Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 4, pp. 1083-1091, Warsaw 2014

EFFICIENCY OF THE CONSTANT INTERACTION FORCE
VIBROISOLATION (WOSSO)

Marian Witalis Dobry

Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland

e-mail: marian.dobry@put.poznan.pl

In this paper, the force efficiency of theWoSSOvibroisolation in reducing forces transferred
into the base bymachines and the equipmentworking at low operating frequencies has been
discussed. To assess that efficiency, a strongly non-linearmathematicalmodel using Lagran-
ge equations of the second kind has been developed. That model takes into account the
specific design of the vibroisolator. Themathematical model has been solved using s digital
simulation method, by developing a special computer program in the MATLAB/simulink
environment. The efficiency of the force vibroisolation has been determined for a speci-
fic application of the WoSSO vibroisolator. The calculated force vibroisolation efficiency
exceeds 32.

Keywords: low-frequency passive vibroisolation, efficiency of force vibroisolation

1. Introduction

All vibration reduction methods still aim at ensuring effective passive vibroisolation. Practi-
cal use of common spring, rubber and pneumatic vibroisolators is very difficult in the case of
vibroisolation from low-frequency excitations generated by vibration sources. Each passive vi-
broisolation is characterised bya specificvibroisolation efficiency limit value.Thatvaluedepends
on the resonance of the vibroisolating system and a (limited) strength of the materials used for
productionof the vibroisolators.Avibroisolating system is characterised byavery low frequency
of proper vibrations. That requires large deflections of vibroisolators – which, in turn, ensures
large dynamic flexibility of the vibroisolation (Goliński, 1979; Harris and Crede, 1976).
That difficulty has been eliminated in the Constant Interaction Force Vibroisolation system

(WoSSO) developed in theDivision of Vibroacoustics and Biodynamics of Systems at the Insti-
tute of Mechanics of Poznan University of Technology (Dobry, 1983, 1998; Dobry and Cempel,
1983).
The substitute spring constant coefficient of that vibroisolation is zero and independent of

the deflection of the vibroisolating system. The stability of characteristics makes that kind of
vibroisolation better than the pneumatic one, the characteristic of which is stiff (exponential)
at large deflections.

In this paper, the application of the WoSSO vibroisolation to force vibroisolated machines
and equipment has been discussed. An important parameter for the assessment of the efficiency
of that specificapplication of theWoSSOvibroisolation is the force efficiencyof thevibroisolation
at low-frequency vibrations.
The efficiency of the WoSSO in application to free masses (for example machines) has not

yet been determined.
Furthermore, the analyzed vibroisolation system has been supplemented with an additional

spring k1 for faster stabilization of motion of the mass M – see Fig. 1. This spring has been
also used to reduce the deflection from the static equilibrium position defined for the beginning
of the motion.



1084 M.W. Dobry

2. Design of the WoSSO vibroisolator

The WoSSO vibroisolation is used for vibroisolating from forces generated by a machine in
operation. It protects the base against strong and variable forces transferred into the base, which
takes place if the vibroisolation is not used. The design of such type ofWoSSO vibroisolation is
presented in Fig. 1 (Dobry, 1983, 1998; Dobry and Cempel, 1983).

Fig. 1. Design of the Constant Interaction Force Vibroisolator (WoSSO) for force vibroisolation
supplemented with an additional spring k1

The properties of theWoSSO vibroisolation are discussed on the basis of theWoSSO vibro-
isolator which is used to vibroisolation of the base from forces exciting the mass M to move.
It is composed of a cam mounted on a working pin guided in the slide bearings mounted in the
vibroisolator body. That cam has been pressed in between two pressure rollers with radius R.
Those rollers are guided by two guiding rollers mounted at both sides of the pressure rollers
(these guiding rollers may also have the same radius R) and by a special guiding shackle. The
rollers are pressed against the cam by two vibroisolator springs (with the spring constant k)
preliminary compressed with the force S0. The shape of the cam curve is the same as the curve
ofmotion of the axis of the pressure and guiding rollers (Dobry, 1983, 1998, Dobry andCempel,
1983)

y=
−S0+

√

S20 +Pkx0+Pkx

k
(2.1)

where S0 is the vibroisolator spring preliminary compression force, P – constant force of vibro-
isolator reaction – it is equal to the weight of mass M, working pin and cam, k – vibroisola-
tor spring constant, x – generalized co-ordinate that describes vertical motion of the reduced
mass M, x0 – initial position of mass M.
The co-ordinate systemused to describe the cam is shown inFig. 1. Theposition (deflection)

of the vibroisolator at themoment t=0 is defined by the co-ordinate x0 of the initial position.

3. Mathematical model of the analysed mechanical system comprising the
WoSSO vibroisolator

Themathematical model of the analysed system comprising theWoSSO vibroisolator has been
prepared using Lagrange’s equation of the second kind (Cannon, 1973)



Efficiency of the constant interaction force vibroisolation (WoSSO) 1085

d

dt

(∂E

∂q̇j

)

−

∂E

∂qj
=
δLj

δqj
−

∂V

∂qj
−

∂Φ

∂q̇j
j=1,2, . . . ,s (3.1)

where s is the number of degrees of freedom, E – kinetic energy of the mechanical system,
qj – generalized co-ordinates, q̇j – generalized velocities, δLj – small increment of the virtual
work at the j-th virtual displacement along the direction not limited by constraints, δqj – small
increment of the j-th generalized co-ordinate along the direction not limited by constraints, V –
potential energy of the mechanical system, Φ – power of energy dissipation in the mechanical
system.
When calculating the kinetic energy, the virtual work of the external active forces, the

potential energy and the power of energy losses and by making the mathematical operations
defined by Lagrange’s equations (3.1), a single equation of forces has been obtained as the
analysed mechanical system has one degree of freedom connected with motion ot the mass M.
The final equation of force is in form

(

M+
0.5P2m2

S20 +Pkx0+Pkx

)

ẍ−
0.75P3km2

(S20 +Pkxo+Pkx)
2
ẋ
2

+
f2

R

[f1

r
P sgn(ẋ)+2

√

S20 +Pkx0+Pkx
]

sgn(ẋ)+k1x=F0 sin(2πfwt)

(3.2)

where x(t) = q1(t) is the coordinate of the position of the mass M, k1 – spring stiffness for
faster stabilization of motion of themass M of a low value, m2 – total mass as a sumofmasses
of pressure rollers, guiding rollers, axis and the guide of the bearings shackle.
The above differential equation of motion has been elaborated taking into account the fol-

lowing simplifying assumptions (taking into account the individual design and operation of the
vibroisolator):

1) weight of the mass M is equilibrated by the constant vibroisolator response force P in
the whole range of deflections,

2) kinetic energy does not comprise the mass of the vibroisolator springs,

3) kinetic energy does not comprise rotation of the rollers,

4) mass of the axis rollers and the roller guiding shackle in the related rectilinear motion
along the y direction depending on the cam shape has been taken into account,

5) variable rolling friction of the rollers along the cam and the guides has been taken into
account,

6) the following structural frictions have not been taken into account: the structural friction
inside the rollers shackle guides, in the rollers bearings and in the slide bearings of the
working pin,

7) it has been assumed that the system is excited by a harmonic force with the amplitude F0
and frequency fW [Hz].

The above assumptions have beenmade considering a negligible impact of disregarded phe-
nomenaandphysical quantities on themotion of the analysed system. Stronglynon-linear effects
have been taken into account in the above differential equation of forces (Minorski, 1967). Those
effects refer to the structural friction of the vibroisolator for the rolling pressure and guiding
rollers with such friction being dependent on the square of the preliminary compression force
of the vibroisolator springs k and on the sign of velocity of the mass M. The variability of
the inertial force during motion of the mass M is also shown in the equation (that variability
is dependent on the deflection of the vibroisolator x(t)). The system also generates strongly
non-linear forces which depend on the square of the velocity of the mass M, on the second
and third square of the permanent vibroisolator reaction force P and on the deflection of the
vibroisolator x(t) along its operating direction.



1086 M.W. Dobry

Dividing equation (3.2) by the dynamic parameter of mass B

B=M+
0.5P2m2

S20 +Pkx0+Pkx
(3.3)

a differential equation (3.4) ofmotion of themass M has been obtained. The equation has been
solved assuming the following initial conditions: x(0)= 0.015m and velocity v(0)= 0m/s

ẍ−
0.75P3km2

B(S20 +Pkx0+Pkx)
2
ẋ2+

f2

RB

[f1

r
P sgn(ẋ)+2

√

S20 +Pkx0+Pkx
]

sgn(ẋ)

+
k1

B
x=
F0

B
sin(2πfwt)

(3.4)

Such complex form of the strongly non-linear equation does not have a closed-form pure
equation solution (Cannon, 1973; Minorski, 1967). To determine motion of the mass M, a
digital simulation of the solution to the differential equation ofmotion has beenapplied. For that
purpose, a special software called SDWoSSO has been developed using theMATLAB/simulink
software.

4. Solution of the differential equation of motion for the mechanical system with
the force vibroisolation WoSSO

The structure of the computer program for simulation of the dynamics of the vibroisolator
WoSSO (SDWoSSO) has been shwn in Figs. 2 and 3. Figure 2 shows the computer program for
calculation of the efficiency of the force vibroisolation (EWS) of WoSSO, i.e. the ratio of the
RMS values of the following forces: the force that excites vibrations of the mass M (source of
vibration) – Fw(RMS) to the dynamic reactive force of the base Rp(RMS) – the protected area.
This efficiency has been expressed by

EWS WoSSO=
Fw(RMS)

Rp(RMS)
(4.1)

Fig. 2. Structure of the computer program for simulation of the dynamics of the force vibroisolation
that uses theWoSSO vibroisolator – calculation of the efficiency

The computer program for digital simulation of theWoSSO vibroisolator dynamics enables
tracking the efficiency during the system start-up, i.e. from a standstill until the steady-state
motion.



Efficiency of the constant interaction force vibroisolation (WoSSO) 1087

Fig. 3. Structure of the subprogram for simulation of the dynamics of the force vibroisolation that uses
theWoSSO vibroisolator – simulation using the mathematical model

Figure 3 shows the structure of the subprogram for simulation of the dynamics of the for-
ce isolation that uses the WoSSO vibroisolator. That program solves, through simulation, the
strongly non-linear mathematical model of the force vibroisolation that uses theWoSSO vibro-
isolator.

The following data (frompractical use of theWoSSOvibroisolator) have beenused in sample
calculations: P = 137.3N, k = 6500N/m, M = 14kg, S0 = 100N, x0 = 0.015, m2 = 0.2kg,
f1 = f2 = 0.0002, R = 0.0095m, k1 = 500N/m, F0 = 300N, fw = 16Hz, simulation time
t=120s.

In the paper, the calculation results have beenpresented for the following case: the sinusoidal
pattern of the exciting force characterised by the amplitude F0 =300Nand the frequency 16Hz
applied to the concentrated mass M =14kg placed on theWoSSO vibroisolator.

In Figs. 4a,b,c the acceleration, velocity and displacement of the reduced mass M for a
sinusoidal exciting forcewith amplitude 300Narepresented.Theacceleration amplitude reaches
21.8m/s2, the velocity amplitude 0.21m/s, and themass M displacement is 0.0021m.

Figure 5 shows the time history of the force of structural friction with amplitude 6.59N.

The variability of the friction force is caused by the resistance of the pressure rollers rolling
along the guides and the resistance of the pressure rollers rolling along theWoSSO cam, which
generate variable pressure forces during theirmotion. Figure 6 shows the structural friction force
during the whole start-up procedure as a function of the displacement x(t) of the mechanical
system.

It shows the impact of variable pressure forces generated by the pressure rollers rolling along
the cam. This variability results from changes in the cam shape for all cam positions [(−)0.025
to approx. 0.0185m] taken during the transitional process, i.e. from time t = 0 to 120s. The
dense concentration of lines (shown in the figure) is the area with the variable friction force for
steady-state motion of the analysed mechanical system.

5. Force efficiency of the WoSSO vibroisolator

An important parameter for assessing the vibroisolation is the vibroisolation efficiency. In the
case in question, theWoSSOvibroisolator uses the force vibroisolation defined by formula (4.1).



1088 M.W. Dobry

Fig. 4. Acceleration (a), velocity (b) and displacement (c) of the mass M in steady-statemotion of the
mechanical system

Fig. 5. Time history of the structural friction force in theWoSSO in steady-statemotion of the
mechanical system

Fig. 6. Structural friction force pattern for theWoSSO vibroisolator as a function of the reduction point
displacement during the start-up of the system comprising theWoSSO vibroisolation



Efficiency of the constant interaction force vibroisolation (WoSSO) 1089

The force efficiency of the above vibroisolator has been calculated for the whole duration of
the start-up phase of the mechanical system, from a standstill to the steady-state motion. To
determine that efficiency, has been also calculated the dynamic base response. The latter was
determined by subtracting, from the total base response, the constantWoSSOvibroisolator load
capacity which equals the weight of mass M. The dynamic response of the base is shown in
Fig. 7. As shown, the force of the response has a constant component of approx. 6.81N, and
its minimum and the maximum value is 7.685 and 4.95N, respectively. The root-mean-square
value (RMS) of the dynamic base response needed for calculation of the force efficiency of the
WoSSO vibroisolator is 6.43N.

Fig. 7. Dynamic response of the base vibroisolated by WoSSO (steady-state motion)

The changes of the efficiency value of the force vibroisolation of the WoSSO vibroisolator
duringthe start-upof themechanical systemare shown inFig. 8.Thefinalvalueof that efficiency
for the steady-state motion after 120s of the system dynamics simulation is 32.94.

Fig. 8. Force vibroisolation efficiency of theWoSSO vibroisolator at the start-up of the
mechanical system

It means that the force exciting motion of the mass M is reduced by the vibroisolator by
more than 32 times. The efficiency of the force vibroisolation is very high, given the fact that it
is a passive vibroisolation.
The specific characteristic of this vibroisolator is that the vibroisolation effectiveness does

not drop below one at the beginning of motion, whichmeans that the force transferred into the
base is amplified. On the contrary, themaximum value of the force efficiency of 31.5 is achieved
within 0.71s from the start of motion. It is a quite significant difference when compared to
standard vibroisolators in which no vibroisolation effect is visible during the start-up phase
(force is amplified), which is connected with passage of the system through the resonance.
The efficiency drop down to approx. 20.3 (as shown in Fig. 8) is caused by the stabilising

spring k1.Because of the impact of that spring, thenatural frequencyof the analysedmechanical
system is equal to 0.94Hz – the passage through the resonance results in a reduced efficiency of
the vibroisolation.



1090 M.W. Dobry

6. Stability of motion of the WoSSO vibroisolator used for force vibroisolation

Practical usefulness of strongly non-linear vibroisolators depends on the stability of motion of
themechanical system. For this purpose, a digital simulation of the dynamics of the system has
been carried out to show themotion of the mass M placed on theWoSSO vibroisolator on the
phase plane (Fig. 9).

Fig. 9. Phase portrait of the transitional process for force vibroisolation that uses theWoSSO
vibroisolator – stability of the motion

The whole transitional process of the mass M motion from t = 0 to t = 120s has been
shown.The phase portrait is a confirmation of the good stability of themechanical system – the
motion of mass M tends to achieve a stable orbit in the steady-statemotion. That result shows
that using theWoSSO vibroisolator for force vibroisolation of various technical objects will not
cause any implementation problems.

7. Conclusions

• The Constant Interaction Force Vibroisolator (WoSSO) supplemented with an additional
spring k1 is passive and strongly non-linear.

• The efficiency of the force vibroisolation ismore than 32.9 for sample parameters used for
calculations: mass M = 14kg, motion exciting force: sinusoidal pattern with amplitude
300N and frequency 16Hz.

• Motion of themass placed on theWoSSO vibroisolator supplemented with the additional
spring k1 is stable, therefore the WoSSO vibroisolator can be efficiently used to force
vibroisolation of machines and the equipment.

The properties described abovemakes theWoSSOvibroisolation suitable for awide range of
applications. The tests on the vibroisolation efficiency in various applications will be continued.

References

1. Cannon R.H. jr., 1973,Dynamics of Physical Systems (in Polish),WNT,Warszawa

2. Dobry M.W., 1983, Dynamics and Stability of the Constant Interaction Force Vibroisolator for
Hand-Held Percussive Tools (in Polish), Ph.D. Thesis, Poznan University of Technology, Depart-
ment of Mechanical Engineering andManagement, Poznań, Poland

3. Dobry M.W., 1998, Optimisation of the Energy Flow within the Human-Tool-Base System (in
Polish),Wyd. Politechniki Poznańskiej, Poznań, Seria: RozprawyNr 330, ISSN 0551-6528



Efficiency of the constant interaction force vibroisolation (WoSSO) 1091

4. Dobry M.W., Cempel C., 1983,Vibroisolator (in Polish), Patent RP nr 120 458, Urząd Paten-
towy RP. Opis patentowy opublikowano (published 25.07.1983)

5. Goliński J.A., 1979,Vibroisolation of Machines and Equipment (in Polish),WNT,Warszawa

6. Harris C.M., Crede C.E., 1976, Shock and Vibration Handbook, chapt. 30-33, McGraw-Hill,
NewYork

7. Minorski N., 1967,Non-linear Vibrations (in Polish), PWN,Warszawa

Manuscript received November 13, 2013; accepted for print June 10, 2014