Parametric vibrations of flexible hoses excited by a pulsating fluid flow, Part II: Experimental research

Journal of Fluids and Structures 55 (2015) 174–190

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Parametric vibrations of flexible hoses excited by a pulsating fluid flow, Part II: Experimental research Andrzej Czerwiński a,n, Jan Łuczko b a Institute of Machine Design, Faculty of Mechanical Engineering, Cracow University of Technology, Jana Pawła II 37, 31-864 Krakow, Poland b Institute of Applied Mechanics, Faculty of Mechanical Engineering, Cracow University of Technology, Jana Pawła II 37, 31-864 Krakow, Poland

a r t i c l e in f o

abstract

Article history: Received 5 June 2014 Accepted 12 March 2015 Available online 14 May 2015

The paper presents the results of experimental studies of vibrations of an elastic hose which are induced by a pulsating fluid flow. It was found that there is a possibility of parametric resonances: principal and combination associated with certain modes of vibrations. The influence of frequency and the amplitude of pulsation, average flow velocity, pressure inside pipe, the length of the hose, and the temperature on the ranges of parametric vibrations were analysed. The character of vibrations in resonance ranges was determined by showing time histories and the results of spectral analyses. A flexible hose applied in high-pressure hydraulic systems was used as an object of research. The values of basic parameters which describe the hose's mechanical properties were identified experimentally. The results of the experiments were compared with the results of numerical simulations conducted on the basis of the methodology proposed in Part I of this paper. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Hydraulic systems Vibrations induced by the flow Parametric resonance

1. Introduction Vibrations and noise in hydraulic systems constitute a significant problem which is studied in theoretical and experimental research. The main hydraulic causes of noise and vibrations are cavitations (Ruchonnet et al., 2012), impact changes of fluid pressure in pumps and motors, and pressure fluctuation in a system (Borisyuk, 2010; Hayashi and Kaneko, 2014). Individual elements of a hydraulic system are tested to minimize their vibro-acoustic activity (Harrison and Edge, 2000; Casoli et al., 2006; Spence and Amaral-Teixeira, 2009). The transient states of vibration are also analysed, especially those connected with the phenomenon of water hammer (Zhao et al., 2005; Zarzycki and Kudzma, 2007). A pressure wave with a great amplitude is a result of the water hammer. This wave generates vibrations and can cause damage in the hydraulic system. Authors frequently focus on the analysis of a hydraulic system with regard to the minimization of pressure fluctuation (Ortwig, 2005; Klop and Ivantysynova, 2011; Kudźma and Stosiak, 2013). A pulsating flow of fluid in hydraulic systems is the source of the vibrations of the individual elements of the system and of the hydraulic hoses which connect them. The hose

n

Corresponding author. Tel.: þ48 123743360, þ48 123743621; fax: þ 48 123743360. E-mail address: [email protected] (A. Czerwiński).

http://dx.doi.org/10.1016/j.jfluidstructs.2015.03.007 0889-9746/& 2015 Elsevier Ltd. All rights reserved.

A. Czerwiński, J. Łuczko / Journal of Fluids and Structures 55 (2015) 174–190

Nomenclature a A E(p0,θ) E0, E1, E2 e0, e1, e2 f f1 f2 fn fp Ip L m mf p0 rn rn-m t

dimensionless flow pulsation amplitude internal sectional area of a pipe Young modulus approximation coefficients E(p0,θ) approximation coefficients E(p0,θ) frequency (spectrum) start frequency in swep-sine test stop frequency in swep-sine test natural frequency pulsation frequency cross section moment of inertia length of a pipe mass of the hose with fluid per unit length mass of fluid per unit length pressure inside the hose for x ¼L main secondary resonance principal resonance time

T0 Tp Tsim Uf(t) Uf0 vRMS w(x,t) x α(fp,p0) α0 δ ε θ Θv ϑ(fp) ϑk ν σ χ(p0) χk

175

axial force for x ¼L excitation period time of simulation flow velocity average flow velocity RMS value of vibration velocity transverse displacement coordinate along pipe internal damping coefficient damping coefficient for fp ¼1 Hz, p0 ¼ 0 logarithmic decrement axial strain temperature vibration index dimensionless stiffness ratio approximation coefficients of ϑ(fp) Poisson ratio axial stress dimensionless inertia ratio approximation coefficients of χ(p0)

vibrations have usually a relatively small amplitude and a frequency equal to the frequency of flow pulsation. However, for some flow parameters, conditions for resonant vibrations with a multiple times bigger amplitude may occur. A problem of the dynamics of hoses with a flowing fluid has been the subject of interest to many authors. The mathematical models of the discussed systems allow us to research physical phenomena in a very wide range. Nevertheless, there are still far fewer papers which present the results of experiments in comparison to the number of theoretical studies. First experimental works referred to determination of flow critical velocity and the influence of internal pressure, flow velocity, and initial tension on the system's natural frequencies. Dodds and Runyan (1965) calculated, and then experimentally verified, natural frequencies and the critical values of flow velocity for an aluminium pipe fixed at both sides. Greenwald and Dugundji (1967) studied the influence of velocity on the possibility of buckling of clamped–pinned and cantilevered elastic pipes, and Yoshizawa et al. (1985) for the case sliding downstream end. Enabling an axial shift of the hose's end allowed the authors of these works to disregard the influence of fluid pressure. Naguleswaran and Williams (1968) determined the values of natural frequencies for various pressures and initial tensions for an elastic pipes fixed at both ends. The authors showed that the increase in pressure can be the reason for the hose buckling even in the case when there is no fluid flow. Jendrzejczyk and Chen (1985) studied polythene and acrylic pipes for various types of their support. They noticed that for the fixed pipe the loss of stability through divergence is impossible. They justified their hypothesis on the grounds of the influence of reactive forces produced as a result of the pipe deformation. Zhang et al. (1999, 2000) studied the influence of a tension force and flow velocity on natural frequencies and on the modes of vibrations of an elastic pipe. Jin et al. (2007) analysed the influence of flow velocity and tension on the dynamic response and the damping properties of a silicon pipe. In the work of Ajiro et al. (2006), the authors considered the case of a silicon hose fixed elastically. They used the image analysis methods. A vast number of experimental works refer to the phenomena which emerge when the fluid flows through cantilevered pipes. Gregory and Païdoussis (1966) discussed a flexible pipe conveying water or air and studied the influence of velocity on natural frequencies and the forms of vibrations (with the use of photography). Païdoussis (1970) analysed the phenomenon of oscillating instability for a cantilevered rubber tubes hanging down or standing upright. Yoshizawa et al. (1988) took into consideration the influence of an additional transverse force which acts on a flexible hose conveying water. Both the pipes with the support localised at a smaller distance than the length of the pipe (Jendrzejczyk and Chen, 1985; Chen and Jendrzejczyk, 1985) and the pipes with the spring support (Sugiyama et al., 1985) were studied. Sugiyama et al. (1996) analysed the influence of the degree of submerging the free end of a silicon pipe on the possibility of losing stability through flutter. The theoretical and experimental research of cantilever pipes aspirating fluid were also conducted by Kuiper et al. (2007), Kuiper and Metrikine (2008), Giacobbi et al. (2012), mainly in order to determine natural frequencies and examine stability. The influence of mass added at the end of an elastic pipe on the type of generated vibrations was studied by Copeland and Moon (1992) and Yoshizawa et al. (1997). They used a technique of the image analysis. The pipes with an additional mass at the end were studied by Païdoussis and Semler (1998) as well as by Ajiro et al. (2006). For sufficiently high flow velocities, the flutter phenomenon for two unstable modes of vibrations was noticed. Sugiyama and Noda (1981) examined the influence of the concentrated mass on the vibrations of articulated pipes horizontally fixed.

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Experiments for the cantilevered and clamped–clamped elastic pipes were carried out by Païdoussis and Issid (1976). They studied the influence of flow velocity, frequency and the amplitude of pulsation on the occurrence of the phenomenon of parametric resonance. Yoshizawa et al. (1986) analysed the vibrations of a silicon pipe fixed vertically (clamped-simply supported) and determined vibration amplitudes within the range of a principal parametric resonance. Semler and Païdoussis (1996) conducted experiments for a cantilevered pipe end and determined the stability ranges. Fujita and Tanaka (1990) studied the influence of flow pulsation on the vibrations of a straight pipe, and Nagai and Hayama (1991) analysed the vibrations of a curved pipe with a fluid flowing inside. The authors of this paper carried out experimental research on a typical hose used in high pressure hydraulics. The hose is built from two layers of synthetic rubber (the inner and the outer) and the reinforcement by a steel braid between these layers. Such a construction results in different properties of this hose in comparison to the properties of a hose made from an isotropic material. The properties also depend on the working conditions of the hose. Van den Horn and Kuipers (1988), Hogenbirk et al. (1988) examined the mechanical properties of this type of hoses. They focused on the analysis of the influence of the internal pressure on stress and strain in a hose. Bregman et al. (1993) proposed a model for a hose with spiral reinforcement. They used the model to determine the influence of the internal pressure on both strain and axial and transverse stresses. They pointed out that stiffness and deformations of hoses reinforced by a steel braid depend on the direction of an acting force. The hose has a higher stiffness in the transverse direction than in the axial direction. You and Inaba (2013) analysed and conducted experiments on the influence of the winding angle on the hoop and axial strains and the stiffness of a carbon-fibre reinforced plastic pipe. Catianccio (2009) studied the influence of the internal pressure on the strain and deflection of hoses reinforced with a braid with orthotropic features. He proved that the pressure may cause the pipe shortening. It also stabilizes the pipe since the deflection is decreased. Such an influence occurs if substitute Poisson's ratio (which depends, for instance, on the angle of wrapping of the reinforcing grid) takes the values greater than 0.5. Vašina and Hružík (2009) determined the modulus of transverse stiffness of elastic hydraulic hoses. They proved a strong dependence between the value of the modulus and the pressure inside the hose. Various types of hydraulic pipes were also examined by Johnston et al. (2010). They experimentally showed that the pipe's axial stiffness increases with the increase in pressure, but decreases when the temperature increases. Measured values of Poisson's ratio for the majority of hydraulic hoses take the values greater than 0.5. This paper refers to the experimental research of the vibrations of an elastic hydraulic hose. Special attention has been paid to the possibility of the occurrence of the parametric resonance phenomenon resulting from a pulsating flow in such hoses. In professional literature, the authors have not found an analysis of such phenomena in hydraulic power transmission systems. The results of the experiments have been compared with the results of numerical calculations. The authors have used the model of a hose described in detail in Part I of this article. The identification of the hose's parameters used in the numerical model has been conducted on the basis of the tests carried out on a real object. 2. The test stand In order to study the behaviour of the hose with a pulsating flow and to confirm the results derived on the basis of numerical simulations presented in Part I of this paper, a measurement test stand has been built. The scheme of the measurement test stand is shown in Fig. 1. The test stand consists of a tested hose fixed at both ends horizontally, a hydraulic system which is used for generation and proper formation of a hydraulic oil flow, and a

Fig. 1. Schematic of the experimental setup: (I) electric motor, (II) pump, (III) hydropneumatic accumulator, (IV) overflow valve, (V) proportional directional control valve, (VI) hydropneumatic accumulator, (VII) investigated hose, (VIII) throttling valve.

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177

Fig. 2. Scheme of the measurement method of hose's axial stiffness.

Fig. 3. Axial stiffness modulus of a hose, experiment results – points, simulation results – lines: (a) influence of pressure for temperature 23 1C (blue solid line) and 52 1C (red dashed line), (b) influence of temperature for pressure p0 ¼ 0 MPa (blue solid line), p0 ¼2 MPa (green dotted line) and p0 ¼ 4.2 MPa (red dashed line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

measurement part comprising sensors and the system of data acquisition. In the system, there are used hydraulic elements controlled by voltage. The average flow velocity, the amplitude, the frequency of velocity pulsation, and the pressure can be controlled. The test stand allows for examining various types of pipes (steel and flexible) of different lengths within 2 m. To determine the parameters connected with the flow (velocity and pressure) and the hose's vibrations, appropriate transducers have been used. The fluid velocity in the hose is calculated on the basis of the information about the flow rate (the flow rate sensor—Qf) and about the area of the hose cross section. The pressure measuring (transducers: p1 and p0) has been carried out in two points directly before and after a section of the hose. Accelerometers (a1/2 and a1/4) localised in ½ and ¼ of the hose's length have been used for measuring hose vibration parameters. The measurement of the acceleration of vibrations is made in two directions perpendicular to the hose's axis. The measurement of the hose's temperature (Th) is carried out with the use of a contact sensor (a thermocouple of the type K). Signals from the sensors are sent to the computer through the converter A/D (NI USB-6009). The program DasyLab 11 has been used for the acquisition and processing of the measured signals. The computer with the card D/A (NI USB-6009) has also been used for controlling the pump displacement (POMP) and the overflow valve (OV), as well as for generating the control signal of the proportional directional control valve (PDCV). The examined hose, placed in a horizontal position, is fixed at both ends with the use of clamps which prevent an axial shift. The preliminary tension force is regulated by the distance between the supports. The configuration of the test stand allowed for the measurements for both the determined flows (free vibrations) and the pulsating flows during the examination of the parametric resonance phenomenon. 3. Determination of the mechanical properties of a hose A flexible hydraulic hose of the 1SN DN12 type has been used as a research object. The hose is built from two layers of synthetic rubber (the inner and the outer) reinforced by a single steel braid. The hose's density is 1870 kg/m3, the outer diameter equals 21.0 mm, and the bore is 12.7 mm. The experiments have been carried out for hoses of the lengths of 1.91 m and 1.45 m. The mechanical properties of the hose are featured by stiffness, damping coefficient, and Poisson's ratio. Due to the hose's composite construction, the values of these parameters cannot be accepted as they are for a homogenous material. Thus, tests allowing the evaluation of their values have been performed. The axial stiffness of the hose is determined on the basis of tensile tests. This parameter is verified using the information about the values of the first four frequencies of free vibrations. The damping properties are determined on the basis of the analysis of damping decrements. Due to the hose's

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Fig. 4. Scheme of the measurement method of natural frequencies and logarithmic decrements (example for the case of the second form of vibration).

Fig. 5. The influence of pressure on the eigenfrequencies (a) and on the logarithmic decrements (b), (L ¼1.91 m, Uf ¼ 0 m/s), experimental results – points, simulation results – lines, blue solid line – 1st mode, green dashed line – 2nd mode, orange dot-dashed line – 3rd mode, red dotted line – 4th mode. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

construction, its stiffness and damping properties depend on the work parameters (pressure, tensile force, and temperature). Taking these facts into consideration, it is necessary to examine the influence of the change of the abovementioned parameters on the mechanical properties of the hose.

3.1. Evaluation of stiffness The stiffness of the hose is one of basic parameters which describe its mechanical properties. Due to the construction of the hose, its stiffness depends on the internal pressure. In order to evaluate stiffness, a test has been conducted. In the test, the elongation of the hose under a given force is measured (Fig. 2). The axial stiffness modulus is calculated by measuring the change in the hose's tension and the corresponding change of elongation. The test has been performed for the internal pressures in a range of 0–10 MPa and the temperatures from 23 1C to 52 1C. Fig. 3(a) illustrates the measurement results of the axial stiffness modulus of a hose dependant on the internal pressure (points on the characteristics). The measurements have been conducted for a hose of the temperature of about 23 1C and 52 1C. The results of the experiment confirm observations included in the works (Vašina and Hružík, 2009; Johnston et al., 2010) about the influence of the internal pressure which increases the stiffness of a hydraulic hose. Fig. 3(b) shows the change in the value of the axial stiffness modulus, which results from the fact that the pipe becomes warmer. The examinations have been performed for three values of the internal pressure. A decrease in the axial stiffness together with an increase in the temperature has an approximately linear character, and the influence of the temperature change on the stiffness decreases together with the increase in pressure. On the basis of the results of the above-mentioned experiments it is assumed that the change in the axial stiffness of the hose takes place (in a simple way) according to this formula: Eðp0 ; θÞ ¼ ðE0  e0 θÞ þðE1 e1 θÞp0 þðE2 e2 θÞp20 :

ð1Þ

Thus the dependence on pressure is described by a quadratic function, and on the temperature—by a linear function. Good approximation of the results of the experiments is provided by the curves obtained for the values E0 ¼142 MPa, E1 ¼21, E2 ¼1.5 MPa  1, e0 ¼2.32 MPa K  1, e1 ¼0.26 K  1 and e2 ¼0 MPa  1 K  1. For example, at the temperature of 23 1C, for the oil pressure p0 ¼4 MPa, the Young modulus determined from Formula (1) equals E ¼172.8 MPa; and at the working temperature of about 52 1C the value of the Young modulus decreases almost two-fold to the value E ¼71.9 MPa. In Fig. 3, the solid lines depict the graphs of the functions described by the Formula (1) for the given values of the appropriate coefficients.

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Fig. 6. The influence of flow velocity on the eigenfrequencies (a) and on the logarithmic decrements (b), (L ¼ 1.91 m, p0 ¼ 3 MPa), experimental results – points, simulation results – lines, blue solid line – 1st mode, green dashed line – 2nd mode, orange dot-dashed line – 3rd mode. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3.2. Determination of the values of natural frequencies and logarithmic decrements. Free vibrations of the hose are examined in order to determine the values of natural frequencies and logarithmic decrements. Each form of the hose's vibration is excited separately. The vibration is forced by applying a momentary force in the places of the maximum of suitable mode. Fig. 4 presents a scheme of the measurement method for the case of the second form of vibration. The acceleration of vibrations is measured with the use of an accelerometer localized in the place of the maximum deflection of a suitable mode. The experiment yielded the time histories of damped vibrations for a few successive forms. Band-pass filtration has been used in the analysis. The filter's middle frequencies are established as equal to the natural frequency for a given form of vibration. Such an operation allowed for minimizing the influence of other forms on the vibration signal of the analysed form. The frequencies of free vibrations are determined as the inverse of the vibration period, and the logarithmic decrement is determined with the use of a classical method by analysing the ratios of the following maximums of the signal. The dependence of natural frequencies and damping decrements on the pressure in the case of lack of a flow (obtained experimentally) is shown in Fig. 5. The dependence of the same values on the flow velocity for the constant pressure p0 ¼ 3 MPa is shown in Fig. 6. The application of the results obtained experimentally for the evaluation of the internal damping coefficient α, introduced in Part I for the assumed damping hypothesis in the form of the relation:
 σ ¼ E ε þ α∂ε=∂t : ð2Þ Between stress, strain, and strain velocity requires an analysis of the proper model of a hose with a flowing fluid. Such an analysis can be conducted on the basis of the linearized equation of the transverse vibration (Eq. (17), Part I), which in dimensional variables takes the form:  4    ∂2 w ∂2 w ∂2 w ∂ w ∂5 w m 2 þ mf 2U f þU 2f 2 þEI p þα 4 4 ∂x∂t ∂x ∂x ∂x ∂t ∂t   2 ∂U f ∂ w ðL  xÞ ¼ 0: ð3Þ þ p0 Að1  2νÞ  T 0 þ mf ∂t ∂x2 In Eq. (3), the coordinate w(x,t) describes the transverse displacement of the hose, Uf(t) is a varying in time flow velocity, and T0 and p0 are respectively an axial force and the pressure at the end of the hose (for x ¼L). The following parameters also influence the solutions of Eq. (3): the elementary mass of fluid mf, the elementary mass of the hose with fluid m, the area of the inner cross section of the hose A, transverse stiffness EIp, Poisson ratio ν, and an assumed dependence (1) of the Young modulus on the pressure and temperature. After the determination of complex eigenvalues, both the dependence of natural frequencies fn and damping decrements on the coefficient α can be described. Assuming a constant value of the damping coefficient α does not allow for obtaining qualitative and quantitative compatibility of the simulation and experiment results, including the results of further analyses which take into account the influence of the changing fluid flow velocity. It has been further accepted that the coefficient α is an assumed function of the excitation frequency fp; and for fp ¼fn the decrements determined theoretically and experimentally should be equal one to another. Additionally, the influence of pressure inside the hose on the values of the α coefficient has been taken into consideration. The hypothesis has been accepted thanks to numerous analyses: αðf p ; p0 Þ ¼ α0 ϑðf p Þχðp0 Þ;

ð4Þ

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Fig. 7. The influence of flow pulsation frequency on RMS value of vibration velocity for different values of flow rate (p0 ¼2 MPa, a ¼0.2): (a) experimental results for Uf0 ¼ 6 m/s, 7 m/s and 8 m/s, and (b) simulation results for Uf0 ¼6 m/s, 7 m/s and 8 m/s. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

with a quadratic function being used for describing the dependence on pressure: χðp0 Þ ¼ ðχ 0 þ χ 1 p0 þ χ 2 p0 2 Þ;

ð5Þ

whereas for describing the influence of frequency, a five-parameter model in the following form has been taken: ϑðf p Þ ¼

4 X

k

ϑk f p :

ð6Þ

k¼0

Similarly defined multi-parameter models were used by Liu and Gorman (1995), and also (for k¼1; a two-parameter model) by Rinaldi and Païdoussis (2010). The introduction of the models was justified on the grounds of better compatibility of the results of simulations and experiments. By comparing the graphs of decrements determined experimentally with those obtained through numerical simulations, the values of the coefficients present in Eqs. (4)–(6) are evaluated. In Figs. 5 and 6, the solid curves illustrate the results of numerical calculations obtained for the values: ϑ0 ¼1, ϑ1 ¼  28 s, ϑ2 ¼ 730 s2, ϑ3 ¼  4.8  103 s3, ϑ4 ¼ 1.1  104 s4, χ0 ¼1, χ1 ¼  0.16 MPa  1, χ2 ¼0.009 MPa  2 and α0 ¼6.5  10  3 s  1. In Figs. 5 and 6, the solid lines show the graphs of functions determined on the basis of the dependence (4) for the assumed values of the appropriate coefficients. The results presented in Fig. 6 are obtained for the constant flow velocity Uf. Since in the experiments the maximum value Uf is significantly smaller than the critical value, the vibration frequencies and damping decrements within the examined velocity range depend on the flow velocity to a small degree (Fig. 6). The pressure p0 has a more profound influence on the eigenvalues. The course of decrement graphs (Fig. 5(b)) depends greatly on the accepted damping model. However, the greatest influence on the frequency graphs (Fig. 5(a)), apart from the stiffness described by Formula (1), has Poisson ratio. For the value ν ¼0.59, the curves determined analytically approximate experimental dependences in the best way. Good compatibility between the results of the experiments and simulations positively acknowledges the accepted damping model. The comparison of the results of the experiments conducted for the changing flow velocity with the appropriate results of the non-linear analysis (Entry 4) constitutes additional verification of the model. 4. The results of the research Experimental research has been conducted to examine physical phenomena produced by a variable in time flow velocity. The obtained results are verified by numerical simulations with the use of a model described by a system of two non-linear partial differential equations with the coefficients varying in time (Eqs. (16 and 17), Part I). During research, the flow pulsation frequency fp is a basic variable parameter. Experiments are conducted for various values of the internal pressure p0, average flow velocity Uf0, and the amplitude of the flow velocity pulsation a. In order to effectively examine the behaviour of the hose in a wide range of excitation frequency, a sinusoidal signal with a linearly variable frequency is used to carry out the assumed change of flow velocity. The signal is retuned according to the formula uðtÞ ¼ A sin 2πðf 1 t þ 0:5kt 2 Þ;

ð7Þ

where k ¼ ðf 2  f 1 Þ=T sim [Hz/s] is the speed of the signal frequency change, f1 and f2 [Hz] are initial and final frequencies, and Tsim [s] is a time duration of frequency change from f1 to f2. The same parameters of the signal are used in all tests to enable the comparison of the obtained results. The speed of the change k is assumed as equal to 0.1 Hz/s, the frequency is changed

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Fig. 8. The influence of flow velocity and pulsation frequency on the index Θv (p0 ¼ 2 MPa, a ¼0.2), ’   - experimental results.

from f1 ¼2 Hz to f2 ¼50 Hz; thus the time of the test is equal to Tsim ¼480 s. An application of such excitation allows for determining the dependence of certain indicators on the excitation frequency fp ¼f1 þkt. Figs. 7, 9, 11, 13–16 show the results of the tests conducted in this way. Vibration intensity is characterized by the following index: RMS values of vibration velocity in the transverse direction to the hose's axis in two measurement points— in the middle of the hose (blue colour) and in ¼ the length from the point of its end (red colour). The results of the experiment have been compared with the results of the simulation of a non-linear model. In numerical calculations, it is assumed that flow velocity changes according to the relation U f ¼ U f 0 ½1 þa sin 2πðf 1 t þ 0:5kt 2 Þ;

ð8Þ

where Uf0 is the average flow velocity, and a is the relative pulsation amplitude referred to the value Uf0. In numerical calculations, the determined values of the parameters characterizing an elastic hose are also used. The values of other parameters connected with the flow and excitation are accepted as the same as in the experiment. Fig. 7 illustrates the influence of the average flow velocity Uf0 in the range from 6 m/s to 8 m/s. In the diagrams are visible the ranges of increased vibration levels in which the phenomenon of parametric resonance occurs most often. In the ranges of simple principal resonances associated with the successive natural frequencies (labelled further as r1-1, r2-2, r3-3 etc., generally rn-n) the nth form of vibrations is observed, with the vibration frequency being twice smaller than the excitation frequency. In combination resonances (r1-2, r2-3, r3-4 etc., generally rn-m, m an), the form of vibrations is more complex and the excitation frequency is a certain combination of at least two natural frequencies (most often fp Efn þfm). For the flow velocity 6 m/s, the ranges of parametric vibrations are relatively narrow, the widest range (about fp E25 Hz) occurs in the resonance r3-3. With an increase in the velocity Uf0, the resonance vibration ranges widen and move slightly to lower frequencies. For the velocity 8 m/s, between the ranges of the resonances r2-2 and r3-3 and between r3-3 and r4-4 (Fig. 7 (b)) additional resonance ranges appear. They can be interpreted as the combination resonances r2-3 and r3-4. In the experimental diagrams (Fig. 7(a)), for Uf0 ¼8 m/s in the range fp E15–27.5 Hz, parametric vibrations of high amplitudes are excited. Then, the interpretation of the adequate resonances is not completely unambiguous. The results of numerical simulations correspond basically with the results of the experiment. The ranges of parametric vibrations and the amplitude values are close to each other. The influence of velocity which widens the ranges or resonances is illustrated properly. There are also some qualitative differences between the results of the experiment and simulations. The parametric resonance r1-1 is not observed in the experiments. The ranges of the combination resonances determined experimentally overlap the ranges of the simple principal resonances, in contrast to those calculated numerically which occur clearly separately. It refers to, among others, the ranges of resonances r2-2 and r2-3 (e.g. 15 Hz ofp o21 Hz for Uf0 ¼8 m/s, Fig. 7). In this range, the experimentally determined vibration amplitude at the centre of the hose (Fig. 7(a)) grows monotonically whereas the calculated amplitude (Fig. 7(b)) starts to noticeably increase only in resonance r2-3 (for fp 418 Hz). This difference can result from non-ideally identical hose properties along the hose's length. The experimentally examined hose (a steel braided rubber hose) is a typical hose used in hydraulic systems. During the experiments it was noticed that even forms of vibrations (e.g. in resonance r2-2) are not ideally symmetrical and the node is not exactly at the centre of the hose. Hence the vibration amplitude in the middle of the hose also grows in resonance r2-2. However, such partially coincidental features of the hose are impossible for taking into account in the model of the system. The influence of frequency on the amplitude at the ¼ of the hose's length is also slightly different. Numerical simulations indicate a monotonic increase of this amplitude whereas, in the experiment, after a relatively rapid increase the amplitude restriction takes place. Vibration amplitudes in resonances are relatively very big. The omission of non-linearity of the

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Fig. 9. The influence of flow pulsation frequency on RMS value of vibration velocity for different values of internal pressure (Uf0 ¼ 8 m/s, a¼ 0.2): (a) experimental results for p0 ¼ 2 MPa, 3 MPa and 4 MPa, and (b) simulation results for p0 ¼2 MPa, 3 MPa and 4 MPa. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

higher than the third degree in the system model can constitute the reason for the appearing differences. However, taking such non-linearities into account would cause too big a complication of the model and relevant simulations would be difficult to conduct in practice. To sum up, the accepted model of the system allows quite good identification of individual resonance ranges. Both in the experiments and in the numerical simulations it has been noticed that occasionally even the smallest change of the test's conditions or parameter values decides about generating or disappearing of resonance vibrations. The experimental research allows for the verification of the parameters of the system's mathematical model. However, they can be conducted only in the limited ranges of the system's parameters. It especially refers to the range of the flow velocity Uf0 usually limited by a certain maximum value which is possible to obtain in the experiment. To receive more complete information about the system's behaviour, numerical calculations in the wider ranges of parameter changes have been performed. Fig. 8 shows the simulation results which illustrate the influence of the flow velocity and the pulsation frequency on the vibration level indicator Θv defined similarly as in Part I. The indicator θv is the root-mean-square value (averaged along the hose's length) of velocity in the transverse direction to the hose's axis: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ðm þ kÞT p Z L 1 _ 2 ðx; tÞdxdt : Θv ¼ w kLT p mT p 0

ð9Þ

The indicator Θv can be determined, due to the orthogonal feature of the shape function, as a sum of the root-mean_ n , which determine the participation of the subsequent modes of vibration. square values of coordinates w The calculations are done in a mesh of points (200  200) with the same initial conditions (close to zero) for each point. The obtained results can be compared with the results presented in Part I of this paper for the relevant non-dimensional parameters (Figure 5 in I Part). Resonance areas widen together with an increase in flow velocity, slightly moving to lower frequencies. For higher velocities, the parametric resonance areas overlap. It results in the occurrence of resonance vibrations in a practically whole range of the analysed frequencies. The widening of the resonance ranges occurs also together with an increase in frequency. The area of the main parametric resonance (the first form of vibrations) is relatively narrow in the whole range of the examined flow velocities and the values of the index Θv are relatively small, which can explain a fact that the resonance of the type r1-1 is not observed in the experiments. The method applied in creating Figs. 8, 10 and 12 distinctly differs from the method used for making diagrams (Figs. 7, 9 and 11). In the swept-sine method, the initial conditions change with the change of pulsation frequency. For subsequent excitation frequencies the initial state is equal to the final state which corresponds to the previous frequency. The result of that is joining together successive resonance ranges. Nevertheless, an attempt can be made to compare the results shown in Figs. 8, 10 and 12 with the experimental research results (Figs. 7(a), 9(a) and 11(a)). In order to do that in Figs. 8, 10 and 12, for chosen values of flow velocity, pressure or amplitude, the ranges of resonances detected experimentally were marked (intermittent lines ended with arrows). These ranges are slightly shifted in the direction of higher frequencies. Moreover, the ranges of simple and combination resonances (e.g. r2-2 and r2-3 or r3-3 and r3-4) are connected together. It should be underlined that the tested system is a non-linear system and in some parameter ranges the solutions depend on the initial conditions. It is also verified by numerical simulations. Fig. 9 shows the influence of the internal pressure p0. While analysing the diagrams obtained experimentally (Fig. 9(a)) and numerically (Fig. 9(b)), it can be noticed that with an increase in pressure a significant shift of the resonance ranges to higher frequencies takes place. As the research into the hose's properties revealed, the pressure increase causes the stiffness

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Fig. 10. The influence of internal pressure and pulsation frequency on the index Θv (Uf0 ¼ 8 m/s, a¼0.2), ’   - experimental results.

Fig. 11. The influence of flow pulsation frequency on RMS value of vibration velocity for different values of pulsation amplitude (Uf0 ¼ 7 m/s, p0 ¼ 2 MPa): (a) experimental results for a¼0.15, 0.20 and 0.25, and (b) simulation results for a ¼0.15, 0.20 and 0.25. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

Fig. 12. The influence of pulsation amplitude and pulsation frequency on the index Θv (Uf0 ¼7 m/s, p0 ¼ 2 MPa), ’   - experimental results.

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Fig. 13. The influence of flow pulsation frequency on RMS value of vibration velocity for hose's length L ¼ 1.45 m and L ¼ 1.91 m (Uf0 ¼6 m/s, p0 ¼ 2 MPa, a ¼0.25): (a) experimental results for, and (b) simulation results. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

Fig. 14. The influence of flow pulsation frequency on RMS value of vibration velocity for different values of hose's temperature (Uf0 ¼ 5 m/s, p0 ¼ 1.4 MPa, a ¼0.22): (a) experimental results for θ¼ 23 1C, 39 1C and 47 1C and (b) simulation results θ¼ 23 1C, 39 1C and 47 1C. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

increase and shortening, which in the case of a both-end fixing is the reason for increasing the axial force. The shift of the resonance ranges can be thus explained by increasing the values of free vibration frequencies with the increase in stiffness and in the axial force. With the pressure increase, the ranges of the resonances determined experimentally (Fig. 9(a)) slightly narrow, and in those obtained numerically (Fig. 9(b)) the ranges of the combination resonances disappear. Despite certain inconsistencies between the results of the experiments and simulations, the obtained characteristics of the system prove that the above-mentioned effects are shown well by the proposed model. The values of the parameters determined on the basis of the identification of the real object are also properly determined. The results of numerical simulations in a wider range of the internal pressure changes are presented in Fig. 10. The observation of the results confirms conclusions derived on the basis of the analysis of the diagrams (Fig. 9). The increase in pressure and the increase in stiffness and the axial force (connected with it) cause a significant shift of the vibration ranges to higher frequencies. Simultaneously, the change of damping and elastic properties results in a lack of the excitation of resonance vibrations for higher pressures. The obtained results can be partially compared with the results presented in Part I of this paper, which illustrate separately the influence of both the non-dimensional parameter Γ connected mainly with the axial force (Figure 11 in I Part) and of the parameter κ which characterizes the change of the hose's stiffness (Figure 15 in I Part). It should be mentioned here that the results from Part I were determined for the constant value of the internal damping coefficient. In Fig. 10, ranges of resonances determined with the use of the swept-sine method in the experiment (Fig. 9(a)) are marked in an analogical way as in Fig. 8. The flow pulsation amplitude connected with the non-dimensional parameter a through Formula (8) is an important parameter which decides about the possibility of appearing of the parametric resonance phenomenon. For the same values of this parameter, the absolute pulsation amplitude increases with the increase in the average flow velocity Uf0. The results obtained for different values Uf0 are influenced by, apart from the average flow velocity Uf0, a proportional change of the pulsation amplitude a  Uf0 (Figs. 7 and 8). It is caused by an accepted definition of the parameter a. The results obtained for the same value of flow velocity (Uf0 ¼7 m/s) and different values of the parameter a are compared in Fig. 11. As a result of the increase in the value a, there is a clear widening of resonance ranges. At the same time,

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Fig. 15. The influence of flow pulsation frequency on RMS value of vibration velocity for two different type of test signals (stepped-sine and swept-sine), (Uf0 ¼ 8 m/s, p0 ¼ 2 MPa, a¼ 0.25): (a) experimental results and (b) simulation results. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

Fig. 16. The influence of flow pulsation frequency on RMS value of vibration velocity for different sweep direction (Uf0 ¼5 m/s, p0 ¼ 1.4 MPa, a¼0.22): (a) experimental results and (b) simulation results. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

for bigger pulsation amplitudes combination resonances between the ranges of simple principal resonances appear. The sequence of the occurrence of these resonances is analogical to the one shown in Fig. 7. Since the pulsation amplitude practically does not influence the natural vibration frequencies, the ranges of parametric vibrations do not shift together with the amplitude increase. Fig. 12 shows the results of numerical simulations which illustrate the influence of the parameter a and the pulsation frequency fp on the vibration index Θv. Since the calculations are conducted for the determined values of pressure (p0 ¼2 MPa) and the flow velocity (Uf0 ¼7 m/s), the influence of the parameter a can be in this case interpreted as the influence of a real pulsation velocity amplitude. The presented results are quite characteristic of the parametric systems. Since the natural frequencies of the system do not depend on the parameters a and fp, the increase in the pulsation amplitude causes only the widening of the ranges of parametric resonances. The areas of simple and combination resonances occur alternately (r1, r1-1, r1-2, r2-2, r2-3, r3-3,…., r5-6), with the first resonance area r1 being a secondary resonance connected with the first natural frequency. Only in this range, the vibration frequency is equal to the excitation frequency. For comparison reasons, Fig. 12 contains also information about the experimentally determined ranges of parametric resonances. The results of research done for a hose of the same structure as the previous one but with a smaller length L ¼1.45 m are included in Fig. 13. The shortening of the hose changes a distribution of natural frequencies. For the string model frequencies are inversely proportional to the hose's length L, and for the Euler beam model (without taking into consideration the influence of the axial force) they are inversely proportional to the square of the length. The model used in simulations is a non-linear beam model with low rigidity and the influence of the length change on the frequencies is more complex. Nevertheless, shortening the length in a significant way increases the values of frequencies, shifting the resonances into higher frequency ranges. In the case of a shorter hose, some ranges of parametric resonances, e.g. of the type r5-5 are localised outside the examined frequency range. Identification examinations showed a significant influence of temperature on the hose's stiffness (Fig. 3) and indirectly on the change of the axial force, hose's damping properties and other factors (e.g. change of density and oil viscosity), not all of them taken into account in the process of modelling. The probability of the excitation of parametric resonance increases with the increase in temperature. This fact is illustrated in Fig. 14 that shows both the results of experiments (Fig. 14a) and the numerical simulations (Fig. 14(b)), which (through Formula (1)) take into account the influence of the temperature on the stiffness. The decrease in the stiffness and axial force causes the shift of resonance ranges to lower frequencies.

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Fig. 17. The spectrogram (the upper part) and RMS value (the lower part) of vibration signal at point x¼ ¼L (Uf0 ¼ 8 m/s, p0 ¼2 MPa, a ¼0.25).

Fig. 18. Time histories of vibration velocity at point x¼ L/4 (Uf0 ¼ 8 m/s, p0 ¼ 2 MPa, a¼ 0.25): (a) fp ¼ 3.9 Hz, (b) fp ¼8.2 Hz, (c) fp ¼11.8 Hz, (d) fp ¼13.2 Hz, (e) fp ¼15.3 Hz, (f) fp ¼ 21.3 Hz, (g) fp ¼ 22.3 Hz, (h) fp ¼26.9 Hz, (i) fp ¼31.1 Hz, (j) fp ¼36.8 Hz, (k) fp ¼40.9 Hz, and (l) fp ¼46.5 Hz.

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Fig. 19. Modes of vibrations in principal resonances: (a) fp ¼8.2 Hz, resonance r1-1, (b) fp ¼ 16.1 Hz, resonance r2-2, (c) fp ¼ 24.5 Hz, resonance r3-3, (d) fp ¼32.2 Hz, resonance r4-4, and (e) fp ¼ 40.8 Hz, resonance r5-5.

Fig. 15 shows the results of experiments and simulations obtained for the same values of the parameters (Uf0 ¼ 8 m/s, p0 ¼ 2 MPa, a ¼0.25) and with the use of two different test signals. The lower characteristics are obtained in the test with a signal of the swept-sine type, and the upper ones with the use of a signal with a stepped change of frequency (steppedsine). In the swept-sine signal, the excitation frequency changed in a continuous (linear) way. That is why the hose is also continuously excited and there are no pauses in flow velocities pulsation. In the stepped signal, a discrete change of frequency every 0.25 Hz is applied. The duration of one step is 10 s, with the signal being generated for 8 s, and for the next 2 s the signal is equal to 0. Such excitation allowed for obtaining the initial conditions close to zero for each next excitation frequency. While comparing the graphs, a significant widening of the ranges of the resonances obtained with the use of a method of continuous frequency change (swept-sine) in relation to the discrete change (stepped-sine) can be noticed. The ranges widen in the direction of the change of the excitation frequency. It is caused by lack of pauses during the test and favourable conditions for the existence of resonance vibrations in the situation when the resonance phenomenon has already occurred. Resonance vibrations practically appear in the whole range between the parametric resonances r2-2 and r5-5. In the case of a test with the use of a signal with a stepped frequency change, the combination resonances r2-3 and r3-4 do not appear. Since in the stepped-sine test the excitation duration is longer, the parametric resonance r1-1 is excited (Fig. 15(a)). This resonance is for the pulsation frequency fp E7.7 Hz, which is twice bigger than the basic frequency of the system. Observations of numerous experiments show that the elongation of the test time duration clearly favours the formation of simple resonances (e.g. r1-1) and, to a lesser degree, combination resonances. The course of the combination resonance phenomenon strongly depends on the initial conditions. It usually excites faster for a swept-sine step. An additional argument for the swept-sine method is a significantly shorter experiment time needed for creating a whole diagram. A shorter time allows for ensuring a roughly constant hose's temperature during the experiment. Since the hose's mechanical properties depend on temperature the swept-sine method provides high repeatability of experiments results. Hence the swept-sine method has been chosen for determining most of diagrams. Comparisons shown in Fig. 15 partially

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Fig. 20. Mode of vibrations in selected time moments, combination resonance r3-4, fp ¼ 26.9 Hz.

inform about the influence of the initial conditions on the course of the resonance phenomenon. They can also signify some non-linear proprieties of the system. Equations (33, 34), which describe a hose model and have been obtained in Part I by using the Galerkin method, contain non-linear segments of the third degree (the Duffing type). Hence, the solutions in resonance ranges should depend on the initial conditions. The experimentally examined system clearly reveals the features typical of a non-linear system with a rigid characteristic. It can be confirmed by examining the influence of the frequency change direction during the test conducted with a swept-sine signal. The relevant results of the experiments and numerical calculations are shown in Fig. 16. Shifting the resonance ranges dependent on the direction of frequency change proves non-linear features of the examined phenomenon. When the excitation frequency is decreased, the probability of the excitation of parametric vibrations is significantly lower. A spectrogram is made for a chosen case in order to get the information about the frequency composition of the vibrations generated in the subsequent ranges of the parametric resonances (the upper part of Fig. 17). It presents the results of the vibration spectrum analysis in the function of frequency excitation. The analysis is performed on the basis of an experimentally measured response of the swept-sine signal system (the lower part of Fig. 17). The line labelled as fp in the spectrogram depicts vibrations of the pulsation frequency. Additionally, first five natural frequencies of the system were marked with horizontal intermittent lines. When the pulsation frequency is close to the basic frequency of the system (resonance r1), the vibration amplitude increase is observed. The frequency of vibrations is then equal to the excitation frequency and, at the same time, to the first natural frequency. This phenomenon can also be noticed for the third (r3), and partially for the second natural frequency (r2). In the ranges of the simple principle resonances (r1-1, r2-2, r3-3, r4-4, r5-5), the frequency of dominant spectral component is twice smaller than the excitation frequency (the line labelled as fp/2). In the combination resonance r2-3 for fp E21–22 Hz, besides vibrations with the frequency fp/2 (third mode), vibrations with the second and the first natural frequency excite. In the resonance r3-4 for fp E26–28.5 Hz, the content of the spectrum proves the excitation of vibrations with all five frequencies of free vibrations. In the range of the combination resonance r4-5 (fp E35–39 Hz), the vibration spectrum contains the fourth and the fifth natural frequency, with their arithmetic average being equal to the half of the excitation frequency. Thus the condition of the combination resonance f4 þf5 ¼fp is fulfilled. In this range, additionally the first form of vibrations becomes conspicuous. The spectrum also comprises multiplicities of the discussed frequencies; with odd harmonics being at large. The combination resonance r5-6 with the dominant fifth and sixth forms of vibrations can be interpreted alike. In all simple principal resonances of the type rn-n, vibrations have a character of polyharmonic vibrations or similar to them because in the spectrum, besides an adequate harmonic, there are its multiples. In the combination resonances (rn-m, n am), due to the occurrence of at least two harmonics of incommensurable frequencies, the vibrations are quasi-periodical.

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In Fig. 18, there are fragments of time histories in the chosen points of the diagram which is shown in the lower part of Fig. 17. The time in the diagrams is referred to the flow pulsation period Tp. The time range in the presented diagrams contains four excitation periods in the case of simple resonances. During the presentation of the time courses for the combination resonances, the time range is increased to 32 periods due to the presence of the components of lower frequencies. In the case of the resonances of the type r1 and r3, generally rn, the course Tn-periodical is most often observed, and for the case of the principal parametric resonances rn-n it is the 2Tn-periodical course. Time histories confirm the fact of the excitation of the quasi-periodical vibrations in the combination resonance ranges (e.g. r4-5 and r5-6), sometimes transients vibrations (r2-3). For the frequencies fn ¼13.2 Hz, the time histories of the vibrations outside the resonance range are presented. Fig. 19 shows photographs of the hose which illustrate the behaviour of the system in simple principal resonances. The distance between the horizontal lines equals 50 mm, the hose's lengths is L ¼1.91 m. Markers on the hose (bright lines) are localised in 1/2 L and symmetrically on both sides at 1/3 and 1/4 of its length. The photographs show first five forms in the following resonances: r1-1, r2-2, r3-3, r4-4 and r5-5. They take the shape of the adequate forms of free vibrations. In Fig. 20, there are photographs of the hose in selected time moments. They illustrate the behaviour of the hose during the combination resonance. A characteristic feature is the fact that the hose changes smoothly in time the forms of vibrations. It takes the forms similar to the basic forms or to their combinations. After analysing these few chosen photos, the existence of all four first forms of vibrations can be acknowledged. The behaviour of the hose during the analysed combination resonance is especially dangerous. It is because of vibration amplitudes and dynamic changes of vibration forms. 5. Conclusions On the basis of the analysis of the results of experimental research and numerical calculations the following conclusions can be drawn:

 Hoses used in high-pressure hydraulic systems can be exposed to resonance parametric vibrations. This phenomenon



  

 



can occur for the determined working conditions of the system; the ranges of these parameters are within the typical boundaries in hydraulic systems. The vibration amplitudes in parametric resonances exceed a multiple times the values of the amplitudes of vibrations excited, e.g. kinematically. Especially dangerous vibrations were noticed in the range of the combination resonance. The stiffness of elastic hoses depends on temperature and internal pressure. The function that approximates the dependence of the stiffness modulus on the mentioned parameters describes the properties of the tested hose in a good way. Since the hose stiffness decreases with the increase in temperature, the probability of the occurrence of the parametric resonance phenomenon in typical working conditions (higher oil temperature) of the hydraulic drive system is significantly higher. Internal damping in the hose depends also on the internal pressure and, additionally, on the frequency of the flow velocity pulsation. A better correspondence between the results of experiments with numerical simulations is provided by multi-parameter models in the form (6). In the case when the hose is fixed in such a way that an axial shift is impossible, an increase in the axial force induced by the pressure increase is an effect of untypical mechanical properties. This fact results in lack of buckling even when the oil pressure reaches high values. Experimental research of elastic hoses confirmed the possibility of the occurrence of parametric resonances of various types, including combination resonances. During resonance vibrations, various forms of vibrations up to the fifth one can be observed. An increase in flow velocity, pulsation amplitude, the length of the hose, and temperature help the excitation of vibrations. Vibrations get excited easier for lower values of pressure. The spectrum analysis revealed that the frequency of vibrations in simple principal resonances is most often equal to a half of the excitation frequency and the signal has a polyharmonic character. In combination resonances, quasi-periodical vibrations can be noticed, and the form of vibrations is a certain combination of two or more forms of free vibrations The application of the signal with a continuous change of frequency in the experiments allowed for the effective examination of the system in a wide range of frequencies. A comparison of frequency characteristics gives a possibility to examine the influence of significant system parameters on the ranges of parametric resonances. The application of the swept-sine signals with various directions of frequency changes and the stepped-sine signal shows that the examined system has definitely the features of a non-linear system. From the scientific point of view, further research into the explanation of interesting physical phenomena is recommended. The comparison of the results of experimental research with the results of numerical simulations verifies that the described model and the process of the identification of its parameters are proper. The methods of the numerical analysis proposed in Part I of this article turned out to be effective in spite of a relatively high complexity of the model and the fact that eight forms of vibrations were taken into account in the approximate solution.

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