Influence of Torsional Motion on the Axial Vibrations of a Drilling Tool
|
|
- Irma Robertson
- 5 years ago
- Views:
Transcription
1 Sergey A. Voronov Alexander M. Gouskov Alexey S. Kvashnin Bauman Moscow State Technical University, 5, 2-nd Baumanskaya, Moscow, Russia Eric A. Butcher University of Alaska, P.O. Box , Fairbanks, AK S. C. Sinha Fellow ASME Auburn University, 202 Ross Hall, Auburn, AL Influence of Torsional Motion on the Axial Vibrations of a Drilling Tool The nonlinear dynamics of a tool commonly employed in deep hole drilling is analyzed. The tool is modeled as a two-degree of freedom system that vibrates in the axial and torsional directions as a result of the cutting process. The mechanical model of cutting forces is a nonlinear function of cutting tool displacement including state variables with time delay. The equations of new surface formation are constructed as a specific set. These equations naturally include the regeneration effect of oscillations while cutting, and it is possible to analyze continuous and intermittent cutting as stationary and nonstationary processes, respectively. The influence of the axial and torsional dynamics of the tool on chip formation is considered. The Poincaré maps of state variables for various sets of operating conditions are presented. The obtained results allow the prediction of conditions for stable continuous cutting and unstable regions. The time domain simulation allows determination of the chip shape most suitable for certain workpiece material and tool geometry. It is also shown that disregarding tool torsional vibrations may significantly change the chip formation process. DOI: / Introduction Deep hole machining is one of the most complex manufacturing processes. Adverse conditions of chip formation, problems of chip removal, low stiffness of tool and its special design, and the impossibility of observing the tool during machining make the deep hole drilling as one of the most difficult operations. The main feature for holes of small diameter machining is the difficulty of removal of chips from the cutting region. Naturally, the chip removal becomes much easier in case of small crushed chip. A few methods of chip fragmentation have been recently discussed 1,2. The vibratory drilling is most efficient among them. This technology presumes installation of a special vibrator that excites the drill bit in its support fixture. The axial oscillations of cutting edges form a discontinuous chip if the correct amplitude and frequency of the vibrator are chosen. The new method of an autoresonant cutting technique has been developed for turning 3 and drilling 4, which implies that the vibration of tool by the self-exited vibration mechanism is inherent to the cutting processes. This method requires a more accurate analysis of tool dynamics and chip formation. This research has been carried out in Russia in the past several years and its features and model analyses are presented in several papers 3,5 8. The main attention was devoted to the modeling of axial tool vibrations as they define the chip formation process. But the tool is loaded by a torque as well as by a thrust force and thus can produce torsional oscillations too, which, under certain conditions, may have sufficient influence on the tool dynamics. The analysis of axial-torsional vibrations of the tool for deep hole drilling is considered in this paper. Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 17, 2006; final manuscript received September 1, Review conducted by Bala Balachandran. Paper presented at the ASME 2005 Design Engineering Technical Conferences and Computers and Information in Engineeromg Conference DETC2005, September 24 28, 2005, Long Beach, California, USA. The frequency analysis of drill vibrations has been presented in many papers. The bending vibrations of twisted beams under the action of an axial force and a moment were studied in 9. The influence of tool parameters on bending and torsional vibrations by the finite element method FEM was analyzed in 10. Rincon and Ulsoy 11 examined the influence of tool vibrations on cutting forces. But these authors did not include the interaction of cutting forces and tool vibrations in the model. Actually, the cutting forces arise due to the intrusion of tool edges into the material being cut and vary in proportion to chip thickness. Thus the cutting process dynamics is not a response to external loading, but to a self-excited process that can be stable or unstable. Gouskov et al. in 5,6 analyzed the stability of self-excited axial and transverse vibrations applying delayed differential equations. The stability lobes of drilling due to transverse 5 or axial drill bit vibrations 12 were determined by an asymptotic method. The research was extended by introducing equations of new surface formation 6,8 into the dynamical model that allow the simulation of nonstationary cutting processes, including discontinuous cutting when cutting edges disengage the machined material due to vibrations. It was shown that the bifurcation diagrams are useful in stability analysis and the prediction of dynamic behavior as a function of system parameters and the shape of the chip as well. The synchronization of tool vibration at the frequency of the external vibrator excitation in the case of vibratory drilling was analyzed in 7. Other researchers have demonstrated that in twisted beams the axial-torsional coupling exists and it can sufficiently influence the process of chip formation 13,14. As the drill vibrates in torsion, it lengthens and shortens periodically, causing a wavy surface formation on the bottom of the drilled hole. Bayly et al. 15 used a model that included torsional-axial coupling for frequency analysis to predict regions of instability. The simulation results were compared with experimental research and a close agreement was found. However, the illustrated model is suitable only for a stationary process of cutting with small amplitudes of tool vibrations when cutting edges are permanently in contact with workpiece 58 / Vol. 2, JANUARY 2007 Copyright 2007 by ASME Transactions of the ASME
2 edges into the workpiece material. The feed rate V and an angular speed are imparted to the workpiece. The full model of the system dynamics may be separated into three components: model of the tool dynamics; model of the new surface formation; model of the cutting forces. 2.1 Model of Tool Motion. The equations of tool motion in axial and torsional direction are obvious and do not differ from those presented in 15,17. These are given by Fig. 1 Photograph of the vibratory drilling setup: 1 vibrator; 2 tool; 3 workpiece material. If the vibration amplitudes are large, then cutting edges may lose contact with the material, introducing nonlinearity into the process 16. A similar model of coupled torsional-axial vibrations in 17 was presented, but the authors introduced a new cutting force model. The cutting force was considered distributed along the cutting edge and was determined as an empirical function of chip thickness at each point of the cutting edge. The chip thickness was determined by a numerical algorithm that disregarded the thickness variation due to tool torsional motion. This mechanical cutting force model was used in the time domain simulation of drilling with a pilot hole for an accurate prediction of vibrations that were compared against experimental results. This model can be used to study the dynamics of tool entry into and exit from the workpiece. Some other models of cutting processes are described in The main difficulty, in tool simulation taking into account the coupled torsional-axial vibration, is that the time delay entering in the model of cutting force varies due to torsional vibrations. This fact requires using a special algorithm employing interpolation for chip thickness calculation, and it can sufficiently increase the calculation time. In the present paper a new algorithm is suggested by introducing the angle of tool rotation as a new independent variable, which is determined in time by an additional equation. In this case the integration by constant step size with constant delay makes calculations faster and more accurate. 2 Model Description The photograph of the vibratory drilling setup for machining holes in four connecting-rods simultaneously is shown in Fig. 1. A drill bit tool is considered a two-degree of freedom system that can vibrate in the axial with displacement w and torsional with twist angle directions. The schematic model of tool is presented in Fig. 2. The tool has a mass m and moment of inertia with respect to the z axis and has axial stiffness k ax and torsional stiffness k rot.an oscillatory motion imposed by an external vibrator as Z 0 t =à 0 sin 0 t, where à 0 and 0 are amplitude and frequency of the vibrator excitation, respectively, is imparted to the left support of the tool. The right end is under the action of cutting thrust force P C and torque M C arising during the penetration of tool cutting mẅ + d ax ẇ + k ax w + c w = mz 0 P C ; + d rot + k rot + c w = M C, 1 where w, are the generalized axial displacement and twisting angle; m, are tool mass and moment of inertia, d ax and d rot are generalized damping factors of axial and torsional vibrations correspondingly, and c w, c are coefficients characterizing axialtorsional coupling. The derivatives of state variables with respect to time t are denoted by dots. We assume that the tool of length l and workpiece are absolutely rigid, but the tool support is flexible. 2.2 Model of New Surface Formation. The tool has n C cutting edges arranged symmetrically at the right end of the tool. The nominal feed rate per one turn is determined as h 0 =VT, where T is the period of rotation T=2 /. The position of cutting edges in axial direction is determined as Z C t = Z 0 t + l + w t 2 The axial coordinate of surface L is measured in a coordinate system attached to the workpiece 8,16. For the following calculations it is more convenient to consider the coordinate of machined surface as a signal that comes to the cutting edge at time t, but it was formed by previous cutting edge J at time t T /n c when it cuts the same surface. For the jth-cutting edge at time t, this distance is equal to the surface coordinate of the cutting edge with number J= j 1 +n C fix 1/j, j=1,n C at time t T /n c, where fix 1/j denotes function that rounds the 1/ j to the nearest integer toward zero. It should be noted that T is not constant and depends on torsional vibrations and is equal to T only in the case when is equal zero or constant. Then the coordinate of the unmachined surface under the j-cutting edge is derived by the following relation: Z j t = Z l + H V t + L J t T /n c, where H is the distance from tool to the workpiece end mean surface at the initial instant and L J t T /n c is the depth of the hole surface under the Jth cutting edge measured from the workpiece left end. Let L j t, j=1,n C designate the depth of the machined hole along the generatrix under the jth cutting edge at the current instant. The surface L j t is formed due to cutting of instant chip thickness h j t of the surface that was formed by previous cutting edge J at time t T /n c : L j t = L J t T /n c + h j t. 4 In order to account for the possible disengagement of cutting edges out of material being cut, we introduce new variables D j t =Z C t Z j t as given by the distance from cutting edges to the unmachined surface 3 Fig. 2 Schematic model of a drill D j t = Z 0 t Z w t H + Vt L J t T /n C h j t =max 0,D j t 5 The second nonlinear relation in Eq. 5 defines the thickness of the undeformed chip. The initial function required to solve Eqs. 4 and 5 is given by Journal of Computational and Nonlinear Dynamics JANUARY 2007, Vol. 2 / 59
3 L j t = L 0J t, t 0. 6 In the case of an initially plane workpiece end that is normal to the hole axis, we should specify L 0j t =0, t 0. Obviously all cutting edges are in similar geometrical conditions. The full kinematics of vibratory drilling for any number of symmetrically disposed cutting edges and arbitrary law of tool axial motion is described by the set of equations 4 6. Inthe case of an absolutely rigid tool without elastic support and with a given law of support excitation, this set is sufficient for a calculation of the machined surface. In order to simplify calculations, the dimensionless variables are introduced by scaling displacement to h 0 and time to the period of tool rotation =t/t, 0 T=2 p, where p is the number of tool vibrations imparted by a vibrator during one workpiece rotation. The following dimensionless variables and parameters are used: = t T, = w h 0, = 2, f rot = T 2 k rot, = d ax 2 kax m, p = 0T 2, Z 0 = Z 0 h 0, A 0 = Ã 0 h 0, f ax = T 2 k ax m, H = H h 0, d rot 2 ; = m f d ax ax C = f ax c w ; C = T2 h 0 h 0 2 c Now the kinematical equations in dimensionless form are given as j = Z 0 Z H + J T /Tn C ; j =max 0, j ; j = J T /Tn C + j ; j = 0j, 0; J = j 1 + n C fix 1/j Mechanical Model of Cutting Forces. There are several mechanical models of orthogonal and oblique cutting in the literature 1,12, In general, the cutting force is taken to be proportional to the uncut chip cross-sectional area. In case of drilling, the geometry of the cutting edge angles and speed are varied along the edge and the usual model of oblique cutting is not accurate. Altintas et al. in 17 used the distributed mechanistic model with empirical coefficients determined from a database using the orthogonal to oblique cutting transformation. This was applied to modeling of drilling with a pilot hole of diameter larger than the chisel edge. However, in this case the conditions along the edge do not vary significantly. In the case of usual drilling, the model used in 16 seems to be more efficient, as well as simpler. It satisfies similarity and dimensions theory and considers tool geometry with two empirical coefficients. Let us represent the axial component of the cutting force as follows see Fig. 2 : P C = k C0 h 0 /q q, 8 where k C 0 =g L Rq h 0 R/n C q 1 is the static axial cutting stiffness, g is a geometrical constant that is approximately unity g 1; L is the characteristic stress of workpiece material; R is drill radius; q is a nonlinear parameter 1 ; and is the reduced uncut chip thickness calculated by the following relation: Fig. 3 Stability diagram in parameters versus f ax = 1 n C n C q 1/q n C j. 9 j=1 Since the vibrator excitation is given by Z 0 =A 0 sin 2 p, the equations of tool motion for the chosen dynamic model disregarding disengagement of cutting edges out of cut material can be shown to be here derivatives by dimensionless time are dotted : +2 2 f ax + 2 f ax 2 + c = 2 p 2 A 0 sin 2 p 2 f ax 2 q ; q f ax + 2 f rot 2 + c = P 2 f 2 ax q, q 10 where c, c are dimensionless coefficients of axial-torsional coupling and =k 0 C /k is dimensionless cutting stiffness. The resultant torque of cutting forces is considered to be proportional to the thrust by a dimensionless coefficient P. The damping factor of torsional vibrations is taken as the damping factor of axial vibrations multiplied by some factor. The full system model is given by a combination of the equations of system dynamics 10, and equations of new surface formation 7. The resulting system is a set of nonlinear differential equations with the varying state dependent delay entering in nonlinear algebraic equations. The analysis of such a system is rather complicated and can be performed only numerically. 3 Stability Analysis of Continuous Cutting The stability analysis of continuous cutting is carried out from the linearized equations of system 7 and 10 that reduce to a set of delayed first order differential equations DDE. It should be noted that for case of an uncoupled system s =0, and s =0 the stability analysis is completely defined by the analysis of the single DOF model. The analysis shows that the stability regions do not significantly differ from those obtained for a single degree of freedom system given in 7,16. This is explained by the fact that in stable cutting the twist angle is constant and does not change the conditions of chip formation. The domains of stability in parameters versus f ax in Fig. 3 are presented s =0, s =0. The torsional-axial coupling does not change the stability region if we vary coefficients c w and c. The only difference between 1-DOF and 2-DOF systems is the small change in eigenfrequencies of the coupled system Dynamical Equation of the Full Model For a numerical analysis of the system model, we transform Eq. 10 into a normal Cauchy form. The main difficulty in numerical 60 / Vol. 2, JANUARY 2007 Transactions of the ASME
4 integration of Eqs. 10 is the variation of time delay due to torsional vibrations, i.e., delay is state dependent. One option is to use a special algorithm of the delayed function approximation at each step of integration, which, unfortunately, also decreases the accuracy of the calculations and increases the time of integration. In order to avoid this problem, we introduced a new independent variable for integration by changing the time t to the full relative angle of tool and workpiece rotation that varies in time as: t = t+ t, and its derivative as: d /dt= +d /dt. Or in terms of dimensionless variables, we have =t/t, =2, =2, T =2, d /d =1+d /d. Further, we consider as the new independent variable. An additional differential equation for the time derivative takes the form = d d = 1 1+ d d =1 d d 11 Therefore Eqs. 10 transform to: x = A x 1 + Bb 12 where the derivative with respect to is denoted by, and the new state vector is represented as x = T and = B 2 p c 2 f rot f ax , 2 A 0 2 f ax 2 /q 0 P 2 f 2 ax /q 0 0 = A 2 f ax 2 c 2 2 f ax 0 0, b = sin 2 p q The equations of new surface formation now include the constant delay and are given as: j = Z 0 x 5 Z 0 x x 1 H + x 5 J 1/n C j =max 0, j j = J 1/n C + j j = 0j, 0; J = j 1 + n C fix 1/j 13 Equations 12 and 13 represent a set of delayed algebraic differential equations DADE with a constant delay. This system is integrated by the Euler method with a constant step and iterations. For numerical analysis the following typical values of system parameters were specified: n C =2, q=0.75, P =1.0, =1.0, H=0, c =c =0. For validation purposes, first, the simplified model of axial and torsional vibrations coupling was analyzed 22. It was assumed that there were imparted stationary torsional vibrations of the tool with dimensionless amplitude and frequency. In this case the second equation in 10 is dropped and new variable becomes a periodic function given by 22 =1/ 1+ cos 2 and the last equation of system 12 can be integrated independently. Thus we obtain the delayed dynamical system with parametric excitation. The delay is constant and it is independent of the period of the parametric excitation. If the amplitude of parametric excitation is equal to zero, we obtain tool rotation with a constant angular speed and system is purely self-vibratory that is unstable under certain values of system parameters Fig. 3. The frequency of tool self-vibrations is a little bit greater than the eigenfrequency of its axial vibrations. Obviously, we expect that the main parametric resonance should arise in the vicinity of 2f ax, regardless of the fact that our system is delayed and nonlinear. Actually, this phenomenon is also observed here to. The only difference is that in our system the self-excited vibrations are strongly nonlinear and, probably, we can regard the system to be synchronized at the frequency of torsional vibration excitation. The Poincaré map of tool axial vibration amplitudes and uncut chip thickness for varying amplitudes of excitation is shown in Fig. 4. The case when torsion excitation frequency is equal to the eigenfrequency of axial vibrations is considered. The results of the excitation amplitude modulation 0,0.8 at = f ax show that in case of constant tool rotation without torsional vibrations =0 we have periodic stationary axial vibrations with chip fragmentation. In case of 0.6 due to parametric excitation, we can have continuous cutting with varying chip thickness. Under given values of parameters the parametric excitation has the largest effect at 0.4 when the amplitude of axial vibrations increases and chip shape becomes irregular. These results confirm the suggestion that torsional motion should be considered for a more correct analysis of the problem. Fig. 4 Poincaré map of the amplitude of a axial vibrations; b chip thickness versus excitation amplitude Journal of Computational and Nonlinear Dynamics JANUARY 2007, Vol. 2 / 61
5 Fig. 6 Poincaré maps of uncut chip thickness versus tool axial stiffness: 1 single-dof system; 2 two-dof system Fig. 5 Poincaré maps of a axial vibration; b torsional vibration versus tool axial stiffness in case of self-exited vibration 5 Results From Numerical Simulation Since the full model is a complicated set of nonlinear equations with a time delay, the most satisfactory method for pictorial presentations of various system motions is through Poincaré maps by plotting the extreme points of the tool position or the reduced thickness of the uncut chip when steady-state vibration is reached. We can pick out the most interesting factors for technology design, such as amplitude of cutting edge vibration, cutting discontinuity, chip dimension, the process of chip segmentation regularity and its synchronization with the frequency of imposed external vibrations. In this paper, only the self-excited vibrations without any external excitation are analyzed, i.e., A 0 =0. The Poincaré map of the tool axial and torsional motion versus axial stiffness of tool is presented in Figs. 5 a and 5 b for the following typical values of the system parameters P =1, =1, = , =0.06, f rot / f ax =7. These parameter values correspond to the point A of stability diagram Fig. 3 close to the stability lobe but a little bit higher * = , i.e., we cross the stability lobes in the horizontal direction through point A. We can observe from Fig. 5 that the regions of stable continuous cutting and interrupted cutting are periodically repeated. The character of axial motion is close to mono harmonic and torsional motion is more complex. The Poincaré map of the chip thickness variation versus tool axial stiffness for two system models: 1 single DOF only axial motion is considered and 2 two DOF are compared in Fig. 6. As it is seen, the plots are similar. Both are with an interrupted chip, but the amplitude of chip thickness variation for single DOF system is lower. The variation of axial displacement and twisting angle versus angle of rotation,, is shown in Fig. 7. These plots are obtained for a system simulation with parameters corresponding to point A f ax =6.6 of the diagram in Fig. 3. It should be noted that for the considered case when f rot / f ax =7, the amplitude of torsional vibrations is small and does not essentially influence chip process formation. The Poincaré maps of the axial displacement and torsional vibrations versus the dimensionless cutting stiffness in Figs. 8 a and 8 b for the two-dof system are presented. The diagram corresponds to the cross section of the stability lobes Fig. 3 in the vertical direction through the point A. The following values of parameters were taken P =1, =1, f ax =6.6, =0.06, f rot / f ax =7. The Poincaré map of the chip thickness variation versus the dimensionless cutting stiffness for two system models: 1 single DOF and 2 two DOF are compared in Fig. 9. We can see here that for the small magnitude of cutting stiffness, continuous cutting is stable, and at point * = , we have a bifurcation point corresponding to an unstable region of the linearized system and the system vibration occurs with amplitude that is Fig. 7 Variation of a axial displacement and b twisting angle versus angle of rotation 62 / Vol. 2, JANUARY 2007 Transactions of the ASME
6 Fig. 8 Poincaré map of a amplitude of axial displacement; b amplitude of torsional vibration; versus dimensionless cutting stiffness sufficient for discontinuous cutting. If we compare results of the single-dof system and two-dof system, it is observed that we have a lower amplitude of chip thickness for the second case and amplitude increases if we increase the cutting stiffness. As a practicing engineer, one would be interested in how the ratio of torsional to axial stiffness f rot / f ax affects tool vibrations. The Poincaré maps of the tool axial and torsional motion versus f rot / f ax are presented in Figs. 10 a and 10 b for the following values of the system parameters: P =1, =1, f ax =2.7, = , * = , =0.02. The Poincaré map of the chip thickness versus f rot / f ax is shown in Fig. 11. We can see that the largest effect of torsional motion occurs when the ratio f rot / f ax is an integer and the influence is greater if the ratio f rot / f ax is lower than 5. In case when the ratio is greater than 5, the tool motion is regular at all ranges of the ratio f rot / f ax variation, except the values when f rot / f ax are close to an integer 5 or6or7, where the axial and torsional vibrations become polyharmonic and the chip Fig. 10 Influence of torsional stiffness on a amplitude of axial displacement; b amplitude of torsional vibration; f ax =2.7, =0.02, n=6000 rpm becomes irregular. At the same time, if we further increase f rot / f ax, then the region of irregular motion vanishes. In the case when the ratio is lower than 5, the tool motion is more complicated. The torsional vibration amplitude sufficiently increases and the amplitude of axial vibrations decreases and for some intervals 3.2, 3.4, 3.5, 3.6, 4.6, 4.8 becomes close to zero. The energy at these points transfers to the torsional vibrations. But the uncut chip amplitude here is small and cutting is intermittent. The result obtained confirms that in the case of autoresonant vibratory drilling when we apply the special tool holder with an additional elastic element we increase the magnitude of the ratio f rot / f ax and thus we make the process more regular with a lower amplitude of torsional vibrations. Usually the high frequency torsional oscillations decrease tool life, and so the autoresonant vibratory drilling becomes more efficient with a suitable choice of system parameters. It should be pointed out that the suggested algorithm does not work at lower values of the ratio f rot / f ax because an increase in torsional vibration amplitude leads to the cases where the absolute angular velocity of the tool becomes negative and the nu- Fig. 9 Poincaré map of chip thickness amplitude versus dimensionless cutting stiffness, : 1 single-dof system; 2 two-dof system Fig. 11 Influence of torsional stiffness on chip thickness amplitude f ax =2.7, =0.02, n=6000 rpm Journal of Computational and Nonlinear Dynamics JANUARY 2007, Vol. 2 / 63
7 merical integration procedure stops as time stops at the constant value. Actually in this case cutting edges are moving in an opposite direction to the tool rotation along the surface that was formed by the same tool edge some time before. Cutting conditions are significantly changed and time delay is not constant in spite of applying the new independent variable. The results obtained here show that the cutting force models disregarding influence of variable delay, used by Bayly et al. 15 and Roukema, Altintas 17 are incorrect in cases when the ratio of torsional to axial frequencies is lower than 5. 6 Conclusions A new model of coupled torsional-axial oscillations of vibratory drilling is presented. The model considers continuous and discontinuous cutting as well as the influence of new surface formation equations. The model permits the simulation of nonstationary processes and the prediction of cutting forces, displacements, and chip shapes for a specified set of system parameters. The introduction of an absolute angle of rotation as the new independent variable, instead of time, increases the system dimension, but leads to a set of equations with constant delay. Therefore, the integration is more accurate and effective. The stability analysis of the 2-DOF system shows that the torsional vibrations do not significantly change the stability lobes of continuous cutting, but can considerably change the character of system vibrations and the shape of chips in the instability region. The same conclusion but for the two-dof model of milling process by Insperger et al. 23 was drawn. The numerical simulation of self-excited vibrations of the 2-DOF systems shows that when torsional to axial frequency ratios are greater than 5, then the character of tool vibration in the region of unstable continuous cutting with respect to torsional vibration is similar to the same case in a 1-DOF system. The region of regular periodic motion decreases and the amplitude of chip thickness is lower. For all cases where f rot / f ax 5 the torsional motion plays an important role, as it essentially affects the process of chip formation. Especially, it should be considered in the case when tool axial stiffness is close to an integer multiple of torsional stiffness. Due to internal resonance, the energy of external excitation redistributes between axial and torsional modes and the chip formation process character changes. The amplitude of the uncut chip decreases and the motion of the system become highly irregular. Acknowledgment This research is financially supported by the National Science Foundation under Grants No. CMS and No. CMS References 1 Stephenson, D. A., and Agapiou, J. S., 1997, Metal Cutting Theory and Practice, Marcel Decker, New York. 2 Poduraev, V. N., 1970, Cutting with Vibrations, Mashinostroenie, Moscow, p Poduraev, V. N., and Kibalchenko, A. V., 1993, The Technology of Defense Industry for Manufacturing of Customer Goods, Rosconversia, Moscow, p Brun-Picard, D., and Gouskov, A., Tête de Perçage à Effets Vibratoires, Patent B036, INPG. 5 Gouskov, A. M., Svetlitsky, V. A., and Voronov, S. A., 1979, Transverse Self-Vibration Excitation of Tool Used For Deep Hole Drilling, Collection of Papers Raschety Na Prochnost, Mashinostroenie, Moscow, 20, pp Gouskov, A. M., Voronov, S. A., and Nikitin, A. S., 1992, Stochastic Regimes in Technologic Cutting Processes, Proc. 2nd International Scientific Technical Conference Actual Problems of Fundamental Science, Technosphera Inform, Moscow, BMSTU, 2, pp. B2 B5. 7 Gouskov, A. M., Voronov, S. A., and Batzer, S. A., 2000, Chatter Synchronization in Vibratory Drilling, Dynamics, Acoustics and Simulations/ASME 2000, 68, pp Batzer, S. A., Gouskov, A. M., and Voronov, S. A., 2001, Modeling Vibratory Drilling Dynamics, ASME J. Vibr. Acoust., 123, pp Voronov, S. A., and Svetlitsky, V. A., 1979, Influence of Natural Twisting on Beam Vibrations, Izvestiya Vuzov. Mashinostroenie., 5, Moscow, pp Telkinap, O., and Ulsoy, A. G., 1990, Effects of Geometric and Process Parameters on Drill Transverse Vibrations, ASME J. Eng. Ind., 112, pp Rincon, D., and Ulsoy, A. G., 1994, Effects of Drill Vibrations on Cutting Forces and Torque, CIRP Ann., 43, pp Voronov, S. A., 1980, Vibratory Drilling Process Optimization, Trans. BM- STU Dynamics and Strength of Materials, 332, Moscow, pp Hodges, D. H., 1980, Torsion of Pre-Twisted Beams Due to Axial Loading, ASME J. Appl. Mech., 47, pp Telkinap, O., and Ulsoy, A. G., 1989, Modeling and Finite Element Analysis of Drill Bit Vibrations, ASME J. Eng. Ind., 111, pp Bayly, P. V., Metzler, S. A., Shaut, A. J., and Young, S. G., 2002, Theory of Torsional Chatter in Twist Drills: Model, Stability Analysis and Composition Test, ASME J. Manuf. Sci. Eng., 123, pp Gouskov, A. M., 2000, Nonlinear Dynamics of Vibratory Drilling. The Significance of the Equations of New Surface Formation, Proc. CSDT-2000, STANKIN, Moscow, pp Roukema, J. C., Altintas, Y., 2004, Kinematic Model of Dynamic Drilling Process, Proc. ASME International Mechanical Engineering Congress and Exposition, November, Anaheim. 18 Montgomery, D., Altintas, Y., 1991, Mechanism of Cutting Force and Surface Generation in Dynamic Milling, ASME J. Eng. Ind., 113, pp Bayly, P. V., Lamar, M. T., and Calvert, S. G., 2002, Low-Frequency Regenerative Vibration and the Formation of Lobed Holes in Drilling, ASME J. Appl. Mech., 124, pp Stepan, G., 2001, Modeling Nonlinear Regenerative Effects in Metal Cutting, Philos. Trans. R. Soc. London, 359, pp Gouskov, A. M., Voronov, S. A., Kvashnin, A. S., Butcher, E. A., and Sinha, S. C., 2005, Influence of Torsional Motion on Vibratory Drilling, Proc. IDETC/CIE 2005, September, Long Beach, California, US. 22 Gouskov, A. M., Voronov, S. A., Batzer, S. A., and Sinha, S. C., 2002, Nonlinear Oscillations of a Tool Used in Deep Hole Drilling, Proc. 4th EURO- MECH Nonlinear Oscillations Conference, August, Moscow, Russia. 23 Insperger, T., Stepan, G., Hartung, F., and Turi, J., 2005, State Dependent Regenerative Delay in Milling Processes, Proc. IDETC/CIE 2005, September, Long Beach, California, US. 64 / Vol. 2, JANUARY 2007 Transactions of the ASME
PERIOD-N BIFURCATIONS IN MILLING: NUMERICAL AND EXPERIMENTAL VERIFICATION
PERIOD-N BIFURCATIONS IN MILLING: NUMERICAL AND EXPERIMENTAL VERIFICATION Andrew Honeycutt and Tony L. Schmitz Department of Mechanical Engineering and Engineering Science University of North Carolina
More informationNumerical simulation of the coil spring and investigation the impact of tension and compression to the spring natural frequencies
Numerical simulation of the coil spring and investigation the impact of tension and compression to the spring natural frequencies F. D. Sorokin 1, Zhou Su 2 Bauman Moscow State Technical University, Moscow,
More informationSTATE-DEPENDENT, NON-SMOOTH MODEL OF CHATTER VIBRATIONS IN TURNING
Proceedings of the ASME 215 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 215 August 2-5, 215, Boston, Massachusetts, USA DETC215-46748
More informationActive control of time-delay in cutting vibration
THEORETICAL & APPLIED MECHANICS LETTERS 3, 633 (213) Active control of time-delay in cutting vibration Pingxu Zheng a) and Xinhua Long b) State Key Laboratory of Mechanical System and Vibration, Shanghai
More informationStructural Dynamics Lecture 2. Outline of Lecture 2. Single-Degree-of-Freedom Systems (cont.)
Outline of Single-Degree-of-Freedom Systems (cont.) Linear Viscous Damped Eigenvibrations. Logarithmic decrement. Response to Harmonic and Periodic Loads. 1 Single-Degreee-of-Freedom Systems (cont.). Linear
More informationEvaluation of active structural vibration control strategies in milling process
Evaluation of active structural vibration control strategies in milling process Monnin, J. (a); Wegener, K. (a) a) Institute of Machine Tools and Manufacturing, Zurich, Switzerland Keywords: Mechatronics,
More informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.
More informationAN EXAMINATION OF SURFACE LOCATION ERROR AND SURFACE ROUGHNESS FOR PERIOD-2 INSTABILITY IN MILLING
AN EXAMINATION OF SURFACE LOCATION ERROR AND SURFACE ROUGHNESS FOR PERIOD-2 INSTABILITY IN MILLING Andrew Honeycutt and Tony L. Schmitz Mechanical Engineering and Engineering Science University of North
More informationTheory and Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 7 Instability in rotor systems Lecture - 4 Steam Whirl and
More informationTheory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 2 Simpul Rotors Lecture - 2 Jeffcott Rotor Model In the
More informationA Robust Semi-Analytical Method for Calculating the Response Sensitivity of a Time Delay System
A Robust Semi-Analytical Method for Calculating the Response Sensitivity of a Time Delay System Mohammad H. Kurdi 1 Postdoctoral Associate e-mail: mhkurdi@gmail.com Raphael T. Haftka Distinguished Professor
More informationTable of Contents. Preface... 13
Table of Contents Preface... 13 Chapter 1. Vibrations of Continuous Elastic Solid Media... 17 1.1. Objective of the chapter... 17 1.2. Equations of motion and boundary conditions of continuous media...
More information202 Index. failure, 26 field equation, 122 force, 1
Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic
More informationWheel Regenerative Chatter of Surface Grinding
Hongqi Li Graduate Research Assistant Yung C. Shin Professor School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 Wheel Regenerative Chatter of Surface Grinding In this paper we
More informationSURFACE PROPERTIES OF THE MACHINED WORKPIECE FOR HELICAL MILLS
Machining Science and Technology, 13:227 245 Copyright 29 Taylor & Francis Group, LLC ISSN: 191-344 print/1532-2483 online DOI: 1.18/191349312167 SURFACE PROPERTIES OF THE MACHINED WORKPIECE FOR HELICAL
More informationCOMPUTER AIDED NONLINEAR ANALYSIS OF MACHINE TOOL VIBRATIONS AND A DEVELOPED COMPUTER SOFTWARE
Mathematical and Computational Applications, Vol. 10, No. 3, pp. 377-385, 005. Association for Scientific Research COMPUTER AIDED NONLINEAR ANALYSIS OF MACHINE TOOL VIBRATIONS AND A DEVELOPED COMPUTER
More informationSimulating Two-Dimensional Stick-Slip Motion of a Rigid Body using a New Friction Model
Proceedings of the 2 nd World Congress on Mechanical, Chemical, and Material Engineering (MCM'16) Budapest, Hungary August 22 23, 2016 Paper No. ICMIE 116 DOI: 10.11159/icmie16.116 Simulating Two-Dimensional
More informationAnalysis of the Chatter Instability in a Nonlinear Model for Drilling
Sue Ann Campbell Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1 Canada e-mail: sacampbell@uwaterloo.ca Emily Stone Department of Mathematical Sciences, The University of
More informationDETC98/PTG-5788 VIBRO-ACOUSTIC STUDIES OF TRANSMISSION CASING STRUCTURES
Proceedings of DETC98: 1998 ASME Design Engineering Technical Conference September 13-16, 1998, Atlanta, GA DETC98/PTG-5788 VIBRO-ACOUSTIC STUDIES O TRANSMISSION CASING STRUCTURES D. Crimaldi Graduate
More information1859. Forced transverse vibration analysis of a Rayleigh double-beam system with a Pasternak middle layer subjected to compressive axial load
1859. Forced transverse vibration analysis of a Rayleigh double-beam system with a Pasternak middle layer subjected to compressive axial load Nader Mohammadi 1, Mehrdad Nasirshoaibi 2 Department of Mechanical
More informationResearch Article The Microphone Feedback Analogy for Chatter in Machining
Shock and Vibration Volume 215, Article ID 976819, 5 pages http://dx.doi.org/1.1155/215/976819 Research Article The Microphone Feedback Analogy for Chatter in Machining Tony Schmitz UniversityofNorthCarolinaatCharlotte,Charlotte,NC28223,USA
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationModelling of lateral-torsional vibrations of the crank system with a damper of vibrations
Modelling of lateral-torsional vibrations of the crank system with a damper of vibrations Bogumil Chiliński 1, Maciej Zawisza 2 Warsaw University of Technology, Institute of Machine Design Fundamentals,
More informationMODELING OF REGENERATIVE CHATTER OF A MILLING PROCESS TO DELINEATE STABLE CUTTING REGION FROM UNSTABLE REGION
International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue, November 208, pp. 748 757, Article ID: IJMET_09 076 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=9&itype=
More information3 Mathematical modeling of the torsional dynamics of a drill string
3 Mathematical modeling of the torsional dynamics of a drill string 3.1 Introduction Many works about torsional vibrations on drilling systems [1, 12, 18, 24, 41] have been published using different numerical
More informationTime domain simulation of chatter vibrations in indexable drills
Int J Adv Manuf Technol (2017) 89:1209 1221 DOI 10.1007/s00170-016-9137-8 ORIGINAL ARTICLE Time domain simulation of chatter vibrations in indexable drills Amir Parsian 1,2 & Martin Magnevall 1,2 & Mahdi
More informationAn improved brake squeal source model in the presence of kinematic and friction nonlinearities
An improved brake squeal source model in the presence of kinematic and friction nonlinearities Osman Taha Sen, Jason T. Dreyer, and Rajendra Singh 3 Department of Mechanical Engineering, Istanbul Technical
More informationRock fragmentation mechanisms and an experimental study of drilling tools during high-frequency harmonic vibration
Pet.Sci.(03)0:05- DOI 0.007/s8-03-068-3 05 Rock fragmentation mechanisms and an experimental study of drilling tools during high- harmonic vibration Li Wei, Yan Tie, Li Siqi and Zhang Xiaoning School of
More informationNonlinear Dynamic Analysis of a Hydrodynamic Journal Bearing Considering the Effect of a Rotating or Stationary Herringbone Groove
G. H. Jang e-mail: ghjang@hanyang.ac.kr J. W. Yoon PREM, Department of Mechanical Engineering, Hanyang University, Seoul, 133-791, Korea Nonlinear Dynamic Analysis of a Hydrodynamic Journal Bearing Considering
More informationLaboratory 4 Topic: Buckling
Laboratory 4 Topic: Buckling Objectives: To record the load-deflection response of a clamped-clamped column. To identify, from the recorded response, the collapse load of the column. Introduction: Buckling
More informationSTICK-SLIP WHIRL INTERACTION IN DRILLSTRING DYNAMICS
STICK-SLIP WHIRL INTERACTION IN DRILLSTRING DYNAMICS R. I. Leine, D. H. van Campen Department of Mechanical Engineering, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands
More informationOrder Reduction of Parametrically Excited Linear and Nonlinear Structural Systems
Venkatesh Deshmukh 1 Eric A. Butcher e-mail: ffeab@uaf.ediu Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775 S. C. Sinha Nonlinear Systems Research Laboratory,
More informationParametric Excitation of a Linear Oscillator
Parametric Excitation of a Linear Oscillator Manual Eugene Butikov Annotation. The manual includes a description of the simulated physical system and a summary of the relevant theoretical material for
More informationMODELLING AND ANALYSIS OF CHATTER MITIGATION STRATEGIES IN MILLING
MODELLING AND ANALYSIS OF CHATTER MITIGATION STRATEGIES IN MILLING by Khaled Saleh Submitted in fulfilment of the degree of Doctor of Philosophy June 2013 Department of Mechanical Engineering Sheffield,
More informationNonlinear effects on the rotor driven by a motor with limited power
Applied and Computational Mechanics 1 (007) 603-61 Nonlinear effects on the rotor driven by a motor with limited power L. Pst Institute of Thermomechanics, Academy of Sciences of CR, Dolejškova 5,18 00
More informationCOPYRIGHTED MATERIAL. Index
Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,
More informationEQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION
1 EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION The course on Mechanical Vibration is an important part of the Mechanical Engineering undergraduate curriculum. It is necessary for the development
More informationAvailable online at ScienceDirect. Procedia CIRP 36 (2015 ) CIRP 25th Design Conference Innovative Product Creation
Available online at www.sciencedirect.com ScienceDirect Procedia CIRP 36 (2015 ) 111 116 CIRP 25th Design Conference Innovative Product Creation Machine stiffness rating: Characterization and evaluation
More informationExpedient Modeling of Ball Screw Feed Drives
S. Frey a A. Dadalau a A. Verl a Expedient Modeling of Ball Screw Feed Drives Stuttgart, February 2011 a Institute for Control Engineering of Machine Tools and Manufacturing Units (ISW), University of
More informationStability of Water-Lubricated, Hydrostatic, Conical Bearings With Spiral Grooves for High-Speed Spindles
S. Yoshimoto Professor Science University of Tokyo, Department of Mechanical Engineering, 1-3 Kagurazaka Shinjuku-ku, Tokyo 16-8601 Japan S. Oshima Graduate Student Science University of Tokyo, Department
More informationThis equation of motion may be solved either by differential equation method or by graphical method as discussed below:
2.15. Frequency of Under Damped Forced Vibrations Consider a system consisting of spring, mass and damper as shown in Fig. 22. Let the system is acted upon by an external periodic (i.e. simple harmonic)
More informationDevelopment of a test apparatus that consistently generates squeak to rate squeak propensity of a pair of materials
Development of a test apparatus that consistently generates squeak to rate squeak propensity of a pair of materials Gil Jun LEE 1 ; Jay KIM 2 1, 2 Department of Mechanical and Materials Engineering, University
More informationTheory & Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory & Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 5 Torsional Vibrations Lecture - 4 Transfer Matrix Approach
More informationContact problems in rotor systems
Contact problems in rotor systems Liudmila Banakh Mechanical Engineering Research Institute of RAS, Moscow, Russia E-mail: banl@inbox.ru (Received 18 July 2016; accepted 24 August 2016) Abstract. We consider
More informationFORCES, VIBRATIONS AND ROUGHNESS PREDICTION IN MILLING USING DYNAMIC SIMULATION
FORCES, VIBRATIONS AND ROUGHNESS PREDICTION IN MILLING USING DYNAMIC SIMULATION Edouard Rivière (1), Enrico Filippi (2), Pierre Dehombreux (3) (1) Faculté Polytechnique de Mons, Service de Génie Mécanique,
More informationROLLER BEARING FAILURES IN REDUCTION GEAR CAUSED BY INADEQUATE DAMPING BY ELASTIC COUPLINGS FOR LOW ORDER EXCITATIONS
ROLLER BEARIG FAILURES I REDUCTIO GEAR CAUSED BY IADEQUATE DAMPIG BY ELASTIC COUPLIGS FOR LOW ORDER EXCITATIOS ~by Herbert Roeser, Trans Marine Propulsion Systems, Inc. Seattle Flexible couplings provide
More informationProcess Damping Coefficient Identification using Bayesian Inference
Process Damping Coefficient Identification using Bayesian Inference Jaydeep M. Karandikar, Christopher T. Tyler, and Tony L. Schmitz Mechanical Engineering and Engineering Science University of North Carolina
More informationStability Analysis of a Hydrodynamic Journal Bearing With Rotating Herringbone Grooves
G. H. Jang e-mail: ghjang@hanyang.ac.kr J. W. Yoon PREM, Department of Mechanical Engineering, Hanyang University, Seoul, 33-79, Korea Stability Analysis of a Hydrodynamic Journal Bearing With Rotating
More informationActive in-process chatter control E.J.J. Doppenberg*, R.P.H. Faassen**, N. van de Wouw**, J.A.J. Oosterling*, H. Nijmeijer**,
Active in-process chatter control E.J.J. Doppenberg*, R.P.H. Faassen**, N. van de Wouw**, J.A.J. Oosterling*, H. Nijmeijer**, *TNO Science and Industry, P.O. Box 155, 2600 AD Delft, The Netherlands 1 **Eindhoven
More informationDetermination of Dynamic Characteristics of the Frame Bearing Structures of the Vibrating Separating Machines
IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS Determination of Dynamic Characteristics of the Frame Bearing Structures of the Vibrating Separating Machines To cite this article:
More informationThe... of a particle is defined as its change in position in some time interval.
Distance is the. of a path followed by a particle. Distance is a quantity. The... of a particle is defined as its change in position in some time interval. Displacement is a.. quantity. The... of a particle
More informationName: Fall 2014 CLOSED BOOK
Name: Fall 2014 1. Rod AB with weight W = 40 lb is pinned at A to a vertical axle which rotates with constant angular velocity ω =15 rad/s. The rod position is maintained by a horizontal wire BC. Determine
More informationChapter a. Spring constant, k : The change in the force per unit length change of the spring. b. Coefficient of subgrade reaction, k:
Principles of Soil Dynamics 3rd Edition Das SOLUTIONS MANUAL Full clear download (no formatting errors) at: https://testbankreal.com/download/principles-soil-dynamics-3rd-editiondas-solutions-manual/ Chapter
More informationLANMARK UNIVERSITY OMU-ARAN, KWARA STATE DEPARTMENT OF MECHANICAL ENGINEERING COURSE: MECHANICS OF MACHINE (MCE 322). LECTURER: ENGR.
LANMARK UNIVERSITY OMU-ARAN, KWARA STATE DEPARTMENT OF MECHANICAL ENGINEERING COURSE: MECHANICS OF MACHINE (MCE 322). LECTURER: ENGR. IBIKUNLE ROTIMI ADEDAYO SIMPLE HARMONIC MOTION. Introduction Consider
More informationNonlinear Rolling Element Bearings in MADYN 2000 Version 4.3
- 1 - Nonlinear Rolling Element Bearings in MADYN 2000 Version 4.3 In version 4.3 nonlinear rolling element bearings can be considered for transient analyses. The nonlinear forces are calculated with a
More informationMechanics of Materials II. Chapter III. A review of the fundamental formulation of stress, strain, and deflection
Mechanics of Materials II Chapter III A review of the fundamental formulation of stress, strain, and deflection Outline Introduction Assumtions and limitations Axial loading Torsion of circular shafts
More informationVibration modelling of machine tool structures
Vibration modelling of machine tool structures F. Haase, S. Lockwood & D.G. Ford The Precision Engineering Centre, University of Huddersfield (UK) Abstract Productivity in modem machine tools is acheved
More informationAnalysis of Tensioner Induced Coupling in Serpentine Belt Drive Systems
2008-01-1371 of Tensioner Induced Coupling in Serpentine Belt Drive Systems Copyright 2007 SAE International R. P. Neward and S. Boedo Department of Mechanical Engineering, Rochester Institute of Technology
More informationSuppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber
Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber J.C. Ji, N. Zhang Faculty of Engineering, University of Technology, Sydney PO Box, Broadway,
More informationNUMERICAL INVESTIGATION OF CABLE PARAMETRIC VIBRATIONS
11 th International Conference on Vibration Problems Z. Dimitrovová et al. (eds.) Lisbon, Portugal, 9-1 September 013 NUMERICAL INVESTIGATION OF CABLE PARAMETRIC VIBRATIONS Marija Nikolić* 1, Verica Raduka
More informationEvaluation of Cutting Forces and Prediction of Chatter Vibrations in Milling
H. B. Lacerda and V. T. Lima H. B. Lacerda and V. T. Lima Federal University of Uberlândia School of Mechanical Engineering Av. João Naves de Ávila, Bl. M 38400-90 Uberlândia, MG. Brazil helder@mecanica.ufu.br
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationNumerical Methods for Solving the Dynamic Behavior of Real Systems
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 1 (2014), 25-34. Numerical Methods for Solving the Dynamic Behavior of Real Systems V. Nikolić,
More informationPrecision Engineering
Precision Engineering 46 2016) 73 80 Contents lists available at ScienceDirect Precision Engineering jo ur nal ho me p age: www.elsevier.com/locate/precision A coupled dynamics, multiple degree of freedom
More informationHELICAL BUCKLING OF DRILL-STRINGS
HELICAL BUCKLING OF DRILL-STRINGS Marcin Kapitaniak 1,, Vahid Vaziri 1,, and Marian Wiercigroch 1 1 Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, Aberdeen, AB24 3UE,
More informationAppendix C. Modal Analysis of a Uniform Cantilever with a Tip Mass. C.1 Transverse Vibrations. Boundary-Value Problem
Appendix C Modal Analysis of a Uniform Cantilever with a Tip Mass C.1 Transverse Vibrations The following analytical modal analysis is given for the linear transverse vibrations of an undamped Euler Bernoulli
More informationThe impact of rotor elastic suspension settings on the acceleration of the automatic balancer compensating mass
The impact of rotor elastic suspension settings on the acceleration of the automatic balancer compensating mass Guntis Strautmanis 1, Mareks Mezitis 2, Valentina Strautmane 3 1, 2 Riga Technical University,
More informationDynamic Green Function Solution of Beams Under a Moving Load with Dierent Boundary Conditions
Transaction B: Mechanical Engineering Vol. 16, No. 3, pp. 273{279 c Sharif University of Technology, June 2009 Research Note Dynamic Green Function Solution of Beams Under a Moving Load with Dierent Boundary
More informationUNCERTAINTY PROPAGATION FOR SELECTED ANALYTICAL MILLING STABILITY LIMIT ANALYSES
UNCERTAINTY PROPAGATION FOR SELECTED ANALYTICAL MILLING STABILITY LIMIT ANALYSES G. Scott Duncan, Mohammad H. Kurdi, Tony L. Schmitz Department of Mechanical and Aerospace Engineering University of Florida
More informationDYNAMICS ASPECT OF CHATTER SUPPRESSION IN MILLING
11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V) 6th European Conference on Computational Fluid Dynamics (ECFD VI) E. Oñate, J. Oliver
More informationPhysics 2001/2051 The Compound Pendulum Experiment 4 and Helical Springs
PY001/051 Compound Pendulum and Helical Springs Experiment 4 Physics 001/051 The Compound Pendulum Experiment 4 and Helical Springs Prelab 1 Read the following background/setup and ensure you are familiar
More informationThis is an author-deposited version published in: Eprints ID: 13508
Open Archive Toulouse Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited
More informationON THE PREDICTION OF EXPERIMENTAL RESULTS FROM TWO PILE TESTS UNDER FORCED VIBRATIONS
Transactions, SMiRT-24 ON THE PREDICTION OF EXPERIMENTAL RESULTS FROM TWO PILE TESTS UNDER FORCED VIBRATIONS 1 Principal Engineer, MTR & Associates, USA INTRODUCTION Mansour Tabatabaie 1 Dynamic response
More informationThe Design of a Multiple Degree of Freedom Flexure Stage with Tunable Dynamics for Milling Experimentation
The Design of a Multiple Degree of Freedom Flexure Stage with Tunable Dynamics for Milling Experimentation Mark A. Rubeo *, Kadir Kiran *, and Tony L. Schmitz The University of North Carolina at Charlotte,
More informationTorsion Spring Oscillator with Dry Friction
Torsion Spring Oscillator with Dry Friction Manual Eugene Butikov Annotation. The manual includes a description of the simulated physical system and a summary of the relevant theoretical material for students
More informationIDENTIFICATION OF NONLINEAR CUTTING PROCESS MODEL IN TURNING
ADVANCES IN MANUFACTURING SCIENCE AND TECHNOLOGY Vol. 33, No. 3, 2009 IDENTIFICATION OF NONLINEAR CUTTING PROCESS MODEL IN TURNING Bartosz Powałka, Mirosław Pajor, Stefan Berczyński S u m m a r y This
More informationEngineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration
Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration Module 15 Lecture 38 Vibration of Rigid Bodies Part-1 Today,
More informationInternational Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: Issue 12, Volume 4 (December 2017)
International Journal of Innovative Research in Advanced Engineering (IJIRAE ISSN: 349-63 Issue, Volume 4 (December 07 DESIGN PARAMETERS OF A DYNAMIC VIBRATION ABSORBER WITH TWO SPRINGS IN PARALLEL Giman
More information142. Determination of reduced mass and stiffness of flexural vibrating cantilever beam
142. Determination of reduced mass and stiffness of flexural vibrating cantilever beam Tamerlan Omarov 1, Kuralay Tulegenova 2, Yerulan Bekenov 3, Gulnara Abdraimova 4, Algazy Zhauyt 5, Muslimzhan Ibadullayev
More informationShape Optimization of Revolute Single Link Flexible Robotic Manipulator for Vibration Suppression
15 th National Conference on Machines and Mechanisms NaCoMM011-157 Shape Optimization of Revolute Single Link Flexible Robotic Manipulator for Vibration Suppression Sachindra Mahto Abstract In this work,
More informationAppendix A Satellite Mechanical Loads
Appendix A Satellite Mechanical Loads Mechanical loads can be static or dynamic. Static loads are constant or unchanging, and dynamic loads vary with time. Mechanical loads can also be external or self-contained.
More informationFirst-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns
First-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns J. A. Loya ; G. Vadillo 2 ; and J. Fernández-Sáez 3 Abstract: In this work, closed-form expressions for the buckling loads
More informationStructural Dynamics A Graduate Course in Aerospace Engineering
Structural Dynamics A Graduate Course in Aerospace Engineering By: H. Ahmadian ahmadian@iust.ac.ir The Science and Art of Structural Dynamics What do all the followings have in common? > A sport-utility
More information, follows from these assumptions.
Untangling the mechanics and topology in the frictional response of long overhand elastic knots Supplemental Information M.K. Jawed, P. Dieleman, B. Audoly, P. M. Reis S1 Experimental details In this section,
More informationStochastic Dynamics of SDOF Systems (cont.).
Outline of Stochastic Dynamics of SDOF Systems (cont.). Weakly Stationary Response Processes. Equivalent White Noise Approximations. Gaussian Response Processes as Conditional Normal Distributions. Stochastic
More informationDynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras
Dynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras Module - 1 Lecture - 10 Methods of Writing Equation of Motion (Refer
More informationFundamentals Physics. Chapter 15 Oscillations
Fundamentals Physics Tenth Edition Halliday Chapter 15 Oscillations 15-1 Simple Harmonic Motion (1 of 20) Learning Objectives 15.01 Distinguish simple harmonic motion from other types of periodic motion.
More informationSAMCEF For ROTORS. Chapter 1 : Physical Aspects of rotor dynamics. This document is the property of SAMTECH S.A. MEF A, Page 1
SAMCEF For ROTORS Chapter 1 : Physical Aspects of rotor dynamics This document is the property of SAMTECH S.A. MEF 101-01-A, Page 1 Table of Contents rotor dynamics Introduction Rotating parts Gyroscopic
More informationParametrically Excited Vibration in Rolling Element Bearings
Parametrically Ecited Vibration in Rolling Element Bearings R. Srinath ; A. Sarkar ; A. S. Sekhar 3,,3 Indian Institute of Technology Madras, India, 636 ABSTRACT A defect-free rolling element bearing has
More informationHB Coupled Pendulums Lab Coupled Pendulums
HB 04-19-00 Coupled Pendulums Lab 1 1 Coupled Pendulums Equipment Rotary Motion sensors mounted on a horizontal rod, vertical rods to hold horizontal rod, bench clamps to hold the vertical rods, rod clamps
More informationMechanical Oscillations
Mechanical Oscillations Richard Spencer, Med Webster, Roy Albridge and Jim Waters September, 1988 Revised September 6, 010 1 Reading: Shamos, Great Experiments in Physics, pp. 4-58 Harmonic Motion.1 Free
More informationCollocation approximation of the monodromy operator of periodic, linear DDEs
p. Collocation approximation of the monodromy operator of periodic, linear DDEs Ed Bueler 1, Victoria Averina 2, and Eric Butcher 3 13 July, 2004 SIAM Annual Meeting 2004, Portland 1=Dept. of Math. Sci.,
More informationDispersion relation for transverse waves in a linear chain of particles
Dispersion relation for transverse waves in a linear chain of particles V. I. Repchenkov* It is difficult to overestimate the importance that have for the development of science the simplest physical and
More informationApplication of Viscous Vortex Domains Method for Solving Flow-Structure Problems
Application of Viscous Vortex Domains Method for Solving Flow-Structure Problems Yaroslav Dynnikov 1, Galina Dynnikova 1 1 Institute of Mechanics of Lomonosov Moscow State University, Michurinskiy pr.
More informationComputational non-linear structural dynamics and energy-momentum integration schemes
icccbe 2010 Nottingham University Press Proceedings of the International Conference on Computing in Civil and Building Engineering W Tizani (Editor) Computational non-linear structural dynamics and energy-momentum
More informationMeasuring the Universal Gravitational Constant, G
Measuring the Universal Gravitational Constant, G Introduction: The universal law of gravitation states that everything in the universe is attracted to everything else. It seems reasonable that everything
More information2018. Nonlinear free vibration analysis of nanobeams under magnetic field based on nonlocal elasticity theory
2018. Nonlinear free vibration analysis of nanobeams under magnetic field based on nonlocal elasticity theory Tai-Ping Chang National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan
More informationForced Oscillations in a Linear System Problems
Forced Oscillations in a Linear System Problems Summary of the Principal Formulas The differential equation of forced oscillations for the kinematic excitation: ϕ + 2γ ϕ + ω 2 0ϕ = ω 2 0φ 0 sin ωt. Steady-state
More informationDynamics of assembled structures of rotor systems of aviation gas turbine engines of type two-rotor
Dynamics of assembled structures of rotor systems of aviation gas turbine engines of type two-rotor Anatoly А. Pykhalov 1, Mikhail А. Dudaev 2, Mikhail Ye. Kolotnikov 3, Paul V. Makarov 4 1 Irkutsk State
More informationMembers Subjected to Torsional Loads
Members Subjected to Torsional Loads Torsion of circular shafts Definition of Torsion: Consider a shaft rigidly clamped at one end and twisted at the other end by a torque T = F.d applied in a plane perpendicular
More information