Linearization and parametric vibration analysis of some applied problems in multibody systems
Van Khang Nguyen
Phong Dien Nguyen
Manh Cuong Hoang
M.C. Hoang Maritime University
, Haiphong City,
The paper deals with the application of the Runge-Kutta method for calculating steady-state periodic vibrations of the parametric vibration systems governed by linearized differential equations. The numerical calculation is also demonstrated by two models of multibody systems and measurements on real objects. Good agreement is obtained between the numerical and experimental results. Consequently, the obtained results can also be applicable to investigate other complicated models of multibody systems which perform the steady-state motions. The differential equations of motion of a multibody system are fully nonlinear in both the independent or dependent coordinates [1-6]:
M(q, t )q + k(q , q, t ) = h(q , q, t ).
The solution of these nonlinear equations is required in order to simulate the dynamic
behavior of multibody systems that undergo large displacements and rotations. It is very difficult
or impossible to find the analytical solution. The efficient way to solve the problem is by the
numerical methods .
Many technical systems work mostly on the proximity of an equilibrium position or,
especially, in the neighborhood of a desired motion which is usually called programmed
motion, desired motion, fundamental motion, inputoutput motion, etc., according
to specific problems. In this study, we use the term desired fundamental motion for this
object. The desired fundamental motion of a robotic system, for instance, is usually
described through state variables determined by prescribed motions of the end-effector. For
a mechanical driver system, the desired fundamental motion can be the motion of working
components of the driver system, in which the driver output rotates uniformly and all
components are assumed to be rigid. It is very convenient to linearize the equations of motion
of this configuration in order to take advantage of the linear analysis tools . In other
words, linearization makes it possible to use tools for studying linear systems to analyze
the behavior of multibody systems in the vicinity of a desired fundamental motion. For this
reason, the linearization of the equations of motion is most useful in the study of control [12,
13], machinery vibrations [14, 16] and the stability of motion . Mathematically, the
linearized equations of motion of a multibody system usually form a set of linear differential
equations with time-varying coefficients. Considering steady-state motions of the multibody
system alone, one obtains a set of linear differential equations having time-periodic
M(t )x + C(t )x + K(t )x = d(t ).
In the steady state of a machine, the working components perform stationary motions [14
17]. Matrices M(t ), C(t ), K(t ) and vector d(t ) in (2) are time-periodic with the least
period T .
Equation (2) can then be expressed in the compact form as
where we use the state variable x such that
x = P(t )x + f(t )
and the matrix of coefficients P(t ), vector f(t ) are defined by
f(t ) = M1d ,
where I is the identity matrix. To calculate the periodic vibrations of mechanical systems
described by differential equations (1) or (2), the harmonic balance method, the shooting
method and the finite difference method are usually used [9, 10, 21, 24]. To find the
steadystate solution of elastic mechanism and machine, we could also use the Newmark or the
Following this introduction, an overview of the numerical algorithms for calculating the
dynamic stability conditions and periodic vibrations of multibody systems in the steady
state is presented in Sect. 2. Sections 3 and 4 are important parts of the paper, in which the
numerical calculation using the RungeKutta method is demonstrated and validated by two
models of multibody systems and measurements on real objects.
where P(t ) is a continuous T -periodic n n matrix. According to Floquet theory
[15, 17, 18], the characteristic equation of (6) is independent of the chosen fundamental
set of solutions. Therefore, the characteristic equation can be formulated in the following
way. Firstly, we specify a set of n initial conditions xi (0) for i = 1, . . . , n, their elements
and [x1(0), x2(0), . . . , xn(0)] = I, where I denotes the n n identity matrix. By
implementing numerical integration of (6) within interval [0, T ] for n given initial conditions, we
obtain respectively n vectors xi (T ), i = 1, . . . , n. The matrix (t ) defined by
(T ) = x1(T ), x2(T ), . . . , xn(T )
2 An overview of the calculation of dynamic stability conditions and periodic vibrations
2.1 Dynamic stability conditions
We shall consider a system of homogeneous differential equations
x = P(t )x
(k = 1, . . . , n).
When the Floquet multipliers or Floquet exponents are known, the stability conditions of
solutions of the system of linear differential equations with periodic coefficients can be
easily determined using the Floquet theorem .
2.2 Periodic vibrations of multibody systems
Now we consider only the periodic vibration of a multibody system in the steady state,
which is governed by a set of linear differential equations with the periodic coefficients. As
is called the monodromy matrix of (6) . The characteristic equation of (6) can then be
written in the form
Expansion of (9) yields an nth-order algebraic equation
n + a1n1 + a2n2 + + an1 + an = 0
where unknowns i (i = 1, . . . , n), called Floquet multipliers, can be determined from (10).
Floquet exponents are given by
x = P(t )x + f(t )
x = P(t )x.
x(0) = x(T ).
already mentioned in the previous section, these differential equations can be expressed in
the compact matrix form
where x is the vector of state variables, matrix P(t ) and vector f(t ) are periodic in time with
period T . The system of homogeneous differential equations corresponding to (12) is
As is well known from the theory of differential equations, if (13) has only nonperiodic
solutions except the trivial solution, then (12) has a unique T -periodic solution. This
periodic solution can be obtained by choosing the appropriate initial condition for the vector
of variables x and then implementing numerical integration of (12) within interval [0, T ].
An algorithm is developed to find the initial value for the periodic solution [9, 10, 16, 17].
Firstly, the T -periodic solution must satisfy the following condition:
The interval [0, T ] is now divided into m equal subintervals with the step-size h = ti ti1 =
T /m. At the discrete times ti and ti+1, xi = x(ti ) and xi+1 = x(ti+1) represent the states of
the system, respectively. Using the fourth-order RungeKutta method, we get a numerical
k(1i1) = h P(ti1)xi1 + f(ti1) ,
k(4i1) = h P(ti ) xi1 + k(3i1)
Substituting (16) into (15), we obtain
where matrix Ai1 is given by
xi = Ai1xi1 + bi1
P2 ti1 + 2
P(ti1) + 2 P(ti )P2 ti1 + 2
(i = 1, . . . , m)
and vector bi1 takes the form
P2 ti1 + 2
Expansion of (17) for i = 1 to m yields
Ai x0 = cm
where c0 = 0, c1 = A0c0 + b0, c2 = A1c1 + b1, . . . , cm = Am1cm1 + bm1. Using the
boundary condition according to (14), the last equation of (20) yields a set of the linear
where I is the n n identity matrix. The solution of (21) gives us the initial value for the
periodic solution of (12). Finally, the periodic solution of (12) with the corresponding initial
value can be calculated without difficulty by using numerical methods.
Based on the described above RungeKutta method, a computer program for calculating
periodic vibrations of multibody systems has been developed at Hanoi University of
Technology. The computer program has been used for the numerical calculation of the following
two application examples.
3 Periodic vibration of the transport manipulator of a forging press
The most common forging equipment is the mechanical forging press. Mechanical presses
function by using a transport manipulator with a cam mechanism to produce a preset at a
certain location in the stroke. The kinematic schema of such mechanical adjustment unit is
depicted in Fig. 1.
The dynamic model of this system is schematically shown in Fig. 2. The mechanical
system of the driver shaft, the flexible transmission mechanism and the operating
mechanism can be considered as rigid bodies connected by massless springdamping elements
with time-invariant stiffness ki and constant damping coefficients ci , i = 1, 2. The rotating
components are modeled by two rotating disks with moments of inertia I0 and I1. Let us
Fig. 1 Kinematic schema of the
transport manipulator of a
forging press, 1the first
gearbox, 2driving shaft, 3the
second gearbox, 4cam
Fig. 2 Dynamic model of the
introduce into our dynamic model the nonlinear transmission function U (1) of the cam
mechanism as a function of the rotating angle 1 of the cam shaft, the driving torque from
the motor M (t ) and the external load F (t ) applied on the system.
The kinetic energy and the potential energy of the considered system can be expressed in
the following form:
By using the generalized coordinates 0, 1 and q2, we get the generalized forces
T = 21 I002 + 21 I112 + 21 m2x22,
= 2 k1(1 0)2 + 2 k2(x2 y)2.
Q0np = M (t ) + c1(1 0),
Q1np = F (t )U c1(1 0),
Q2np = F (t ) c2q2
U ( t + q1) = U + U q1 + 21 U q12 +
where we used the notations
Since the system performs small vibrations, i.e. there are only small vibrating amplitudes
q1 and q2, substituting (30) into (26) and (27) and neglecting nonlinear terms, we obtain the
linear differential equations of vibration for the system:
where the prime represents the derivative with respect to the generalized coordinate 1.
Substitution of (22)(24) into the Lagrange equation of the second type yields the differential
equations of motion of the system
I00 c1(1 0) k1(1 0) = M (t ),
I1 + m2U 2 1 + m2U q2 + m2U U 1 + c1(1 0) + k1(1 0)
= F (t )U ,
m2U 1 + m2q2 + m2U 1 + c2q2 + k2q2 = F (t ).
When the angular velocity of the driver input is assumed to be constant in the steady
one leads to the following relation:
where q1 is the difference between rotating angles 0 and 1 due to the presence of the spring
element k1 and the damping element c1, resulted from the flexible transmission mechanism.
If we assume that 1 varies little from its mean value during the steady-state motion, then
the transmission function y = U (1) depends essentially on the input angle 0 = t . Using
the Taylor series expansion around t , we get
We consider now the function U (), called the first grade of the transmission function
U (), where the angle is the rotating angle of the cam shaft. In steady-state motion of the
cam mechanism, function U () can be approximately represented by a truncated Fourier
series [14, 16]
U () =
(ak cos k + bk sin k).
The functions U , U , U in (34) can then be calculated using (35) for = t . The
following parameters are used for numerical calculations: rotating speed of the driver input n = 50
(rpm) corresponding to = 5.236 (1/s), stiffness k1 = 7692 N m; k2 = 106 N/m, damping
coefficients c1 = 18.5 N m s; c2 = 2332 N s/m, mass moment of inertia I1 = 1.11 kg m2 and
mass m2 = 136 kg. The Fourier coefficients ak in (35) with K = 12 are given in Table 1
for two different cases and coefficients bk = 0. We consider only periodic vibrations which
are a commonly observed phenomenon in the system. The periodic solutions of (34) can be
obtained by choosing appropriate initial conditions for the vector of variables q.
In most cases, the force F (t ) can be approximately a periodic function of the time or a
constant. Thus, (32) and (33) form a set of linear differential equations with periodic
coefficients. Finally, the linear vibration equations of the mechanical adjustment unit can be
expressed in the compact matrix form as
Table 1 The Fourier coefficients
of the function U ()
Table 2 The ||max values of
the characteristic equation
Fig. 3 Calculating results for q1, (a) curve 1using the proposed numerical method, curve 2using WKB
method, (b) frequency spectrum corresponding to curve 1
To verify the dynamic stable condition of the vibration system, the maximum of absolute
value ||max of the solutions of the characteristic equation, according to (34), is now
calculated. The obtained values corresponding to both cases are given in Table 2. It can be
concluded that these values satisfy the stable condition ||max < 1 and the system is
dynamically stable for both cases.
Some calculating results for periodic vibrations of the mechanical adjustment unit are
shown in Figs. 35. The periodic solutions of (34) obtained by the RungeKutta method
are compared with the calculating results using the WKB method  in Figs. 3a and 4a.
Comparing both, a considerable difference in the amplitude can be recognized. In addition,
the frequency spectra show harmonic components of the rotating frequency, such as 2 ,
4 , 6 (Fig. 3b) or , 3 , 5 (Fig. 4b). These spectra indicate that the considered system
performs stationary periodic vibrations only.
To validate the calculating results using the RungeKutta method, the dynamic load
moment of the mechanical adjustment unit was measured on the driving shaft (see also Fig. 1).
A typical record of the measured moment is plotted in Fig. 5, together with the curves
calculated from the dynamic model by using the WKB method, the kinesto-static calculation and
the RungeKutta method. Comparing the curves displayed in this figure, it can be observed
that the results using the RungeKutta method more closely agree with the experimental
results than the results obtained by the WKB method agree with the kinesto-static calculation.
Fig. 4 Calculating results for q2, (a) curve 1using the proposed numerical method, curve 2using WKB
method, (b) frequency spectrum corresponding to curve 1
Fig. 5 Dynamic moment acting on the driving shaft of the mechanical adjustment unit
4 Parametric vibration of a gear-pair system with faulted meshing
Dynamic modeling of gear vibrations offers a better understanding of the vibration
generation mechanisms as well as the dynamic behavior of the gear transmission in the presence
of gear tooth damage. Since the main source of vibration in a geared transmission system is
usually the meshing action of the gears, vibration models of the gear-pair in mesh have been
developed, taking into consideration the most important dynamic factors such as effects of
friction forces at the meshing interface, gear backlash, the time-varying mesh stiffness and
the excitation from gear transmission errors .
Fig. 6 Dynamic model of the gear-pair system with faulted meshing
From experimental works, it is well known that the most important components in gear
vibration spectra are the tooth-meshing frequency and its harmonics, together with sideband
structures due to the modulation effect. The increment in the number and amplitude of
sidebands may indicate a gear fault condition, and the spacing of the sidebands is related to their
source [22, 26]. However, according to our knowledge, there are in the literature only a few
theoretical studies concerning the effect of sidebands in gear vibration spectrum, and the
calculating results are usually not in agreement with the measurements. Therefore, the main
objective of the following study is to unravel modulation effects which are responsible for
generating such sidebands.
Figure 6 shows a relatively simple dynamic model of a pair of helical gears. This kind of
the model is also considered in Refs. [23, 24, 27, 28]. The gear mesh is modeled as a pair of
rigid disks connected by a springdamper set along the line of contact.
The model takes into account influences of the static transmission error which is
simulated by a displacement excitation e(t ) at the mesh. This transmission error arises from
several sources, such as tooth deflection under load, nonuniform tooth spacing, tooth
profile errors caused by machining errors, as well as pitting, scuffing of teeth flanks. The mesh
stiffness kz(t ) is expressed as a time-varying function. The gear-pair is assumed to operate
under high torque condition with zero backlash and the effect of friction forces at the
meshing interface is neglected. The viscous damping coefficient of the gear mesh cz is assumed
to be constant. The differential equations of motion for this system can be expressed in the
where i , i , i (i = 1, 2) are rotation angle, angular velocity, angular acceleration of the
input pinion and the output wheel, respectively. J1 and J2 are the mass moments of inertia
of the gears. M1(t ) and M2(t ) denote the external torque loads applied on the system, rb1
and rb2 represent the base radii of the gears. By introducing the composite coordinate
q = rb11 + rb22,
(36) and (37) yield a single differential equation in the following form:
mredq + kz(t )q + czq = F (t ) kz(t )e(t ) cze(t )
F (t ) = mred
Note that the rigid-body rotation from the original mathematical model in (36) and (37) is
eliminated by introducing in (39) a new coordinate q(t ). Variable q(t ) is called the dynamic
transmission error of the gear-pair system . Upon assuming that when 1 = 1 = const,
2 = 2 = const, cz = 0, kz(t ) = k0, the dynamic transmission error of the gear-pair system
q is equal to the static tooth deflection under constant load q0 as q = rb11 + rb22 = q0.
Equation (39) yields the following relation:
F (t ) F0(t ) = k0q0 + k0e(t ).
Equation (39) can then be rewritten in the form
mredq + kz(t )q + czq f (t ) = 0
where f (t ) = k0q0 [kz(t ) k0]e(t ) cze(t ).
In steady-state motion of the gear system, the mesh stiffness kz(t ) can be approximately
represented by a truncated Fourier series 
kz(t ) = k0 +
where z is the gear meshing angular frequency which is equal to the number of gear teeth,
times the shaft angular frequency, and N is the number of terms of the series.
In general, the error components are not identical for each gear tooth and will produce
displacement excitation that is periodic with the gear rotation (i.e. repeated each time the
tooth is in contact). The excitation function e(t ) can then be expressed in a Fourier series
with the fundamental frequency corresponding to the rotation speed of the faulted gear. For
instance, when the errors are situated at the teeth of the pinion, e(t ) may be taken in the
e(t ) =
Therefore, the vibration equation of gear-pair system according to (42) is a differential
equation with the periodic coefficients.
In this example, ten dominant coefficients in the Fourier series of the mesh stiffness
expressed in (43) are taken into account:
where z = z11 and the excitation function e(t ) is expressed by the six first terms of its
Fourier series as follows:
Table 4 Fourier coefficients and
phase angles of the mesh stiffness
According to the experimental setup which will be described later, the following
parameters of the model are used for numerical simulation: J1 = 9.3 102 (kg m2); J2 =
0.272 (kg m2). Other parameters of the gears are shown in Table 3. A nominal pinion speed
of 1800 rpm (1 = 60 or f1 = 30 Hz) is chosen. The mesh stiffness of the test gear-pair at
particular meshing position was obtained by means of a FEM software . The static tooth
deflection is estimated to be q0 = 1.2 105 (m). The values of Fourier coefficients of the
mesh stiffness with corresponding phase angles are given in Table 4. So, the mean 0 value
of the undamped natural frequency can be determined as 0 = k0/mred 5462 (1/s),
corresponding to f0 = 0/2 869 (Hz). Based on the experimental work , the mean
value of the Lehr damping ratio = 0.024 is used for the dynamic model. The damping
coefficient cz can then be determined by cz = 2 0 mred.
The numerical calculation for finding the periodic solutions of (42), based on the Runge
Kutta method, is realized with the computing program MATLAB. The calculated dynamic
transmission errors are shown in Figs. 7 and 8, corresponding to different excitation
functions e(t ).
The spectra in Fig. 8, (a) and (b), show clearly the meshing frequency and its
harmonics with sideband structures. As expected, the sidebands are spaced by the rotational
frequency f1 of the pinion. Comparing amplitudes of these sidebands in both spectra, it can
be concluded that the excitation function e(t ) caused by the tooth errors is responsible for
Fig. 7 Modeling result: dynamic
transmission error q(t)
Fig. 8 Modeling result: frequency spectrum of dq/dt , (a) excitation function e(t) with coefficients
e1 = 0.0015, e2 = 0.0035, e3 = 0.0027, e4 = 0.0011, e5 = 0.0005, e6 = 0.0013 (mm) and phase angles
1 = 0.049, 2 = 1.7661, 3 = 0.7286, 4 = 0.5763, 5 = 0.7810, 6 = 1.8172 (radian), (b)
excitation function e(t) with larger coefficients e1 = 0.01, e2 = 0.003, e3 = 0.0018, e4 = 0.0011, e5 = 0.0009,
e6 = 0.0003 (mm) and phase angles 1 = 1.047, 2 = 1.4521, 3 = 0.5233, 4 = 1.457, 5 = 0.8622,
6 = 1.1966 (radian)
The experiment was carried out at an ordinary back-to-back test rig (Fig. 9). The major
parameters of the test gear-pair are given in Table 3. The load torque was provided by a
hydraulic rotary torque actuator which remains the constant external torque for any motor
speed. The test gearbox operates at a nominal pinion speed of 1800 rpm (30 Hz), thus the
meshing frequency fz is 420 Hz. A Laser Doppler Vibrometer was used for measuring the
Fig. 9 Gearbox test rig
Fig. 10 Test rig and measuring configuration (schema)
oscillating parts of the angular speed of the gear shafts (i.e. oscillating parts of 1 and 2) in
order to experimentally determine the dynamic transmission error (Fig. 10).
The measurement was taken with two noncontacting transducers mounted in
proximity to the shafts, positioned at the closest place to the test gears. The vibration signals
were sampled at 10 kHz. The signal used in this study was recorded at the end of
12hour total test time. At that time a surface fatigue failure occurred on some teeth of the
Figure 11 shows the frequency spectrum of the first derivative of the dynamic
transmission error q (t ) determined from the experimental data. The spectrum presents sidebands at
the meshing frequency and its harmonics. In particular, the dominant sidebands are spaced
by the rotational frequency of the pinion and characterized by high amplitude. This gives a
clear indication of the presence of the faults on the pinion. By comparing the spectra
displayed in Figs. 12 and 13, it can be observed that the vibration spectrum calculated by the
RungeKutta method (Fig. 12) and the spectrum of the measured vibration signal (Fig. 13)
show the same sideband structures.
Fig. 11 Experimental results: frequency spectrum of dq/dt
Fig. 12 Calculating results: frequency spectrum of dq/dt
Fig. 13 Experimental results: zoomed frequency spectrum of dq/dt from Fig. 11
Some early calculating results for the same problem of gear vibrations could also be
found in the previous study , in which the harmonic balance method is employed for
solving (42). In study , however, only two first terms of the excitation function e(t )
in (46) and three coefficients k1, k2, k3 of the Fourier series of the mesh stiffness in (45)
were taken into account since the computational cost increases highly with the number of
harmonic terms. On the contrary, by using the RungeKutta method, six terms of the
excitation function and up to ten coefficients of the mesh stiffness could be taken in the
calculation with an acceptable computational cost. Consequently, the calculating results with the
RungeKutta method (Fig. 12) more closely agree with the experimental data than the results
calculated by the harmonic balance method in . The obtained results seem promising and
an extension to the more complicated geared systems.
The calculation of dynamic stable conditions and periodic vibrations of elastic mechanisms
and machines present an important problem in mechanical engineering. In the this paper, the
RungeKutta method is used for calculating the steady-state vibration of the parametric
vibration systems governed by linearized differential equations. This numerical method is also
demonstrated and tested with two dynamic models of multibody systems and measurements
on real objects. Good agreement is obtained between the numerical solutions and
experimental results. In the first example, the agreement between the experimental and calculated
results with RungeKutta method is closer than the results calculated by WKB method. In
the second example, the concordance of the results is obtained for both the amplitude and the
frequency content of the parametric vibration of the gear-pair system. The obtained results
can also be applicable to investigate other complicated multibody systems which perform
the steady-state motions.
The application of the RungeKutta method for calculating nonlinear periodic vibrations
and bifurcations of multibody systems will be the subject of our future investigation.
Acknowledgements This paper was completed with a research grant from the Flemish Inter-university
Council for University Development Cooperation (VLIR UOS). An additional support was given by the
National Foundation for Science and Technology Development of Vietnam. The authors gratefully acknowledge
the generous support of these sponsors.
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