AN EVALUATION OF A SIMPLE DYNAMICAL MODEL FOR IMPACTS BETWEEN RIGID OBJECTS

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1 XIX IMEKO World Congress Fundamental and Alied Metrology Setember 6, 009, Lisbon, Portugal AN EVALUATION OF A SIMPLE DYNAMICAL MODEL FOR IMPACTS BETWEEN RIGID OBJECTS Erik Molino Minero Re, Mariano Lóez, Antoni Mánuel, Alfonso Carlosena, Xavier Roset Polytechnic University of Catalonia (SARTI), Vilanova i la Geltrú, Sain, molino@eel.uc.edu Public University of Navarre, Pamlona (Navarra), Sain, carlosen@unavarra.es Abstract The main urose of this work is to roose a dynamical model for simulating the resonse of different metallic objects when imacted by another rigid body. In addition, a methodology for estimating the model arameters is resented and discussed. Results from real exeriments shows that by assuming certain characteristics on imacting objects, the dynamic model can reroduce the transient dynamics during contact time. Keywords: Imacts, dynamic model, arameters estimation. INTRODUCTION There are different works among literature that consider the study of collisions between bodies and roose different kind of models for analysing imacts [-8]. Most of these investigations concentrate on estimating the ost-imact conditions of the objects, or monitoring comosite materials, rather than analyzing the behaviour of objects during the imact eriod. Breckenridge [] resents an exerimental study of different transient sources and their characteristic resonses. Wildes [] roose a dynamical model to study the characteristics of materials by hearing the sound roduced by an imact, but does not consider the initial transient, and concentrates only on the steady state resonse. On [3], an interesting study on the dynamics of a onedegree-of-freedom system subjected to an imact is resented, focused from the oint of view of control, for comensating the forces released during collision. In another kind of works, [4]-[5] resent a different aroach, they use collisions for evaluating the health-state of comosite materials based on the coin-ta test method. [6]-[7] are interesting references for modelling and studying the effects of imacts on structures. The motivation for develoing this work arise from a revious investigation [9] where the authors resented the design of an inverse filter based on Neural Networks and a transient excitations. On that work, a model of imacts dynamics was required for training the network, and its imrovement was left as a future develoment. In this aer, we roose a mechanical model that describes the dynamics of two rigid objects during a collision. On Section, we begin by describing the model, roosing a methodology for measuring the arameters, and exlaining the characteristics of the imacting device that has been used for the exeriments. On Section 3, we rovide details of the exerimental setu, and on final sections we resent results and conclusions.. MODEL DESCRIPTION AND METHODOLOGY The mechanical model we are roosing describes the collision between two rigid bodies, as shown in Fig., where object is a sensorised hammer that acts as the imactor, and object is the samle that receives the imact, which for exerimental uroses consist on a set of cylinders of different sizes and materials. During imact, there is a small eriod of time, τ P, where both objects are in contact and some amount of energy transfers from one body to the other. Modeling the dynamics of the samle during this eriod is not easy, mainly because the collision deends largely on three factors: the geometrical characteristics of the objects; the tye of imact (direct imact, lateral, etc); and the velocities involved during the collision [3]. In order to simlify the statement of the model and equations, some assumtions are considered: Collisions are between very hard materials that have regular geometries, as cylinders. Objects will have a quasistatic behavior during imact: this means that all stress releases instantaneously to both objects. Velocities and forces during imact are low enough that they will roduce an elastic imact: lastic deformations are negligible and kinetic energy is conserved. The imact develos in line with the center of mass of each body, and the movement of each object can be described by the dislacement of their centre of mass. These assumtions ermit considering each object as a sring-mass-damer system, as shown in Fig., where it is also ossible to observe that only during imact both objects are in contact and act as a second-order sring-mass-damer system, while rior and after imact they behave as individual first-order single-degree-of-freedom systems. Equations () and () describe this rocess, where the first one illustrate the collision during contact time, τ P, and the second secify the dynamics of the system once objects are no longer in contact. Initial conditions at t = 0, (3), consider that object is at rest, while object swings an angle Δθ, during a time Δt, where, ω is the average angular-velocity of

2 the imactor, and L is the length of the suorting holder, as shown in Fig.. m&& x = c ( ) k( x x ) 0 t τ () m&& x = F( t) + c( ) + k( x x ) c kx m&& x = c kx m && x = c k x t > τ () x i = 0, v i = ω L (3) t = 0 xi = 0, vi = 0 Parameters, c,, and, k,, are the daming and stiffness of each body resectively, and are materials constants that deend directly on the tye of material and geometrical roerties of the imactor and the samle [7], while m, are the masses of each body. An imortant variable in () is the imacting force, F(t). Because imacts are transients that release a large amount of energy in a short eriod of time, this force is modeled as a single ulse of very short duration, that last only the time that both objects are in contact, τ P. The duration and the shae of this ulse deends on the tye of deformations roduced by the collision, which are strongly relate to the seed and force level during imact, the tye of materials M M Electromagnetic holder Δt = 07 ms Δθ=3.7 Fig.. Imacting device. f(t) K C M K C V i K C X K C K C K C X M L = 65 mm Imactor: M Samle: M Sensor M M M IMPACT Contact time (τ P ) Fig.. Mechanical model of imact rocess. (soft, hard), the geometry of bodies, and the location of the imact [8]. In order to describe the force F(t), the imact has been divided in two arts, being the first the comression stage and the second the restitution stage; each having a characteristic shae that deends on how materials react during imact. In the case of an elastic collision, materials resond as linear srings, comressing and recovering their initial shae without deformations, with a symmetric ulse, similar to a versed-sine ulse [7], as in (4). F π t π t f ( t) = ( cos ) = F 0 sin, τ τ f ( t) = 0, t > τ 0 t τ 0 (4) The eriod τ P is defined in (5), where m eq and k eq are the relative mass [3] (6a), and relative stiffness [7] (6b), of the second-order system. m m m eq m + m m τ = π k =, eq eq k k k eq k + k (5) = (6a, b). Proosed methodology for measuring model arameters In order to calculate the arameters of the model roosed in () and (), and to comare it with real data, the imacting device shown in Fig. was built and tested using cylindrical samles of different materials and dimensions. The device consists on a sensorized imact hammer laced as a endulum and embraced in a way that movement is ossible only in one direction. The hammer is the imactor, and the end that gets in contact with the samles has an aluminium ti located inline with the center of mass of the hammer and the samles. The hammer starts its swing from an angle Δθ from the vertical; to assure that the angle and the imacting velocities are constant on all the exeriments, an electromagnetic holder suorts the hammer on a fixed osition, as shown in Fig.. Samles are located over hard foam, in a osition that allows a direct imact to one of the flat surfaces of the cylindrical samles. To measure the resonse of the samles, another sensor, a iezoelectric accelerometer, is attached at the centre of the other flat surface, inline with the imact. A. Measuring the ulse-duration: τ P From the oint of view of an elastic collision and from (5), it is clear that the ulse-duration is a arameter related to the mass and the stiffness of the imacting bodies. This means that for every collision, between the imactor and each cylindrical samle, there will be a ulse of a articular length. In order to measure these ulses, exeriments are carried out (one for every samle): each samle is imacted a certain number of times with the hammer, and then the hammer and samles mean time-resonses are calculated. The measurement of the ulse-duration, τ P, can be a difficult task because it is not easy to locate the actual starting and ending oints on these tyes of ulses. As an alternative, it is convenient to use the hammers resonse as the reference signal, due its resemblance to a versed-sine like in (4), and to measure its average ulse-width, τ av, by

3 simly measuring the ulse width at the half-amlitude of the mean time-resonse, and then calculate τ P alying: τ P = τ av [7]. B. Measuring stiffness k and k The next ste is to calculate the arameter k (hammer s stiffness) by means of a control exeriment. This requires that both bodies, the imactor and the samle, have similar characteristics: as to be made from the same material and to have similar dimensions and masses, m Hammer m Samle. Considering this, we can assume that both objects have similar stiffness, k = k k, and from (6b) it is easy to observe that in this case, k = k eq. Once k has been determinated, the value of k for each cylinder is calculated alying (5) and (6a-b). C. Measuring daming constants c and c The daming is a arameter that has a larger effect on the system resonse once the imact excitation has finished and the system enters in a steady state. It is related to the amlitude decay of the oscillating resonse of the system. Its measurement is not always easy, articularly when daming ratio is equal or larger than, ξ. For cases where the system is under-damed, ξ, the ratio ξ can be estimated by measuring the exonential decay of the imulse resonse of the system, using (7a) and (7b), where U n and U n+ are consecutive eak values of the oscillating signal. U Δ = ln U n, ξ = Δ n+ 4π ξ = c k m Δ (7a, b) We roose to measure the exonential decay, Δ, using U as the maximum eak-acceleration roduced by the imact, and use U as the next eak found in the steady state oscillation. The relation between ξ and c is described in (8), where k and m are the articular arameters for each samle. (8) bronze, and brass; in three sizes: all cylinders have a diameter of, d c = 30 mm, and the three lengths are: L = 0 mm, L = 30 mm, and L 3 = 50 mm, designated as the small, medium and large samles, resectively. Table, shows the mass value of all samles. Each cylinder has in one of their flat surfaces a small holder for suorting a iezoelectric accelerometer (8309 from Brüel & Kjær) for recording the imacting time-resonse. Also, the imacting hammer ( from Brüel & Kjær) has an internal accelerometer that is used for measuring its time-resonse. Signals from both sensors were conditioned and then acquired with a -bit, National Instruments NI-PCI-65, digitizer board. The samling rate has been set to.5 MHz, and the recording time to ms. Signals were filtered at 50 khz, digitally, to reduce noise and high frequency comonents. For the exeriments, each samle was located in horizontal osition on hard foam, as shown in Fig.. Using the electromagnetic holder for the hammer, each swing was reroduced under similar conditions. The hammer swings an angle of Δθ = 3.7, in, Δt =07 ms, which gives an angular velocity of ω=0.3 Rad/s. The length of the hammer, from the centre of mass to the suorting holder, is L = 65 mm, which leads to a linear velocity of v i = 8.9 cm/s. This seed is constant for all imacts. Table. Samles weight in [gr] Without accelerometer * Aluminium Steel Bronze Brass Hammer effective seismic mass: m = 00 gr * Accelerometer (8309) mass: 3 gr 3. EXPERIMENTAL SET-UP The samles used for testing the model are cylinders made from four different materials: aluminium, steel, Fig. 3. Pulse-width Vs Relative mass. Fig. 4. Maximum amlitude Vs Pulse-width. Average values. Samles sizes: small (squares), medium (triangles), large (stars).

4 To measure the hammer stiffness k, the large aluminium cylinder has been selected as the reference samle for the control exeriment. From Table, it is ossible to observe that both masses are similar, m Hammer m L-alum ; and that hammer s ti and cylinder are made from the same tye of material (aluminium), and both have similar dimensions. 4. RESULTS The model described in () and (), and all the equations used for calculating the arameters, were imlemented using MATLAB. MSE % MSE % 0.30% 0.5% 0.0% 0.5% 0.0% 0.05% 0.00% 3.00%.50%.00%.50%.00% 0.50% 0.00% Fig. 5. Model and real cylinders resonses. a) MSE - Hammer resonse b) MSE- Cylinders resonse Aluminium St eel Bronze Brass Aluminium St eel Bronze Brass Small Medium Large Small Medium Large Samles Size Fig. 6. Mean Square Error (MSE) between model and real data: a) for hammer resonse, b) for cylinders resonses. Model arameters were estimated following the rocedure described in Section and 3. For every samle, 0 imacts were roduced and average time-resonses were calculated for the hammer and the cylinders. At the end of the exeriments, a grou of 4 mean time-resonses are available for analysis ( from the hammer, and from the cylindrical samles). For measuring the ulse-duration, the average signals of the hammer were used, and results are shown in Table. As it can be observed, there is a relationshi between the ulsewidth and the mass of the samles. For every material, small samles have the shortest ulse-duration. This relation can be areciated as well in Fig. 3, which shows the lot ulse-width Vs relative-mass, where imacts from all exeriments are shown. It is ossible to areciate the variability of the ulse-width among different exeriments considering individual samles. Also, it is clear that there is a relationshi among ulse-duration, tye of material and the relative mass. Table. Samles ulse-duration, τ P [μs] Aluminium Steel Bronze Brass Table 3 shows hammer and samles stiffness arameter values. Here, it is also ossible to areciate that stiffness changes according to the size of the samles. Another interesting lot is the maximum amlitude Vs ulsewidth, shown in Fig. 4, where average values are shown, and again, it is ossible to areciate the relation between the acceleration level, the mass of the samles and the ulsewidth. Table 3. Samles Stiffness k, [MN/m] Aluminium Steel Bronze Brass Hammer stiffness: k =.54 [MN/m] Table 4 shows the values of the daming constants, and hammer estimated daming value. These were obtained using the logarithmic decay method assuming that all samle have and under-damed resonse. Fig. 5 shows a comarison between the model simulated resonse and some real resonses. For clarity, only three

5 cases are shown, those corresonding to the small brasscylinder (To), the medium steel-cylinder (Middle), and the large aluminium-cylinder (Bottom). Signals from the model and real exeriments were normalized and shifted in time for comarison uroses. Solid lines corresond to the model resonse and the dashed lines are the real resonses. Table 4. Samles daming: c, [N s/m] Aluminium Steel Bronze Brass Hammer: c = 909 [N s/m] The mean square error (MSE) was calculated to comare these results. Fig. 6a show the MSE, in ercentage, between simulated and real hammer resonse, for every samle, and Fig. 6b shows the MSE between the simulated and the real cylinders resonse. Fig. 6a shows that the roosed versedsine ulse describes roerly the hammer s resonse, where the maximum errors corresond to the small samles, robably due to asymmetries in the ulse. In all cases, MSE < 0.5%, which is a reasonable good result considering that the majority have similar MSE. Fig. 6b shows that the errors related to cylinders resonses are a bit larger, being the worst (~.8%) for the large bronze cylinder. All other errors are below %, which is also a reasonable good result, considering as well that the majority have similar MSEs. REFERENCES [] F. R. Breckenridge, T. M. Proctor, N. N. Hsu, S. E. Fick and D. G. Eitzen, "Transient sources for acoustic emission work," in 990, [] R. P. Wildes and W. A. Richards, "Recovering materials roerties from sound," in Natural Comutation W. Richards, Ed. Cambridge, MA: MIT Press, 988, [3] A. Tornambé, "Modelling and controlling one-degree-offreedom imacts," IEE Proc. Control Theory Al., vol. 43, , 996. [4] S. Baglio and N. Savalli, "Fuzzy ta-testing sensors for material health-state characterization," IEEE Transactions on Instrumentation and Measurement, vol. 55, , 006. [5] F. Tong, S. K. Tso and M. Y. Y. Hung, "Imact-acousticsbased health monitoring of tile-wall bonding integrity using rincial comonent analysis," Journal of Sound and Vibration, vol. 94, , 6/ [6] A. K. Chora, Dynamics of Structures: Theory and Alications to Earthquake Engineering. Prentice Hall, 995, [7] C. M. Harris and A. G. Piersol, Harris' Shock and Vibration Handbook.,5th ed.mcgraw-hill, 00, [8] R. M. Brach, Mechanical Imact Dynamics, Rigid Body Collisions. John Wiley and Sons, 99, [9] E. Molino, M. Loez, J. del Río and A. Manuel, "Inverse filtering aroximation for imacting signals estimation using a multilayer neural network," Conference of the IEEE Industrial Electronics Society (IECON) 006, CONCLUSIONS This aer rooses a method for modelling the transient of an imact between two rigid bodies and for estimating the arameters of this model. An exerimental imacting device has been designed for testing different samles, which are cylinders made of aluminium, steel, bronze and brass; in three sizes, which allows testing the rocedures with samles of different characteristics. To roduce reeatable imacts, a electromagnetic holder was used to release an imacting hammer always from the same osition. Contact time, stiffness and daming arameters of each samle were calculated using the roosed method. These values were used in the roosed model to generate simulated results. The mean square error between the simulated resonses and the real data shows that the model has a reasonably good behaviour during the transient of an imact. The MSE is less than 5% for samles simulated resonse, and less than 0.5% for the hammer simulated resonse. ACKNOWLEDGMENTS We acknowledge the financial suort of the Sanish Ministerio de Educación y Ciencia, under grant DIP C0-0; the Mexican Consejo Nacional de Ciencia y Tecnología ; and el suort del Comissionat er a Universitats i Recerca del Deartament d Innovació, Universitats i Emresa de la Generalitat de Catalunya i del Fons Social Euroeu.