Numerical study of the hydraulic excavator overturning stability during performing lifting operations [original]

Research Article

Advances in Mechanical Engineering

2019, Vol. 11(5) 1–14

ÓThe Author(s) 2019

DOI: 10.1177/1687814019841779

journals.sagepub.com/home/ade

Numerical study of the hydraulic

excavator overturning stability during

performing lifting operations

Rosen Mitrev

and Dragan Marinkovic

Abstract

This article presents a numerical study of the stability of a hydraulic excavator during performing lifting operations. A pla-

nar dynamic model is developed with six degrees of freedom, which considers the base body elastic connection with the

terrain, the front digging manipulator links, and the presence of the freely suspended payload. Differential equations

describing the excavator dynamic behavior are obtained by using the Lagrange formalism. Numerical experiments are

carried out to study the excavator dynamic stability under different operating conditions during the motion along a verti-

cal straight-line trajectory. It is shown that the arising inertial loads during the movement of the links along the vertical

trajectory, combined with the payload swinging and the motion of the base body, decreases the excavator stability. It

was found that the excavator stability during following vertical straight-line trajectory decreases considerably in the

lower part of the vertical trajectory. If the stability coefficient is close to 1, the payload swinging can cause the separation

of a support from the terrain; nevertheless, the excavator stability can be restored. A method for tire stiffness and

damping coefficients estimation is presented. The validation of the dynamical model is performed by the use of a small-

scale elastically mounted manipulator.

Keywords

Hydraulic excavator, lifting operations, modeling, overturning stability

Date received: 10 June 2018; accepted: 12 March 2019

Handling Editor: James Baldwin

Introduction

Due to the considerable efforts in the industry in the

past decades, the hydraulic excavators have trans-

formed from specialized heavy-duty earth-moving

machines, whose primary function is the digging, into

multifunctional devices. By the use of specialized

attachments and working tools, they can perform a

wide variety of technical tasks that are specific for

other types of machines. This has allowed many opera-

tions at the construction and mining sites to be per-

formed by a smaller number of machines and, thus, to

increase the economic efficiency.

One of the widespread activities is the use of the

hydraulic excavators as cranes.

The typically per-

formed operations are lifting, moving, and positioning

of freely suspended payloads, such as concrete pipes

and beams, industrial equipment, and construction ele-

ments. The payload is attached by slings to the lifting

points, hooks, lifting eyes, lashing points, or other spe-

cial attachment that has been provided and approved

by the excavator manufacturer.

Faculty of Mechanical Engineering, Technical University of Sofia, Sofia,

Bulgaria

FG Strukturmechanik und Strukturberechnung, Institut fu¨r Mechanik,

Technische Universita

¨t Berlin, Berlin, Germany

Corresponding author:

Dragan Marinkovic

´, FG Strukturmechanik und Strukturberechnung,

Institut fu¨r Mechanik, Technische Universita

¨t Berlin, Straße des 17. Juni

135, Berlin 10623, Germany.

Email: dragan.marinkovic@tu-berlin.de

Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License

(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without

further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/

open-access-at-sage).

Despite the undeniable convenience when using an

excavator as a crane, the lifting operations performed

have some peculiarities, leading to the reduction of the

excavator exploitation reliability. Basically, excavators

are not designed for lifting operations and the operator

is trained to perform earth-moving operations but not

to manipulate freely suspended payloads. Even in the

case of a very skilled operator, this leads to an undesir-

able payload swinging and, as a consequence, to the

increase of dynamical loads, which negatively influ-

ences the excavator exploitation reliability. The reduced

ability of the excavators to perform lifting operations

increases the risk of accidents with the staff near the

excavator, and especially in the vicinity of the freely

suspended payload. The statistical reports show a con-

siderable risk of injuries caused by the strike by the

bucket or swinging boom.

Edwards and Holt

have

studied the hazards associated with using construction

excavators as cranes and have investigated the cata-

strophic failure of a link. The same authors have stud-

ied excavator overturn accidents.

According to Lim

et al.,

38% of the excavator accidents in Korea are

due to performing lifting operations. Some require-

ments for the safety of the excavators are presented in

the standards.

6,7

The literature survey reveals different aspects of the

excavator exploitation as a crane. Some documents

emphasize on the practical aspects of the problem,

while other papers aim at the design and development

of a research methodology and simulation models. Lim

et al.

presented the application of the Zero Moment

Point theory to the computation of the excavator stabi-

lity during performing lifting operations. Yu et al.

introduced the Static Compensation Zero Moment

Point algorithm that has a more accurate output com-

pared to the general algorithm. Heep et al.

proposed

a method for payload estimation by the use of a multi-

body simulation.

Reduced overturning stability is the most critical

indication of reduced exploitation reliability as it may

even lead to accidental situations. The problem of using

excavators as cranes and investigations of their over-

turning stability during performing lifting operations in

the vertical plane of the front manipulator motion has

not received enough attention over the years, mainly

because such type of work operations are not typical

for this type of machines, but are performed in prac-

tice. In this aspect, more attention has been given to

some other machines that are similar to the excavators.

Although ensuring the stability of the mobile machin-

ery is an up-to-date subject and intensive research is

carried out to improve it, much of the recent research

does not take into account the dynamic nature of the

loading due to the payload swinging, elastic supports,

and moving links.

Rauch et al.

developed a

dynamic model of a mobile boom crane and studied

the tip-over stability by a simplified semi-dynamic

approach to account for the payload swing and inertia

forces. Since the dynamic stability is influenced by the

foundation–soil interaction, Messioud et al.

used

complex mathematical models, but Towarek

obtained

adequate results for the dynamic stability of a boom

crane by the use of a two-parameter rheological model

for the representation of the soil. Abo-Shanab and

Sepehri

presented the development of a model that

can adequately simulate the dynamic stability of

manipulators mounted on moveable platforms.

Janosevic et al.

investigated the influence of the

hydraulic transmission oil temperature on the excava-

tor dynamic stability. The presented mathematical

model takes into account the hydraulic cylinders com-

pliance due to the hydraulic fluid compressibility.

Another research by Mitrev et al.

resulted in a com-

plex multibody model of a classical hydraulic excava-

tor, based on the use of Lagrange multipliers, which

allows the study of dynamic behavior during digging

operations. The approaches for dynamic modeling and

control of different types of cranes were presented in

the very detailed and extensive study by Abdel-

Rahman et al.

Fodor et al.

presented a very interest-

ing approach based on an input shaping control tech-

nique for oscillation reduction in the vertical plane of a

forestry crane in order to improve the overturning sta-

bility of the machine.

Along with the widely used free or commercial com-

puter codes for simulation of multibody systems, the

dynamic models of a wide range of physical systems are

formulated and studied by use of analytical

21–23

numerical techniques.

Setting the mathematical model

represents the very core of dynamic modeling of such

type of systems composed of hydraulic and mechanical

subsystems.

Recent investigations

show the feasi-

bility of such an approach to study dynamic behavior

of excavators, despite the fact that in certain cases, the

resulting dynamic equations are quite demanding to

resolve. However, the literature review also shows that

the available dynamical models of excavators and simi-

lar mobile machines cannot be used reliably for all types

of performed operations because the models do not

include the peculiarities of the hydraulic excavator kine-

matic structure in performing lifting operations and

especially the presence of a freely suspended payload

attached to the bucket and elastic support. For this rea-

son, the main aim of this article is to numerically inves-

tigate the dynamic behavior and overturning stability

of the hydraulic excavator during performing lifting

operations. For this purpose, kinematic and dynamic

models of the excavator are developed, and numerical

experiments are carried out to study dynamic behavior

of the system in different operating conditions.

This article is organized as follows: ‘‘Introduction’’

section lists some peculiarities during the performance

2Advances in Mechanical Engineering

of lifting operations by hydraulic excavators. Some

studies concerning different aspects of the dynamic

modeling of the excavators and similar machines are

reported. ‘‘Kinematic model and trajectory planning’’

section presents the solution to the inverse kinematics

task and describes the trajectory planning. ‘‘Dynamic

model’’ section is focused on dynamic modeling of the

excavator and the derivation of the system of differen-

tial equations. ‘‘Overturning stability of the excavator’’

section points out some peculiarities concerning the

excavator overturning stability. The results of the

numerical experiments are shown in ‘‘Numerical experi-

ments and discussions’’ section. In ‘‘Parameter estima-

tion and validation of the mathematical model’’ section,

a method for determination of the tire characteristics

and experimental model validation are presented. The

article concludes with a summary in ‘‘Conclusion’’

section.

Kinematic model and trajectory

planning

Figure 1 shows a hydraulic excavator performing typi-

cal lifting operation—lifting/lowering of the payload R

along the vertical straight-line trajectory. The cutting

edge, denoted by H, serves as a reference point for the

start (xH,ys

H) and end (xH,yf

H) points of the vertical lift-

ing trajectory. The payload is attached to the lifting

point Pby slings, allowing its free swinging in the verti-

cal plane. There are two options for controlling the

movement of the front digging manipulator.

In the first case, the cutting edge Hmoves along a

straight line and the angle of the bucket tilt is kept con-

stant relative to the stick. In this case, the manipulator

is considered as a system with two controllable degrees

of freedom—the boom and the stick together with the

bucket—and one uncontrollable degree of freedom—

the free payload swinging.

The motion of the cutting edge along the trajectory

leads to the change of the bucket orientation uwith

respect to the horizontal line (see Figure 1), resulting in

a change of direction of the payload weight force

according to the lifting point causing its unfavorable

loading. The other drawback due to the bucket rota-

tion is the occurrence of the horizontal movement of

the lifting point Pwhich induces a payload swinging

and, as a consequence, poor positioning of the payload

and increased risk of accidents, especially in confined

spaces.

In the second case, during the motion along the ver-

tical trajectory, the bucket hydraulic cylinder is addi-

tionally controlled to maintain a constant bucket angle

u. The system has four degrees of freedom, three of

which are controlled and one is uncontrolled. The

above described motion along a predefined trajectory

can be considerably facilitated by the use of the widely

used

and well-studied excavator automatic digging

system.

29–31

The current article investigates the second case. The

values of the angles u2,u3, and u4(see Figure 1) deter-

mine the horizontal and vertical position of the cutting

edge in the plane. In order to find the angle values cor-

responding to the motion of the cutting edge along the

straight line with start point with coordinates (xH,ys

and end point with coordinates (xH,yf

H), one must solve

the inverse kinematics task for a three link planar

manipulator.

The angular velocities and accelerations

of the links are determined by using the mechanical sys-

tem Jacobian. Due to the constant orientation of the

bucket _

u=€

u=0, the velocity of the lifting point Pis

the same as the velocity of the cutting edge H. The

determination of the values of the angles u2,u3, and u4

and their derivatives as a function of the cutting edge

position and bucket orientation completely ends the

inverse kinematics task solution.

The goal of the trajectory planning is to transport

the payload from the initial point ys

Hto the final point

H(see Figure 1) keeping the motion of the bucket ref-

erence point Halong a vertical straight line with a pre-

defined constant position xHinside the excavator

workspace. The desired trajectory for the vertical

motion of point His proposed in the form of a fifth-

order polynomial.

Imposing zero initial and final con-

ditions for the velocity and acceleration of the point H,

the polynomial takes the following form

HtðÞ=ys

H+

t3yf

Hys



6t215ttf+10t2



ð1Þ

where t

denotes the duration of the motion along the

trajectory.

Figure 1. Schematic view of an excavator performing lifting

operations.

Mitrev and Marinkovic

´3

Dynamic model

Figure 2 shows a geometrical layout of the excavator

with the consideration of the payload swinging. The

excavator mechanical system is modeled as an open

kinematic chain undergoing planar motion and consist-

ing of a base body (composed of traveling and swing

bodies, represented as a single body), boom, stick,

bucket, and payload. The links are interconnected by

rotational joints and are subjected to forces generated

by the hydraulic actuators and gravity. Each link is

characterized by its geometrical and inertia parameters.

The base body is connected to the ground by spring-

damper elements which represent the elastic and damp-

ing properties of the excavator compliant tires, outrig-

gers, or undercarriage. The presented system has six

degrees of freedom, and thus, six generalized coordi-

nates are used to define the system geometrical config-

uration. The base body can move in the vertical

direction—generalized coordinate s(heave)—and

rotate according to its mass center C

—generalized

coordinate u

(pitch). The boom, stick, bucket, and

payload can rotate according to the connecting joints

(points E, F, G, and P)—the corresponding generalized

coordinates are u

, and u

. All angles are mea-

sured counterclockwise.

An up-to-date analysis of methods suitable for deri-

vation of the mathematical models of cranes is pre-

sented in Abdel-Rahman et al.

In order to derive a

dynamic model suitable for the study of the lifting

operations, in this article, we make the following

assumptions: (1) the payload is considered as a point

mass; (2) the stiffness of the excavator elements is

neglected, and they are considered as rigid bodies; (3)

the inertia properties of the hydraulic cylinders and lin-

kages are included into the inertial parameters of the

boom, stick, and bucket; (4) the backlashes and the

friction forces in the joints are neglected; and (5) the

compliance of the hydraulic cylinders is neglected.

The dynamic equations of motion of the excavator

during the lifting operations are derived in a systematic

way using the Lagrange formalism

∂La

∂_

∂La

∂qi

+∂D

∂_

=Qii=1=6ðÞð2Þ

where the Lagrangian L

represents the difference

between the kinetic Tand potential Penergies of the

system studied, Ddenotes the dissipation energy of the

system, and Q

are the generalized forces associated

with the generalized coordinates.

The kinetic and potential energies of the system are

positive definite quadratic forms of the, respectively,

generalized velocities _

q=½_

u5_

sT

and generalized coordinates q=½u1u2u3u4

u5sT.

Kinetic energy of the system

The total kinetic energy of the system comprises the

kinetic energies of the base body T

, boom T

, stick T

bucket T

, and swinging payload T

. Using the nota-

tions in Figure 2 and according to Ko

¨nig’s theorem,

the total kinetic energy of the mechanical system is

obtained as follows

T=X

i=1

Tið3Þ

where 2T1=m1_

s2+J1

1,2T2=m2(_

C2+_

C2)+J2

12,

2T3=m3(_

C3+_

C3)+J3

123,2T4=m4(_

C4+_

C4)+

1234, and 2T5=m5(_

R+_

R). Equation (12) uses the

Figure 2. Dynamic model of the hydraulic excavator.

4Advances in Mechanical Engineering

following short notations: _

12 =(_

u1+_

u2)2,

123 =(_

u1+_

u2+_

u3)2, and so on; _

xCi and _

yCi denote

the absolute velocities of the mass centers of the links.

Similarly, _

xRand _

yRdenote the absolute velocities of

the payload. All velocities are obtained by a time differ-

entiation of the corresponding position vectors accord-

ing to the global coordinate system (CS).

As is shown in Figure 2, a local CS {x

} is con-

nected to every link of the kinematic chain. The CS

} is connected to the gravity center C

of the base

body, the CS {x

}, {x

}, and {x

} are con-

nected to the joints E, F, G, and P, respectively. The

transition between the local CS is performed by the use

of transformation matrices in the following form

iTa,ax,ay



cos asin aax

sin acos aay

001

5ð4Þ

where iand jdenote the numbers of the adjacent CSs,

ais the angle between the CSs, ½axayTrepresents

the vector from the origin of CS ito the origin of CS j

expressed in the CS i. The kinetic energy of the system

depends on the system geometrical configuration and

the velocities of the links.

Potential and dissipative energies of the system

The potential energy of the system is computed as a

sum of the potential energies of the links P

and the

deformations of the springs P

P=PG+Pkð5Þ

The potential energy of the spring deformation is

easily determined by use of Figure 3, where the dashed

line depicts the position of the base body when the

springs are undeformed. In this position, the gravity

center C

of the base body coincides with the beginning

Oof the global CS. The normal line shows the position

of the base body when the springs are deformed due to

the vertical translation sand rotation u

If Dy

denotes the vertical spring deformation in

point B,then

DyB=0y00

B0y0

Bð6Þ

where 0y00

Band 0y0

Bare ycoordinates of point Bin the

global CS for a deformed and undeformed spring,

respectively.

Denoting the local coordinates of point Bin the base

body local CS by 1xBand 1yBand taking into account

that

0y00

B=s+1yBcos u1+1xBsin u1ð7Þ

and

0y0

B=1yBð8Þ

then for the spring deformation, one can write

DyB=s+1yBcos u11ðÞ+1xBsin u1ð9Þ

In a similar manner, for the deformation of the

spring in point A, we have

DyA=s+1yAcos u11ðÞ+1xAsin u1ð10Þ

Then, the total potential energy of the spring defor-

mation is

2Pk=kDy2

B+Dy2

 ð11Þ

The potential energy of a single excavator link is

computed as the work required to raise its gravity cen-

ter according to the xaxis of the global CS, that is, it is

a function of the system geometrical configuration. The

potential energy of the links is the sum of the potential

energies of the base body P1

G, boom P2

G, stick P3

bucket P4

G, and payload P5

Gand is given as

PG=X

i=1

Gð12Þ

where P1

G=m1gs,P2

G=m2g(L1xs1+L1yc1+LC2xs12 +

LC2yc12 +s), P3

G=m3g(L1xs1+L1yc1+L2xs12 +LC3xs123

+LC3yc123 +s), P4

G=m4g(s+L1xs1+L1yc1+L2xs12 +

L3xs123 +LC4xs1234 +LC4yc1234), P5

G=m5g(L1xs1+L1yc1

+L2xs12 +L3xs123 +L5xsd1234 +L6xsd12345 +s), sd1234 =

sin(d+u1+u2+u3+u4), and so on.

The dissipative energy of the system has the same

structure as the potential energy (equation (11)) of the

spring deformation, thus

Figure 3. Scheme for determining spring deformation.

Mitrev and Marinkovic

´5

2D=bD_

B+_

Dy2

 ð13Þ

Using the already determined deformations of the

spring-damper elements and their first derivatives, it is

possible to determine the dynamic forces in the spring-

damper elements, attached to the points Aand B

FABðÞ

=kDyABðÞ

+bD_

yABðÞ ð14Þ

One can take into account that during the normal

operating conditions, the computed forces must have

only negative or zero values, which corresponds to the

negative or zero values of the spring-damper element

deformations. In case of positive values of the spring-

damper element deformations, a separation of the tires

from the terrain is observed and the spring-damper ele-

ments are not active anymore.

Differential equations of the system

Using the derived equations for the kinetic, potential,

and dissipative energies and performing the mathemati-

cal operations in equation (2), the nonlinear dynamic

equations of motion of the excavator are obtained in

the form of

MqðÞ

€

q+Vq,_

qðÞ+GqðÞ=Qð15Þ

The notations used are as follows: inertia matrix

M(q)636, consisting of inertia terms, it should be noted

that the inertia matrix is symmetric and positive defi-

nite; vector V(q,_

q)631, consisting of centrifugal and

Coriolis terms; vector G(q)631, consisting of terms pro-

portional to the weight of the links; vector Q631, con-

sisting of generalized forces and moments, associated

with the respective generalized coordinates: Q1=0,

Q2=t2,Q3=t3,Q4=t4,Q5=0, and Q6=0where

tidenotes the driving torques applied to the corre-

sponding joints. The elements of the matrices are not

presented here due to their rather large size.

Due to the already available inverse kinematics solu-

tion for the generalized coordinates u

, and u

and

their corresponding velocities and accelerations (see

‘‘Kinematic model and trajectory planning’’ section),

the system of differential equations (15) are divided into

two sets.

The first set of equations consists of the three equa-

tions which describe the motion of the system for coor-

dinates u

, and s:

€

qI=MqðÞ

1

IVq,_

qðÞ

I+GqðÞ



ð16Þ

where €

qI=½€

€

u5€

sTdenotes the vector of the gen-

eralized accelerations and M(q)I,V(q,_

q)I, and G(q)I

denote the matrices, corresponding to the first set of

equations. The system of equations (16) is considered

as an initial-value problem and is solved by a standard

numerical routine.

The second set consists of the rest of the equations

used to compute the necessary driving torques t

and t

QII =MqðÞ

€

qII +Vq,_

qðÞ

II +GqðÞ

II ð17Þ

where QII =½t2t3t4Tdenotes the vector of driv-

ing torques and €

qII =½€

€

u4Tdenotes the vector

of the known accelerations. M(q)II ,V(q,_

q)II , and G(q)II

denote the matrices, corresponding to the second set of

equations.

Overturning stability of the excavator

As Alexandrov

states, the overturning stability of the

mobile machine is defined as its ability to resist the

overturn. The excavator performing lifting operations

will overturn around the tipping axis (point B, see

Figure 2) if the overturning moment is greater than the

restoring moment for a sufficient period of time. The

excavator overturning stability is estimated by the over-

turning stability coefficient

us=Mres

Mov

ð18Þ

whose value must be equal or greater than 1.15. In

equation (18), M

res

and M

denote the sums of the

restoring and the overturning moments around the

excavator tipping axis, respectively. When u

= 1, the

balanced condition is reached and the reaction in the

left support (point A) becomes equal to 0.

The arising inertial loads during the movement of

the links, combined with the payload swinging and

motion of the base body due to the elastic mounting

and soft ground, can contribute to the decreasing of

the excavator overturning stability. That is why during

the performance of lifting operations, the dynamic

character of M

res

and M

must be considered and the

dynamic stability coefficient udyn

smust be used to deter-

mine the stability of the excavator.

Numerical experiments and discussions

The developed system of differential equations (15) is

used in order to perform numerical experiments for dif-

ferent operating conditions and study the excavator

static and dynamic stability. The inertia and geometri-

cal data are obtained by CAD modeling of a real-life

excavator design. The numerical values used for the

inertia, geometrical, stiffness, and damping data of

the excavator elements are shown in Table 1, where the

measurement units are as follows: for mass (kg), for

mass moment of inertia (kg m

), for length (m), for

6Advances in Mechanical Engineering

angles (degrees), for spring constant (N/m), and for

damping coefficient (Ns/m).

To study the dynamic behavior of the system during

the different operating conditions, a few simulations

are carried out for the following parameters of the tra-

jectory: ys

H=4m,yf

H=4:5m, x

=4m, t

=12s,

and the bucket angle is kept constant at u=180°. The

simulated motion corresponds to the payload lowering.

The length of the simulations is divided into two time

intervals. The first interval is [0, t

] and it corresponds

to the motion along the trajectory. The second interval

+ 3] is intended to capture the dynamic effects in

the system after the end of the motion along the trajec-

tory. The initial values of the coordinates u

(0) and s(0)

correspond to the stable equilibrium position of the

system, determined by the gradient of the potential

energy (equation (5)) with respect to the generalized

coordinates.

In the first simulation, the initial angle of the pay-

load is set to u

(0) = 20°(0°from the vertical position),

which corresponds to the motion along the vertical tra-

jectory without a payload swinging. In the second

simulation, the initial angle of the payload is set to

(0) = 10°(10°from the vertical position), which cor-

responds to the presence of a payload swinging.

To clarify the dynamical behavior of the excavator,

frames from the motion animation of the two consid-

ered cases are shown in Figure 4. The following nota-

tions are used: 1—mechanical structure; 2—trajectory

of the gravity center CG of the whole structure; 3, 4, 5,

and 6—the trajectories of the gravity centers C

, and Rof the boom, stick, bucket, and payload,

respectively. It is remarkable that despite the constant

distance between the payload Rand the right support

Bduring the vertical motion along the trajectory 7, the

gravity center CG of the whole system moves toward

the right support, but still remains between the two

supports. The reasons for that are the variable boom

and stick x-positions (curves 3 and 4), also the payload

swinging and base body motion, thus reducing the

Table 1. Values for the inertia and geometrical data.

Link Value

Base body m1=183103,J1=7:533103,L1x=0:51, L1y=0:98, LC1x=0, LC1y=0,

1xB=1:5, 1yB=1:04, 1xA=1:1, 1yA=1:04

Boom m2=2:523103,J2=4:2083103,L2x=4:5, LC2x=2:2, LC2y=1:19

Stick m3=1:833103,J3=1:263103,L3x=3, LC3x=0:93, LC3y=0:19

Bucket m4=540, J4=148, L4x=1, L5x=0:6, LC4x=0:6, LC4y=0:52, d=70

Payload m5=2:53103,L6x=1:5

Spring and damping constants k=4:613106,b=45:013103

Figure 4. Frames of the motion animation and trajectories of the specific points: (a) in the absence of payload swinging and (b) in

the presence of payload swinging.

Mitrev and Marinkovic

´7

stability reserve. This fact must be taken into account

during the excavator exploitation and used to create

the mobile machine load charts for the considered case.

For the case without a payload swinging, in

Figure 5(a), the static coefficient of stability ust

sis pre-

sented jointly with the dynamic one udyn

sas a function

of the position yd

Halong the trajectory. The static stabi-

lity coefficient is calculated without considering the

support deformations and by taking into account only

the weights of the links, the payload, and the base

body. As can be seen from the presented results, the

dynamic coefficient is slightly smaller than the static

one. The main reason for that is the deformation of the

supports in the equilibrium position leading to the

machine tilting to the right (see Figure 2) and, as a con-

sequence, an increase of the overturning moments and

decrease of the restoring moment is observed. Figure

5(b) shows the graph of the dynamic stability coeffi-

cient in the case of the presence of a payload swinging.

Its values have oscillatory behavior, induced by the

interaction of the payload swinging (see Figure 4(b),

curve 6) and the oscillatory structure. For example, for

H=2:5m, the value of the dynamic coefficient of the

overturning stability is smaller than the static one by

approximately 8%. The reduction of the excavator sta-

bility due to the presence of payload swinging should

be taken into account during the performance of lifting

operations, especially when the coefficient of stability is

close to its limit value.

To study the system dynamic behavior when the stabi-

lity coefficient is close to 1, additional numerical experi-

ments are conducted. For this purpose, the mass of the

payloadischosentobem

= 4300 kg and the simulation

model is modified to allow the separation of the left sup-

port (see Figure 2, point A) from the ground. The total

length of the simulation is set to 17 s. The first numerical

experiment is performed without a payload swinging.

Figure 6(a) shows the time evolution of the deformation

of the spring in the left support Dy

, computed according

to equation (10). The shape of the deformation graph is

influenced by the vertical motion of the payload com-

bined with the vibrations due to the elastic mounting.

The values of the deformation are negative during the

simulation interval. Around 7.5 s, the deformation is very

close to 0 which means that balance point is reached, but

the left support is still in contact with the ground. The

time evolution of the dynamic stability coefficient for the

considered case is shown in Figure 6(b).

The second experiment is performed in the presence

of payload swinging, achieved by setting the initial

value of the payload angle equal to 7°, measured from

the vertical position. Figure 7(a) clearly shows the

separation of the left support from the ground—the

maximum value of Dy

is 0.08 m. After the first separa-

tion, one can see three consecutive impacts between the

base body and the terrain. Although the left support is

separated for a certain period of time, the excavator

initial position has been restored. Figure 7(b) shows the

graph of the dynamic overturning stability coefficient

corresponding to the case studied. Compared to Figure

6(b), the overturning stability coefficient is smaller and

at t= 8.5 s, its value is less than 1.

A typical real-world case is the lifting of the payload

laying on the ground with loose slings, that is, the load

Figure 5. Static ust

sand dynamic udyn

sstability coefficients: (a) in the absence of payload swinging and (b) in the presence of payload

swinging.

8Advances in Mechanical Engineering

is lifted with a non-zero initial bucket speed. The simu-

lation results for motion with loose slings for 2 s accom-

panied by the payload swinging are shown in Figure 8.

In this case, during the sling tightening due to the verti-

cal motion, a sudden displacement of the system gravity

center to the right support (see Figure 2, point B)is

occurred, accompanied by the left support separation

(see Figure 8(a)) and a sudden overturning stability

decrease which is restored later (Figure 8(b)).

Another operating condition causing the change in

the excavator dynamic behavior is the sudden change of

the payload weight due to the payload spillage or slings

and hanging elements failure. In Figure 9, the results of

the simulation during the payload raising and sudden

change of the payload weight are shown, and in Figure

10, the same during the payload lowering is shown. As

one can observe, both cases lead to oscillatory motions

of the mechanical system, but without extremal values

and support separation.

Parameter estimation and validation of

the mathematical model

Tire stiffness and damping characteristics estimation

The accuracy of the simulation results strongly depends

on the accuracy of the dynamic model parameter

Figure 6. Time evolution of the system characteristics (in the

absence of payload swinging): (a) spring deformation and (b)

dynamic stability coefficient.

Figure 7. Time evolution of the system characteristics (in the

presence of payload swinging): (a) spring deformation and (b)

dynamic stability coefficient.

Figure 8. Time evolution of the system characteristics (in the

presence of payload swinging): (a) spring deformation and (b)

dynamic stability coefficient.

Mitrev and Marinkovic

´9

values. While the inertia and geometrical parameters

are determined relatively easy by proper measurements,

calculations, and CAD modeling, the stiffness and

damping properties have to be determined by an experi-

mental research. Although the excavator tires are com-

plex structures with nonlinear characteristics that

depend on a number of factors,

under certain load

conditions, their stiffness and damping properties in the

vertical direction can be adequately modeled as a linear

spring connected in parallel combined with a viscous

damper. A number of methods are used to estimate the

numerical values of the stiffness and damping coeffi-

cients, among which is the well-known Free-Vibration

Logarithm Decay Method. Typical usage of this

method includes the application of an impulse loading

to a single tire and the use of the measured free vibra-

tions signal for estimation of the stiffness and damping

coefficients values. The main advantage of this method

is that it allows estimating the stiffness and damping

coefficients of the tire together with the contacting sur-

face. Due to difficulties to test the single tire in this

study, a new method is presented here to carry out the

study of the tire elastic and damping characteristics.

The excavator base body jointly with the fixed in a

certain configuration digging manipulator without pay-

load (see Figure 2) is represented as a single rigid body

with two degrees of freedom, mounted on linear springs

and viscous dampers in parallel. Two generalized coor-

dinates represent the oscillatory motion of the

excavator—vertical motion sof the gravity center C

and in-plane rotation uaccording to the same point

(see Figure 11(a)).

An impulse loading is applied to the oscillatory

structure by the excavator boom lowering at maximal

velocity followed by a sudden stopping caused by the

corresponding hydraulic valve closing. The appeared

free decaying oscillations of the structure are measured

by a two-axis accelerometer mounted on the chassis in

the point M, shown in Figure 11(a). Under the assump-

tion of small oscillations, the following equation is used

to determine the generalized acceleration €

seof the grav-

ity center (see Figure 11(b))

€

se=axxM+ayyM

ð19Þ

whereby a

and a

are denoted the x

and y

accelera-

tions, captured by the accelerometer.

Based on the theory of the free decaying oscillations,

the motion of the body along the vertical translation

coordinate is presented in the following form

€

stðÞ=X

k=1

keBktcos vdktck

ðÞð20Þ

Figure 9. Time evolution of the system characteristics (in the

presence of payload swinging): (a) spring deformation and (b)

dynamic stability coefficient.

Figure 10. Time evolution of the system characteristics (in the

presence of payload swinging): (a) spring deformation and (b)

dynamic stability coefficient.

10 Advances in Mechanical Engineering

where v

is the kth damped natural angular frequency,

kis the initial amplitude, B

is the decay rate, and c

is the initial phase. The theoretical solution (equation

(20)) is fitted to the experimental data (equation (19))

by the use of the least-squares method and the follow-

ing numerical values of the parameters are found:

= 26.29, A

= –6.04, B

= 2.37, B

= 0.55, v

21.9, v

= 10.55, c

= –0.11, c

= –0.05. The coeffi-

cient of multiple determination is R

’0.97, which can

be explained by the noise of the measured signal.

Figure 11(b) depicts the experimental and the fitted

curves. The obtained numerical values are used to com-

pose the roots of the characteristic polynomial

1,2=Bk6ivdk ð21Þ

and to calculate the characteristic polynomial f

(l) coef-

ficients ae

felðÞ=X

j=0

jlj=Y

k=1

llk



llk



ð22Þ

The system of differential equations, describing the

free decaying oscillations of the presented two degrees

of freedom system, is

M€

q+B_

q+Cq =0ð23Þ

where M=m0



,C=2kkL

BLA

ðÞ

BLA

ðÞkL

A+L2





B=2bbL

BLA

ðÞ

BLA

ðÞbL

A+L2





,mis the mass of the

body, Jis the mass moment of inertia of the body

according to the point C, k is the tire stiffness coeffi-

cient, and bis the damping coefficient.

Comparing the experimentally determined coeffi-

cients ae

jto the derived from the theoretical model

(equation (23)) characteristic polynomial coefficients

j, it is possible to estimate the numerical values of the

stiffness and damping coefficients.

Inserting notations L1=LBLAand L2=L2

A+L2

the theoretically derived coefficients at

jfor the system

(equation (23)) are as follows: at

0=k2(2L2L2

1)=Jm,

1=2bk(2L2L2

1)=Jm,at

2=½k(mL2+2J)b2(L2

12L2)=Jm,

3=b(2J+L2m)=Jm, and at

4=1. Thus, comparing at

to ae

0,aswellasat

3to ae

3, the following equations for cal-

culation of kand bare obtained

k=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

0Jm

2L2L2

sð24Þ

b=ae

3Jm

2J+L2mð25Þ

For m= 22,890 kg, J= 154,320 kg m

= 2.5 m,

and L

=0.5 m, the calculated values are k=4610 kN/m

and b= 45,098 Ns/m.

Validation of the mathematical model

The mathematical model is validated by the compari-

son between the simulation and the experimentally

obtained data.

For this purpose, a small-scale elasti-

cally mounted manipulator, depicted in Figure 12, with

known geometrical and inertial parameters and having

the same kinematic structure as the considered excava-

tor is used. The manipulator consists of an arm and

forearm and allows free suspension of the load to the

forearm distal end. It consists of the following main

elements: a rotating column 1, an arm 2, a forearm 3,

Figure 11. (a) Dynamic model of the excavator, represented as a single rigid body, and (b) experimental and fitted curves of the

gravity center vertical motion acceleration.

Mitrev and Marinkovic

´11

and a hydraulic power unit 4. The structure is equipped

with sensors for measuring six different quantities as

follows: (1) the pressures at the cap end and the rod

end of the arm 5 and forearm 6 driving cylinders 5 and

(2) the displacement of the arm and the forearm driving

cylinders 7 and 8. For the displacement measurement,

two Linear Variable Inductance Transducers are used,

attached to the hydraulic cylinders, while for the pres-

sure measurement, four pressure sensors to measure

absolute pressure are utilized. Analog outputs from the

sensors are measured, converted, and transmitted to

the computer measurement system 9 by an analog-to-

digital converter.

The lifting process starts with a fixed in a certain

forearm, payload laying on the ground and loose slings.

The process corresponds to the following sequence of

motions: (1) arm raising—from initial angle u

= –12°

to angle u

= 20.5°; (2) dwell phase in the upper posi-

tion; and (3) arm lowering to the initial position.

The experimental data for the measured quantities

are shown in Figure 13(a): (1) the pressure at the cap

end of the arm, (2) the pressure at the rod end of the

arm, (3) the pressure at the cap end of the forearm, (4)

the pressure at the rod end of the forearm, (5) linear dis-

placement of the arm cylinder, and (6) the linear displa-

cement of the forearm cylinder. As can be expected, the

payload swinging inserts an additional low-frequency

variation in the pressures—see Figure 13(a), curves 1–4.

The following phases with time length t

can clearly be

distinguished in the displacement of the arm cylinder

(see Figure 13(a), line 5): 1—arm raising without pay-

load (loose slings), 2—arm raising with payload (tight

slings), 3—fixed arm (dwell phase), and 4—arm

lowering.

The measured and filtered linear displacement signal

of the arm hydraulic cylinder (Figure 13(a), line 5) is

used to perform a simulation by the use of the theoreti-

cal mathematical model (equation (15)) with the corre-

sponding known numerical values of the small-scale

manipulator parameters. In Figure 13(b), the obtained

by the simulation torques of the arm and forearm are

used to determine the driving hydraulic cylinders forces

(arm—line 1, forearm—line 3) shown together with the

experimental forces (arm—line 2, forearm—line 4),

determined by the use of the measured hydraulic

cylinders pressures, presented in Figure 13(a), lines 1–4.

Although some differences are observed during

the duty cycle, the good correlation between the experi-

mental data and simulation results allows us to con-

clude that the adopted mathematical model may

confidently be used to simulate the dynamic problems

presented.

Figure 12. View of the small-scale manipulator and sensor

placement.

Figure 13. (a) The experimental and (b) the comparison between the experimental and the simulation data.

12 Advances in Mechanical Engineering

Conclusion

In this article, we have studied the dynamic stability of

a hydraulic excavator during performing lifting opera-

tions. The developed dynamic model with six degrees of

freedom considers the base body elastic connection

with the terrain, the front digging manipulator links,

and the presence of the freely suspended payload swing-

ing. A system of non-linear differential equations

describing the dynamic behavior is obtained by using

the Lagrange formalism. Numerical experiments are

carried out to study the excavator dynamic overturning

stability during the motion along a vertical straight-line

vertical trajectory under various operating conditions.

Finally, the mechanical system model has been vali-

dated by a small-scale experimental model.

The revealed basic insights allow a deeper under-

standing of the dynamic behavior of the excavator dur-

ing performing lifting operations along the vertical

trajectory. It is shown that the arising inertial loads due

to the movement of the links, combined with the pay-

load swinging, motion of the system gravity center, and

the motion of the base body, decrease the excavator

overturning stability. It was found that the excavator

overturning stability while following a vertical straight-

line trajectory decreases during the motion from the

higher to the lower part of the trajectory. If the stability

coefficient is close to 1, the payload swinging can cause

the separation of a support from the terrain; neverthe-

less, it was found that the excavator overturning stabi-

lity can be restored. The low-frequency vibration,

induced in the hydraulic cylinders by the payload

swinging and the base body elastic mounting, should

be taken into account during the hydraulic system

design and its improvement. The proposed calculation

method for estimation of the elastic and damping char-

acteristics of the supports is based on the experimental

data fitting and can be used reliably.

The decrease of the overturning stability coefficient

due to the payload swinging and moving gravity center

of the system is very important from the exploitation

point of view and this knowledge should be used for the

following:

Correction of the existing and creation of new

mobile machine load diagrams in order to keep

the overturning stability reserve demanded by

the standards;

Education of the excavator operators to account

for the specific aspects of the excavator dynamical

behavior during performing lifting operations.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with

respect to the research, authorship, and/or publication of this

article.

Funding

The author(s) disclosed receipt of the following financial sup-

port for the research, authorship, and/or publication of this

article: We acknowledge support by the German Research

Foundation and the Open Access Publication Fund of TU

Berlin.

ORCID iD

Dragan Marinkovic

´https://orcid.org/0000-0002-3583-

9434

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14 Advances in Mechanical Engineering