Sommerfeld effect in a vibrocompaction process

International Journal of Mechanical Sciences 254 (2023) 108407 Available online 2 May 2023 0020-7403/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci Sommerfeld effect in a vibrocompaction process Javier González-Carbajal∗, Daniel García-Vallejo, Jaime Domínguez Department of Mechanical Engineering and Manufacturing, University of Seville, Spain ARTICLE INFO Keywords: Vibrocompaction Sommerfeld effect Nonideal excitation Granular materials ABSTRACT This study addresses the influence of the Sommerfeld effect on the behaviour of vibrocompacting machines driven by unbalanced electric motors. Based on a simplified model with 4 degrees of freedom previously published by the authors, we investigate the dynamics of quasistatic vibrocompaction processes, i.e. those in which the compaction is achieved through a sufficiently slow increase in the input power of the electric motors. The main novel contribution of the paper is the characterization of the evolution of the system state through a quasistatic torque curve, which is analytically derived. This curve provides the maximum compaction that can be quasistatically achieved and the amount of motor torque required to reach a desired compaction level, while also revealing the specific way in which the Sommerfeld effect takes place in the system under study. The analytical results are used to give a clear physical interpretation to the numerical results previously published by the authors. Good agreement is found between numerical and analytical results. 1. Introduction The need to compact granular materials is often encountered in different engineering fields, such as geotechnics [1], aerospace engineering [2] and manufacturing [3–5]. Vibrating machines constitute an efficient and widely used means to meet this objective, by producing a denser packing of the material grains [6,7]. We also note that the propagation of nonlinear elastic waves in granular materials has received considerable scientific attention in the last years [8–10]. Specifically, this work has been motivated by the authors’ interest in the manufacturing of quartz agglomerates, one of whose most crucial stages is the vibrocompaction of a quartz-resin mixture by using unbalanced electric motors, together with a vacuum system [11,12]. The present paper can be seen as an extension of a previous work published by the authors, where a nonlinear model for the vibrocompaction of quartz agglomerates was presented and justified [13]. This model was numerically investigated in [13] by conducting representative simulations that allowed assessing the effect of different machine parameters on the final result of the compaction process. It is worth noting that relevant nonlinear behaviour was observed in these simulation results resembling the typical features of the well-known Sommerfeld effect, which occurs due to reciprocal interaction between a vibrating system and its energy source [14]. However, the physical interpretation of these phenomena was not straightforward due to the increased complexity of the model with respect to those generally used to investigate the Sommerfeld effect. This article intends to provide significant insight into the vibrocompaction dynamics through a detailed analytical investigation of the model. ∗Corresponding author. E-mail addresses: [email protected] (J. González-Carbajal), [email protected] (D. García-Vallejo), [email protected] (J. Domínguez). It is worth stressing the high interest of this type of research from an industry perspective. The ability to better understand and predict the behaviour of vibrocompacting machines would allow to optimize their design and to properly tune their parameters for each specific scenario. On the other hand, the originality of this work is attested by the fact that no previous investigation on the Sommerfeld effect in vibrocompaction systems is available in the literature, to the best of the authors’ knowledge. This introduction is divided into two subsections. In the first one, a description of the industrial vibrocompaction process of interest is presented. The second subsection provides a summary and literature review on the Sommerfeld effect. 1.1. The vibrocompaction of quartz agglomerates Quartz agglomerates are very frequently used as an artificial stone for countertops in kitchens and bathrooms [15,16]. The production of a slab of this material includes the mixing of the quartz grains with a polyester resin, the vacuum vibrocompaction of the mixture, the polimerization of the resin in a kiln and the machining of the slab after cooling [17]. It should be stressed that the compaction step is crucial for the result of the manufacturing process, since it avoids the presence of pores on the surface of the final countertop. These pores tend to accumulate dirt and are particularly difficult to clean. In fact, the detection of a pore of sufficient size during the quality control of the process implies the rejection of the slab. https://doi.org/10.1016/j.ijmecsci.2023.108407 Received 23 November 2022; Received in revised form 12 April 2023; Accepted 23 April 2023 International Journal of Mechanical Sciences 254 (2023) 108407 2 J. González-Carbajal et al. In order to better understand the nature of the compaction, note that the mixture is initially composed of three different phases: solid (the quartz grains), liquid (the resin) and gas (air). The air is present in two ways: as bubbles within the resin and as gaps among the quartz grains that the resin has not been able to fill. The aim of the compaction is to evacuate this air, considerably reducing the thickness of the slab. The vibrocompaction process takes place as follows. First, a heavy piston on which the unbalanced motors are mounted descends onto the mould containing the material to be compacted. The piston, which has the dimensions of the slab, exerts initially a static pressure on the mixture due to its own weight and to an external air pressure applied on it. Then, the air pressure inside the mould is reduced by means of a vacuum system. Finally, the electric motors are switched on, producing the vibration needed to complete the compaction. During this dynamic process, separations and impacts can occur between the piston and the mixture, and also between the mixture and the mould [18]. It is easy to note the high complexity of the physics involved in this process, with the final result of the compaction depending on a large number of factors: granulometry of the quartz, rheological properties of the resin, amount of unbalance, available power of the electric motors, weight of the piston, etc [19,20]. Notice also that there are several sources of nonlinearity in the vibrocompacting system, the most relevant of them being: •The constitutive behaviour of the mixture. The mechanics of granular materials is a very complex subject that has received wide attention in the scientific literature [21–24]. Clearly, any model that intends to represent the mechanical response of a quartz-resin mixture being compacted needs to be nonlinear, in order to account for irreversible deformation. In fact, a material model based on first principles would need to include the motion of the bubbles through the mixture, the friction between quartz particles, the interaction between quartz and resin, etc [25]. Thorough investigations on the behaviour of three-phase mixtures, granular media and porous materials can be found elsewhere [26– 30]. As a powerful alternative to models based on first principles, it is often useful to resort to phenomenological models that can capture the global behaviour of the material, as has been done in [13]. •The possibility of having separations and impacts at the mixturepiston and mixture-mould contacts. •The dynamic interaction itself between the machine vibration and the rotation of the motors. This particular nonlinearity is usually referred to as the Sommerfeld effect, and is briefly described in the next subsection. 1.2. The Sommerfeld effect In the analysis of forced vibrations, it is customary to implicitly assume that the external excitation is a known function of time. This is generally known as the assumption of ideal excitation. However, since the experimental work of Sommerfeld in 1904 [31], it is known that some kinds of behaviour found in mechanical systems cannot be explained upon the ideality hypothesis. In other words, there are situations in which the excitation has to be considered as nonideal, meaning that it is influenced by the motion of the system [32]. Sommerfeld’s findings were based on a setup consisting in an unbalanced electric motor mounted on an elastic support. During the tests, the motor input power was gradually increased/decreased, making the rotor speed pass through the system’s resonance frequency in both directions [33]. The results of a numerical simulation of Sommerfeld’s test are displayed in Fig. 1. The points on the graph represent the obtained stationary states when the motor input power is varied linearly with time in small equal steps (after each step, the motor control is kept constant until a stable motion is attained). The figure shows the interesting phenomenon that certain range of rotor speeds cannot be Fig. 1. Numerical simulation of Sommerfeld’s experiment. The motor input power is varied linearly with time in small equal steps. After each step, the motor control is kept constant until the system reaches a stationary state. These points of stable motion are represented on the graph. The natural frequency of the vibrating system is denoted as 𝜔𝑛. Fig. 2. Graphical explanation for the Sommerfeld Effect according to Kononenko [34]. Curves 𝐿𝑚𝑖 (𝑖= 1,2,3) represent the motor characteristics at different time instants, while 𝐿𝑣corresponds to the torque on the rotor due to vibration. Jump phenomena occur from point 𝐵to 𝐶for increasing input power and from point 𝐷to 𝐸for decreasing input power. reached. The reason is that, at some point in the process, the system experiences a sudden jump, both in rotor speed and oscillation amplitude (note that the jump is different for increasing and decreasing input power). This is the essence of what is usually called the Sommerfeld effect. Years later, Kononenko used averaging techniques to obtain the stationary motions of Sommerfeld’s system as the intersections between two functions: the torque–speed curve of the motor, 𝐿𝑚, and a curve representing the torque on the rotor due to vibration, 𝐿𝑣(see Fig. 2) [34]. Assuming that the motor is controlled by displacing its torque–speed curve parallel to itself, Fig. 2 shows how Kononenko’s representation accounts for Sommerfeld’s experimental results: the state of the system experiences a jump phenomenon when tangency point 𝐵or 𝐷is reached for increasing or decreasing input power, respectively [34–37]. It is interesting to note that the previous description assumes that all solutions between points 𝐴and 𝐵, and between 𝐶and 𝐷, are stable. However, it was proved in [38,39] that some of the solutions between 𝐶and 𝐷can actually be unstable under certain conditions. This means that the jump phenomenon for decreasing input power can occur before predicted in Fig. 2. International Journal of Mechanical Sciences 254 (2023) 108407 3 J. González-Carbajal et al. After the works of Sommerfeld and Kononenko, substantial research effort has been devoted to the study of vibrating systems excited by nonideal power sources, and the particular dynamics that they can exhibit. Sghaier et al. addressed the coupling of bending and torsional vibrations of rotors under nonideal excitation [40]. Varanis et al. [41] used the continuous wavelet transform to study the Sommerfeld effect on an oscillator with fractional damping. Avanço et al. applied bifurcation analysis to control the appearance of chaotic behaviour on a nonideally driven electromechanical pendulum [42]. Palacios Félix et al. demonstrated that nonideal systems can be used for energy harvesting purposes [43]. The mitigation of nonideally excited oscillations by linear vibration absorbers was addressed by Piccirillo [44]. Drozdetskaya et al. investigated the case of an elastically supported electric motor with two coaxial unbalances [45]. Bharti et al. coined the term Sommerfeld effect of second kind [46], referring to a variant of the classical Sommerfeld effect associated with the instability of the rotor whirl, and presented a numerical study of the phenomenon. A methodology to mitigate the Sommerfeld effect by properly tuning the external damping of the system was described by Jha and Dasgupta [47]. Kong et al. analysed the particular problem of a simply supported beam excited by two nonideal unbalanced motors [48]. They also focused on the self-synchronization between both motors during the beam oscillation. The wavelet synchrosqueezed transform was used by Varanis et al. to investigate the dynamics of nonideally excited systems [49]. Bharti et al. performed a numerical investigation on the appearance of the Sommerfeld effect in the torsional vibrations of a double-Cardan joint driveline [50]. Although most analytical works on nonideally excited oscillations used averaging techniques to simplify the equations of motion [33,35, 38,51], the method of direct separation of motions, first introduced by Blekhman [52], has also proved to be a powerful analytical tool for the study of nonideal systems [53,54]. In terms of experimental works, it is worth mentioning some recent investigations by Varanis et al. where different setups—a cantilever beam, a portal frame, a shear-building structure—are used for experimental validation of theoretical results [55,56]. Also relevant are the investigation of the nonideal behaviour of electrodynamic shakers in [57] and the experimental demonstration offered by Kossoski et al. on the mitigation of the Sommerfeld effect by means of shape memory alloys [58]. For a more detailed literature review, the interested reader is referred to [59,60]. As is clear from the above lines, the Sommerfeld effect is a relevant nonlinear phenomenon that can significantly affect the behaviour of such varied engineering systems as beams, pendulums, energy harvesting devices, rotors, Cardan joints, portal frames or shape memory alloys. However, to the best of the authors’ knowledge, there are no previous works in the literature addressing the impact of the Sommerfeld effect on vibrocompacting machines. In this work, the dynamics of vibrocompacting machines driven by unbalanced electric motors will be analysed using a simplified analytical model. This will provide a good understanding of the appearance of the Sommerfeld effect in the system and a useful physical interpretation to the numerical results previously published by the authors [13]. The paper is structured as follows. Section 2presents a simplified mathematical model for vibrocompaction that can be treated analytically. Section 3makes use of this model to study the appearance of the Sommerfeld effect in quasistatic vibrocompaction processes. In Section 4, the obtained analytical results are used to provide a clear physical interpretation to the numerical results previously published by the authors. The good agreement between analytical and numerical results also serves as a validation of the presented analytical developments. Finally, Section 5puts forth the main conclusions of the study. 2. Simplified model and torque–speed curves The present investigation is based on a 4-degree-of-freedom (4- DOF) vibrocompaction model previously published by the authors [13], which is briefly described in Appendix A. Our analysis also builds upon some classical results on the response of linear systems to the nonideal excitation provided by unbalanced motors, which are summarized in Appendix B. Regarding the reasons for choosing a 4-DOF model, some comments are pertinent. It is clear that modelling the dynamics of a vibrocompaction machine driven by an unbalanced motor requires at least 2 DOFs: one for the rotation of the motor and one for the displacement of the piston on which the motor is mounted. However, if the model is intended to consider separations and impacts at the top and at the bottom of the mixture being compacted, two additional DOFs are needed (one for each of the contacts). Hence, 4 is the minimum number of DOFs that allows modelling the system of interest allowing for separations. Of course, more complex and realistic models could be built with additional DOFs. For example, more DOFs could be included to model several vibration modes of the mixture. The philosophy behind this work is to start the analysis of the vibrocompacting system with a low-dimension tractable model that facilitates the interpretation of results. Future steps in this line of research will include considering additional DOFs for more accurate predictions. Even though the 4-DOF vibrocompaction model is, of course, very simplified with respect to real vibrocompacting machines, it is still too complex for an analytical treatment of the equations of motion. Section 2.1 shows that, under reasonable assumptions, the 4-DOF model can be reduced to a more tractable 2-DOF model. Furthermore, it is shown that the results presented in Appendix B can be applied to the 2-DOF vibrocompaction model. Particularly useful is the possibility to obtain the stationary motions of the machine as the intersections between the motor torque curve and the vibration torque curve, as described in Appendix B. Section 2.2 addresses the main particularity of the vibration torque curves for the vibrocompaction system: since the mixture stiffness is not fixed, but increases as the compaction progresses, the system will exhibit a family of infinite curves, one for each possible value of the mixture stiffness. 2.1. Simplified model In order to simplify the model summarized in Appendix A, consider the following assumptions: •The motion of interest occurs without separations between mixture and piston nor between mixture and mould. This implies 𝑦𝑡=𝑦𝑝,𝑦𝑏= 0 in Eq. (A.1), which means that the number of DOFs is reduced to 2 (𝑦𝑝,𝑦𝑡and 𝑦𝑏represent the vertical positions of the piston, the top surface of the mixture and the bottom surface of the mixture, respectively). •Since the simplified model will be used to investigate the stationary motions of the machine, it is reasonable to assume also that the nonlinear spring representing the mixture (see Fig. A.1) is only subjected to elastic deformation. The reason is that, once the system has reached a stationary state of motion, there is no further compaction and the spring force, 𝐹𝑚, follows a straight line in the force–displacement graph, such as segment 𝐴𝐵 in Fig. A.2, with certain stiffness 𝑘𝑖∈ [𝑘0, 𝑘𝑓]. Parameters 𝑘0and 𝑘𝑓 represent the stiffness of the uncompacted and fully compacted mixture, respectively (see Appendix A for more details). With these two assumptions, system (A.1) can be written as 𝑚 𝑥 +𝑏 𝑥 +𝑘𝑖𝑥=𝑚1𝑟( 𝜙2cos 𝜙+ 𝜙sin 𝜙) 𝐼 𝜙=𝐿𝑚( 𝜙) + 𝑚1𝑟sin 𝜙(𝑥 +𝑔)(1) where 𝑚=𝑚𝑝+𝑚𝑚∕3 + 𝑚1and 𝐼=𝐼0+𝑚1𝑟2. Parameter 𝑚𝑝stands for the mass of the piston and the motor, 𝑚𝑚is the mass of the mixture, 𝑚1 International Journal of Mechanical Sciences 254 (2023) 108407 4 J. González-Carbajal et al. is the unbalanced mass, 𝐼0is the rotor inertia, 𝑟is the eccentricity of the unbalance, 𝑏is the damping coefficient, 𝑔is the gravity constant, 𝑥represents the displacement of the piston with respect to its static equilibrium position, 𝜙represents the angular position of the rotor and an overdot represents differentiation with respect to time. 𝐿𝑚( 𝜙)is the driving torque provided by the motor, which is assumed to be a linear function of the rotor speed: 𝐿𝑚( 𝜙) = 𝐶+𝐷 𝜙, (2) We note that the derivation of system (1) simply requires performing the substitutions 𝑦𝑡=𝑦𝑝,𝑦𝑏= 0,𝑦𝑝=𝑥,𝑦𝑝=𝑥 and 𝐹𝑚+𝑚𝑔 =𝑘𝑖𝑥on system (A.1). Then, adding up the first two equations in system (A.1) provides the first equation in system (1), while the fourth equation in system (A.1) becomes the second equation in system (1). The gravity term in Eq. (1) can be shown to have no significant effect for the purposes of this investigation. The reason is that the stationary motions of the system will be obtained using the method of direct separation of motions, proposed by Blekhman [52]. This approach, which has some similarities with classical averaging techniques [61], includes the averaging of the equations of motion in order to find the mean values of rotor speed and vibrating amplitude. Since the gravity term, 𝑚1𝑟𝑔 sin 𝜙, becomes zero when averaged over angle 𝜙, it does not affect the mean stationary rotor speed or oscillation amplitude (see [52,62] for details). Hence, Eq. (1) can be rewritten as 𝑚 𝑥 +𝑏 𝑥 +𝑘𝑖𝑥=𝑚1𝑟( 𝜙2cos 𝜙+ 𝜙sin 𝜙) 𝐼 𝜙=𝐿𝑚( 𝜙) + 𝑚1𝑟 𝑥 sin 𝜙(3) Note that system (3) is exactly the same as system (B.1), which allows applying the general results presented in Appendix B to the vibrocompaction model. In the remainder of the paper, we will refer to systems (A.1) and (3) as the full model and the simplified model, respectively. 2.2. Torque–speed curves for variable stiffness An important outcome of the presented simplification is the fact that the torque–speed curves described in Appendix B can be used to represent the stationary motions of the compacting machine. As a relevant particularity of the simplified vibrocompaction model, note that the stiffness 𝑘𝑖is not a fixed parameter, but can take different values between 𝑘0and 𝑘𝑓, depending on the degree of compaction achieved (Fig. A.2). Then, instead of a single curve for the vibration torque, there will be a family of curves, one of them for each possible value of the mixture stiffness. Although, in principle, the system can exhibit any stiffness between 𝑘0and 𝑘𝑓, the range of possible values for this parameter is actually narrower during the vibrocompaction. The reason lies in the fact that there is already some static compaction before the motor is switched on, due to the weight of the masses located above the spring—mainly the weight of the piston. Using 𝑘𝑠𝑡 to denote the spring stiffness after the static compaction, it is clear that, during the dynamic compaction process, the stiffness can take values 𝑘𝑖∈ [𝑘𝑠𝑡, 𝑘𝑓]. The vibration torque curves for the vibrocompacting machine can be obtained by using Eq. (B.6), which is rewritten here for the case of variable stiffness: 𝐿𝑣(𝑘𝑖, 𝛺0)≡𝑘𝑖𝜉𝑖(𝑚1𝑟 𝑚)2(𝛺0 𝜔𝑛𝑖 )5 [(𝛺0 𝜔𝑛𝑖 )2 − 1]2 +[2𝜉𝑖 𝛺0 𝜔𝑛𝑖 ]2 for 𝑘𝑖∈ [𝑘𝑠𝑡, 𝑘𝑓], (4) with 𝜔𝑛𝑖 =√𝑘𝑖∕𝑚and 𝜉𝑖=𝑏∕(2𝑚𝜔𝑛𝑖). A careful analysis of Eq. (4) reveals that the resonance peak of the curve grows approximately in proportion to 𝑘3∕2 𝑖, while the resonance frequency 𝜔𝑛is proportional to 𝑘1∕2 𝑖. This is reflected in Fig. 3, where several vibration torque Fig. 3. Torque–speed curves for the vibrocompaction process. 𝐿𝑚represents the motor characteristic, while curves 𝐿𝑣(𝑘𝑖)represent the vibration torque, given in Eq. (4), for different possible values of the stiffness of the mixture (this stiffness increases as the mixture gets compacted). The possible stationary motions are marked with dots. curves are plotted for growing values of the spring stiffness. Note in this figure, as will generally be the case for the rest of the article, that the dependence of 𝐿𝑣and 𝐿𝑚on 𝛺0is not explicitly written, for convenience in the notation. The collection of curves in Fig. 3 is a representative set of the whole family of curves for 𝑘𝑖∈ [𝑘𝑠𝑡, 𝑘𝑓]. If the torque–speed curve of the motor is depicted on the same plot, the possible stationary motions of the machine are graphically obtained as the intersections between the motor curve and each one of the vibration curves. As shown in Fig. 3, there are many feasible stationary motions for the system. In fact, the amount of possibilities is infinite if all the curves for continuously increasing stiffness are considered. The next section will address the problem of finding, analytically, which of these infinite possible solutions is actually exhibited by the machine during a compaction process. 3. Analytical investigation of the Sommerfeld effect in a quasistatic vibrocompaction process This section is intended to analyse a particular type of vibrocompaction process, where the control settings of the motor are very slowly varied. Then, it is first convenient to say a word about the motor control. An electric motor of any kind is controlled by means of one or more input magnitudes (current, voltage, frequency...). For example, the speed of a 3-phase induction motor is usually controlled by modifying the amplitude and frequency of the stator voltage through a variable frequency drive. For each setting of the motor control, the motor characteristic is fixed. In the presented model, where the motor curve is assumed to be given by a straight line, the motor control would allow changing parameters 𝐶and 𝐷(see Eq. (2)). Suppose that the effect of the chosen control method consists in displacing the motor characteristic parallel to itself, i.e. changing parameter 𝐶while keeping a fixed slope 𝐷. As mentioned in Section 1.2, this is actually the case in the simplest control approach for an induction motor, known as the V/f control [63]. Now, consider a situation where the motor characteristic is very slowly displaced upwards, occupying at different time instants the positions 𝐿𝑚1,𝐿𝑚2,𝐿𝑚3... as represented in Fig. 4. This change in the motor control is assumed to occur quasistatically, in the sense that it does not produce any transient effects in the machine, but only a succession of stationary states. Let us focus on one particular time instant of the process, say the one at which the motor curve is given by 𝐿𝑚2. As observed in Fig. 4, several stationary solutions are possible in principle. Note that there are actually infinite potential stationary motions for 𝐿𝑚2, since the mixture stiffness, 𝑘𝑖, varies in a continuous manner. However, for clarity of exposition, we will consider only 3 stiffness values and, therefore, 3 possible stationary solutions for 𝐿𝑚2(marked with squares in Fig. 4). The question that will be addressed in this section is the International Journal of Mechanical Sciences 254 (2023) 108407 5 J. González-Carbajal et al. Fig. 4. Torque–speed curves along a generic quasistatic compaction process. 𝐿𝑚𝑖 (𝑖= 1,2,3) represent the motor characteristic at successive time instants, while 𝐿𝑣(𝑘𝑖) (𝑖= 1,2,3) are the vibration torque curves for increasing mixture stiffness. The squares mark the possible stationary motions for 𝐿𝑚2. The yellow dots represent the quasistatic stationary motion for each considered stiffness value. following: which of the 3 mentioned solutions is actually reached by the system during the quasistatic compaction process? In other words, how can we analytically predict which of the possible stationary motions, for each specific position of the motor characteristic, is actually exhibited by the machine during the quasistatic process? Before entering the core of the discussion, an additional consideration is relevant. We assume that the evolution of the state of the system during the quasistatic compaction is of the same type as in the classical Sommerfeld’s test, described in Section 1.2 and represented in Fig. 1: the amplitude of the oscillations grows monotonically until reaching a point where a jump phenomenon occurs, making the vibration amplitude decrease abruptly. This assumption will be confirmed by the results presented in Section 4. This Section is divided in two subsections. In the first one, we develop the concept of quasistatic torque curve and present a method to calculate it. This curve provides the succession of stationary states of the machine during the quasistatic process. The second subsection investigates the maximum compaction that can be quasistatically achieved and presents the specific way in which the Sommerfeld effect occurs in the vibrocompaction system under study. 3.1. The quasistatic torque curve In a quasistatic compaction process, the system is initially at rest, under the static load produced by the weight of all its elements. In the force–displacement graph of Fig. 5, this initial point is marked with a triangle (see Appendix A for details on this constitutive law). Then, the motor is switched on and its input power is slowly increased, producing a gradual increment in the mixture stiffness as it gets compacted. Fig. 5 represents the stationary oscillation for the 3 stiffness values considered in Fig. 4. The motion is depicted as a straight segment, with slope 𝑘1,𝑘2or 𝑘3, delimited by the maximum and minimum displacements reached during the system vibration (circular markers in the figure). Note the important feature that, according to Fig. 5, the minimum displacement reached by the spring during the stationary vibration for stiffness 𝑘𝑖coincides with the tangency point between the parabola and the elastic line corresponding to stiffness 𝑘𝑖. This requires some detailed justification. For each specific stiffness value, the stationary vibration necessarily takes place around the position of static equilibrium of the system, as highlighted in Fig. 5 with square markers (the static equilibrium position can always be obtained as the intersection between the elastic line of interest and the horizontal line given by 𝐹𝑚= −𝑚𝑔). Then, parameter 𝐴∗(𝑘𝑖)is defined as the displacement between the static Fig. 5. Quasistatic compaction represented on the force–displacement graph of the nonlinear spring that models the mixture behaviour. 𝑘𝑖(𝑖= 1,2,3), represents 3 increasing stiffness values of the mixture. 𝐴∗(𝑘𝑖)represents the quasistatic oscillation amplitude for each considered stiffness. 𝐹𝑚𝑖 represents the maximum compressive force experienced by the spring during the quasistatic oscillation for each considered stiffness. The term 𝑚𝑔 represents the weight of all the masses in the model located above the spring. The static equilibrium position for each considered stiffness is marked with square. The initial state of the system, before the motor is switched on, is marked with a triangle. equilibrium position and the tangent point of the elastic line with the parabola for stiffness 𝑘𝑖(see Fig. 5): 𝐴∗(𝑘𝑖)≡−𝑚𝑔 −𝐹𝑚𝑖 𝑘𝑖 (5) where 𝐹𝑚𝑖 represents the value of 𝐹𝑚at which the parabolic curve of the spring exhibits a slope 𝑘𝑖. Let us now justify that, for stiffness 𝑘𝑖, the stationary oscillation amplitude is given by 𝐴∗(𝑘𝑖). First, it is clear that this oscillation amplitude cannot be greater than 𝐴∗(𝑘𝑖), because that would imply following some length of the parabolic path, with the subsequent increase in the spring stiffness. On the other hand, a stationary oscillation amplitude lower than 𝐴∗(𝑘𝑖), for stiffness 𝑘𝑖, would require a decrease in oscillation amplitude over time. This is, however, not possible under the hypotheses of this study, since we assume the vibration amplitude to grow monotonically until experiencing a jump phenomenon (as was put forth previously in this section). Hence, if we focus on the portion of the process previous to the jump, a reduction in oscillation amplitude cannot take place. We have shown that the stationary oscillation amplitude for stiffness 𝑘𝑖cannot be greater nor lower than 𝐴∗(𝑘𝑖), if the attention is restricted to the fraction of the compaction before the jump phenomenon. The obvious conclusion is that the spring stiffness grows continuously with the amplitude of the oscillation being given, for each instantaneous 𝑘𝑖, by 𝐴∗(𝑘𝑖). Since the oscillation amplitude for each 𝑘𝑖is known, the rotor speed 𝛺∗ 0(𝑘𝑖)can also be computed by using Eq. (B.4), which is rewritten here for convenience: 𝐴∗(𝑘𝑖) = 𝑚1 𝑚𝑟(𝛺∗ 0(𝑘𝑖) 𝜔𝑛𝑖 )2 √ √ √ √[(𝛺∗ 0(𝑘𝑖) 𝜔𝑛𝑖 )2 − 1]2 +[2𝜉𝑖 𝛺∗ 0(𝑘𝑖) 𝜔𝑛𝑖 ]2 ,(6) where 𝜔𝑛𝑖 =√𝑘𝑖∕𝑚and 𝜉𝑖=𝑏∕(2𝑚𝜔𝑛𝑖). Notice that there may be two possible rotor speeds for the same vibration amplitude, as observed in Fig. B.2(a). Then, care should be taken to choose the pre-resonant solution, which is the one of interest for the process under analysis (see Fig. 4). The conclusion from all the above considerations is that, when the system reaches stiffness 𝑘𝑖through the quasistatic procedure being International Journal of Mechanical Sciences 254 (2023) 108407 6 J. González-Carbajal et al. Fig. 6. Quasistatic torque curve, 𝛤(𝛺0), representing the complete sequence of motions along the quasistatic compaction. Each point of the curve is associated with a different value of the mixture stiffness (the stiffness increases upwards along the curve). 𝐿𝑚𝑖(𝛺0) represents a motor characteristic that would allow reaching stiffness 𝑘𝑖, corresponding to the intersection point between 𝐿𝑚𝑖(𝛺0)and 𝛤(𝛺0). described, it will do so with rotor speed 𝛺∗ 0(𝑘𝑖), which can be computed using Eqs. (5) and (6). Then, going back to Fig. 4, we can represent the stationary solution corresponding to each considered stiffness value (yellow dots in the graph). Recall now that we were trying to find which of the 3 possible stationary solutions for the motor curve 𝐿𝑚2, marked with squares in Fig. 4, is actually reached by the machine during a quasistatic compaction process. By comparing these 3 square markers with the 3 yellow dots in Fig. 4, the answer becomes clear. The stationary motion of the system, when the motor curve is 𝐿𝑚2, is the one corresponding to stiffness 𝑘2. This is the only stiffness value whose corresponding vibration torque curve 𝐿𝑣(𝑘𝑖)intersects 𝐿𝑚2at speed 𝛺∗ 0(𝑘𝑖). Going one step further, very relevant information can be extracted by focusing on the sequence of yellow dots in Fig. 4: a new curve 𝛤(𝛺0) can be constructed by plotting the quasistatic stationary solutions for the different stiffness values, representing the complete sequence of motions followed by the machine during the quasistatic compaction. In what follows, this new curve, 𝛤(𝛺0), will be denoted as the quasistatic torque curve of the system. The proposed procedure to compute the quasistatic torque curve is as follows: •Select a set of increasing, equally spaced, stiffness values 𝑘𝑖(𝑖= 1,2,…, 𝑁)with 𝑘1=𝑘𝑠𝑡 and 𝑘𝑁=𝑘𝑓. •Obtain, for each stiffness 𝑘𝑖, the corresponding quasistatic rotor speed 𝛺∗ 0(𝑘𝑖), by using relations (5) and (6). •Obtain, for each pair of values {𝑘𝑖, 𝛺∗ 0(𝑘𝑖)}, the quasistatic torque as 𝛤(𝛺∗ 0(𝑘𝑖)) = 𝐿𝑣(𝑘𝑖, 𝛺∗ 0(𝑘𝑖)) by using Eq. (4). •Represent the points given by (𝛺∗ 0(𝑘𝑖), 𝛤 (𝛺∗ 0(𝑘𝑖))) on a torque– speed graph and connect the points using straight lines or splines. This gives rise to a graph such as shown in Fig. 6. It should be stressed that the quasistatic torque curve gives very relevant information from a practical perspective. Each point of the curve corresponds to a particular stiffness and, therefore, to a particular degree of compaction. Then, the curve reveals the required driving torque to achieve the desired degree of compaction through a quasistatic process. For instance, reaching stiffness 𝑘𝑖in Fig. 6 would require a motor characteristic such as 𝐿𝑚𝑖(𝛺0). 3.2. Maximum quasistatic compaction and Sommerfeld effect It is interesting to analyse the end points of the quasistatic torque curve because, as will be seen, they provide very meaningful information about the possibilities of compaction. First, it is clear that, for stiffness 𝑘𝑠𝑡, the stationary solution is given by 𝛺0= 0,𝐿𝑣= 0, corresponding to the initial state of the machine, with the motor Fig. 7. Quasistatic compaction represented on a torque–speed graph. The collection of black curves represent the vibration torque curves for a set of increasing stiffness values, from 𝑘𝑠𝑡 to 𝑘𝑓. The yellow dots represent the quasistatic stationary motion for each considered stiffness. The orange curve represents the quasistatic torque curve, 𝛤(𝛺0). (a) Case 𝑘max =𝑘𝑓 (b) Case 𝑘max < 𝑘𝑓. at rest. Thus, curve 𝛤(𝛺0)starts at (0,0). In order to investigate the final point of the quasistatic torque curve, consider the torque–speed graphs in Fig. 7. Note that, in this figure, the quasistatic torque curve is represented together with the collection of vibration torque curves for a set of 10 equally spaced stiffness values between 𝑘𝑠𝑡 and 𝑘𝑓. In principle, we should expect the quasistatic torque curve to end at the point associated with the stiffness of the totally compacted mixture, 𝑘𝑓(this is the situation represented in Fig. 7(a)). However, another possibility also exists. Consider the case shown in Fig. 7(b), where it can be observed that the quasistatic point associated with stiffness 𝑘8is already close to the maximum of the corresponding vibration torque curve. For 𝑘9and 𝑘𝑓, no quasistatic point is marked because its corresponding vibration amplitude 𝐴∗(𝑘𝑖)(obtained through Eq. (5)), turns out to be greater than that of the resonance peak. Then, when 𝛺∗ 0(𝑘𝑖)is tried to be computed through relation (6), no solution is found. In physical terms, it can be said that, for stiffness 𝑘9or 𝑘𝑓to be reached through a quasistatic process, the system would need to stationarily oscillate with greater amplitude than that of the resonance peak, which is not possible. Then, for the situation depicted in Fig. 7(b), the end point of the quasistatic torque curve is given by certain stiffness 𝑘max ∈ (𝑘8, 𝑘9)such that the corresponding quasistatic vibration amplitude, 𝐴∗(𝑘max)coincides with that of the resonance peak. Generalizing the above considerations, we can define parameter 𝑘max as the maximum stiffness between 𝑘𝑠𝑡 and 𝑘𝑓such that the system of Eqs. (5),(6) has a solution. This critical value 𝑘max has a very meaningful interpretation: it is the maximum stiffness that can be achieved through a quasistatic compaction process. Observe that, since the motor parameters do not appear in Eqs. (5) and (6),𝑘max has the significant feature of being completely independent of the available power of the electric motors. Clearly, two situations are possible: if 𝑘max =𝑘𝑓, the quasistatic torque curve is qualitatively as represented in Fig. 7(a) and a complete compaction can be quasistatically achieved. Conversely, if 𝑘max < 𝑘𝑓, International Journal of Mechanical Sciences 254 (2023) 108407 7 J. González-Carbajal et al. Fig. 8. Sommerfeld effect in a quasistatic vibrocompaction process. As the motor characteristic is displaced from 𝐿𝑚1to 𝐿𝑚2, the evolution of the state of the system is given by the black arrows. When tangency point 𝐵is reached, the system experiences a jump towards the post-resonant point 𝐶. (a) Case 𝑘max =𝑘𝑓 (b) Case 𝑘max < 𝑘𝑓. the quasistatic torque curve is qualitatively as shown in Fig. 7(b) and the mixture cannot be totally compacted through a quasistatic procedure. The definition of 𝑘max can be directly extended to the level of compaction achieved, 𝛾, defined in Eq. (A.6): 𝛾max =𝑘max −𝑘0 𝑘𝑓−𝑘0 .(7) At this point, we already have all the required tools to describe how the Sommerfeld effect occurs in the vibrocompacting system under study. Consider a quasistatic compaction process in which the motor characteristic is displaced well beyond the point at which the mixture reaches stiffness 𝑘max. As represented in Fig. 8, the system initially follows the quasistatic torque curve as the mixture stiffness gradually increases. After stiffness 𝑘max is reached, the state of the system necessarily evolves on the vibration torque curve 𝐿𝑣(𝑘max)—as has been justified in the above paragraphs, the stiffness cannot keep increasing—until reaching the point at which the motor characteristic is tangent to 𝐿𝑣(𝑘max)(point 𝐵in Fig. 8). Past this point, no more stationary solutions are found near the resonance peak, which makes the system jump towards the only remaining stationary solution, corresponding to a post-resonant motion with small oscillation amplitudes (jump from point 𝐵to point 𝐶in Fig. 8). This is precisely the jump phenomenon observed by Sommerfeld in 1904 [31], and one of the most well-known effects of nonideal excitations, as described in Section 1.2. It should be noted that the portion of the process between reaching stiffness 𝑘max and finding the jump phenomenon, which is clearly observable for 𝑘max =𝑘𝑓(see path from 𝐴′to 𝐵in Fig. 8(a)), is almost nonexistent for 𝑘max < 𝑘𝑓(see Fig. 8(b)). The reason is that, when 𝑘max < 𝑘𝑓, the quasistatic point corresponding to 𝑘max coincides with the maximum of the amplitude–frequency curve, which is very close to the maximum of curve 𝐿𝑣(𝑘max)and, consequently, also very close to the tangency point (point 𝐵in Fig. 8). By comparing Fig. 8 to Fig. 2 in the introduction of the paper, the peculiar way in which the Sommerfeld effect takes place in the vibrocompaction system becomes apparent. For a vibrating system with fixed stiffness, such as the one studied by Sommerfeld and Kononenko [31, 34], the state of the system evolves on the vibration torque curve until encountering a jump, as represented in Fig. 2. However, in the vibrocompaction system under study, part of this evolution does not take place on a vibration torque curve, but on the quasistatic torque curve of the system, as shown in Fig. 8. This is due to the fact that, during this portion of the process, the stiffness of the vibrating system is not fixed, but gradually increases as the mixture gets compacted. Let us summarize the main conclusions of this section. The concept of quasistatic torque curve has been presented, together with a procedure for its computation. This curve characterizes the evolution of the state of the vibrocompaction system along a quasistatic compaction process. It also provides the amount of motor torque required to reach a desired compaction level, together with the maximum compaction that can be quasistatically attained. This maximum value was found to be independent of the available power of the electric motors. Finally, the quasistatic torque curve reveals the specific manner in which the Sommerfeld effect takes place in the vibrocompacting machine. After the analysis of the quasistatic compaction presented in this section, it is reasonable to wonder what the differences are when the process is not quasistatic. This would be the case, for instance, if the motor was switched on with the driving torque curve being directly in its final position, or if the control settings of the motor were abruptly modified. In these situations there can be significant transient effects influencing the evolution of the system state and, therefore, affecting the level of compaction achieved. Some of these transient effects will be discussed in the next section. 4. Results and discussion As was commented in the Introduction, this paper builds upon a previous publication of the authors [13], where the model presented in Appendix A was described in detail, justified and investigated by means of various simulations. The research presented in [13] was based on a numerical approach to the problem: the equations of motion (A.1) were numerically integrated for different sets of parameters values and the results were discussed and analysed. The present section intends to complement these simulations with useful analytical insight into the model. Section 4.1 summarizes the numerical results that were obtained in [13]. In Section 4.2, a clear physical interpretation for these results is provided, based on the developments of Sections 2and 3. Analytical predictions are also verified by comparing them to the numerical results. 4.1. Summary of previous numerical results The simulations presented in [13] were conducted by numerically solving system (A.1) for 𝑡∈ [0, 𝑡𝑓]. The total time 𝑡𝑓includes three different stages (𝑡𝑓=𝑡1+𝑡2+𝑡3): •Stage 1: 𝑡∈ [0, 𝑡1]. Parameter 𝐶of the motor control is linearly increased from 𝐶0to 𝐶𝑓, while keeping slope 𝐷constant (see Eq. (2)). Note that this can correspond to the quasistatic process described in Section 3, as long as 𝑡1is long enough. The motor characteristic at the beginning and at the end of this phase are denoted as 𝐿𝑚0and 𝐿𝑚𝑓 , respectively. •Stage 2: 𝑡∈ [𝑡1, 𝑡1+𝑡2]. Parameter 𝐶is kept constant (𝐶= 𝐶𝑓). During this phase, the vibrocompacting machine reaches a stationary motion. •Stage 3: 𝑡∈ [𝑡1+𝑡2, 𝑡𝑓]. The system returns to rest after the motor is switched off at 𝑡=𝑡1+𝑡2. International Journal of Mechanical Sciences 254 (2023) 108407 8 J. González-Carbajal et al. Table 1 Parameter values that remain constant through the different simulations: 𝑚𝑚stands for the mass of the mixture, 𝑚𝑝is the mass of the piston and the motor, 𝑟 is the eccentricity of the unbalance, 𝑏is the damping coefficient, 𝑑𝑓and 𝐹𝑓represent, respectively, the spring deformation and spring force corresponding to a complete compaction, 𝑅𝑘is the ratio between the minimum and maximum values for the spring stiffness and 𝑡2and 𝑡3represent the time length of the second and third stages of the simulation, respectively. 𝑚𝑚(kg) 240 𝑚𝑝(kg) 1500 𝑟(m) 0.1 𝑏(Ns∕m) 4 ⋅103 𝑑𝑓(m) −0.1 𝐹𝑓(N) −105 𝑅𝑘0.1 𝑡2(s) 15 𝑡3(s) 15 It is useful to define a new parameter, 𝜔𝑛𝑝, as the natural frequency exhibited by the system during stage 2. It can be calculated as 𝜔𝑛𝑝 =√𝑘𝑝 𝑚𝑝+𝑚𝑚∕3 + 𝑚1 ,(8) where 𝑘𝑝denotes the stiffness exhibited by the nonlinear spring during the stationary vibration attained at stage 2. Five different simulations are considered. Some of the system parameters are the same for the 5 cases, and their values are displayed in Table 1. The parameters that vary among the different simulations are shown in Table 2, together with the result of each simulation quantified through the final level of compaction 𝛾𝑠(see definition in Eq. (A.6), subscript sstands for simulation). The obtained results are graphically represented in Fig. 9 for Cases 1–4 and in Fig. 10 for Case 5, which is the only one in which separations and impacts occur between piston and mixture. 4.2. Discussion and comparison with analytical results In this subsection, we use the developments of Sections 2and 3 to interpret the presented numerical results. In particular, it will be shown how the quasistatic torque curve, 𝛤(𝛺0), constitutes a useful tool to understand and quantify the behaviour of the vibrocompacting machine. Note that, among the varying parameters in Table 2, the only one that affects the vibration torque curves of the system, 𝐿𝑣(𝑘𝑖), is the unbalanced mass 𝑚1, as can be observed in Eq. (B.6). Since 𝑚1is the same for Cases 1–4 and takes a different value for Case 5 (see Table 2), there will be one set of vibration torque curves for Cases 1–4 and a different set of curves for Case 5. These curves are computed through relation (B.6) for 20 equally spaced stiffness values from 𝑘𝑠𝑡 to 𝑘𝑓and are represented in Fig. 11 together with the motor characteristics for the different simulations. Once the vibration torque curves are obtained, the procedure described in Section 3can be followed to compute the quasistatic stationary motion corresponding to each considered stiffness value (yellow dots in Fig. 11). By connecting these points, the quasistatic torque curve, 𝛤(𝛺0), is obtained (orange line in Fig. 11). Finally, we can use Eq. (7) to calculate parameter 𝛾max, defined as the maximum compaction level that can be reached quasistatically: 𝛾Cases 1–4 max = 48.2% 𝛾Case 5 max = 71.8% (9) Based on these representations and computations, the global behaviour of the system can be described as follows. If the motor characteristic passes through the end point of the quasistatic torque curve, then the mixture will reach the compaction level given in Eq. (9), as long as the motor control is sufficiently slow. This will happen as the system experiences an increasingly growing vibration amplitude followed by a jump phenomenon towards a post-resonant state of motion. On the other hand, if the motor characteristic does not reach the end point of the quasistatic torque curve, the final compaction level of the mixture will be lower than given in Eq. (9). The level of compaction achieved can be computed by introducing the maximum stiffness exhibited by the nonlinear spring into relation (A.6). This stiffness can in turn be obtained as the stiffness corresponding to the intersection point between curves 𝐿𝑚𝑓 (𝛺0)and 𝛤(𝛺0)(see square markers in Fig. 11). The application of the described procedure to the five considered simulations yield the analytical predictions presented in Table 3. It is interesting to note that a comparison between Case 1 and Cases 2–4 provides useful insight into the effect of parameters 𝐶and 𝐷, which define the motor characteristic, on the system behaviour. As Fig. 11(a) shows, the motor characteristic passes through the end point of the quasistatic torque curve in Case 1, but not in Cases 2–4. Therefore, the analytical prediction is that the mixture attains its maximum quasistatic compaction in Case 1, but not in Cases 2–4 (see Table 3). Likewise, the system experiences a jump towards a post-resonant motion in Case 1 (see Fig. 9(a) and (b)), but not in Case 2 (see Fig. 9(c) and (d)) or Case 4 (see Fig. 9(g) and (h)). The numerical results for Case 3 do display the jump (see Fig. 9(e) and Fig. 9(f)), but in this scenario the analytical prediction does not apply because the motor control is not quasistatic (the ramp-up time, 𝑡1, has been reduced to zero, as shown in Table 2). The effect of parameters 𝐶and 𝐷on the system dynamics will be discussed in more detail in the next paragraphs, by separately analysing each of the simulation scenarios. For Case 1, it is clear from Fig. 11 that the motor characteristic 𝐿𝑚𝑓 passes over the end point of the quasistatic torque curve. Hence, we should expect the system to exhibit a gradually increasing vibration amplitude followed by a jump phenomenon towards a post-resonant motion with smaller amplitude. Note that this is precisely the behaviour found in the simulation results plotted in Fig. 9(a) and Fig. 9(b). Regarding the analytical prediction for the level of compaction achieved in Case 1, a comparison between Tables 2 and 3shows some difference with respect to the numerical result. This indicates that the process may not have been slow enough to be considered as quasistatic. To validate this hypothesis, the simulation was repeated after increasing parameter 𝑡1to 50 s. This produced a level of compaction 𝛾𝑠= 48.0%, which is significantly closer to the analytical value given in Table 3. This supports the idea that, the slower the displacement of the motor characteristic, the more similar the process is to the ideal quasistatic case. Case 2 constitutes a compacting process where the motor characteristic does not pass through the end point of the quasistatic torque curve. This means that we should expect an increasing vibration amplitude during stage 1 (without any jump phenomena), followed by a stabilization during stage 2 into a stationary oscillation with the rotor speed being slightly below the system resonance frequency. This is in accurate agreement with the numerical results depicted in Fig. 9(c) and Fig. 9(d). Note also the precise accordance between the analytical and numerical evaluations of the final compaction level (compare Tables 2 and 3for Case 2). Case 3 has the same parameter values as Case 2 except for time 𝑡1, which is now set to 0(see Table 2). This means that there is no motor control: the motor characteristic is set to its final position from the beginning of the process. Thus, the simulation cannot be considered as quasistatic, which is why no analytical prediction is given in Table 3 for this case. In any case, it is interesting to compare the numerical results of Cases 2 and 3, in order to assess the effect that a reduction of the rampup time can have on the machine behaviour. As clearly shown in Fig. 9, two different types of behaviour arise in these two simulations. While International Journal of Mechanical Sciences 254 (2023) 108407 9 J. González-Carbajal et al. Table 2 Parameter values that vary among the different simulations: 𝑚1is the unbalanced mass, 𝐼0is the rotor inertia, 𝑡1is the time length of the first simulation stage, 𝐷is the slope of the motor characteristic and 𝐶0and 𝐶𝑓represent the initial and final values for the motor stall torque. The final compaction level 𝛾𝑠obtained in each case is also shown. Case 1 Case 2 Case 3 Case 4 Case 5 Variable Parameters 𝑚1(kg) 20 20 20 20 40 𝐼0(kgm2)0.84 0.84 0.84 2.52 0.84 𝑡1(s) 10 15 0 0 15 𝐷(Nm∕s) −5−1.5 −1.5 −1.5 −2.3 𝐶0(N m) 20 22 22 22 40 𝐶𝑓(N m) 140 43 43 43 80 Numerical Results 𝛾𝑠46.5% 47.4% 44.6% 47.4% 60.3% Fig. 9. Simulation results from numerical integration of system (A.1). Graphs on the left represent the displacement of the piston, while graphs on the right represent the rotor speed. From top to bottom, the results correspond to Cases 1, 2, 3 and 4. The dashed line represents the natural frequency of the system during the stationary oscillation, 𝜔𝑛𝑝, as defined in Eq. (8).