Effects of Vibration on a Coulomb Friction System

Transcription

1 Effects of Vibration on a Coulomb Friction System M. Holland and D. Tran School of ACME, Victoria University, P O Box 14428, MCMC, Vic 8001, AUSTRALIA Abstract Deformable systems that rely on dry friction for their proper functioning are widely found in many engineering applications such as bolted components, railway ballast. The design of these systems is usually considered from static point of view and employing Coulomb model of friction. This paper studies the effects of dynamic loading on the behavior of a two dimensional system by finite element modeling and by experiments. It was found that the functioning of the system can break down depending on the value of the set of amplitude of vibration, frequency and preload. There are domains in the space of these characteristics that the contact surfaces would be momentarily separated resulting in accumulative sliding, leading to failure in the system s functioning. 1. Introduction Friction is often viewed as a nuisance and many efforts are devoted to reduce it to a minimum, for example by using hydrodynamic and compressed air lubrication. Friction-induced vibration, tool chattering, brake squealing are cases in which friction interferes with proper functioning of engineering systems. On the other hand, friction is taken for granted in many engineering applications by designers. When such friction-dependent systems are subjected to dynamic loading, the interaction between vibration and dry friction is a highly non-linear problem. It is well known that under dynamic loading, threaded fasteners can become loosened leading to malfunctioning, costly maintenance or even catastrophic failures of systems in automotive, aeronautical and electronic industries [1]. Railway lines are laid on sleepers which in turn rest on a foundation of ballast consisting of crushed hard rocks. At high speeds in excess of 400 km/hours it has been found that dynamic vibrations can set the ballast in motion and the ballast would offer hardly any resistance to the dynamic load leading to unacceptable fast sinking of the railway lines. This does impose a limit on the speed of trains using ballasted railway lines [2]. These failures can be attributed to vibration induced loss of contact of friction surfaces. The deformable bodies are usually under some preloading force besides gravity. Loosening is supposed to take place when the friction forces that exist between surfaces are eliminated in the course of interaction between friction and vibration. This paper focuses on the possibility of two surfaces momentarily loosing contact, resulting in loss of friction leading to relative movement between them. This loss of friction has been accepted as causing threaded fasteners loosening of bolted assemblies and has prompted efforts in the last sixty years, mainly in USA to understand the phenomenon and to prevent vibration induced bolt loosening. A brief survey is presented on these studies as the findings can be applied to other systems that rely on friction for their proper functioning. The first study on threaded fastener loosening can be attributed to Goodier and Sweeney [3] in 1945 who proposed that in the course of dynamic loading and unloading, initial loosening is caused by loading on the assembled components and tightening caused by unloading, the net effect being a small rotation in the nut. Sauer [4] employed a Sonnatag universal fatigue machine that could produce 1000 lbs of 30Hz sinusoidal axial load superimposed upon a bolt and nut assembly and concluded that thread loosening can be prevented by keeping the dynamic to static preloading to less then 0.7. Kerley [5] showed that low frequency excitation caused bolts to loosen, whereas high frequencies did not. However it was noted that high frequencies could create subharmonics, which could cause loosening and eventual failure. Baubles et al [6] found that the impacts or shocks imparted on the assembly would make the components to resonate and hence loosen, that the low frequencies excited the high resonant frequencies of the fasteners; these authors attributed the accumulative wear induced by impacts eventually lead to loss in pre-load tension followed by loosening. This conclusion is in line with Koga [7] who considered the effects of stress waves induced by impact as the main cause of thread loosening. Because the thread angle is usually between 55 to 60 o, the direction of vibration excitation with respect to the thread axis would make a difference to how the vibration would affect the thread loosening. The effect was confirmed by Junker [8] who subjected threaded assemblies to transverse loading conditions and concluded that transverse vibrations have more adverse effects than axial vibrations. Hess and associates [9-10] carried out extensive experiments on a simple nut and bolt assembly and found that for a single bolt assembly, given a moderate pre-load by a spring, that vibration causes both loosening and tightening and that system parameters can be tuned to cause either action to occur: no loosening for amplitude of vibration below certain level for all

2 frequencies studied ( Hz), loosening tendency is strong at low frequencies but as frequency increases, the amplitude of vibration to induce loosening is decreased, in fact for large amplitudes, tightening can take place. Dong and Hess [11] showed that in structures assembled by threaded fasteners, shear force plays a dominant role in thread loosening. The research over almost sixty years have brought about progress in understanding the vibration induced thread loosening, albeit conclusive agreement among researchers has yet to be reached. The main reason is such a simple problem may involve up to 100 parameters, most of them are very hard to control [12]. In order to focus on a number of fundamental parameters, a number of analytical models that have been proposed. 2 Two-dimensional models of frictional surfaces in contact Real systems relying on friction for their functioning usually have very complex geometry. Threaded surfaces of bolts and nuts have regular defined geometry but due to inaccuracy in manufacturing there exists variation in surface finish and dimensions resulting in changing degree of interference and preloading between two mated threads. It is also well known that only a few threads are actually taking the load and responsible for the deformation. Crushed rocks employed in a ballast system are polyhedral surfaces of various sizes and orientations coming into contact in a random fashion during the pouring, tampering and loading processes. They all pose as a formidable three dimensional problems. The approach adopted here is to employ a two dimensional model. A simple model of discrete rigid bodies was proposed by Daadbin and Chow in 1992 [13]: the thread surface of the nut is represented by a rigid body supported on a spring and dashpot, initially in contact with a rigid incline surface representing the bolt. The spring is initially compressed to represent the pre-load. When an impact force is applied, the spring reaction will force the mass upwards. If the mass is displaced more than the spring s initial compression due to pre-load, the system will be separated from the inclined plane. When the system returns to the surface of the inclined plane, it will fall on the spring and dashpot. Daadbin and Chow were able to formulate equations to relate pre-load to displacement. Their numerical results showed that friction coefficient, duration of impact force, lead angle and initial pre-load were the main factors when considering the loosening effect. An earlier and more complex model had been studied by Vinogradov and Huang in 1989 [14], the nut is replaced by a discrete five-mass system which are inter-connected by springs and dashpots with Coulomb dry friction model used for the friction between two mated surfaces. They found that once macro slip has occurred, the rate of selfloosening increases with the reduction in pre-load, that loosening was more prominent in coarse thread fasteners; but a larger thread angle and a more uniform distribution of preloading would result in less loosening. A more recent model has been proposed by Hess [10] in which it is assumed that the mated surfaces are in intimate contact and can be developed into an incline plane representing the bolt and a block representing the nut. These components are under preloading and are treated as rigid bodies, with interaction between surfaces modelled by springs and dashpots, dry friction contact and under compressive preloading force. The analytical results produced by the model correlate well with experimental results [9, 10]. Finite element method (FEM) has also been used to investigate the threaded fastener loosening [15, 16]. In this paper, both the block and incline are treated as deformable continuum bodies and the interaction between friction and vibration in a block-incline under preloading is studied by experiments and by FEM. 3 Finite element modelling (FEM) using two-dimensional block-incline model A number of assumptions was made to simplify the system: the thread width is small compared to the diameter of the thread; ignoring small variation in dimensions and surface finish due to manufacturing and use. A twodimensional model of the frictional contact surfaces can be used, in which the threaded surface of the bolt is developed into an incline plane and that of the nut is modelled as a block in contact with the incline. The system is under gravity, frictional force and preload force. This block-incline model with both considered as rigid bodies has been successfully used in textbooks on design and theory of machines [17, 18] for static analysis. In this paper both bodies are treated as deformable bodies and effects of dynamic loading on the system are considered. The dry friction between the surfaces is modelled as Coulomb friction with a modified steep ramp of coefficient of friction rather than a sudden jump in the region of small relative displacements. The actual surface interaction under dynamic loading is very complicated with effects of Hertz contact stress, ploughing and wearing mechanisms, stick-slip, chattering impact between surfaces. Various sources of energy dissipation may be lumped together as equivalent damping in dynamic consideration.

3 3.1 FEM Model The block-incline was modeled by a two-dimensional FEM model using ANSYS finite element software with CONTAC48 as the element type to depict the interface between the block and the incline, as shown in Figure 1. The block and incline themselves were modeled by plane stress elements, PLANE42. Figure 1: FEM two dimensional model of the block-incline A CONTAC48 element has three nodes, one connected to the block and the other two to the incline surface. The parameters that define the characteristics of contact surface are: interface angle, friction coefficient, normal stiffness and sticking stiffness, displacement interference, initial element status, gap status, gap size, force components for both normal and tangential directions and relative displacement in the tangential direction. The block is under gravity loading, preloading and friction. The problem is highly non-linear because of large displacements and changing direction of friction force and a non-linear implicit solver was used. 3.2 Simulation of vertical harmonic excitation To simulate the harmonic vibration, excitation is input to the incline, all nodes on the vertical edge of the incline are restrained from moving horizontally and the nodes at the base are supported on very soft springs, as shown in Figure 2 and subjected to harmonic displacements as function of times. Resonant frequencies of the block-incline were far much higher than the natural frequency of the equivalent support spring. Frequencies studied were in the range Hz at acceleration amplitudes up to 70 Gs. Varying Coincident Figure 3: Simulation of harmonic excitation

4 3.3 FEM results Displacement components along and normal to the incline plane, termed X and Y, as functions of time can be extracted for any node or pair of nodes. A typical set of results for 200 Hz excitation of 10 Gs is shown in Figure 3, the upper curve shows the relative normal displacement, a zero value indicating intimate contact while a positive value indicating loss of contact. It can be seen that there is a random sequence of momentary loss of contacts interspersed by re-establishing instantaneous contact, which can be roughly described as clattering. This phenomenon is very interesting as it shows that when deformation of block-incline as a continuum is taken into account the surface contacts show impact behavior even though the excitation is purely harmonic. The lower curve represents the tangential relative displacement, where a negative value represents moving up the incline plane against gravity (or equivalent to thread tightening) G For a Frequency of 200Hz G X-Dir 10G Y-Dir Time(sec) Figure 3: Displacement components of block relative to incline While the pattern of normal displacement clattering is very similar for other frequencies and excitation amplitudes, the pattern of the time signal of tangential component is affected by both the level of excitation force and frequency. Figures 4 and 5 show the tangential relative displacement at 300 and 550 Hz respectively for various Gs level, where a positive value indicates movement down the incline (loosening). It can be seen that at low levels of Gs, after a very short initial tightening, the tendency of loosening may take over, this tendency increases with excitation frequency. The distance traveled in the initial tightening is barely observable. As the Gs level of amplitude of vibration excitation force increases, the tendency to tightening increases and for sufficient Gs level, the resulting relative motion is entirely tightening. In Figure 5, the curves for 20 Gs and 40 Gs clearly show the same behavior stated previously: after a very short period and very small distance of traveling up (which would be not be practically observable) the movement is traveling down (loosening). For higher levels of Gs, the difficulty of numerical convergence of the implicit solver did not produce sufficient numerical results but the trend of eventual loosening is very clear at high frequencies. 4. Experimental studies of a block incline model 4.1 Development of experimental apparatus The incline consisted of three components bolted together: an incline plate, base plate, side plate. The block has a thin layer of steel sheeting to simulate a steel surface. The base is screwed onto the shaker. The block and the incline (Figure 6) have a combined mass of 6 kilograms. The total mass was limited by the maximum force tolerated by the shaker used to impart the vibration.preloading was effected by using a number of rare earth small magnets as shown in Figure 7. The base of the block had countersunk holes to accommodate up

5 to 11 small magnets of diameter 10mm and depth 5mm. The arrangement allows the preloading to be varied by changing the number of magnets while ensuring a symmetrical pattern to create uniform distribution of magnetic flux over the block base. A sliver of steel sheet or shim of 0.3 mm thick was placed over the base of the block to provide a smooth surface. Horizontal Displacement for Frequency of 300Hz G X-Dir 20G X-Dir 30G X-Dir 60G X-Dir 70G X-Dir Time(sec) Figure 4: Relative tangential displacement component of block-incline node pair at 300 Hz for various Gs levels Horizontal Displacement for Frequency of 550Hz G X-Dir 40G X-Dir 60G X-Dir 80G X-Dir Time(sec) Figure 5: Relative X displacement of a block-incline node pair at 550 Hz.

6 Figure 6: Block incline system for experimental studies Figure 7 Block Assembly. 4.2 Experimental Set-up The components of the experimental set up are shown in Figure 8. Two accelerometers were required to get acceleration signals from the base and the incline to compare the amplitudes and phase difference. The signals were conditioned by charge amplifiers and fed into a Data-Physics data acquisition card, and then analyzed using HPVEE software. The sampling rates for all of the experiments were taken at 1000Hz for a duration of five seconds. The state of the excitation force measured by Gs and its frequency were set and monitored for each test. Maximum acceleration that could be achieved was in the order of 20 Gs. Different combinations of preload were used to investigate the effects of preload on the dynamic behavior of the blockincline interface. The preload was varied by changing the pattern of the magnets in the base of the block, ensuring uniform and symmetrical distribution of magnetic force. Three patterns were adopted: Minimum (6 Newtons), Medium (13 Newtons) and Maximum Preload (23 Newtons). Figure 8: Experimental set up for the block-incline model

7 Acceleration (G s) Acceleration (G s) Acceleration (G s) Frequency (Hz) Figure 9a: Minimum Preload Frequency (Hz) Figure 9b: Medium Preload Frequency (Hz) Figure 9c: Maximum Preload Movement Up Movement Down No Effect Figure 9: Three domains of behaviour 4.3 Experimental Data and Analysis It was observed that when the block-incline was subjected to harmonic vibration, for the same level of preload, depending on the combination of harmonic excitation amplitude measured in G and frequency (Hz), one of three scenarios could happen: the block is stationary with respect to the incline (termed as no effect) corresponds to no relative twist of threaded surfaces, the block moves up the incline (movement up) for thread tightening, and the block moves down the incline (movement down) for thread loosening. The results for three levels of preloading are shown in Figures 9a-c. These states form clear regions on the plane of acceleration amplitude-frequency. The representation of each region is as follows: dark-grey for movement of the block up the incline surface, grey for movement down, and dark for no effect. In polychromatic plot the colours are red, blue and green respectively. With the level of

8 preload increases, the region of tightening and the region of no effect, to a lesser extent, expand at the expense of the loosening region. Figure 10: Frequency distribution of magnitude and phase of accelerations in loosening at 500 Hz Figure 11: Frequency distribution of magnitude and phase of acceleration in tightening at 150 Hz The time signals of accelerations of the block and incline were recorded and processed by FFT to yield the relative amplitude and phase difference plot in frequency domain and are shown in Figure 10 and 11, the former encapsulates cases of loosening at 500 Hz and the latter tightening at 150 Hz. During loosening, a changing phase difference is found between the acceleration of the block and the incline; tightening on the other hand happens with hardly any phase difference between the two signals. Evidence of clattering at various frequencies is very clear in Figure 10 with overall loosening, while in Figure 11, the tightening effect ensures intimate contact, only well defined super-harmonics (frequencies of higher or multiples of excitation frequency) were observed.

9 5. Conclusions Experimental studies and FEM modeling show that dynamic vibration affects in an intriguing fashion the behavior of deformable bodies that are statically in contact and held by Coulomb friction, gravity, friction and preload. There are three regions of behavior depending on the combination of amplitude of vibration, frequency and amount of preloading. Designers of threaded fasteners and ballast system would avoid the loosening region, a breakdown of the system, but the possibility of using the region of moving up against gravity for material handling up an incline without using moving conveyor belts is very interesting. Loosening is a strong indication of the breakdown of the contact of surfaces and friction force. The no-effect region is also of interest but it requires small threshold of amplitude of vibration which can limit the options available to the designer. Further studies are needed to expand the domain studied in terms of amplitudes of vibration, frequency and the preloading. However both FEM simulation and experimental studies require a lot of time and effort. It is planned to propose a comprehensive analytical model using shake-down analysis [19] conducive to numerical programming that would require less computing resources to study this phenomenon. References [1] Hess, D. P., Vibration- and Shock- Induced Loosening, in Handbook of Bolts and Bolted Joints, Edited by J. H. Bickford and S. Nassar, Marcel Dekker, New York, [2] Ricci, L, Nguyen, V.H., Sab, K., Duhamel, D., Schmitt, L., Dynamic behaviour of ballasted railway tracks: A discrete/continuous approach, Computers & Structures, 82, , [3] Sauer, J. A., Lemmon, D. C., Lynn, E. K., Bolts: How to Prevent Their Loosening, Machine Design, 22, , [4] Goodier, R.J. Sweeney, R. J., Loosening by Vibration of Threaded Fastenings, Mechanical Engineering, 67, , [5] Kerley, J. J. An Application of Retroduction to Analyzing and Testing the Backing Off of Nuts and Bolts During Dynamic Loading, NASA Technical Memorandum 4001, Goddard Space Flight Center, Maryland, [6] Baubles, R. C., Mccormick, G. J., Faroni, C. C., Loosening of Fasteners by Vibration, Report No. ER , Elastic Stop Nut Corporation of America (ESNA), Union, New Jersey, [7] Koga, K., Loosening by Repeated Impact of Threaded Fastenings, Bulletin of the JSME, 13, , [8] JUNKER, G. H., New Criteria for Self Loosening of Fasteners Under Vibration, Society of Automotive Engineers Transactions, 78, , [9] Hess, D., Sudhirkashyap, S. V., Dynamic Analysis of Threaded Fasteners Subjected to Axial Vibration, Journal of Sound and Vibration, 193 (5), , [10] Hess, D. Sudhirkashyap, S. V., Dynamic Loosening and Tightening of a Single-Bolt Assembly, ASME Journal of Vibration and Acoustics, 119, , [11] Dong, Y., Hess, D., The Effect of Thread Dimensional Conformance on Vibration-Induced Loosening, Journal of Vibration and Acoustics, 121, , [12] Ramsey, G. E., Jenkins, R. C., Experimental analysis of thread movement in bolted connections due to vibrations, Final report for NASA research project NAS , Marshall Space Flight Center, Alabana, [13] Daadbin, A., Chow, Y. M., Theoretical Model to Study Thread Loosening, Mech. Mach. Theory, 27, 69-74, [14] Vinogradov, O., Huang, X., On a High Frequency Mechanism of Self Loosening of Fasteners, Proceedings of the 12 th ASME Conference on Mechanical Vibration and Noise, Montreal, Quebec, , [15] Holland, M., Tran, D., Two-dimensional Modelling of Interaction between Threaded Surfaces due to Dynamic Loading, Proceedings of the Fourth International Conference on Modelling and Simulation, Melbourne, Australia, , [16] Pai, N. G., Hess, D., Three-dimensional finite element analysis of threaded fastener loosening due to dynamic shear load, Engineering Failure Analysis, 9, , [17] Green, W.G., Theory of Machines, Blackie & Son Limited, London, [18] Shigley, J. E., Mischke, C. R., Mechanical Engineering Design, McGraw-Hill, London, [19] Nguyen, Q. S., Instability and Friction, Comptes Rendus Mecanique, 331, , 2003