- Email: [email protected]
Method of identifying nonlinear characteristic of energy dissipation in dynamic systems with one degree of freedom
archives of civil and mechanical engineering 14 (2014) 354–359
Available online at www.sciencedirect.com
ScienceDirect journal homepage: http://www.elsevier.com/locate/acme
Original Research Article
Method of identifying nonlinear characteristic of energy dissipation in dynamic systems with one degree of freedom M. Bocian *, M. Kulisiewicz Wroclaw University of Technology, Institute of Materials Science and Applied Mechanics, Smoluchowskiego 25 Str., 50-370 Wroclaw, Poland
article info
abstract
Article history:
A method of determining the shape of the vibration damping characteristic in systems with
Received 15 May 2013
one degree of freedom in the case when the characteristic depends not only on the velocity
Accepted 13 November 2013
but also in an unknown way depends on the displacement has been developed. The method
Available online 15 December 2013
is intended for determining the specific form of the mathematical function describing this dependence. The method utilizes an appropriate analysis of experimentally determined
Keywords:
traces of free vibrations of the system. The method has been verified on a few selected
Nonlinear mechanical systems
computer systems.
Dynamics
# 2013 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All
Vibrations
rights reserved.
1.
Introduction
The dynamic properties of most construction materials are usually described by a stiffness parameter and a viscous damping parameter determining the rate of energy dissipation. The parameters stem from the traditionally adopted rheological Kelvin model [1], which was the basis for developing, among other things, the well-known method of analyzing dynamic mechanical systems, called experimental modal analysis [2–4]. However, the analytical description of the dynamic properties of novel construction materials as well as biological materials (biomaterials) often poses difficulties due to the fact that the strain–stress dependence in these
materials is not linear. This is observed already for quasi-static loads inducing deformations with given constant velocities. Moreover, the strain–stress dependences obtained at constant velocities often depend on the velocity with which specific strength tests (tensile, compressive, torsional, etc.) are conducted while the moduli of elasticity determined in this way often significantly depend on the rate of deformation and change depending on the deformation level (nonlinear dependence). This is observed not only in the case of many non-metallic construction materials (plastics and composites), but also for the deformation of biological materials, e.g. human and animal bones [5–9]. This indicates that the linear Hooke model which is traditionally adopted in the mechanics of materials is unreliable for a very wide range of materials.
* Corresponding author. Tel.: +48 71 320 27 54. E-mail address: [email protected] (M. Bocian). 1644-9665/$ – see front matter # 2013 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved. http://dx.doi.org/10.1016/j.acme.2013.11.006
archives of civil and mechanical engineering 14 (2014) 354–359
355
Fig. 1 – Example of dynamic system in which tested material acts on concentrated mass m in accordance with Eq. (1).
Hence there is a need to look for new rheological models of materials and to develop new identification methods for them. The idea put forward by the authors of this paper is to build such a (possibly simplest) dynamic system whose elastodamping element would be wholly made of the tested material while the movement of concentrated mass m attached in a specified point to this element would be described by a simple differential equation in the form € þ sð?Þ ¼ pðtÞ mx
(1)
where x stands for time function x(t) describing the movement of mass m, and p(t) is the exciting force. Moreover, it is assumed that the action of the tested elasto-damping element on mass m is described by unknown force S which is a function of displacement x and velocity v. An example of such solution can be a beam made of the tested material that is rigidly constrained at one end and a concentrated mass m is attached to the other end (Fig. 1). If mass m jest is sufficiently large in comparison with the mass of the beam, then one can assume that Eq. (1) sufficiently accurately describes the vibrations of mass m. In this paper it is assumed, a priori, that force S(x,v) has the form Sðx; vÞ ¼ ½k þ kðxÞv þ f s ðxÞ
(2)
where k(x) is a certain unknown displacement function satisfying the condition kðx ¼ 0Þ ¼ 0
(3)
It is apparent that for k(x) = 0 and fs(x) = cx relation (2) defines force S in the Kelvin model, i.e. the model commonly used in engineering practice to describe the mechanical vibrations of dynamic systems. Term k(x) (in any form) was introduced because damping properties often depend on the level (state) of material deformation. For example, the resistance of the fluids moving in the canaliculi of a deformed bone is greater than in an undeformed bone since the canaliculi in the former bone are narrowed. To sum up, a dynamic model of a system described by the equation € þ ½k þ kðxÞx_ þ f s ðxÞ ¼ pðtÞ mx
(4)
was adopted in this paper. A scheme of the model is shown in Fig. 2. The aim of the method described below is to determine constant k and function k(x) for arbitrarily nonlinear elasticity characteristic fs(x) describing pure elastic interactions. Eq. (4)
Fig. 2 – Scheme of adopted model.
leads to a qualitatively new process modeling of vibration of mechanical systems similarly as it have place today in mechatronics [10,11].
2.
Theoretical basis of the method
Differential equation (4) describes the motion of concentrated mass m in the adopted dynamic model having the form shown in Fig. 2. In the case of nonlinear and unknown functions k(x), fs(x), it is impossible to obtain solution x(t) of this equation. One can notice, however, that for oscillating €ðtÞ motion there exist such instants ti in which acceleration x _ regardless of is equal to zero. In such instants velocity v ¼ x, the form of the solution, must reach extreme values since € ¼ dv=dt. x In order to facilitate the analysis let us assume that p(t) = 0, which corresponds to free vibrations. Thus for instant t = ti one gets: €ðti Þ ¼ 0; xðt _ i Þ ¼ vðti Þ ¼ vi ; xðti Þ ¼ xi x
(5)
According to differential equation (4), values vi, xi for p(t) = 0 must satisfy the following algebraic equation: ½k þ kðxi Þvi þ f s ðxi Þ ¼ 0
(6)
Hence the following relation is obtained:
f s ðxi Þ ¼ k þ kðxi Þ ¼ yðxi Þ vi
(7)
where y(x) is a certain unknown function of variable x. As one can notice, the graph of this function corresponds to that of the characteristic k(x) shifted parallel by constant k. Relation (7) can be used to determine constant k and function k(x) in a given interval of variable x if: a certain set of data xi, vi is experimentally determined (by measurement) for values sufficiently densely filling the given interval of variable x, function fs(x) is known (e.g. has been determined by static measurements) in the given interval of variable x.
356
archives of civil and mechanical engineering 14 (2014) 354–359
Knowing the shape of the diagram of yi versus xi f ðx Þ yi ¼ s i vi
f s ðxÞ ¼
pðxÞ ½k þ kðxÞv0
(8)
one can adopt a certain class of function k(x) (a linear function, a polynomial of any degree, an exponential function, etc.) and approximate it for the experimental data in the given interval of variable x.
It is apparent that if the form of function k(x) is unknown, function fs(x) cannot be determined from quasi-static measurements. However, the lower the given velocity v0, the more similar the relation p(x) to sought function fs(x) since according to relation (11) the following holds true: limv0 ! 0 pðxÞ ¼ f s ðxÞ
3.
Description of the application of the method
The basis for the use of the described method is the precise determination of static characteristic fs(x) in the given interval of changes in displacement x. In the case of static tests consisting in successively applying static loads pv = const (v = 1, 2, . . ., n) and recording the constant displacements xv at which x_ v ¼ 0, one directly gets f s ðxv Þ ¼ pv
(9)
whereas in the case of quasi-static tests in which such load p(t) is applied at which displacements increasing at constant rate v0 are obtained (as in typical strength tests), one gets: xðtÞ ¼ v0 t
(10)
½k þ kðxÞv0 þ f s ðxÞ ¼ pðxÞ
(11)
hence
(12)
(13)
Hence the best solution for estimating characteristic fs(x) in this case seems to be to use the static method and relation (9). Having obtained the static characteristic fs(x) one should make the system vibrate freely by setting different (e.g. randomly selected) initial conditions: xð0Þ ¼ x0 ; vð0Þ ¼ v0
(14)
€ðtÞ; xðtÞ; _ xðtÞ recorded for and then from the time functions x them one should select the values of vi, xi in time instants ti, for €ðti Þ ¼ 0, to obtain the following two-column matrix: which x 2 3 v1 x1 6 v2 x2 7 7 E0 ¼ 6 (15) 4 . . . . . . 5 ¼ ½vi ; xi ði ¼ 1; 2; . . . ; sÞ vs xs (see Fig. 3). Substituting the values in the successive rows of this matrix into relation (8) one gets appropriate values yi of variable y, forming the following matrix Ec
Fig. 3 – Illustration of method of determining values xi, vi from free vibration diagrams.
357
archives of civil and mechanical engineering 14 (2014) 354–359
Fig. 4 – (a) Exemplary free vibration diagrams for System 1: x0 = 1.5, c3 = 0. (b) Exemplary free vibration diagrams for System 2: x0 = 1.5, c3 = 10,000. (c) Exemplary free vibration diagrams for System 3: x0 = 1.5, c3 = 0, ABS(kappa). (d) Exemplary free vibration diagrams for System 4: x0 = 1.5, c3 = 10,000, ABS(kappa).
2
y1 6y 2 Ec ¼ 6 4... ys
3 x1 x2 7 7 ¼ ½y ; xi i ...5 xs
ði ¼ 1; 2; . . . ; sÞ
(16)
Examining relation yi(xi) in the form of a two-dimensional diagram of dependent variable y versus independent variable x one should: a. select a particular shape of function k(x), b. calculate the constant parameters of the selected function (including parameter k), using the well-known methods of parametric optimization (e.g. nonlinear regression analysis [12]).
4.
Computer verification of the method
The practical functioning of the above method was tested on four dynamic systems described by Eq. (4). The verification tests for the systems were carried out for both linear and nonlinear functions fs(x), k(x) and the numerical data are shown in Table 1. For the tests it was assumed that the elasticity characteristics for all the systems were precisely determined. The aim of the tests was to determine constant k and function k(x). The systems were built using the Matlab Simulink software.
Table 1 – Numerical data on tested systems.
System System System System
1 2 3 4
m [kg]
c1 [N/m]
c3 [N/m3]
k [Ns/m]
k0
k(x)
20 20 20 20
800 800 800 800
0 0 10,000 10,000
40 40 40 40
600 600 600 600
k0 x v k0 x jxj v k0 x v k0 x jxj v
Table 2 – Results of identification.
System System System System
1 (c3 = 0) 2 (c3 = 0) 3 4
k [N/m]
k0
40.00 0.01 40.00 0.01 40.00 0.01 40.00 0.01
599.98 0.13 603.12 14.79 600.07 0.16 590.87 16.24
358
archives of civil and mechanical engineering 14 (2014) 354–359
Fig. 5 – Relations yi(xi) for tested systems and their approximations (continuous lines).
Different initial conditions for displacement x0 in the systems were set at v0 = 0 and p(t) = 0. In total, 15 different initial conditions for x0 ranging from x0 = 0.01 to x0 = 0.15 at a step of 0.01 were set for each of the systems. Exemplary time diagrams for selected initial conditions for all the tested systems are shown in Fig. 4. Four values of xi in instants ti determined in the previous section of this paper were selected for each of the initial condition, using a dedicated software. In this way about 60 values xi (i = 1, 2, . . ., 60) and extreme velocity values vi were obtained for which the corresponding values of yi were calculated from relation (8). Diagrams of relation yi(xi) for each of the systems are shown in Fig. 5. By approximating these relations by means of the regression analysis method the results presented in Table 2 were obtained.
5.
Conclusions
The tests have shown that the proposed method functions properly. The values obtained for the systems with linear characteristics k(x) are practically equal to the given values, regardless of the shape of function fs(x). Slightly larger scatter of results was obtained for the systems with nonlinear characteristics k(x) (systems 2 and 4). It should be noted that the precise synchronization of the displacement, velocity and acceleration traces in time is critical for the accuracy of the
obtained results. Such synchronization can be difficult in the case of real measurements performed on real objects, which requires further research. It would be also interesting to extend the developed method for the so-called degenerated systems which are used, among other things, in the process of piercing ballistic shields [13].
references
[1] H.A. Barnes, J.F. Hutton, K. Walters, An Introduction to Rheology, Elsevier Science B.V., Amsterdam, Lausanne, New York, Oxford, Shannon, Singapore, Tokyo, 1989. [2] D.J. Ewins, Modal Testing, Theory, Practice and Application, second ed., Research Studies Press Ltd., Baldock, Hertfordshire, England, 2000 . [3] Structural Testing. Part 1. Mechanical Mobility Measurements, Bruel & Kjaer, 1988. [4] Structural Testing. Part 2. Modal Analysis and Simulation, Bruel & Kjaer, 1988. [5] R. Skalak, S. Chien, Handbook of Bioengineering, McGrawHill Book Company, New York, 1979. [6] F.G. Evans, Mechanical Properties of Bone, Charles C. Thomas, Springfield, IL, 1973. [7] D.T. Reilly, A.H. Burstein, V.H. Frankel, The elastic modulus for bone, Journal of Biomechanics 7 (3) (1974) 271–275. [8] J.H. McElhaney, Dynamic response of bone and muscle tissue, Journal of Applied Physiology 21 (1966) 1231–1236.
archives of civil and mechanical engineering 14 (2014) 354–359
[9] S.S. Sarva, S. Deschanel, M.C. Boyce, C. Weinong, Stress– strain behavior of polyurea and polyurethane from low to high strain rates, Polymer 48 (2007) 2208–2213. [10] A. Buchacz, M. Płaczek, Development of mathematical model of a mechatronic system, Solid State Phenomena 164 (2010) 319–322. [11] A. Buchacz, D. Galeziowski, Synthesis as a designing of mechatronic vibrating mixed systems, Journal of
359
(no. of Vibroengineering 14 (June (2)) (2012) 553–559 publication 781). [12] J.J. Johnston, Econometric Methods, McGraw-Hill Book Company Inc., New York, 1963. [13] K. Jamroziak, Analyis of a degenerated standard model in the piercing process, Journal of Achievements in Materials and Manufacturing Engineering 22 (1) (2007) 49–52.
Available online at www.sciencedirect.com
ScienceDirect journal homepage: http://www.elsevier.com/locate/acme
Original Research Article
Method of identifying nonlinear characteristic of energy dissipation in dynamic systems with one degree of freedom M. Bocian *, M. Kulisiewicz Wroclaw University of Technology, Institute of Materials Science and Applied Mechanics, Smoluchowskiego 25 Str., 50-370 Wroclaw, Poland
article info
abstract
Article history:
A method of determining the shape of the vibration damping characteristic in systems with
Received 15 May 2013
one degree of freedom in the case when the characteristic depends not only on the velocity
Accepted 13 November 2013
but also in an unknown way depends on the displacement has been developed. The method
Available online 15 December 2013
is intended for determining the specific form of the mathematical function describing this dependence. The method utilizes an appropriate analysis of experimentally determined
Keywords:
traces of free vibrations of the system. The method has been verified on a few selected
Nonlinear mechanical systems
computer systems.
Dynamics
# 2013 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All
Vibrations
rights reserved.
1.
Introduction
The dynamic properties of most construction materials are usually described by a stiffness parameter and a viscous damping parameter determining the rate of energy dissipation. The parameters stem from the traditionally adopted rheological Kelvin model [1], which was the basis for developing, among other things, the well-known method of analyzing dynamic mechanical systems, called experimental modal analysis [2–4]. However, the analytical description of the dynamic properties of novel construction materials as well as biological materials (biomaterials) often poses difficulties due to the fact that the strain–stress dependence in these
materials is not linear. This is observed already for quasi-static loads inducing deformations with given constant velocities. Moreover, the strain–stress dependences obtained at constant velocities often depend on the velocity with which specific strength tests (tensile, compressive, torsional, etc.) are conducted while the moduli of elasticity determined in this way often significantly depend on the rate of deformation and change depending on the deformation level (nonlinear dependence). This is observed not only in the case of many non-metallic construction materials (plastics and composites), but also for the deformation of biological materials, e.g. human and animal bones [5–9]. This indicates that the linear Hooke model which is traditionally adopted in the mechanics of materials is unreliable for a very wide range of materials.
* Corresponding author. Tel.: +48 71 320 27 54. E-mail address: [email protected] (M. Bocian). 1644-9665/$ – see front matter # 2013 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved. http://dx.doi.org/10.1016/j.acme.2013.11.006
archives of civil and mechanical engineering 14 (2014) 354–359
355
Fig. 1 – Example of dynamic system in which tested material acts on concentrated mass m in accordance with Eq. (1).
Hence there is a need to look for new rheological models of materials and to develop new identification methods for them. The idea put forward by the authors of this paper is to build such a (possibly simplest) dynamic system whose elastodamping element would be wholly made of the tested material while the movement of concentrated mass m attached in a specified point to this element would be described by a simple differential equation in the form € þ sð?Þ ¼ pðtÞ mx
(1)
where x stands for time function x(t) describing the movement of mass m, and p(t) is the exciting force. Moreover, it is assumed that the action of the tested elasto-damping element on mass m is described by unknown force S which is a function of displacement x and velocity v. An example of such solution can be a beam made of the tested material that is rigidly constrained at one end and a concentrated mass m is attached to the other end (Fig. 1). If mass m jest is sufficiently large in comparison with the mass of the beam, then one can assume that Eq. (1) sufficiently accurately describes the vibrations of mass m. In this paper it is assumed, a priori, that force S(x,v) has the form Sðx; vÞ ¼ ½k þ kðxÞv þ f s ðxÞ
(2)
where k(x) is a certain unknown displacement function satisfying the condition kðx ¼ 0Þ ¼ 0
(3)
It is apparent that for k(x) = 0 and fs(x) = cx relation (2) defines force S in the Kelvin model, i.e. the model commonly used in engineering practice to describe the mechanical vibrations of dynamic systems. Term k(x) (in any form) was introduced because damping properties often depend on the level (state) of material deformation. For example, the resistance of the fluids moving in the canaliculi of a deformed bone is greater than in an undeformed bone since the canaliculi in the former bone are narrowed. To sum up, a dynamic model of a system described by the equation € þ ½k þ kðxÞx_ þ f s ðxÞ ¼ pðtÞ mx
(4)
was adopted in this paper. A scheme of the model is shown in Fig. 2. The aim of the method described below is to determine constant k and function k(x) for arbitrarily nonlinear elasticity characteristic fs(x) describing pure elastic interactions. Eq. (4)
Fig. 2 – Scheme of adopted model.
leads to a qualitatively new process modeling of vibration of mechanical systems similarly as it have place today in mechatronics [10,11].
2.
Theoretical basis of the method
Differential equation (4) describes the motion of concentrated mass m in the adopted dynamic model having the form shown in Fig. 2. In the case of nonlinear and unknown functions k(x), fs(x), it is impossible to obtain solution x(t) of this equation. One can notice, however, that for oscillating €ðtÞ motion there exist such instants ti in which acceleration x _ regardless of is equal to zero. In such instants velocity v ¼ x, the form of the solution, must reach extreme values since € ¼ dv=dt. x In order to facilitate the analysis let us assume that p(t) = 0, which corresponds to free vibrations. Thus for instant t = ti one gets: €ðti Þ ¼ 0; xðt _ i Þ ¼ vðti Þ ¼ vi ; xðti Þ ¼ xi x
(5)
According to differential equation (4), values vi, xi for p(t) = 0 must satisfy the following algebraic equation: ½k þ kðxi Þvi þ f s ðxi Þ ¼ 0
(6)
Hence the following relation is obtained:
f s ðxi Þ ¼ k þ kðxi Þ ¼ yðxi Þ vi
(7)
where y(x) is a certain unknown function of variable x. As one can notice, the graph of this function corresponds to that of the characteristic k(x) shifted parallel by constant k. Relation (7) can be used to determine constant k and function k(x) in a given interval of variable x if: a certain set of data xi, vi is experimentally determined (by measurement) for values sufficiently densely filling the given interval of variable x, function fs(x) is known (e.g. has been determined by static measurements) in the given interval of variable x.
356
archives of civil and mechanical engineering 14 (2014) 354–359
Knowing the shape of the diagram of yi versus xi f ðx Þ yi ¼ s i vi
f s ðxÞ ¼
pðxÞ ½k þ kðxÞv0
(8)
one can adopt a certain class of function k(x) (a linear function, a polynomial of any degree, an exponential function, etc.) and approximate it for the experimental data in the given interval of variable x.
It is apparent that if the form of function k(x) is unknown, function fs(x) cannot be determined from quasi-static measurements. However, the lower the given velocity v0, the more similar the relation p(x) to sought function fs(x) since according to relation (11) the following holds true: limv0 ! 0 pðxÞ ¼ f s ðxÞ
3.
Description of the application of the method
The basis for the use of the described method is the precise determination of static characteristic fs(x) in the given interval of changes in displacement x. In the case of static tests consisting in successively applying static loads pv = const (v = 1, 2, . . ., n) and recording the constant displacements xv at which x_ v ¼ 0, one directly gets f s ðxv Þ ¼ pv
(9)
whereas in the case of quasi-static tests in which such load p(t) is applied at which displacements increasing at constant rate v0 are obtained (as in typical strength tests), one gets: xðtÞ ¼ v0 t
(10)
½k þ kðxÞv0 þ f s ðxÞ ¼ pðxÞ
(11)
hence
(12)
(13)
Hence the best solution for estimating characteristic fs(x) in this case seems to be to use the static method and relation (9). Having obtained the static characteristic fs(x) one should make the system vibrate freely by setting different (e.g. randomly selected) initial conditions: xð0Þ ¼ x0 ; vð0Þ ¼ v0
(14)
€ðtÞ; xðtÞ; _ xðtÞ recorded for and then from the time functions x them one should select the values of vi, xi in time instants ti, for €ðti Þ ¼ 0, to obtain the following two-column matrix: which x 2 3 v1 x1 6 v2 x2 7 7 E0 ¼ 6 (15) 4 . . . . . . 5 ¼ ½vi ; xi ði ¼ 1; 2; . . . ; sÞ vs xs (see Fig. 3). Substituting the values in the successive rows of this matrix into relation (8) one gets appropriate values yi of variable y, forming the following matrix Ec
Fig. 3 – Illustration of method of determining values xi, vi from free vibration diagrams.
357
archives of civil and mechanical engineering 14 (2014) 354–359
Fig. 4 – (a) Exemplary free vibration diagrams for System 1: x0 = 1.5, c3 = 0. (b) Exemplary free vibration diagrams for System 2: x0 = 1.5, c3 = 10,000. (c) Exemplary free vibration diagrams for System 3: x0 = 1.5, c3 = 0, ABS(kappa). (d) Exemplary free vibration diagrams for System 4: x0 = 1.5, c3 = 10,000, ABS(kappa).
2
y1 6y 2 Ec ¼ 6 4... ys
3 x1 x2 7 7 ¼ ½y ; xi i ...5 xs
ði ¼ 1; 2; . . . ; sÞ
(16)
Examining relation yi(xi) in the form of a two-dimensional diagram of dependent variable y versus independent variable x one should: a. select a particular shape of function k(x), b. calculate the constant parameters of the selected function (including parameter k), using the well-known methods of parametric optimization (e.g. nonlinear regression analysis [12]).
4.
Computer verification of the method
The practical functioning of the above method was tested on four dynamic systems described by Eq. (4). The verification tests for the systems were carried out for both linear and nonlinear functions fs(x), k(x) and the numerical data are shown in Table 1. For the tests it was assumed that the elasticity characteristics for all the systems were precisely determined. The aim of the tests was to determine constant k and function k(x). The systems were built using the Matlab Simulink software.
Table 1 – Numerical data on tested systems.
System System System System
1 2 3 4
m [kg]
c1 [N/m]
c3 [N/m3]
k [Ns/m]
k0
k(x)
20 20 20 20
800 800 800 800
0 0 10,000 10,000
40 40 40 40
600 600 600 600
k0 x v k0 x jxj v k0 x v k0 x jxj v
Table 2 – Results of identification.
System System System System
1 (c3 = 0) 2 (c3 = 0) 3 4
k [N/m]
k0
40.00 0.01 40.00 0.01 40.00 0.01 40.00 0.01
599.98 0.13 603.12 14.79 600.07 0.16 590.87 16.24
358
archives of civil and mechanical engineering 14 (2014) 354–359
Fig. 5 – Relations yi(xi) for tested systems and their approximations (continuous lines).
Different initial conditions for displacement x0 in the systems were set at v0 = 0 and p(t) = 0. In total, 15 different initial conditions for x0 ranging from x0 = 0.01 to x0 = 0.15 at a step of 0.01 were set for each of the systems. Exemplary time diagrams for selected initial conditions for all the tested systems are shown in Fig. 4. Four values of xi in instants ti determined in the previous section of this paper were selected for each of the initial condition, using a dedicated software. In this way about 60 values xi (i = 1, 2, . . ., 60) and extreme velocity values vi were obtained for which the corresponding values of yi were calculated from relation (8). Diagrams of relation yi(xi) for each of the systems are shown in Fig. 5. By approximating these relations by means of the regression analysis method the results presented in Table 2 were obtained.
5.
Conclusions
The tests have shown that the proposed method functions properly. The values obtained for the systems with linear characteristics k(x) are practically equal to the given values, regardless of the shape of function fs(x). Slightly larger scatter of results was obtained for the systems with nonlinear characteristics k(x) (systems 2 and 4). It should be noted that the precise synchronization of the displacement, velocity and acceleration traces in time is critical for the accuracy of the
obtained results. Such synchronization can be difficult in the case of real measurements performed on real objects, which requires further research. It would be also interesting to extend the developed method for the so-called degenerated systems which are used, among other things, in the process of piercing ballistic shields [13].
references
[1] H.A. Barnes, J.F. Hutton, K. Walters, An Introduction to Rheology, Elsevier Science B.V., Amsterdam, Lausanne, New York, Oxford, Shannon, Singapore, Tokyo, 1989. [2] D.J. Ewins, Modal Testing, Theory, Practice and Application, second ed., Research Studies Press Ltd., Baldock, Hertfordshire, England, 2000 . [3] Structural Testing. Part 1. Mechanical Mobility Measurements, Bruel & Kjaer, 1988. [4] Structural Testing. Part 2. Modal Analysis and Simulation, Bruel & Kjaer, 1988. [5] R. Skalak, S. Chien, Handbook of Bioengineering, McGrawHill Book Company, New York, 1979. [6] F.G. Evans, Mechanical Properties of Bone, Charles C. Thomas, Springfield, IL, 1973. [7] D.T. Reilly, A.H. Burstein, V.H. Frankel, The elastic modulus for bone, Journal of Biomechanics 7 (3) (1974) 271–275. [8] J.H. McElhaney, Dynamic response of bone and muscle tissue, Journal of Applied Physiology 21 (1966) 1231–1236.
archives of civil and mechanical engineering 14 (2014) 354–359
[9] S.S. Sarva, S. Deschanel, M.C. Boyce, C. Weinong, Stress– strain behavior of polyurea and polyurethane from low to high strain rates, Polymer 48 (2007) 2208–2213. [10] A. Buchacz, M. Płaczek, Development of mathematical model of a mechatronic system, Solid State Phenomena 164 (2010) 319–322. [11] A. Buchacz, D. Galeziowski, Synthesis as a designing of mechatronic vibrating mixed systems, Journal of
359
(no. of Vibroengineering 14 (June (2)) (2012) 553–559 publication 781). [12] J.J. Johnston, Econometric Methods, McGraw-Hill Book Company Inc., New York, 1963. [13] K. Jamroziak, Analyis of a degenerated standard model in the piercing process, Journal of Achievements in Materials and Manufacturing Engineering 22 (1) (2007) 49–52.