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Planetary vertical concrete mixers: Simulation and predicting useful life in steady states and in perturbed conditions M.C. Valigi a,*, I. Gasperinib a Department of Industrial Engineering (DIIN), University of Perugia, Via G. Duranti, 63 06125 Perugia, Italy b SICOMA Societa Italiana Costruzione Macchine, Via Brenta, 16 06078 Ponte Valleceppi (PG), Italy Received 9 March 2006; received in revised form 30 May 2007; accepted 23 July 2007 Available online 25 August 2007 Abstract A simulation environment for the dynamics of planetary concrete mixers has been developed, it is a model based method using the lumped parameters analysis and integrating the theory of classical mechanics and life analysis. The aim of this work is to give a fast and easy to use tool capable of predicting the behaviour and the useful life of concrete mixers through geometrical and physical parameters. Simulations were conducted in steady states and in per- turbed conditions. The gear reduction unit is the part of the mixer under most stress. Results were compared with data obtained from costumers on building sites. ? 2007 Elsevier B.V. All rights reserved. Keywords: Simulation; Lumped parameter model; Gear reduction unit; Concrete mixer; Useful life 1. Introduction The two main categories of mixers are: batch mixers and continuous mixers. The fi rst type produces con- crete one batch at a time, while the second produces concrete at a constant rate. The fi rst type needs to be emptied completely after each mixing cycle, cleaned and then reloaded with the raw materials for the next batch. In the second type, as the name indicates, the raw materials are continuously loaded at one end as the fresh concrete exits the other end. Batch mixers are the most common type. Diff erent types of batch mixers can be distinguished by looking at the orientation of the axis rotation: horizontal or inclined (drum mixers) or vertical (pan mixes) 1. Long term usage of a mixer leads to wear on the blades and/or scraper, the build-up of materials (hardened mortar or cement paste) on the blades, the container, and/or the scraper. To avoid this situation, the concrete mixer should be thoroughly cleaned at the end of each day of opera- tion and the blades and scraper should be changed on a regular basis. 1569-190X/$ - see front matter ? 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2007.07.010 * Corresponding author. Tel.: +390755753724. E-mail address: mc.valigiunipg.it (M.C. Valigi). Available online at Simulation Modelling Practice and Theory 15 (2007) 12111223 /locate/simpat The mixing process also includes the mixing energy that is the energy needed to mix a concrete batch; it is determined by the product of the power consumed during cycle and the duration of cycle 1. The mixer presented in this paper is a batch mixer with a vertical axis. The working rotation motion is plan- etary and its aim is to physically and chemically mix multiple components in order to make a homogeneous mixture. Actually the mixer distributes all the constituents uniformly in the tank without favouring one or the other. The power of the mixer is 10 Hp and the duration of mixing time (cycle) is 70 s. In this paper, a description of the fundamental components of the machine is given followed by the devel- opment of a model based method that focuses on the most stressed part of the mixer: the gear reduction unit. On the basis of this model a simulation environment was developed with the aim of evaluating the useful life of a mixer. In addiction a test rig, reproducing working conditions was built to make experimental simulations using destructive tests to evaluate the useful life of mixers. Experimental tests are expensive and accidents are almost impossible to reproduce. Therefore, the early detection, diagnosis, and prognosis of gear-tooth fatigue cracks has always been one of the major technical challenges for safe and cost-eff ective operation of concrete mixers. The combination of a test rig and a numerical simulation environment represents a powerful instrument during the design process. The simulation tool should be fl exible, easy to use and inexpensive in order to constantly improve the design of machines. In addiction, the behaviour of the mixer under diff erent working conditions, introducing suitable parameters regarding the concrete mix could be simulated. Given the geometrical and physical param- eters of mixers, the tool has to be able to predict which gears are more stressed and supply information about their life span. 2. Concrete mixer: description The concrete mixer analysed in this paper (Fig. 1) has a vertical axis, three rotating mixing arms and a fi xed tank. The machine is made up of two principal elements: the planetary mixers; and loading bucket (skip). Fig. 1. Concrete mixer. 1212M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 12111223 The loading bucket is an individual and separate circuit. The shape of the bucket is a truncated pyramid, ideal for containing the raw material. A safety device prevents the bucket from falling. The loading bucket can be fi xed to the mixer, as a pre-loading storage unit, or it can have a balance attached for batching and weigh- ing the aggregates. Discharge is controlled by a oleodynamic or pneumatic cylinder. The mixing tank (Fig. 2) is constructed in extremely thick sheet steel mounted on a channel section frame arranged in such a way as to allow for several discharge openings. The entire tank is protected by a casing to prevent dust escaping and the mixing operation will be interrupted by a micro-switch if the door at the front is opened. The mixing arms (Fig. 3) are peripheral and constructed with steel drill rods and can be adjusted to allow regulation of the blades. A box in spheroidal graphite cast iron containing three gear pairs (Fig. 4) and a bridge frame is mounted on the top of the tank so that it is suspended at a suitable height to support the arms and blades. One shaft of the gear reduction unit is attached to the motor and another is moving the epicycloi- dal step-up gearing that moves the arms and generates the mixing. The fi rst and third gear pair are helicoidal and the second is a straight-tooth gear. This is because the second pair is under less stress. The gears work in an oil bath so that the best lubrication is guaranteed on the tooth contour. 3. Main features of the simulation process One of the most common aspects of concrete mixer design regards the gear reduction unit, because many major faults can be classifi ed into shaft and tooth-fatigue cracks due to cyclic loading in diff erent process phases 2,3. For this reason, the work presents a simulation instrument which is useful for analysing the behaviour of the gear reduction unit and for predicting its useful life. There are many conceptually diff erent ways in which the gear reduction unit could be modelled. The problem was dealt with using various methods: FEM for predicting structural stress. Lumped parameters model in order to simulate dynamic forces acting on the system. Heuristic approach for simulating the mechanical eff ect of the mixing process. The above mentioned methods are necessary in designing the gear unit; this paper focuses on the latter two. Fig. 2. Mixing tank. M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 121112231213 4. Mechanical modelling of the gear box The simulation instrument has to be practical and easy to apply due to industrial demands. A rotational lumped parameter modelling was chosen for the main components of the mixer that need to be analysed. Therefore, the motor characteristics were supplied and the kinematic parameters, the inertial parameters, the stiff ness and damping parameters were evaluated using geometrical and material properties. The gear reduction system can be modelled by considering the gears as lumped masses connected by fl exible mass-less shafts. The confi guration of the model is shown in Fig. 5. with three degrees-of-freedom (the angular displace- ment of three stages) describing the vibrations of the gear system 4. The analysis of the system is greatly facil- itated by assuming a linear hypothesis. In the model a rigid contact between teeth is hypothesized and each gear is reduced to its pinion. According to the classical mechanical approach, the dynamic equations of motions were written using the Lagrangian formulation. The equations could also be applied for a damping system where viscous damping and external forces are considered as nonconservative. Fig. 3. Mixing arms. Fig. 4. Gear reduction unit. 1214M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 12111223 The Lagrangian function L is L T ? U where the kinetic energy T of the system is T 1 2 J1 s2 12J2 ? _ #2 1 1 2 J3 s2 34J4 ? _ #2 2 1 2 J5 s2 56J6 ? _ #2 3 and the potential energy U is U ? 1 2 K0#2 1? 1 2 K1#2? #1s12?2? 1 2 K2#3? #2s34?2 So that L ? 1 2 K0#2 1? 1 2 K1#2? #1s12?2? 1 2 K2#3? #2s34?2 The virtual work dW for non-conservative forces is dW ?C1s12_#2? _ #1s12 ? Co_#1 ?d# 1 ?C1 _ #2? _ #1s12 C2_#3? _ #2s34s34 ?d# 2? C2 _ #3? _ #2s34 ?d# 3 Mmd#1 Mrs56d#3 where #i(i = 1,2,3) are the angular positions that represent generalised co-ordinates set related to an equiva- lent system for three degrees-of-freedom, sijare gear ratios, Jiare reduced moments of inertia of the gear sys- tem including the inertia properties of the shafts, Ki and Cj are the reduced stiff ness and damping coeffi cients, respectively. Mmand Mrare the motor torque and resisting torque, respectively 5. Using the Lagrange formulation d dt oL o_#i ? ? oL o#i Qi where Qi dW d#i Q1 C1s12_#2? _ #1s12 ? Co_#1 Mm Q2 ?C1_#2? _ #1s12? C2_#3? _ #2s34?s34 Q3 ?C2_#3? _ #2s34? Mrs56 8 : Fig. 5. Lumped parameter model of the gear reduction unit. M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 121112231215 So that J1 s2 12J2 ? 00 0J3 s2 34J4 ? 0 00J5 s2 56J6 ? 2 6 4 3 7 5 #1 #2 #3 8 : 9 = ; Co C1s2 12 ?C1s120 ?C1s12C1 C2s2 34 ?C2s34 0?C2s34C2 2 4 3 5 _ #1 _ #2 _ #3 8 : 9 = ; K0 K1s2 12 ?K1s120 ?K1s12K1 K2s2 34 ?K2s34 0?K2s34K2 2 4 3 5 #1 #2 #3 8 : 9 = ; Mm 0 Mrs56 8 : 9 = ;: 4.1. Motor torque and resisting torque Motor and resisting torque were modelled as input for the environmental simulation. The model regards steady states and perturbed conditions and the related simulations are useful for evaluating the useful life of a mixer. The mixer is moved by synchronous motor with four poles. The motor torque is linearized in its working zone as Mm Ms? b ? _ #1: Table 1 Mixing schedule Time (s)Components/actionAbsorption % motor torque 05Idling20% 520Aggregate30% 2040Cement50% 4050Water80% 5070Mixing period8090% 7080Discharge periodGradual reduction until 20% Fig. 6. Ideal mixing at four diff erent points in the cycle. 1216M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 12111223 In order to model the resisting torque, the dead load components of the mixer were included. The instan- taneous entry of each component is hypothesized 1,6 for a standard cycle that has a time of mixing equal to 70 s (excluding discharge time). The ingredients are loaded and mixed according to the following sequence (Table 1): at the beginning the mixer works loadless but then the aggregate (in increasingly smaller sizes), the concrete and fi nally the water are introduced. In Fig. 6, the ideal kind of mixing at four diff erent points in the cycle is simulated and it is possible to see that the arms reach every area of the tank bottom. A test rig with a mechanical disc brake (Fig. 7) and six pliers driven by an oleodynamic circuit was built to simulate Fig. 8. Modelling of resisting toque estimated by ammeter clamps. Fig. 7. Test rig: mechanical disk brake. M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 121112231217 resisting torque. The planetary reduction unit assembled with its motor is placed on the copes of the frame. When motor reaches a steady state, the step-down gearing and its shafts will rotate. The action of the six pliers, manually controlled and measured by an oil pressure gauge which is situated at the back, simulates the resisting torque. Brake tests are destructive and randomly conducted therefore, a model-based simulation was set-up in order to reduce the number of experimental tests required and to eval- uate the useful life of the planetary reduction unit. The values of resisting torque were estimated by ammeter clamps during a mixing cycle (Fig. 8) and expressed as a percentage of regime motor torque that is 103 KN. In the resisting torque model also took into consideration the infl uence of damping force due to the mixture according to Mrt Mr? t0 6 t 6 40 Mr? t Cc _ #340 ? t 6 100 ? : The damping eff ect is increased when water is introduced to the mixture, in the fi rst 40 s. there are only dry ingredients. 5. Heuristic approach for simulating the mechanical eff ect of the mixing process The resisting torque is also modelled in perturbed conditions and obtained considering a situation in which the tank is not suffi ciently cleaned at the end of the working day so that a number of concrete clots are on the bottom of the tank. Fig. 9. Comparison between perturbed and steady-state torque. Fig. 10. Procedure for simulation. 1218M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 12111223 In this case, the model of the mixing cycle was evaluated using the following additional hypothesis: (i) just one clot accumulates in the bottom of the tank; (ii) the mixer has three mixing arms and consequently three contacts are made with the clot in each turn; and (iii) progressive elimination of the clot following an almost linear law so that the resistance torque perturbed by the clot is stepped. The diagram of perturbed torque is shown in Fig. 9 (in light gray) along with the steady-state torque (the diff erence regards the fi rst 10 min of the mixing cycle). Fig. 11. Simulating scheme. Fig. 12. Torque load acting on the second shaft. M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 121112231219 6. Simulation and results The proposed instrument must be inexpensive, fl exible and user friendly in order for it to be adopted by fi rms. The physical laws and relationships that described the system were exploited to obtain the model structure using the following three equations of motion in matrix form 5 Jrid?f#g Crid?f_#g Krid?f#g fMg Fig. 13. Oscillation of torque acting on the third shaft. Fig. 14. Stress of pitting on the surface teeth. 1220M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 12111223 where # is the angular displacement vector of the three pinions; JridCridKrid are reduced inertial matrix to pinions, reduced damping factor including the damping of material and lubricant oil and reduced stiff ness ma- trix, respectively. Elements of vector M are motor torque (asynchronous motor with four poles) at its regime value (103 km) and resisting torque (due to concrete mixing and applied to the gear unit by the arms) shown in Figs. 8 and 9 as percentage of motor torque. The mathematical model is implemented through a software in such a way as to simulate the behavior of mixer in standard states and in perturbed conditions. The procedure for simulation is outlined in Fig. 10. Parameters including inertias, damping and stiff ness regarding the gear reduction unit of concrete mixers are elaborated by a commercial code (Matlab + Simulink) that was used to create this instrument for simu- lating the dynamic behaviour of concrete mixers. Fig. 15. Angular pinion speed of the fi rst pair in steady states (in light gray) and in perturbed conditions (in black). Fig. 16. Angular pinion speed of the second pair in steady states (in light gray) and in perturbed conditions (in black). M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 121112231221 Fig. 11 shows the simulating scheme: motor and resisting torque are system input. Numerical. The output are: angular displacement of pinions and their speeds, torsional torque on shafts, fl exional ten- sion, gear teeth contact forces, load on bearings, torsional stress, stress of fl exure acting on teeth due to pitting. Fig. 12 shows the torque load acting on the second shaft, Fig. 13 shows the oscillatory peak of torsional stress, this acts on the third shaft and is the maximum torsional stress reached. Fig. 14 shows the stress of pitting on the surface teeth. Simulations have shown that the third gear is under more stress than the others and therefore, as confi rmed by experience, the failure of the gear reduction unit is almost always caused by a fault in the third pair. Simulations were also carried out in perturbed mixing conditions and results were com- pared to simulations in steady-state mixing conditions. The results are shown in the following fi gures. Fig. 15 shows the angular pinion speed of the fi rst pair, Fig. 16 shown the angular pinion speed of the second pair and Fig. 17. Angular pinion speed of the third pair in steady states (in light gray) and in perturbed conditions (in black). Fig. 18. Stress of pitting on the surface teeth of the third pair in steady states (in light gray) and in perturbed conditions (in black). 1222M.C. Valigi, I. Gasperini / Simulation Modelling Practice and Theory 15 (2007) 12111223 Fig. 17 shows the angular pinion speed of the last pair. In Fig. 18, the stress of pitting on the surface teeth of the third pair can be seen. Transient oscillation occurs on each impact with the concrete clot and when the ingredients are introduced tension values are more or less the same as the