERISTICS AND
COMPARISON WITH THE EXPERIMENTAL RESULTS
Software has been developed for the analysis of the test crane. In this
software, one can take any number of ?nite elements and modal variables for
Body 1.
Experimental studies have been carried out by Balkan for the working range
of the boom in which the boom was moved in 30 s [2]. This speed was selected in
order to minimize the effect of ˉexibility. In that study, the pressure in the
hydraulic actuator and the angular positions of the boom were measured. The
oscillations in the pressure resulting from the boom oscillations are ?ltered out
800
400
0
20
40
60
80
Boom angular position (degree)
Figure 6. Piston force with respect to boom angular position (32á4 kN hook load and 10 s
boom upward motion).
Piston force (kN)
Piston force (kN)
0.00
–0.08
–0.16
MOBILE CRANES
Node 3
8
12
653
20
40
60
80
Boom angular position (degree)
Figure 7. Transverse deˉections of nodes 3, 8 and 13 with respect to boom angular position
(32á4 kN hook load and 30 s boom upward motion).
in the control system, hence they are not seen in the measured data. The test
crane was moved with a 32á4 kN hook load in the upward direction, and the
variations of the pressures in the hydraulic actuator with respect to the boom
angular positions are obtained for the 30 s motion of the boom. Therefore, the
variations of the piston force with respect to the boom angular positions for the
30 s boom upward motion can be calculated for the 32á4 kN hook load.
The variations of the piston force with respect to the boom angular positions
for the 32á4 kN hook load are simulated for the 30 s boom upward motion by
using the computer code and given in Figure 5.
Experimental results for the 30 s motion of the boom are also shown in Figure
5. The data do not include piston acceleration and deceleration intervals.
Moreover, since the boom oscillations are ?ltered out, they are not seen in the
Node 3
0.00
8
–0.08
13
–0.16
20
40
60
80
Boom angular position (degree)
Figure 8. Transverse deˉections of nodes 3, 8 and 13 with respect to boom angular position
(32á4 kN hook load and 10 s boom upward motion).
Transverse deflection (m)
Transverse deflection (m)
654
0.00
S. KILIC?ASLAN ET AL.
–0.08
–0.16
0
2
4
6
8
10
0
4
8
Time (s)
Frequency (Hz)
Figure 9. (a) Time response of transverse deˉection of node 13. (b) FFT of transverse deˉec-
tion of node 13.
?gure. It is seen from the ?gure that simulation and experimental results for the
30 s boom motion are close to each other.
Similarly, the variations of the piston force with respect to the boom angular
positions for the 32á4 kN hook load are simulated for the 10 s boom upward
motion by using the computer code in order to make the effect of ˉexibility
more signi?cant as shown in Figure 6.
In the simulations, the boom is discretized by 12 ?nite elements. Two of them
are taken on A0G where the cross-sectional area is increasing from A0to G
linearly and ten of them are taken on GC where the cross-sectional area is
decreasing from G to C linearly. Damping is included for Body 1 by using a 2%
damping ratio for the ?rst two modes. It is assumed that the ?rst 1á5 s is used
for the acceleration and the last 1á5 s is used for the deceleration of the boom for
the 30 s boom motion. In the case of 10 s boom motion, acceleration and
deceleration intervals are assumed to be 1 s.
A0
FA01B0
FA
02
F
B01
FB02
B1A1
FA
A
FC
A
C
A2
B2
B
FB
Figure 10. Free body diagram of the crane chassis. Dimensions in m: A 5á50; C 3á37; A12á57;
A24á60; B11á77; B22á25.
Tranverse deflection (m)
Arbitrary units
160
120
80
40
0
MOBILE CRANES
655
6
10
Radius (m)
14
18
Load (kN)
656
S. KILIC?ASLAN ET AL.
The magnitudes at small frequencies correspond to the trend due to the
excitation of the system. As the boom moves upwards, it goes towards the
vertical position causing the boom transverse deˉections to decrease. The
frequency due to the load oscillations also falls into this frequency range. The
frequencies over 1á5 Hz are due to the boom oscillations at its natural
frequencies. The variation of the natural frequency with time is a characteristic
of multibody systems and results in a chirp signal as seen in Figure 9(a).
5. SIMULATION OF THE LIFTING CAPACITY ON THE HOOK
Tipping simulation is performed in the blocked condition of the crane to see
when tipping occurs as the boom moves in the upward and downward
directions. When one of the reaction forces coming from the ground to the
vertical jacks becomes zero, tipping occurs. Using the free body diagram of the
crane chassis, shown in Figure 10, equation (21) is written for the tipping case
as
where FA01, FA02 and FB01, FB02 are components of the reaction forces exerted by
the boom and the cylinder on the crane chassis; FAand FBare the reaction
forces exerted by the ground to the jacks and FCis the body force of the crane
chassis.
When FAis smaller than or equal to zero, tipping condition occurs. For the
30 s and 10 s boom motions, the boom angular positions where FAbecomes zero
are determined by using the developed software for different hook loads. The
simulation results for the test crane are given in Figure 11. The allowable load
speci?ed by the manufacturer of the test crane is also shown in Figure 11 where
radius is de?ned as the horizontal distance from the vertical axis of rotation of
the crane to the tip of the boom at the tipping position, calculated as
It can be seen from Figure 11 that when the boom motion time is decreased,
the allowable load for the same radius decreases. Although there is no
information about the conditions such as boom motion time while the allowable
load data are being obtained, the plot of the allowable load is very similar to
30 s boom motion time simulation. In addition to this, it is noted by the
manufacturer that these allowable load data should be used with a safety factor
of 1-5.
6. CONCLUSION
In this study, the mobile crane characteristics are determined by using ˉexible
multibody analysis. In order to achieve this goal software has been developed
which is capable of carrying out dynamic analysis of the crane.
The coupled rigid and elastic motions of the system are formulated by using
absolute co-ordinates and modal variables [3, 4]. Then, joint connections and
prescribed motions are imposed as constraint equations. The ˉexible body is
MOBILE CRANES
657
modelled by the ?nite element method and the modal variables are used as the
elastic variables by utilizing modal transformation.
The variations of piston force with respect to the boom angular positions for
32á4 kN hook load are simulated for both 30 s and 10 s boom upward motions
by using the computer code for the velocity pro?le with cycloidal acceleration
and deceleration. 30 s boom motion simulations are compared with the
experimental results. Moreover, transverse deˉections of node 3, node 8 and
node 13 are obtained with respect to the boom angular positions for both 30 s
and 10 s boom upward motion. Finally, load curves are generated for the 30 s
motion and 10 s motion and compared with those of the manufacturer.
It is seen from the analysis that the boom motion time affects the crane
dynamics considerably. For lower piston speeds (i.e., 30 s motion of the boom),
the effect of ˉexibility is very small. Thus, the boom can be taken as a rigid
body. However, when the piston speed is increased (i.e., 10 s motion of the
boom), the effect of ˉexib
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