The paper presents a new way to determine some dynamical properties of materials modeled by the so-called degenerate system. The system is an element (subsystem) of any complex multidegree-of-freedom system. This subsystem follows from assumption of standard rheological model of stress–strain law of the materials. It is assumed that on the complex system act a set of random excitation forces. For this coincidence, a so-called energy balance equation was developed and was used to create a suitable identification method. The equations were derived for any differentiable function of elasticity. The stationary random process of the system response was assumed in the whole algorithm. As it was proved, in this case, instead of calculating appropriate fields of the hysteresis loop of suitable signals, an application of average values of the input and output signals and their proper combinations can be used. It is assumed that the elastic damping interaction force in the complex dynamical subsystem is described by the function $F(x,x˙)$, in which x is a deformation of the identified degenerated element and denotes a relative displacement between some appropriate neighboring masses of the system. Some numerical examples of the application are shown.

References

References
1.
Drozdov
,
A. D.
, and
Christiansen
,
J. deC.
,
2007
, “
Cyclic Viscoplasticity of Solid Polymers: The Effects of Strain Rate and Amplitude of Deformation
,”
Polymer
,
48
(
10
), pp.
3003
3012
.
2.
Evans
,
F. G.
,
1973
,
Mechanical Properties of Bone
,
Charles C Thomas Publisher
,
Springfield, IL
.
3.
Jamroziak
,
K.
,
2013
,
An Identification of the Material Properties in the Terminal Ballistic
,
Publishing House of Wroclaw University of Technology
,
Wroclaw, Poland
.
4.
Lenci
,
S.
,
2004
, “
Elastic and Damage Longitudinal Shear Behavior of Highly Concentrated Long Fiber Composites
,”
Meccanica
,
39
(
5
), pp.
415
439
.
5.
McElhaney
,
J. H.
,
1966
, “
Dynamic Response of Bone and Muscle Tissue
,”
J. Appl. Physiol.
,
21
(
4
), pp.
1231
1236
.
6.
Reilly
,
D. T.
,
Burstein
,
A. H.
, and
Frankel
,
V. H.
,
1974
, “
The Elastic Modulus for Bone
,”
J. Biomech.
,
7
(
3
), pp.
271
275
.
7.
Sarva
,
S. S.
,
Deschanel
,
S.
,
Boyce
,
M. C.
, and
Chen
,
W.
,
2007
, “
Stress–Strain Behavior of Polyurea and Polyurethane From Low to High Strain Rates
,”
Polymer
,
48
(
8
), pp.
2208
2213
.
8.
Skalak
,
R.
, and
Chien
,
S.
,
1979
,
Handbook of Bioengineering
,
McGraw-Hill Book Company
,
New York
.
9.
Wrobel
,
A.
,
Placzek
,
M.
,
Buchacz
,
A.
, and
Majzner
,
M.
,
2015
, “
Study of Mechanical Properties and Computer Simulation of Composite Materials Reinforced by Metal
,”
Int. J. Mater. Prod. Technol.
,
50
(
3/4
), pp.
259
275
.
10.
Kosobudzki
,
M.
,
2014
, “
The Use of Acceleration Signal in Modeling Process of Loading an Element of Underframe of High Mobility Wheeled Vehicle
,”
Ekspl. Niezawodnosc—Maint. Reliab.
,
16
(
4
), pp.
595
599
.
11.
Kosobudzki
,
M.
, and
Stanco
,
M.
,
2016
, “
The Experimental Identification of Torsional Angle on a Load-Carrying Truck Frame During Static and Dynamic Tests
,”
Ekspl. Niezawodnosc—Maint. Reliab.
,
18
(
2
), pp.
285
290
.
12.
Placzek
,
M.
,
2015
, “
Modelling and Investigation of a Piezo Composite Actuator Application
,”
Int. J. Mater. Prod. Technol.
,
50
(
3/4
), pp.
244
258
.
13.
Masri
,
S. F.
, and
Caughey
,
T. K.
,
1979
, “
A Nonparametric Identification Technique for Nonlinear Dynamic Problems
,”
ASME J. Appl. Mech.
,
46
(
2
), pp.
433
445
.
14.
Masri
,
S. F.
,
Sassi
,
H.
, and
Caughey
,
T. K.
,
1982
, “
Identification and Modeling of Nonlinear Systems
,”
Nucl. Eng. Des.
,
72
(
2
), pp.
235
270
.
15.
Ibanez
,
P.
,
1973
, “
Identification of Dynamic Parameters of Linear and Nonlinear Structural Models From Experimental Data
,”
Nucl. Eng. Des.
,
25
(
1
), pp.
30
41
.
16.
Natke
,
H. G.
,
Juang
,
J. N.
, and
Gawroniski
,
W.
,
1988
, “
A Brief Review on the Identification of Nonlinear Mechanical Systems
,”
6th International Modal Analysis Conference (IMAC VI)
,
D. J.
DeMichele
, ed., Orlando, FL, pp.
1569
1574
.
17.
Rice
,
H. J.
, and
Fitzpartick
,
J. A.
,
1991
, “
The Measurement of Nonlinear Damping in Single-Degree-of-Freedom Systems
,”
ASME J. Vib. Acoust.
,
113
(
1
), pp.
132
140
.
18.
Warminsky
,
J.
,
Lenci
,
S.
,
Cartmell
,
M. P.
,
Rega
,
G.
, and
Wiercigroch
,
M.
,
2012
,
Nonlinear Dynamics Phenomena in Mechanics
,
Springer
,
Heidelberg, Germany
.
19.
Jang
,
T. S.
,
Choi
,
H. S.
, and
Han
,
S. L.
,
2009
, “
A New Method for Detecting Non-Linear Damping and Restoring Forces in Non-Linear Oscillation Systems From Transient Data
,”
Int. J. Non-Linear Mech.
,
44
(
7
), pp.
801
808
.
20.
Roberts
,
J. B.
,
Dunne
,
J. F.
, and
Debonos
,
A.
,
1995
, “
A Spectral Method for Estimation of Non-Linear System Parameters From Measured Response
,”
Probab. Eng. Mech.
,
10
(
4
), pp.
199
207
.
21.
Kulisiewicz
,
M.
,
1983
, “
A Nonparametric Method of Identification of Vibration Damping in Non-Linear Dynamic Systems
,”
Int. J. Solids Struct.
,
19
(
7
), pp.
601
609
.
22.
Bialas
,
K.
,
2013
, “
Mechanical Subsystem as Implementation of Active Reduction of Vibration
,”
Solid State Phenom.
,
198
, pp.
657
662
.
23.
Bialas
,
K.
, and
Sekala
,
A.
,
2013
, “
Vibration Analysis of Mechanical Systems With the Discrete-Continuous Distribution of Parameters
,”
Solid State Phenom.
,
198
, pp.
698
703
.
24.
Kulisiewicz
,
M.
,
Iwankiewicz
,
R.
, and
Piesiak
,
S.
,
1997
, “
An Identification Technique for Non-Linear Dynamical Systems Under Stochastic Excitations
,”
J. Sound Vib.
,
200
(
1
), pp.
31
40
.
25.
Kulisiewicz
,
M.
,
Piesiak
,
S.
, and
Bocian
,
M.
,
2001
, “
Identification of Nonlinear Damping Using Energy Balance Method With Random Pulse Excitation
,”
J. Vib. Control
,
7
(
5
), pp.
699
710
.
26.
Jarczewska
,
K. A.
,
Koszela
,
P.
,
,
P.
, and
Korzec
,
A.
,
2011
, “
Identification of the Structure Parameters Using Short-Time Non-Stationary Stochastic Excitation
,”
J. Sound Vib.
,
330
(
14
), pp.
3352
3367
.
27.
Sieniawska
,
R.
,
,
P.
, and
Żukowski
,
S.
,
2009
, “
Identification of the Structure Parameters Applying a Moving Load
,”
J. Sound Vib.
,
319
(
1–2
), pp.
355
365
.
28.
Bocian
,
M.
,
Jamroziak
,
K.
, and
Kulisiewicz
,
M.
,
2014
, “
An Identification of Nonlinear Dissipative Properties of Constructional Materials at Dynamical Impact Loads Conditions
,”
Meccanica
,
49
(
8
), pp.
1955
1965
.
29.
Kulisiewicz
,
M.
,
2005
,
Modeling and Identification of Nonlinear Mechanical Systems Under Dynamic Complex Loads
,
Publishing House of Wroclaw University of Technology
,
Wroclaw, Poland
.
30.
Piesiak
,
S.
,
2003
,
Identification of Mechanical Systems in Domain Nonlinear and Degenerated Dynamical Models
,
Publishing House of Wroclaw University of Technology
,
Wroclaw, Poland
.
31.
Jamroziak
,
K.
,
Bocian
,
M.
, and
Kulisiewicz
,
M.
,
2013
, “
Energy Consumption in Mechanical Systems Using a Certain Nonlinear Degenerate Model
,”
J. Theor. Appl. Mech.
,
51
(
4
), pp.
827
835
.
32.
Jamroziak
,
K.
, and
Bocian
,
M.
,
2014
, “
Analysis of Non-Classical Models Which Have Been Subjected to Percussive Loads Using Equations of Energy and Power
,”