A Study of Forced Vibrations with Nonlinear Springs and Dry Friction: Application to a Mechanical Oscillator with Very Large Vibrating Blades for Soil Cutting

Abstract

To cut a clod of soil containing the roots of trees in nurseries, a semi-circular vibrating blade digging machine with diameters up to 1.2 m is increasingly used. The heart of the machine is the mechanical oscillator that produces an excitation torque supplied to the blade together with the cutting torque of the soil. The advantage of the vibrating blade is a reduction in the cutting torque up to 70%. This advantage led to the present study of the extension to blades of even 1.8 m for the digging of very large trees. To build an oscillator suitable for all blade sizes (from 0.6 to 1.8 m), it was necessary to equip it with nonlinear (quadratic) springs, since with traditional linear springs, it would not be versatile. The design and simulation of its operation required the development of a new mathematical model. Therefore, an approximate solution of the differential equation of the forced vibration with quadratic springs and dry friction between the blade and soil was developed, aimed at calculating the maximum blade displacement and the phase lag. These quantities, together with the cutting time, had to satisfy certain values to ensure the maximum reduction in the cutting torque (−70%). After the construction of the oscillator, it was coupled with all the blades (0.6, 0.9, 1.2, and 1.8 m) for experimental tests. For all diameters, the oscillator was able to optimally vibrate the blades, preventing the springs from reaching the end of the stroke when cutting the soil. Measuring the maximum blade displacement compared with the calculated one provided a good accuracy of the mathematical modeling, resulting in a mean error of 5.6% and a maximum error of 7.2%.

1. Introduction

Reducing the cutting force of materials through vibrating blades is now a widespread practice in many sectors. Even cutting the soil during mechanical operations (ploughing, subsoiling, potato harvesting, etc.), can be achieved using tools with vibratory movement.

As far as is known, the first studies on the effect of vibration in soil cutting were by Gunn and Tramontini [1], who found that with the blade vibration, the cutting energy is practically equal to that required with the static blade. A few years later, Egenmuller [2] published his results on the positive influence of high ratios between the peak velocity of the oscillation $\dot{A_0}$ and the cutting velocity v_cut ($\dot{A_0}/v_{cut})$, in reducing the cutting force. He found that the maximum displacement of the blade A₀ had to be equal or greater than 6 mm and that with $\dot{A_0}/v_{cut}=6$, the ratio between the cutting force with and without vibration was reduced to 0.4. Many years later, Brixius and Verma [3] and Verma [4] confirmed these results.

A 50% reduction in cutting force with the velocity ratio $\dot{A_0}/v_{cut}>1$, at a frequency of 50 Hz and A₀ = 8 mm, was obtained by Butson and McIntyre [5]. With these data, Butson and Rackham [6] also produced a mathematical model for predicting the cutting force. Then, a similar mathematical model was also developed by Narayanarao and Verma [7], based on their experiments with harmonic vibrations.

Al-Jubouri and McIntyre [8] applied vibration to the blade of a potato digger. They also found a reduction in cutting force that was greater the higher the velocity ratio $\dot{A_0}/v_{cut}$.

Shkurenko [9] and Wolfson and Shkurenko [10] found that vibrations in the cutting direction were much more effective in reducing the cutting force than transverse vibrations.

Compared to the harmonic vibrations used up to that time, Smith and Al [11,12], analyzed the square wave and saw-tooth wave, finding that these do not bring advantages.

In cutting clay soil with the vibrating blade, a significant reduction of the cutting force was obtained, but also a relative increase of the power required compared to the static blade occurred [13].

Compared to Egenmuller [1], Szabo [14] pushed the velocity ratio $\dot{A_0}/v_{cut}$ up to 17, obtaining an even lower ratio between the cutting force with and without vibration equal to 0.3.

Shahgoli et al. [15,16,17] studied a vibrating ripper for hard compacted soil, optimizing the vibration frequency to obtain a ratio of the cutting forces with and without vibration equal to 0.36.

Tang et al. [18] concluded through experimentation that a vibrating subsoiler reduced the cutting force compared to both a static and a rotating one, while Shchukin et al. [19] found that a vibrating subsoiler improved the soil structure. Razzaghi and Sohrabi [20] performed a novel analysis of the interaction between vibrating tools and soil based on polar coordinates.

Rao et al. [21,22,23,24,25] designed a vibrating tillage cultivator after developing optimized design algorithms. Keppler et al. [26] used the discrete element method (DEM) to simulate the effect of vibration of a tillage cultivator on the cutting force. Biris et al. [27] investigated the relationships between vibration, cutting force, and blade–soil friction.

Dzhabborov et al. [28] concluded that increasing the frequency produced both a reduction in cutting force and an improvement in the soil structure. This result was confirmed by Wang et al. [29], even though they found an increase in the power required with increasing frequency.

The synthesis of all these studies is the possibility of obtaining an optimal force ratio (vibration/non-vibration) of 0.3, provided that the maximum displacement of the oscillation is at least 6 mm and the velocity ratio $\dot{A_0}/v_{cut}$ is at least 17.

This latest information has given rise to the first optimization of an oscillator designed to vibrate the semicircular blade (Figure 1) in the tree digging machines [30,31] used to cut a hemispherical clod of soil containing the root system of the tree in the nursery. The blade cutting the soil must move with a rotary motion for an arc of π rad and, at the same time, vibrate thanks to the mechanical oscillator mechanically powered in turn by the engine of the tree digger machine. After cutting the clod, this, together with the tree, is transported and planted in orchards or gardens.

The first tree digger machine had small and medium blades with diameters of 0.6 and 0.9 m; consequently, a study and experimentation [30] were conducted on the oscillator of this medium digger machine. An attempt was also made to mount and use a large blade of 1.2 m diameter, but the mechanical oscillator proved inadequate, showing two limitations. The first problem was that the maximum displacement A₀ was only 3 mm, which is half of the optimum one that would have allowed the minimum ratio (0.3) between the vibrating and non-vibrating cutting torque. The second problem was the dangerous transmission of vibrations to the frame of the digger machine, which reverberated on the operator’s safety [32,33] due to the springs reaching the end of stroke when the blade sank into the soil during cutting.

Consequently, by applying the Den Hartog method [34], based on the approximate solution of the differential equation of the vibratory motion in the presence of dry Coulomb friction between the blade and the soil, a new oscillator was studied and built [31], which was capable of mounting blades with a minimum diameter of 0.6 m and a maximum of 1.2 m. The tests carried out with three blades (0.6, 0.9, and 1.2 m) demonstrated that the predictions obtained with the Den Hartog approximate method are very good, with an average error of 4.4% and a maximum error of 6.4%. Furthermore, the experimental tests showed that, in all cases, the maximum blade displacement A₀ was always equal to or greater than 6 mm and that the velocity ratio $\dot{A_0}/v_{cut}$ was always at least equal to 17.

A 1.2 m diameter blade can work on medium and large trees. For very large trees, the traditional digger machine with spades is still used, where the spades are mounted on a frame arranged around the tree and driven into the soil with hydraulic cylinders without the aid of vibrations. These traditional machines are slow and expensive, so tree nursery operators have asked to replace them with digger machines with very large vibrating blades (1.8 m diameter). Den Hartog’s theory was then applied for this case of a very large blade coupled with the mechanical oscillator. It turned out that it was necessary to increase the excitation torque through an increase in the eccentric masses and an increase in the stiffness of the springs, but this, always according to Den Hartog’s modeling, would have led to the oscillator + blade system leaving the optimal operating condition when the blade was smaller than 1.8 m. In short, the tree digger machine with this hypothetical oscillator with larger eccentric masses and stiffer cylindrical coil springs would not have been universal, that is, it would not have been usable for the entire range of blade diameters.

The following work is dedicated to the solution to this problem. A modified oscillator was designed that would work well with all the blade diameter sizes because it could adapt the excitation torque and the spring stiffness to the blade size. The excitation torque would be adapted through the variation of the excitation frequency, that is, the rotation velocity of the eccentric masses. Spring stiffness would be adapted through the abandonment of linear springs such as those with a cylindrical coil and the adoption of conical coil springs with variable pitch, that is, springs with a relationship between the force F and the shortening x of the type: $F=k·x^2$ (quadratic springs).

With linear springs ($F=k·x$) and dry Coulomb friction, Den Hartog’s theory based on the approximate solution of the equation of vibratory motion was valid. With nonlinear (quadratic) springs Den Hartog’s theory was no longer valid and therefore it was necessary to find, as a first step, a new approximate solution of the new differential equation of motion. This solution and the geometric and dynamic study of the conical coil springs with variable pitch, necessary to have the quadratic relationship F–x, constitute the mathematical modeling of the new oscillator-blade system. This modeling allowed us to design and simulate the behavior of the system with blades of four different diameters (0.6, 0.9, 1.2, and 1.8 m).

Following the construction of these conical coil springs and the modification of the oscillator that provided for the increase in the eccentric masses, an experiment was carried out to verify the operation of the system and confirm the proposed mathematical modeling.

2. Materials and Methods

2.1. Brief Review of the Previous Linear Spring Oscillator and Den Hartog’s Approximate Solution

In previous work [31], a mechanical oscillator with cylindrical coil springs, i.e., linear springs, was studied. It was able to vibrate in an optimal way with the semicircular blades of the tree digger machines with diameters of 0.6, 0.9, and 1.2 m during the cutting of the soil clod. In an optimal way, this means it has a maximum blade displacement A₀ of at least 6 mm, respecting the ratio between the peak velocity of oscillation and the cutting velocity $\dot{A_0}/v_{cut}\ge 17$ and avoiding the transmission of vibrations to the frame of the tree digger machine due to the springs reaching the end of stroke when the blade sinks into the soil during cutting.

Figure 2 shows the vibrating system, consisting of a semicircular blade for cutting the soil and the optimized oscillator. It has a set of five gear wheels inside. The central wheel (in black) is moved by an external hydraulic motor placed on the left. The motion is transmitted via the two intermediate gear wheels (1) to the outermost gear wheels, where each carry an eccentric mass (2).

During the rotation of the gear wheels, when the two masses are at an angle of π/2 with respect to how they appear in Figure 2, they produce a maximum excitation torque T_em, while in the position in Figure 2, they are in opposition, and therefore, their centrifugal forces are balanced. Therefore, the excitation torque varies according to harmonic law and induces the entire gear housing (3) to an oscillating motion, which is transmitted to the shaft (4) and therefore to the horizontal butterfly bush (5) and the blade (6). It is important to note that the splined shaft to the left of the horizontal butterfly bush (5) is not connected to the central gear wheel (in black) but is connected to the gear housing (3). Therefore, the latter transmits the oscillation to the splined shaft and hence to the blade (6) via the shaft (4). The horizontal butterfly bush has a system of springs (7) supported by a vertical butterfly bush (8). However, this bush is not fixed but can rotate because it is controlled by a worm screw (9) when the operator wants to impart the rotating cutting movement to the blade.

This mechanical oscillator was studied [31] through mathematical modeling borrowed from Den Hartog [34] to optimize the design up to blade diameters of 1.2 m. Den Hartog’s modeling was based on the approximate solution of the equation of forced vibrations with dry friction and linear springs. To obtain it, Den Hartog used an artifice, that is, he considered the differential equation of motion adapted here to the angular motion of the blade + oscillator system:

$$J·\ddot{\alpha }+k·\alpha \pm T_F=T_{em}·\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)$$

(1)

where J is the moment of inertia; k is the angular springs rate (angular stiffness); T_em is the maximum excitation torque; ±T_F is the dry friction torque; α is the angular displacement; $\ddot{\alpha }$ is the angular acceleration; ω is the excitation frequency. The quantities J, k, and T_em are constant and correlated with the geometric and dynamic characteristics of the system.

Then, he rewrote Equation (1) as follows:

$$\ddot{\alpha }+{\omega }_n^2·\left(\alpha -{\alpha }_f\right)=\mathrm{a}·{\omega }_n^2·\mathrm{c}\mathrm{o}\mathrm{s}(\omega t+\varphi )$$

(2)

where $\mathrm{a}=T_{em}/k$ is the static displacement under maximum excitation torque; ${\alpha }_f=T_F/k$ is the static displacement under maximum friction torque; ${\omega }_n=\sqrt{{\displaystyle \frac{k}{J}}}$ is the natural frequency.

In this way, Equation (2) takes the simplified form in which the square wave of the dry Coulomb friction torque ±T_F is translated into a constant load capable of producing, for example, when T_F is negative, a static deformation equal to −x_f. In this way, the friction torque “disappeared”, facilitating the solution, which appears in a harmonic form with a zero-phase lag ϕ between the motion and the excitation torque. However, the actual motion is not harmonic, and the phase lag cannot be zero. Den Hartog resolved these discrepancies by introducing the unknown phase lag ϕ into the cosine function of the excitation torque, and answering that, with the practical aim of arriving at the knowledge of the maximum angular displacement α₀ and the phase lag ϕ, the fact that the solution resulted in a harmonic form did not matter. His procedure concluded with the determination of the two integration constants of the harmonic solution and of the two unknowns (α₀ and ϕ) through the four boundary conditions, two concerning the angular displacement α and two concerning the angular velocity $\dot{\alpha }$:

• $t=0\to \alpha ={\alpha }_0\to \dot{\alpha }=0$
• $t=\pi /\omega \to \alpha =-{\alpha }_0\to \dot{\alpha }=0$

The final equations for the calculation of α₀ and ϕ obtained by him are reported in previous work [31]. Den Hartog also developed the exact solution with the use of Fourier series, finding that the approximate solution of α₀ and ϕ were identical to the exact ones. In his book [35] (pp. 740–741), Den Hartog also proposed another method of solving Equation (1), that is, to replace the square wave of the Coulomb friction torque ±T_F with a harmonic wave in phase with the velocity $\dot{\alpha }$ representing the viscous friction torque $\pm T_F\approx C·\dot{\alpha }$, where C is the damping constant, with the condition that the work dissipated during a cycle of the harmonic function (ωt from 0 to 2π) was the same as the cycle of the square wave of Coulomb. Den Hartog obtained the equivalent damping constant C for the motion along an x-axis, where he had the maximum excitation force F and the maximum displacement x₀: $C=4F/\left(\pi \omega x_0\right)$. For the angular motion of our blade + oscillator system, it becomes the following:

$$C={\displaystyle \frac{4{·T}_F}{\pi ·\omega ·{\alpha }_0}}$$

(3)

where C is the equivalent damping constant, α₀ is the maximum angular displacement, and T_F is the maximum friction torque. The solution of the equation: $J·\ddot{\alpha }+k·\alpha +C·\dot{\alpha }=T_{em}·\cos \left(\omega t\right),$ is easy to obtain and produces values like the exact solution with an average deviation of 4%.

2.2. The New Nonlinear Springs

As already said in the Introduction, nursery operators ask for blades up to 1.8 m in diameter for the uprooting of the largest trees that today are extracted with the clod of soil using traditional digger machines with slower and more expensive spades.

A theoretical analysis of the oscillator system in Figure 2 with linear springs and a 1.8 m blade, based on the mathematical modeling just summarized in paragraph 2.1. and subsequent experimental verification, has shown that this oscillator does not meet the requirements for an optimal cut. In fact, the springs reached the end of their travel, transmitting the vibration to the machine frame and therefore to the operator.

As briefly presented in the Introduction, the same mathematical modeling suggested that modifying the oscillator with an increase in the stiffness of the linear springs and an increase in the eccentric masses would allow it to become suitable for the 1.8 m blade but no longer be suitable for smaller blades due to the excessive stiffness of the springs. The solution then was to make the springs nonlinear, i.e., quadratic. Therefore, the first step was the study and realization of springs with nonlinear behavior. The next step was the development of a new mathematical modeling based on the approximate solution of the differential equation of the system forced to vibrate by the eccentric rotating masses, with quadratic springs and dry friction between blade and soil.

The new nonlinear springs must have quadratic behavior, i.e., corresponding to a function $F\propto K·x^2$ and, therefore, for the angular motion of the oscillator + blade system, $T_{spring}\propto K·{\alpha }^2$, where F is the elastic force, and x is the shortening or stretching of the spring, $T_{spring}=F·b_s$ is the elastic torque, b_s is the lever arm of the spring (Figure 2), and α is the angular displacement during the vibratory motion. To have a quadratic behavior, the conical coil spring with variable pitch p was considered. Figure 3 shows this type of spring with the design dimensions.

Each active turn in spring is subjected to torsion, so the approximate relationship between the spring rate k_l and the dimensional and steel characteristics can be written as follows:

$$k_l={\displaystyle \frac{G·d^4}{64·R_m^3}}$$

(4)

where G is the modulus of elasticity in terms of shear; d is the diameter of wire section; R_m is the mean radius of the turn in spring measured from spring axis to center of section.

Equation (4) is approximate because it does not consider the influence of the pitch angle. Young [36] (p. 386) proposes the exact relation which, compared with relation (4), gives an error of less than 1% for the case of the spring in Figure 2. Equation (4) is also further approximated because it does not consider the continuous variation of the radius of the turn in the conical spring. In fact, in Equation (4), the arithmetic mean radius appears: $R_m={\displaystyle \frac{R_1+R_2}{2}}$, where R₁ is the smaller radius, and R₂ is the larger radius of the single turn. Belluzzi [37] (pp. 203–204) proposes the formula for the exact calculation of the mean radius: $R_m=\sqrt[3]{{\displaystyle \frac{R_1^2+R_2^2}{2}}·{\displaystyle \frac{R_1+R_2}{2}}}$ whose result, with the geometric data of the spring in Figure 3, gives an error of only 0.4%.

2.2.1. The Conical Coil Spring with Variable Pitch Subjected to Shortening

If the turns do not touch each other during the spring shortening, the set of six turns of the conical coil spring in Figure 3 presents an overall spring rate equal to k_{l spring6}, which is the harmonic mean of the stiffness of each ith single turn K_li:

$$k_{lspring6}={\displaystyle \frac{1}{\sum _{i=1}^6\frac{1}{k_{li}}}}$$

(5a)

On the other hand, Equation (4) shows that as the radius R_m of the turns increases, the constant k_l decreases; therefore, the larger turns of the spring in Figure 3 are more compliant. If the spring is subjected to an increasing force F, the 6th turn, at the top, which has the largest mean radius R_m, deforms rapidly until it touches the 5th turn. Therefore, starting with a certain value of force F, the 6th turn no longer participates in the formation of the spring’s stiffness. From then on, the increasing force F is absorbed by the deformation of the five remaining turns:

$$k_{lspring5}={\displaystyle \frac{1}{\sum _{i=1}^5\frac{1}{k_{li}}}}$$

(5b)

The stiffness k_{l springs5} is greater than k_{l springs6}. With the further increase in F, the contact of the 5th turn with the 4th is also reached. Therefore, a new higher stiffness k_{l springs4} will occur since only four turns are left active:

$$k_{lspring4}={\displaystyle \frac{1}{\sum _{i=1}^4\frac{1}{k_{li}}}}$$

(5c)

By continuing to increase the force F, the other turns will also come into contact one after the other, with further increases in stiffness. Figure 4 shows the resulting force F vs. shortening x diagram of the conical coil spring of Figure 3. A regression quadratic parabola (R² = 0.992) was constructed over the monotone polygonal chain:

$$F=K_l·x^2=184·x^2$$

(6)

where K_l is the quadratic stiffness or quadratic spring rate (N mm⁻²), and x is the shortening of the spring (mm).

2.2.2. The Conical Coil Spring with Variable Pitch Subjected to Stretching

If the spring is pulled with a force F (N), the progressive end of stroke phenomenon is missing, so the spring has a constant stiffness equal to that given by Equation (5a). Therefore, it behaves linearly according to the following equation:

$$F=k_l·x=1129·x$$

(7)

where k_l = k_{l spring6} is the spring rate (N mm⁻¹), and x is the stretching of the spring (mm).

2.2.3. The Behavior of Conical Coil Springs in the Oscillator

Figure 2 shows the oscillator with four springs that, during the oscillation, work alternately in shortening and stretching. When two are shortened, the other two are stretched. Given the rotary motion of the blade and the oscillator, it is necessary to transform the quadratic spring rate K_l and the spring rate k_l of the two Equations (6) and (7), respectively, into the quadratic angular spring rate K (Nm rad⁻²) and the angular spring rate k (Nm rad⁻¹). The first was achieved through the following formula:

$$K=2·K_l·b_S^3$$

(8)

where coefficient 2 is the number of springs in shortening.

The second relationship was already reported in [31]:

$$k=2·k_l·b_S^2$$

(9)

where coefficient 2 is the number of springs experiencing stretching, and b_s is the lever arm of the spring (Figure 2).

Ultimately, the blade and the horizontal butterfly in Figure 2 oscillate with respect to the neutral position, shortening two springs and stretching the other two. Therefore, the total spring torque T_springs is given by the sum of the torque due to the two springs in stretching and the torque due to the two springs experiencing shortening:

$$T_{springs}=k·\alpha +K·{\alpha }^2$$

(10)

when the operator activates the worm gear (9) (Figure 2), it rotates the vibrating blade to cut the soil. Therefore, when the blade is cutting the soil, it reacts with a cutting torque T_cut, which produces a static angular displacement α_cut of the four springs, but two are shortened and two are lengthened. In this situation, which is the most critical for operation, the springs offer the total spring torque as follows:

$$T_{springs}=k·\left(\alpha +{\alpha }_{cut}\right)+K·{\left(\alpha +{\alpha }_{cut}\right)}^2$$

(11)

2.3. Approximate Solution of the Differential Equation of the Vibrating System with Nonlinear Springs and Dry Coulomb Friction

Figure 5 shows the dynamic scheme of the oscillator + blade system during the cutting of the soil. The scheme shows the excitation torque T_em, due to the motion of the eccentric masses (Figure 2), which varies sinusoidally and has the maximum value T_em:

$$T_{em}=n·m·{\omega }^2·y_G·b_G$$

(12)

where ω is the angular velocity of the eccentric masses and, hence, the angular frequency of the excitation torque; n stands for the number of eccentric masses (2 masses in Figure 2, but they can be 4 or 6); m denotes the eccentric mass value; y_G represents the mass eccentricity; and b_G refers to the lever arm of the masses (i.e., the distance between the eccentric mass shaft and the blade shaft).

Again, in Figure 5, there is the total spring torque T_springs due to the shortening and stretching of the nonlinear conical coil springs. It is expressed by Equation (11).

Furthermore, in Figure 5, there is the cutting torque T_cut, which is a function of the blade diameter D according to the equation already found in [31]:

$$T_{cut}=6733.3·D^2-3693.3·D+1060$$

(13)

This cutting torque produces a static angular displacement of the springs equal to α_cut:

$$T_{cut}=k·{\alpha }_{cut}+K·{\alpha }_{cut}^2$$

(14)

Finally, in Figure 5, there is the dry friction torque ±T_F described by Coulomb’s law and, therefore, variable according to a rectangular wave in phase with the angular velocity $\dot{\alpha }$ of the blade oscillation.

As proposed by Den Hartog [34] and already examined at the end of Section 2.1, the rectangular wave ±T_F, representing dry friction, is replaced by a sinusoidal wave in phase with the angular velocity of the oscillation $\dot{\alpha }$, representing viscous friction $\pm T_F\approx C·\dot{\alpha }$, where C is the damping constant. Under the condition that the two waves are equivalent with respect to the work dissipated by the blade–soil friction, Den Hartog obtained a formula for the damping constant C that has already been adapted to the case of angular motion in paragraph 2.1, which is shown in Equation (3). The value of the dry friction torque T_F can be calculated with the following equation developed in previous work [31]:

$$T_F=\mu ·\rho ·{\displaystyle \frac{D^4}{6}}·g$$

(15)

where ρ is the soil density, D represents the diameter of the blade and is therefore the diameter of the hemispherical soil clod, μ is the external friction coefficient, and g denotes the gravity acceleration.

By imposing the equilibrium of all the torques just described and represented in Figure 5 with the inertia torque $J\ddot{\alpha }$, where J is the moment of inertia and $\ddot{\alpha }$ is the angular acceleration, the differential equation of the blade motion is obtained:

$$J·\ddot{\alpha }+k·\left(\alpha +{\alpha }_{cut}\right)+K{·\left(\alpha +{\alpha }_{cut}\right)}^2+C·\dot{\alpha }-T_{cut}=T_{em}·\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)$$

(16)

Developing the binomial to the square and simplifying thanks to Equation (14), Equation (16) becomes

$$J·\ddot{\alpha }+\left(k+2K·{\alpha }_{cut}\right)\alpha +K·{\alpha }^2+C·\dot{\alpha }=T_{em}·\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)$$

(17)

It is interesting to note that with the linear cylindrical springs used in the old oscillator [31], i.e., with K = 0 and with $\pm T_F$ instead of $C·\dot{\alpha }$, the differential equation of the vibratory motion (17) is reduced to Equation (1), the one from which, following the Den Hartog method, Equation (2) was obtained.

The presence of the variable α squared, together with the presence of the viscous friction torque $C·\dot{\alpha }$, makes Equation (17) not integrable in an exact way [35] (p. 732, par. 8.8). Also, the application of the approximate Den Hartog method already used in previous work [31] and briefly recalled in Section 2.1 is impossible.

On the other hand, the aim of the present work is to find the formulas to calculate the maximum angular displacement α₀ and the phase lag ϕ during the vibration of the oscillator + blade system, and for this purpose, a simplifying assumption can be made, that is, to accept that the resulting motion is harmonic with the same excitation frequency ω of the excitation torque T_em. This hypothesis is the same one adopted by Den Hartog, in proposing his approximate solution [34], stating that, if the aim is only to find the formulas to calculate the maximum angular displacement α₀ and the phase lag ϕ during the vibration of the system, no error is committed by adopting a harmonic resulting motion with the same frequency of the excitation torque T_em. Obviously, the motion is not harmonic for two reasons: (1) the springs are nonlinear (quadratic); (2) the friction torque is not a viscous friction torque as it appears in Equation (17), but it is a dry Coulomb friction torque ±T_F. The assumption of a harmonic motion with the same excitation frequency as the excitation torque but with phase lag ϕ gives the following solution for the angular displacement α:

$$\alpha =\mathrm{a}·\sin \omega t+\mathrm{b}·\cos \omega t$$

(18)

where a and b are unknown coefficients of integration.

Differentiating the first time, the angular velocity is obtained:

$$\dot{\alpha }=\mathrm{a}·\omega ·\cos \omega t-\mathrm{b}·\omega ·\sin \omega t$$

(19)

Differentiating a second time, the angular acceleration is obtained:

$$\ddot{\alpha }=-\mathrm{a}·{\omega }^2·\sin \omega t-\mathrm{b}·{\omega }^2·\cos \omega t$$

(20)

Combining Equations (17)–(20) and imposing that Equation (17) is satisfied in the two orthogonal conditions of motion, ωt = 0 and ωt = π/2, gives the following system of two equations in the unknowns a and b:

$$\omega t=0:-J{·\omega }^2·\mathrm{b}+\left(k+2K·{\alpha }_{cut}\right)·\mathrm{b}+K·{\mathrm{b}}^2+C·\omega ·\mathrm{a}=T_{em}$$

(21)

$$\omega t=\pi /2:-J{·\omega }^2·\mathrm{a}+\left(k+2K·{\alpha }_{cut}\right)·\mathrm{a}+K·{\mathrm{a}}^2-C·\omega ·\mathrm{b}=0$$

(22)

The system of Equations (3), (21) and (22), which gives the dumping constant C, is solvable with iterative methods. However, Equation (3) needs the value of the angular maximum displacement α₀. The latter can be easily obtained from the following equation:

$${\alpha }_0=\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}$$

(23)

Therefore, it is necessary to solve with an iterative method, for example, that of a spreadsheet, the system of Equations (3) and (21)–(23).

Finally, the values obtained for the quantities a and b allow for deriving the phase lag of the motion with respect to the excitation torque T_em. The inversion of the equation $\tan \varphi =a/b$, considering that the function tan ϕ is periodic, gives

$$\varphi =\arctan {\displaystyle \frac{a}{b}}+j\pi \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}:j=0,1,2,\dots \mathrm{\infty }$$

(24)

In this case, j = 1.

It should be noted that the solution found, i.e., the values of a, b, α₀, and ϕ, satisfy the differential Equation (17) at the conditions ωt = 0 and ωt = π/2, i.e., when the oscillatory motion has reached the central and extreme points. This is sufficient to determine the correct values of α₀ and ϕ but excludes that Equation (17) is satisfied at all other points of the motion since the result was obtained under the simplifying hypothesis of harmonic motion.

2.4. Condition to Avoid the Transmission of Vibrations to the Tree Digger Machine Frame

When the operator activates the screw gear (9) (Figure 2), it rotates the vibrating blade to cut the soil. The reaction of the soil induces the cutting torque T_cut on the blade (Figure 5) and, therefore, a static deformation of the springs α_cut. Equation (14) highlights the effect of the cutting torque T_cut, known from Equation (13), on the angular shortening α_cut of two springs, while the other two undergo angular stretching equal to α_cut.

To the static angular shortening α_cut of the first two springs, the dynamic one of the oscillations is added, which has a maximum value equal to the maximum angular displacement α₀. Therefore, these two springs are subjected to a total angular shortening α_cut + α₀. This angular shortening becomes a shortening expressed in mm, x_cut + x₀:

$$x_{cut}+x_0=\left({\alpha }_{cut}+{\alpha }_0\right)·b_s$$

(25)

where b_s (mm) is the lever arm of the springs. The quantity x_cut + x₀ must be less than the total available space S_t between the turns of the springs when these are unloaded, otherwise, the springs reaching the end of stroke and the vibration are transmitted to the tree digger machine frame and, therefore, also to the operator:

$$S_t\ge \left({\alpha }_{cut}+{\alpha }_0\right)·b_s$$

(26)

Having written that the total angular displacement is the sum of α_cut with α₀, one might think that the principle of superposition of effects has been applied, which with linear systems, when there are multiple forces or torques simultaneously causing the motion, consists of analyzing each single force or torque separately and the dynamics of each single consequent motion. Finally, we can superpose the effects, i.e., the motions obtained. This is not true for Equations (25) and (26), because the value of the angular displacement during the oscillation α, obtained through the solution of Equations (3) and (21)–(23), is not independent of the value of the static angular displacement α_cut. The latter is present in Equations (21) and (22) because in writing the differential Equation (16), from which (17) was obtained, the cutting torque T_cut and the consequent elastic reaction of the springs were also included, which cannot be separated due to the excitation torque T_em, since the springs are nonlinear.

Instead, it was possible to apply the principle of the superposition of effects in the previous work [31], where the differential equation of the oscillatory motion was written without the cutting torque T_cut and the consequent effect on the springs k·α_cut, because the linear springs allowed the dynamic analysis to be carried out, keeping the causes of the motion, respectively, T_em and T_cut, separate.

2.5. Soil Cutting Time and Energy Required

In previous works [2,30,31], the optimal value of at least 17 for the velocity ratio ${\dot{A}}_0/v_{cut}$ between the maximum oscillation velocity of the blade, ${\dot{A}}_0=\dot{{\alpha }_0}·D/2={\alpha }_0·\omega ·D/2,$ and the cutting velocity, $v_{cut}=(\pi ·D/2)/t_{cut},$ was found. Therefore, respecting the velocity ratio equal to 17, it is possible to calculate the cutting time t_cut of the soil corresponding to the semicircular stroke of diameter D completed by the blade with the following expression:

$$t_{cut}={\displaystyle \frac{17·\pi }{{\alpha }_0·\omega }}$$

(27)

The power required for the oscillation P_o is given by the following equation:

$$P_o={\left(T_{em}\right)}_{eff}·{\left({\dot{\alpha }}_0\right)}_{eff}·\cos \left(\psi \right)={\displaystyle \frac{1}{2}}·T_{em}·{\alpha }_0·\omega ·\cos \left(\varphi -{\displaystyle \frac{\pi }{2}}\right)$$

(28)

where the excitation torque ${\left(T_{em}\right)}_{eff}$ and the angular velocity of the oscillation of the blade ${\left({\dot{\alpha }}_0\right)}_{eff}$ are the effective values, equivalent to the maximum values $T_{em}$ and ${\dot{\alpha }}_0$ multiplied by $\sqrt{2}/2$; the maximum angular velocity of oscillation is ${\dot{\alpha }}_0={\alpha }_0·\omega$; and the phase lag ψ is the angle between the excitation torque T_em and the angular velocity ${\dot{\alpha }}_0$. As this velocity is ahead by π/2 in comparison with the angular displacement ${\alpha }_0$, and this is behind by $\varphi$ in comparison with T_em, the result is $\psi =\varphi -\pi /2$.

The total power P_t can be obtained by adding the oscillating power P_o to the soil cutting power P_cut, which is

$$P_{cut}={\displaystyle \frac{T_{cut}·\pi }{t_{cut}}}$$

(29)

The cutting power P_cut is supplied by the external hydraulic motor to the worm screw (9 in Figure 2). The oscillating power P_o is supplied by another external hydraulic motor for the central gear wheel (in black) of the housing (3).

Assuming a transmission efficiency of 100%, the theoretical total energy W_t required during the vibratory cutting operation is the sum of the oscillating energy W_o and the cutting energy W_cut. These energies can be obtained by multiplying the power by cutting time t_cut. Equations (27)–(29) combined give the following total energy equation:

$${W_t=W_o+W_{cut}=P}_o·t_{cut}+P_{cut}·t_{cut}={\displaystyle \frac{1}{2}}·T_{em}·17\pi ·\cos \left(\varphi -{\displaystyle \frac{\pi }{2}}\right)+T_{cut}·\pi$$

(30)

2.6. Experimental Evaluation of Maximum Blade Displacement

To measure the maximum blade displacement A₀ while cutting the soil, a laser Doppler vibrometer was used, which measured the maximum displacement x_G0 of the Gear housing (Figure 2). The experimental value of A₀ is $A_0=x_{G0}·D/\delta$, where δ is the diameter of the gear housing and D is the blade diameter.

The cutting tests were performed using a tree digger machine with the modified oscillator coupled with the four different blades’ diameter D equal to 0.6, 0.9, 1.2, and 1.8 m, respectively. The cutting operation, repeated five times, was made on a typical tree-nursery soil, that is, a medium-textured soil with 20.9% moisture, a soil density ρ of 1605 kg m⁻³, and an external friction coefficient with the steel blade μ of 0.52.

3. Results

The mathematical modeling used to obtain the maximum angular displacement α₀ and the phase lag ϕ is constituted by Equations (21)–(24), to which the following must be added: Equation (3) to calculate the equivalent damping constant C, Equation (15) to calculate the dry friction torque T_F, Equation (12) to calculate the excitation torque due to the eccentric masses T_em, Equation (13) to calculate the cutting torque T_cut, Equation (14) to calculate the static angular shortening of the springs α_cut due to T_cut, Equations (8) and (9) to determine the angular quadratic spring rate (angular quadratic stiffness) K and the angular spring rate (angular stiffness) k. The equations just listed then become the design guidelines [38] for any other oscillator equipped with nonlinear springs.

Additionally, some data of the oscillator and the blade are needed, such as the moments of inertia J, the number of springs z and their lever arm b_s, the value of the eccentric masses m and their number n, their eccentricity y_G and their lever arm b_G, the quadratic spring rate K_l and the spring rate k_l, and the excitation frequency ω of the excitation torque. These data are shown in

Table 1. Geometric and dynamic features of the oscillator with nonlinear springs for very large blades.

Quantity	Symbol	Value
Spring rate, Equation (7)	kl (N m−1)	1,129,072
Quadratic spring rate, Equation (6)	Kl (N m−2)	184,963 × 103
Total number of springs	z	4
Lever arm of the springs	bS (m)	0.108
Angular springs rate, Equation (9)	k (N m rad−1)	26,339
Angular quadratic spring rate, Equation (8)	K (Nm rad−2)	466,000
Number of masses	n	4
Mass	m (kg)	6.0
Eccentricity	yG (m)	0.028
Mass lever arm	bG (m)	0.20
Moment of inertia	JO (kg m2)	2.5

. It is important to note that the lever arms b_s and b_G and the eccentricity y_G are the same as in the previous oscillator presented in [31].

Table 2. Geometric and dynamic features of the blades and systems (blades + oscillator).

Quantity	Symbol	Values
Blade Diameter	D (m)	0.6	0.9	1.2	1.8
Moment of inertia (blade)	JB (kg m2)	0.32	1.02	3.40	11.40
Moment of inertia (oscillator)	JO (kg m2)	2.50	2.50	2.50	2.50
Moment of inertia (system b + o)	J (kg m2)	2.82	3.52	4.70	13.90
Excitation frequency	ω (rad s−1)	220	420	420	420
Excitation torque, Equation (12)	Tem (Nm)	6505	23,708	23,708	23,708
Friction torque, Equation (13)	TF (Nm)	177	895	2830	14,325
Cutting torque, Equation (15)	Tcut (Nm)	1268	3190	6324	16,228
Coefficient of solution (18)	a (rad)	0.00280	0.00213	0.00383	0.00512
Coefficient of solution (18)	b (rad)	−0.0596	−0.0423	−0.0246	−0.00424

shows the data relating to the oscillator coupled with all the blade diameters available for the tree digger machine, including the very large one of 1.8 m, for which this new oscillator with quadratic springs was designed and built.

The choice to use these quadratic springs, that is, with a non-constant spring rate and increasing with the shortening of the springs, was made to allow the new oscillator to be versatile and therefore combinable with all blade sizes, from the smallest diameter of 0.6 m to the largest of 1.8 m, where the latter represents a novelty in cutting soil clod in tree nurseries.

Table 2 also shows the value of the excitation angular frequency ω, which is constant and equal to 420 rad s⁻¹ for all blades, except the smallest of 0.6 m, which instead has ω = 220 rad s⁻¹. In fact, in this case, the low values of both the moment of inertia and the cutting torque and the friction torque also require reducing the excitation torque through a reduction in the excitation frequency ω, as suggested by Equation (12). To obtain this, it is sufficient to reduce the rotation velocity of the external hydraulic motor which moves, through a set of gear wheels, the eccentric masses (Figure 2).

The values of the maximum angular displacement α₀ obtained by applying Equations (21)–(23) for the various blade diameters are shown in the histogram in Figure 6.

Figure 7 shows the histogram of the maximum blade displacement values $A_0={\alpha }_0·D/2$, where the maximum angular displacement α₀ is calculated with Equation (23). It is important to note that the largest blade has a value of 6 mm, which is the optimal minimum [2,30,31]. The other blades have displacement values A₀ above the optimal minimum.

The phase lag ϕ calculated with Equation (24) between the excitation torque T_em, intended as a rotating vector, and the maximum angular displacement α₀, also as a rotating vector, is shown in the histogram in Figure 8. The values are 2.265 rad (129.8°), 2.988 rad (171.2°), 3.091 rad (177.1°), and 3.094 rad (177.3°), respectively. The small blade (0.6 m) and the medium blade (0.9 m) have phase lags close to π rad (180°) due to the low values of the friction torque T_F compared to the excitation torque T_em (Table 2).

Figure 9 shows the histogram of the maximum spring shortening $x_{cut}+x_0$, obtained with Equation (25), vs. the total available space S_t equal to 21 mm, which corresponds to the sum of the free spaces between the coils when the springs are unloaded. It is important to note that inequality (26) is always respected. In no case do the springs reach the end of stroke, thus ensuring that there is never any transmission of vibrations to the digger machine frame and therefore to the operator.

The soil cutting time t_cut, calculated with Equation (28), is shown in the histogram of Figure 10. The assumption underlying the calculation of the cutting time with Equation (28) is the absence of asperities, such as stones or roots, along the semicircular path of the blade.

In the case where the blade encounters an obstacle, such as a large stone or a root, the cutting torque increases to a value such that the oil pressure of the hydraulic motor (Figure 2) reaches the safety pressure (17 MPa). The blade, with its vibration, slowly breaks or moves the stone or slowly cuts the root and then resumes its path at a constant soil cutting velocity. In these cases, the cutting time increases compared to the values of the histogram.

Figure 11 shows a comparison between the four blade diameters, 1.8, 1.2, 0.9, and 0.6 m, of the theoretical powers (i.e., net of transmission efficiencies) required for the oscillation P_o and for soil cutting P_cut. The power required for oscillation P_o was calculated with Equation (28); the power required to push the blade into the soil while cutting P_cut was calculated with Equation (29); the total power P_t was the sum of the previous two. It is interesting to note the high oscillation power P_o required for 0.9 and 1.2 m blade diameters. This is due to the high excitation frequency that influences, with its square, the excitation torque T_em (see Equation (12)), on whose value P_o directly depends (Equation (28)). It is possible to reduce the power P_o for the 0.9 and 1.2 m blades by lowering the excitation frequency, as has been achieved for the 0.6 m blade, without excessive loss of maximum blade displacement A₀. In fact, the reduction in ω determines its approach to the value of the natural frequency and, therefore, an increase in A₀, which compensates for the reduction in A₀ produced by the lower value of T_em. In the case of a very large blade of 1.8 m, its natural frequency is approximately ten times lower than the value of ω = 420 s⁻¹. However, as much the latter is reduced, it always remains very far from the value of the natural frequency; therefore, the recovery of A₀ is very limited so as to not compensate for the decrease in A₀ due to the reduction T_em.

Finally, Figure 12 shows the histogram of the theoretical total energy W_t (i.e., net of transmission efficiencies) required for the operation vs. the blade diameter. The total energy was calculated with Equation (30) and is the sum of the energy required to push the blade into the soil during cutting W_cut and the energy required for the oscillation W_o. In Equation (30), the energy W_cut is constant as a function of excitation frequency ω. Also, in Equation (30), the energy required for the oscillation is $W_o=P_o·t_{cut}=0.5·T_{em}·17\pi ·\cos \left(\varphi -{\displaystyle \frac{\pi }{2}}\right)$, where the excitation torque T_em depends directly on ω², while cos(ϕ−π/2) is almost inversely proportional to ω² in the range of ω values considered. Therefore, the total energy W_t required is also substantially invariant with respect to ω while, as seen in the discussion of Figure 11 about the reduction in ω for the 1.2 and 0.9 m blades, it produces a decrease in the oscillation power P_o and, hence, a decrease in the total power P_t, as well as an increase in the cutting time t_cut, as suggested by Equation (27).

The calculated maximum blade displacements A₀, shown in Figure 7, and those measured with the procedure seen in Section 2.6 are compared in

Table 3. Calculated and measured maximum blade displacements using the tree digger machine with the new oscillator with nonlinear springs.

	Quantity	Symbol	Values
	Blade diameter	D (m)	0.6	0.9	1.2	1.8
	Excitation frequency	ω (s−1)	220	420	420	420
Calculated	max. displacement	A0 (mm)	17.9	19.1	15.0	6.0
Measured	max. displacement	A0 (mm)	17.1	18.0	13.9	5.7
	Relative error	RE (%)	4.5	5.8	7.2	5

. For all diameters, the experimental value of the maximum blade displacement A₀ is lower than the theoretical ones, with a mean relative error of 5.6%. This difference can be explained partially by the damping due to the lubricant inside the oscillator and partially by the approximation introduced when the dry friction torque ±T_F was substituted with the viscous friction torque $C·\dot{\alpha }$.

4. Conclusions

To cut the hemispherical soil clod containing the roots system of small, medium, and large trees in nurseries, a semi-circular vibrating blade digger machine with diameters of up to 1.2 m can be conveniently used. The ease of use and speed of operation performed by these machines have led nurserymen to request machines with blades of up to 1.8 m in diameter for digging very large trees, which would otherwise be dug with slower and more expensive traditional digger machines with spades.

The heart of the vibrating blade digger machine is the mechanical oscillator, which has been subjected to a dynamic analysis for blades up to 1.2 m in previous work [31]. This analysis was performed by applying the approximate solution method of the differential equation of the forced vibration with Coulomb friction proposed by Den Hartog [34].

Before starting the present work, the same dynamic analysis was performed on 1.8 m blades suitable for very large trees. It was found that increasing the eccentric masses and the spring stiffness allowed the oscillator to vibrate the 1.8 m blade optimally without the springs reaching the dangerous end of the stroke. However, it was found that it was not suitable for smaller blades. Therefore, a modified oscillator was designed that would work well with all blade diameters because it was able to match the excitation torque and spring torque to the blade size.

The excitation torque T_em would have adapted through the variation of the excitation frequency ω, that is, the rotation velocity of the eccentric masses. The spring torque T_springs would have adapted using quadratic springs ($T_{springs}\propto K·{\alpha }^2$) instead of linear ones ($T_{springs}\propto k·\alpha$), where α is the angular displacement.

To mathematically describe the modified oscillator with nonlinear (quadratic) springs coupled with the blade on which the soil acts with dry friction, a new approximate solution of the differential equation of motion of the blade + oscillator system was found. Furthermore, a study was performed to identify the geometry and dynamics of the quadratic spring, finding that the conical coil spring with variable pitch was the most suitable.

In seeking the approximate solution of the new differential equation with quadratic springs and dry friction, the suggestion of Den Hartog [34] was adopted when he developed the approximate solution for systems with linear springs and dry friction. In fact, he said that if the aim is to find the formulas to calculate the maximum angular displacement α₀ and the phase lag ϕ during the vibration of the system (in this case, of the oscillator + blade), a harmonic resulting motion with the same frequency of the excitation torque T_em, can be assumed.

Starting with this suggestion, the harmonic solution, with the two unknown coefficients of integration, was implemented in the differential equation. This was written for two special points of the forced oscillatory motion, the central one and the terminal one, thus providing a system of two equations that, once solved, gave the value to the two unknown coefficients.

In this way, the harmonic solution satisfies the differential equation of the forced vibration of the oscillator + blade system in these two points of the motion. This is sufficient to precisely determine the two unknown coefficients, which, in turn, easily provide the maximum angular displacement α₀ and the phase lag ϕ.

Another approximation introduced to solve the differential equation was the replacement of the dry Coulomb friction torque ±T_F with a viscous friction torque $C·\dot{\alpha },$ under the condition that the work dissipated by the two different friction torque was equal, as suggested by Den Hartog himself [35].

Ultimately, the result was mathematical modeling that allowed us to design and simulate the behavior of the system with blades of four different diameters, namely, 0.6, 0.9, 1.2, and 1.8, covering the entire range. For all diameters, the oscillator was able to vibrate the blades optimally, ensuring a maximum blade displacement A₀ no less than about 6 mm and a velocity ratio ${\dot{A}}_0/v_{cut}$ of 17 and preventing the springs from going to the end of the stroke during soil cutting.

Following the construction of these conical coil springs and the modification of the oscillator with the increase in the eccentric masses, experimental tests were carried out to measure the maximum blade displacement A₀ during the cutting of the soil. They confirmed the good accuracy of the proposed mathematical modeling of the forced vibrations with nonlinear quadratic springs and dry Coulomb friction. In fact, the maximum and mean error calculated between the maximum blade displacement A₀ predicted by the modeling and the measured one were 7.2% and 5.6%, respectively.

In conclusion, the study carried out in this work will allow the use of vibrating blade tree diggers in nurseries even for very large trees, making the operation faster, more efficient, and economical. Tree digger machine users will probably ask for a further blade enlargement beyond 180 cm in diameter for extremely large trees in the future. It remains to be seen whether this type of oscillator will be able to operate or whether its configuration will have to be radically rethought.