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Mathematical and Computer Modelling of Dynamical Systems, 2015 Vol. 21, No. 1, 23–37, http://dx.doi.org/10.1080/13873954.2013.871564 Wojciech Sochacki* Institute of Mechanics and Machine Design Foundations, Częstochowa University of Technology, Ul. Dąbrowskiego 73, 42–201 Częstochowa, Poland (Received 18 December 2012; final version received 29 November 2013) The study presents a formulation on the basis of Hamilton’s principle and solution for the problem of damped vibration in hydraulic cylinders. The physical model took into consideration the energy dissipation in a vibrating cylinder as a result of external viscous damping and internal damping of viscoelastic material in beams used to construct a model of a cylinder (rheological model by Kelvin–Voigt). Constructional damping in the points of the cylinder connection with the components of the basic structure was also considered. The example computation was made for a cylinder used in a mining prop. The results of the computations concern the determination of the relationships between the first eigenvalue of transverse vibration in the cylinder and its extension length with two values of load and determination of the amplitude decay factor for the first and the second eigenvalue versus the extension length. Furthermore, the study also focused on determination of the relationships between the amplitude decay factor of the first and second mode of vibration for a particular length of the cylinder. Keywords: damped vibration; hydraulic cylinder; amplitude decay factor 1. Introduction A hydraulic cylinder as an object of research studies on dynamics of mechanical systems has been extensively discussed in the literature. Much research has been done on the interactions between the cylinder tube and piston rod and the effect of sealing or the medium on the cylinder’s dynamics. It is often important to analyse the effect of the environment on the cylinder. The above effects have been usually modelled as interactions of particular forces (either constant or variable), point masses or the elasticity components added as discrete elements to the system studied. They provide the basis for building models used for computation of the frequency of cylinder’s vibration, interacting forces of piston rod and cylinder tube and computation of critical forces of particular structures. One of the early studies in this field was study [1]. In the work, the importance of initial imperfection due to misalignment in the interface between cylinder tube and piston rod and its influence in accordance with the stroke was pointed out. In the study [2] was presented the results obtained from the investigations of elastic load bearing capacity in a hydraulic support. Furthermore, the paper [3] presented the results of the investigations of the dynamic response of the model of a cylinder to axial impulse. In the work [4], an analysis of the effect of initial inaccuracy of connection between the piston and cylinder tube on critical loading force in the cylinder was presented. In the study [5], a method of analysis which permits the investigation of stresses and deflections in a hydraulic cylinder, *Email: sochacki@imipkm.pcz.czest.pl © 2014 Taylor & Francis 24 W. Sochacki using a finite element analytical model, was presented. The problem of the stability and free vibration of a hydraulic cylinder with an elastically fixation modelled by rotational springs has been formulated and solved in the paper [6]. In the work [7], a theoretical model that involves the analysis of boundary conditions and presents loads in actual applications was proposed. The results of model have been validated experimentally in a test bench, designed to determine the behaviour of cylinders under load. In the paper [8], the effects caused by friction and clearance in the connection between the rod and the cylinder for hydraulic actuators of the ‘plunger rod’ kind was investigated. Friction at the restrained ends, for the pin connected ends case, was also evaluated. Another publication where calculations of free vibration frequencies were extended with the investigations of the dynamic stability of the cylinder by means of determination of geometrical parameters and load at the time of losing the stability was the study [9]. Analysis of vibration frequency of the cylinder versus the extension of the piston rod was discussed in [10]. Models of cylinders’ vibrations are usually beam-based. Analysis of vibration damp- ing in such structures should also be supplemented with the analysis of vibration damping in beams that model a cylinder tube and piston rod. An interesting publication concerning the effect of external damping on vibration of beams with stepped cross section is the study [11]. Similar study was presented in [12], where the effect of external damping on vibration of a support beam with stepped cross section and a mass attached to a free end of the beam was analysed. The example studies concerning the investigations of the effect of internal damping on beam vibration are [13–16]. In the study [13], the effect of internal damping on vibrations of a support beam with a mass attached to a free end of the beam was analysed. Furthermore, a replacement (discrete) model in the form of spring-damper-mass system was proposed. The effect of small internal and external damping on stability of non-conservative beam systems was presented in [14]. Similar analysis of the effect of internal damping on stability of a support column loaded with a follower force was presented in the study [15]. A mathematical model of bending of non-prismatic beams with damping was presented in [16]. Being typically parts of other structures, hydraulic cylinders often operate under a variety of different conditions of the environment. Fixation of the cylinder to the base structure typically occurs through joints. Variable operating conditions (elevated friction), effect of the environment (corrosion) and variable load cause that modelling of cylinder’s mountings using only joints might not reflect actual fixation points in the system. In order to consider the effect of the above factors on vibration motion of the cylinder, rotational dampers in the fixation model was introduced by the author of presented study. This solution was discussed in the study [17] where modal analysis of a simple Bernoulli–Euler beam with rotational viscous dampers at the ends was carried out. This study presents a formulation and solution for the problems of damped vibration in hydraulic cylinders. The physical model considered dissipation of energy of cylinder’s vibration caused by external viscous damping and internal damping of viscoelastic material in beams used in a model of a cylinder tube and piston rod in hydraulic cylinder (rheological model by Kelvin–Voigt). Constructional damping in the locations of the cylinder’s connection with components of the basic structure was also considered. The example computation was made for a cylinder used in a mining prop. The results of the computations concern the determination of the relationships between the first eigenvalue of transverse vibration in the cylinder and its extension length with two values of load and determination of the amplitude decay factor for the first and the second eigenvalue versus the extension length. The effect of the constructional damping in fixation points on the Mathematical and Computer Modelling of Dynamical Systems 25 first eigenvalue of the system with extreme lengths of the cylinder was also presented. Furthermore, the study also focused on the determination of the relationships between the decay factor for the vibration amplitude of the first and second mode of vibration for a particular length of the cylinder. The considered model of damped vibrations of hydraulic cylinder (including both internal and external damping as well as constructional damping by the cylinder clamping system) is an original achievement. Based on the obtained results, originality of the structure is confirmed with respect to determining cylinder length optimised against reduction in amplitude of vibration. 2. Mathematical model The model of a hydraulic cylinder, presented in the diagram in Figure 1(a), is composed of four beams that model a cylinder tube (l and l ) and piston rod (l and l ) in the 11 12 21 22 cylinder (Figure 1(b)). The model considers essential parameters of the actual system, such as loading the system with force P and damping in individual parts of hydraulic cylinder modelled with viscous damping coefficients E*. Furthermore, the external damping of the system modelled with coefficient c has been taken into consideration. The constructional damp- ing in the form of rotational viscous dampers attached at the ends of beams was modelled with coefficient c . The liquid in the cylinder was adopted as the medium of load transfer between the piston and the cylinder along the length filled with liquid (see [2]). The boundary problem connected to the free vibrations of the considered non-con- servative (due to damping) system was formulated on the basis of Hamilton’s principle [18] in the following form: W (x,t) * * E ,E 21 12 W (x,t) = W (x,t) 12 21 c W (x,t) (a) (b) Figure 1. Diagram (a) and beam model (b) of a hydraulic cylinder with damping. l l = l 11 12 21 x , x x P 12 21 22 26 W. Sochacki t t 2 2 ð ð δ ðT VÞdt þ δW dt ¼ 0 (1) t t 1 1 where T is the kinetic energy, V is the potential energy and δW is the virtual work of non- conservative forces originating from damping. Kinetic and potential energies and virtual work of non-conservative forces are defined by the following relationship: mn 1 @W ðx ; tÞ mn mn T ¼ ρ A dx (2) mn mn mn 2 @t m;n¼1 l l mn mn ð ð 2 2 2 X X 1 @ W ðx ; tÞ 1 @W ðx ; tÞ mn mn mn mn V ¼ E J dx P dx (3) mn mn mn mn 2 2 @x @x mn mn m;n¼1 m;n¼1 0 0 mn 2 2 @ @W ðx ; tÞ @ W ðx ; tÞ mn mn mn mn δW ¼ E J δ dx þ N mn mn mn @x @t @x mn mn m;n¼1 mn @W ðx ; tÞ @W ðx ; tÞ @W ðx ; tÞ (4) mn mn 11 11 11 11 c δW ðx ; tÞdx þ c δ e mn mn mn R @t @t @x x ¼0 m;n¼1 11 x ¼l 22 22 @W ðx ; tÞ @W ðx ; tÞ 22 22 22 22 c δ @t @x where m,n = 1,2 (c = 0 for m = 2 and n =1) W (x , t) – transverse displacement of beams that model cylinder and piston rod, mn mn E – Young’s modulus for individual beams, mn E* – material viscosity coefficient, mn J – moment of inertia in beam cross-sections, mn A – cross-sectional areas of the beams, mn ρ – beam material density, mn – viscous damping coefficient, P – cylinder loading force (at the length l of the cylinder tube coverage with the piston rod in the cylinder P =0), x – spatial coordinates, mn t – time. Geometrical boundary conditions of the considered system are as follows: x ¼l 12 12 2 2 @ W ðx ; tÞ @ W ðx ; tÞ x ¼l 12 12 21 21 22 22 W ðx ; tÞj ¼ W ðx ; tÞj ¼ ¼ ¼ 0 11 11 22 22 x ¼0 11 2 2 @x @x 12 21 x ¼0 (5a–d) Mathematical and Computer Modelling of Dynamical Systems 27 x ¼l 11 11 W ðx ; tÞj ¼ W ðx ; tÞj ¼ W ðx ; tÞj 11 11 12 12 21 21 x ¼0 x ¼0 12 21 (6a–d) x ¼l 12 12 W ðx ; tÞj ¼ W ðx ; tÞj ¼ W ðx ; tÞj 12 12 21 21 22 22 x ¼0 x ¼0 21 22 x ¼l x ¼l 11 11 21 21 @W ðx ; tÞ @W ðx ; tÞ @W ðx ; tÞ @W ðx ; tÞ 11 11 12 12 21 21 22 22 ¼ ; ¼ @x @x @x @x 11 12 21 22 x ¼0 x ¼0 12 22 (7a; b) By substituting relationships (2), (3) and (4) into Hamilton’s principle (1) after taking into account the geometrical boundary conditions, we obtained the differential equation of motion in the transversal direction in the form 4 5 2 @ W ðx ; tÞ @ W ðx ; tÞ @ W ðx ; tÞ mn mn mn mn mn mn E J þ E J þ P þ mn mn mn mn 4 4 2 @x @x @ t @x mn mn mn (8) @W ðx ; tÞ @ W ðx ; tÞ mn mn mn mn þ c þ ρ A ¼ 0 e mn mn @t @ t and natural boundary conditions x ¼l 11 11 2 2 @ @ W ðx ; tÞ @ @ W ðx ; tÞ 12 12 11 11 E þ E J ¼ E þ E J (9) 12 12 11 11 12 11 2 2 @t @t @x @x 12 11 x ¼0 @ W ðx ; tÞ @ @W ðx ; tÞ 11 11 11 11 E J ¼ c (10) 11 11 R @x @t @x 11 11 x ¼0 x ¼0 11 x ¼l 22 22 x ¼l 2 22 22 @ W ðx ; tÞ @ @W ðx ; tÞ 22 22 22 22 E J ¼c (11) 22 22 R @x @t @x 22 22 x ¼l 21 21 2 2 @ @ W ðx ; tÞ @ @ W ðx ; tÞ 22 22 21 21 E þ E J ¼ E þ E J (12) 22 22 21 21 22 2 21 2 @t @x @t @x 22 x ¼0 21 x ¼l x ¼l 3 11 11 11 11 @ @ W ðx ; tÞ @W ðx ; tÞ 11 11 11 11 E þ E J þ P 11 11 11 3 @t @x @x 11 11 3 3 @ @ W ðx ; tÞ @ @ W ðx ; tÞ 12 12 21 21 E þ E J E þ E J (13) 12 12 21 21 12 3 21 3 @t @x @t @x 12 x ¼0 21 x ¼0 12 21 @W ðx ; tÞ 21 21 P ¼ 0 @x 21 x ¼0 x ¼l x ¼l 12 12 21 21 3 3 @ @ W ðx ; tÞ @ @ W ðx ; tÞ 12 12 21 21 E þ E J þ E þ E J 12 12 21 21 12 21 3 3 @t @x @t @x 12 21 (14) @ @ W ðx ; tÞ 22 22 E þ E J ¼ 0 22 22 @t @x 22 x ¼0 22 28 W. Sochacki Solution for Equation (8) is given by iω t W ðx ; tÞ¼ w ðx Þe (15) mn mn mn mn pﬃﬃﬃﬃﬃﬃﬃ where ω* is the complex eigenvalue, i ¼ 1. By substitution of (15) into Equation (8) after separation of variables one can obtain IV 2 II w ðxÞþ β w ðxÞ γ w ðxÞ¼ 0 m; n ¼ 1; 2 (16) mn mn mn mn mn sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ρ A c P mn e mn 2 where γ ¼ ω iω , β ¼ mn mn J ðE þ E iω Þ ρ A J ðE þ E iω Þ mn mn mn mn mn mn mn mn and an order of partial derivative in relation to a space coordinate is denoted by Roman numeral. Similarly, by substitution of the solution (15) into the boundary conditions (9–14), they can be written in the following form: I I I I w ðl Þ¼ w ð0Þ; w ðl Þ¼ w ð0Þ 11 21 11 12 21 22 II II w ð0Þ¼ w ðl Þ¼ w ð0Þ¼ w ðl Þ¼ 0 11 12 22 22 12 21 w ðl Þ¼ w ð0Þ¼ w ð0Þ 11 11 12 21 w ðl Þ¼ w ðl Þ¼ w ð0Þ 12 12 21 21 20 II II ðE þ E iω Þj w ð0Þ¼ ðE þ E iω Þj w ðl Þ 12 12 11 11 11 12 12 11 11 II I E J w ð0Þ¼ c iω w ð0Þ 11 11 R 11 11 (17) II I E J w ðl Þ¼ c iω w ðl Þ 22 22 22 R 22 22 22 II II ðE þ E iω Þj w ð0Þ¼ ðE þ E iω Þj w ðl Þ 22 22 21 21 21 22 22 21 21 II I II ðE þ E iω Þj w ðl Þþ PW ðl ÞðE þ E iω Þj w ð0Þþ 11 11 11 11 12 12 11 11 11 12 12 II I ðE þ E iω Þj w ð0Þ¼ PW ð0Þ¼ 0 21 21 21 21 21 II II II ðE þ E iω Þj w ðl ÞþðE þ E iω Þj w ðl ÞðE þ E iω Þj w ð0Þ¼ 0 12 12 12 21 21 21 22 22 12 12 21 21 22 22 The solution for Equation (16) is given by λ x λ x iλ x iλ x mn mn mn mn w ðxÞ¼ C e þ C e þ C e þ C e (18) mn 1mn 2mn 3mn 4mn rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 4 2 4 β β β β mn mn mn mn where λ ¼ þ þ γ , λ ¼ þ þ γ mn mn mn mn 2 4 2 4 After substitution of Equation (18) into condition (17), one can obtain the system of equations with respect to unknown constants C . In the matrix form, this system can be kmn given by ½Aðω Þ C ¼ 0 (19) Mathematical and Computer Modelling of Dynamical Systems 29 where ½Aðω Þ¼ ½a ,(p, q = 1,2…16) i C ¼½C ,(k = 1,2…4) pq kmn The non-zero elements a of matrix A(ω*) are given in Appendix. pq The system of Equation (19) has a non-trivial solution if the determinant of the matrix A(ω*) equal zero: det Aðω Þ¼ 0 (20) Equation (20) is solved numerically for the eigenvalues ω*. The roots ω* are complex numbers that represent the damped vibration frequencies and damping in the system. Depending on the adopted solution, the real and imaginary parts ω* can specify damped vibration frequency or vibration amplitude decay coefficient of the tested system. Obtained roots may accept positive or negative values of the real part Re (ω*) or imaginary part Im (ω*) at a constant module jj ω . In this paper, the results for the positive values of Re (ω*) and Im (ω*) were presented. 3. Results of numerical simulations The computations were carried out in consideration of the material and geometrical data of the hydraulic cylinder used in a mining prop. The data adopted for computations are presented in Table 1. When presenting the results, dimensionless values of damping coefficients given by – for internal damping mn η ¼ (21) hE mn ρ A þ ρ A c p c p where h ¼ L , E J þ E J c c p p A – cross-sectional area of the cylinder tube, A – cross-sectional area of the piston rod, – for external damping c L # ¼ (22) 2 2 where d ¼ L ðρ A E J þ ρ A E J Þ c c c p p p c p – for constructional damping Table 1. Geometrical and material data adopted in the study. Cylinder tube – external diameter D = D = D = 290 (mm) c 11 12 Cylinder tube – internal diameter d = d = d = 250 (mm) c 11 12 Piston rod – external diameter D = D = D = 160 (mm) p 21 22 Piston rod – internal diameter d = d = d = 120 (mm) p 21 22 Cylinder tube and piston rod density ρ = ρ = 7.86 × 103 (kg/m ) c p Young’s modulus E =Ec = E = 2.1 × 10 (Pa) mn p 30 W. Sochacki μ ¼ (23) were adopted. Figure 2 presents the relationships between the first two free vibration frequencies without damping in the cylinder and its length that ranged from L = 2.6 m to L = 4.0 m and for two variants of loading, which was p = 0 and p = 0.3 (p = P/P , where P is the c c critical force of the cylinder extended to L = 4 m). The other figures present the effect of vibration damping on cylinder’s dynamics (eigenvalues). Figure 3 presents the relationships for real and imaginary parts of the first eigenvalue of the cylinder for selected values of damping coefficients. The computations were carried out for the cylinder extended from L = 2.6 m to L = 4 m for two values of loading. In order to determine the effect of individual types of damping on vibration frequen- cies in the system studied and the degree of decay of vibration amplitudes, individual analysis for different types of damping determined by η, # and µ coefficients were carried out. The range of the parameters changes, allowing to determine the nature of the parameter impact on the eigenvalues of the studied system. 1200 2 400 1 3.2 3.6 4 2.8 L (m) Figure 2. Relationships between the first and second free vibration frequency for the cylinder without damping (p =0, p = 0.3). Re (ω ) η = 0.01 ϑ = 0.5 μ = 0.5 Im (ω ) 2.8 3.2 3.6 4 L (m) Figure 3. The relationships between the first eigenvalue (real and imaginary parts) for the cylinder and the extension length L for selected values of damping coefficients (p =0, p = 0.3). ω (rad/s) ω (rad/s) Mathematical and Computer Modelling of Dynamical Systems 31 The analysis measured the effect of coefficients that characterised a particular type of damping on the first eigenvalue in the cylinder (the damped frequency in the system and the degree of decay of vibration amplitudes). The computations were carried out in each case for the cylinder extended to the length of L = 3 m. Further, the results of the investigations of the effect of the cylinder extension length (from L to L ) on its min max first eigenvalue for selected values of vibration damping coefficients were presented. Figure 4 presents the effect of internal damping on the damped frequency and amplitude decay factor of the cylinder’s vibration. Figure 5 presents the change in the first eigenvalue of the cylinder depending on the extension length from L = 2.6 m to L = 4 m without loading and loaded with the force p = 0.3. The investigations were carried out for internal damping value of η = 0.01. Similarly, Figures 6 and 7 present the results of numerical simulations of the effect of external damping (# coefficient) on the first eigenvalue of the cylinder. When analysing the effect of damping on the damped frequency and the amplitude decay factor, damping coefficient of # = 0.5 and the system load of (p = 0.3) was adopted. The curves Im (ω *) for p = 0 and p = 0.3 in Figure 7 overlap. Re (ω ) Im (ω ) 0 0.01 0.02 0.03 0.04 0.05 Figure 4. The real part and imaginary part of the first eigenvalue of the cylinder as functions of the internal damping coefficient η for L =3m. η = 0.01 Re (ω ) Im (ω ) 2.8 3.2 3.6 4 L (m) Figure 5. The relationships between the first eigenvalue (real and imaginary parts) for the cylinder and the extension length L for selected values of damping η (p =0, p = 0.3). ω (rad/s) ω (rad/s) 1 32 W. Sochacki ω ) Re ( Im (ω ) 0 0.2 0.4 0.6 0.8 1 Figure 6. The relationships between the first eigenvalue (real and imaginary parts) of the cylinder and the external damping coefficient # for L =3m. ϑ = 0.5 Re (ω ) 100 Im (ω ) 2.8 3.2 3.6 4 L (m) Figure 7. The relationships between the first eigenvalue (real and imaginary parts) for the cylinder and the degree of its extension L for the selected value of the coefficient # (p =0, p = 0.3). Further investigations focused on consideration of cylinder vibration damping that resulted from motion resistance which might be generated in the points of cylinder fixations. This resistance was characterised by means of constructional damping coeffi- cient μ. Figures 8 and 9 present the relationships between the cylinder’s eigenvalues (parts Re (ω *) and Im (ω *)) and the constructional damping coefficient μ. The computations 1 1 were carried out for different values of the μ coefficient (Figure 8) and for different lengths of the L cylinder (Figure 9). Similar to the previous cases, the cylinder was additionally loaded with a force of p = 0.3. Changes in the first damped frequency of cylinder vibration (Re (ω *)) in the case of the increase in the value of coefficient μ (Figure 8) ranges from the values representing the vibration frequency for the cylinder with joint mountings without damping (μ = 0) to the values that represent a cylinder which is rigidly fixed in both supports, with μ → ∞. The imaginary part of the first eigenvalue increased from zero to the value of (Im (ω * ) for 1 max μ ≈ 0.45 – maximum damping), and then Im (ω *) → 0 when μ → ∞. Changes in the eigenvalues for the cylinder versus the extension length for selected value of the ω (rad/s) 1 ω (rad/s) 1 Mathematical and Computer Modelling of Dynamical Systems 33 Re (ω ) Im (ω ) 0 0.4 0.8 1.2 1.6 2 Figure 8. The relationships between the first eigenvalue (real and imaginary parts) of the cylinder and the constructional damping coefficient μ dla L =3 m. μ = 0.5 Re (ω ) Im (ω ) 2.8 3.2 3.6 4 L (m) Figure 9. The relationships between the first eigenvalue (real and imaginary parts) for the cylinder and the degree of its extension L for the selected value of the coefficient μ (p =0, p = 0.3). parameter μ = 0.5 are presented in Figure 9. The higher the value of Im (ω *) the more the amplitudes of a particular (n) mode of vibration are damped. With respect to the above, the next figure (Figure 10) presents the maximum values of Im (ω * ) for the two first n max modes of vibration in the system studied depending on the cylinder length. 4. Conclusions This study presents the model of transverse vibration in a hydraulic cylinder with damping. The model of damping took into consideration the internal damping of the beams that modelled a cylinder tube and a piston rod and external damping and construc- tional damping that modelled motion resistance in the location of the cylinder’s mounting. The computations were carried out for the model of a hydraulic cylinder used in mining props. It can be concluded based on the calculations that introduction of the internal and external damping causes only insignificant changes in the damped frequencies of the ω (rad/s) ω (rad/s) 1 1 34 W. Sochacki Im ω 1 max Im ω 2 max 2.6 2.8 3 3.2 3.4 3.6 3.8 4 L (m) Figure 10. The relationships between the maximum values of Im (ω* ) for the two first modes max of vibration in the cylinder and the extension length L (for η = 0 and # = 0). system across the range studied (Re (ω*) Figures 4–7). An increase in the damping coefficients μ i # causes faster decay of vibration amplitudes in the system studied according to the curves (Im (ω*) in Figures 4 and 6). Loading the system with the force p = 0.3 changes only the damped frequencies, but it does not affect changes in vibration amplitude decay (Im (ω*) in Figures 5 and 7). It can be also concluded that introduction of the constructional damping of cylinder vibrations (from mounting parts) causes significant changes in the eigenva- lues of the cylinder. An increase in the damping coefficient μ in the points of mounting causes a constant increase in the damped frequency of vibration and an increase in the value of (Im (ω *)) to the maximum value that depends on the extension length, followed by a reduction in these values towards 0 with μ → ∞. This considerable change in the damped frequency of vibration (Re (ω *) in Figure 8) is caused by the substantial intervention in boundary conditions in the system studied. The increase in the value of μ coefficient leads to ‘locking’ the rotational motion in the mounting joints. In the limiting case, the change in the mounting conditions occurs (from joint-based into rigid ones). When extending the cylinder (Figure 9), its damped vibration frequency decreases, whereas the maximum damping effect occurs when the cylinder is extended to the value of L = 3.11 m (Im (ω*) reaches its maximum value). A considerable load to the system (p = 0.3) causes the reinforced damping effect throughout the length of the cylinder studied and, obviously, a reduction in the value of the damped frequency of the system studied. The amplitude decay factor for the vibration of the first mode adopts the maximum values for μ = 0.42, when the cylinder is extended to the length of L = 3.31 m. In the case of the second mode of vibrations, Im(ω * ) reaches its maximum for μ = 0.06 with the length of the 2 max cylinder of L = 2.94 m. 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Appendix The non-zero elements a of matrix A(ω*) (19) are given as follows: pq a ¼ 1; a ¼ 1; 12 14 a ¼ c ω iλ ; a ¼ E λ ; a ¼ c ω iλ ; a ¼E λ 21 R 22 23 R 11 24 11 11 11 11 11 2 2 2 2 a ¼ λ sinhðλ Þ ; a ¼ λ coshðλ Þ ; a ¼λ sinðλ Þ ; a ¼λ cos ðλ Þ ; 35 36 37 38 12 12 12 12 12 12 12 12 a ¼ λ ; a ¼ λ ; 410 412 21 21 a ¼ sinh ðλ Þ ; a ¼ cosh ðλ Þ ; a ¼ sin ðλ Þ ; a ¼ cos ðλ Þ ; 513 514 515 516 22 22 22 22 36 W. Sochacki 2 2 a ¼ E λ sinh ðλ Þ c ω iλ cosh ðλ Þ ; a ¼ E λ cosh ðλ Þ c ω iλ sinh ðλ Þ ; 613 22 R 614 R 22 22 22 22 22 22 22 22 22 2 2 a ¼E λ sin ðλ Þ c ω iλ cos ðλ Þ ; a ¼E λ cos ðλ Þþ c ω iλ sin ðλ Þ 615 R 616 R 22 22 22 22 22 22 22 22 22 22 (A1) a ¼ sinh ðλ Þ ; a ¼ cosh ðλ Þ ; a ¼ sin ðλ Þ ; a ¼ cos ðλ Þ ; a ¼1; a ¼1; 71 72 73 74 76 78 11 11 11 11 a ¼ 1 ; a ¼ 1 ; a ¼1; a ¼1; 86 88 810 812 a ¼ sinh ðλ Þ ; a ¼ cosh ðλ Þ ; a ¼ sin ðλ Þ ; a ¼ cos ðλ Þ ; 95 96 97 98 12 12 12 12 a ¼ sinh ðλ Þ ; a ¼ cosh ðλ Þ ; a ¼ sin ðλ Þ ; a ¼ cos ðλ Þ ; 99 910 911 912 21 21 21 21 a ¼ sinh ðλ Þ ; a ¼ cosh ðλ Þ ; a ¼ sin ðλ Þ ; a ¼ cos ðλ Þ ; a ¼1; 109 1010 1011 1012 1014 21 21 21 21 a ¼1; 2 2 a ¼ðE þ iωE Þλ sinh ðλ Þ ; a ¼ðE þ iωE Þλ cosh ðλ Þ ; 1101 1102 11 11 11 11 11 11 11 11 2 2 a ¼ðE þ iωE Þλ sin ðλ Þ ; a ¼ðE þ iωE Þλ cos ðλ Þ ; 1103 1104 11 11 11 11 11 11 11 11 a ¼ðE þ iωE Þλ ; a ¼ðE þ iωE Þλ ; 1106 1108 12 12 12 12 12 12 a ¼ λ cosh ðλ Þ ; a ¼ λ sinh ðλ Þ ; a ¼ λ cos ðλ Þ ; a ¼ λ sin ðλ Þ ; 1201 1202 1203 1204 11 11 11 11 11 11 11 11 a ¼λ ; a ¼λ ; 1205 1207 12 12 a ¼ λ cosh ðλ Þ ; a ¼ λ sinh ðλ Þ ; a ¼ λ cos ðλ Þ ; a ¼ λ sin ðλ Þ ; 1309 1310 1311 1312 21 21 21 21 21 21 21 21 a ¼λ ; a ¼λ ; 1313 1315 22 22 2 2 a ¼ðE þ iωE Þλ sinh ðλ Þ ; a ¼ðE þ iωE Þλ cosh ðλ Þ ; 1409 1410 21 21 21 21 21 21 21 21 2 2 a ¼ðE þ iωE Þλ sin ðλ Þ ; a ¼ðE þ iωE Þλ cos ðλ Þ ; 1411 1412 21 21 21 21 21 21 21 21 a ¼ðE þ iωE Þλ ; a ¼ðE þ iωE Þλ ; 1414 1416 22 22 22 22 22 22 a ¼ðE þ iωE Þλ cosh ðλ Þþ Pλ cosh ðλ Þ; 11 11 11 11 11 11 a ¼ðE þ iωE Þλ sinh ðλ Þþ Pλ sinh ðλ Þ ; 11 11 11 11 11 11 a ¼ðE þ iωE Þλ cos ðλ Þþ Pλ cos ðλ Þ ; 11 11 11 11 11 11 a ¼ðE þ iωE Þλ sin ðλ Þ Pλ sin ðλ Þ ; 11 11 11 11 11 11 a ¼ðE þ iωE Þλ ; a ¼ðE þ iωE Þλ ; 1505 1507 12 12 12 12 12 12 a ¼ðE þ iωE Þλ Pλ ; a ¼ðE þ iωE Þλ Pλ ; 1509 1511 21 21 21 21 21 21 21 21 Mathematical and Computer Modelling of Dynamical Systems 37 3 3 a ¼ðE þ iωE Þλ cosh ðλ Þ; a ¼ðE þ iωE Þλ sinh ðλ Þ; 1605 1606 12 12 12 12 12 12 12 12 3 3 a ¼ðE þ iωE Þλ cos ðλ Þ ; a ¼ðE þ iωE Þλ sin ðλ Þ ; 1607 16 08 12 12 12 12 12 12 12 12 a ¼ðE þ iωE Þλ cosh ðλ Þþ Pλ cosh ðλ Þ; 21 21 21 21 21 21 a ¼ðE þ iωE Þλ sinh ðλ Þþ Pλ sinh ðλ Þ ; 21 21 21 21 21 21 a ¼ðE þ iωE Þλ cos ðλ Þþ Pλ cos ðλ Þ ; 21 21 21 21 21 21 a ¼ðE þ iωE Þλ sin ðλ Þ Pλ sin ðλ Þ ; 21 21 21 21 21 21 a ¼ðE þ iωE Þλ Pλ ; a ¼ðE þ iωE Þλ Pλ ; 1613 1615 22 22 22 22 22 22 22 22 where: λ ¼ λ l ; λ ¼ λ l ; E ¼ E J ; E ¼ E J ; m; n ¼ 1; 2 mn mn mn mn mn mn mn mn mn mn mn mn
Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis
Published: Jan 2, 2015
Keywords: damped vibration; hydraulic cylinder; amplitude decay factor
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