Grass trimmer handle vibration reduction by imposing node method using vibration absorber

Abstract

The purpose of this investigation is to apply imposing node technique for vibration reduction of petrol engine grass trimmer handle using vibration absorber. The algorithm is devised to find the required resonance frequency of the absorber to impose node at desired location on beam. The numerical simulations are performed to find the usefulness of absorber for imposing nodes at the handle attachment location on timer pipe. The experimental test is conducted to validate the numerical results and to evaluate the vibration suppression during no cutting and cutting conditions. The vibration total value measured at handle of grass trimmer shows that absorber is useful in handle vibration reduction.

Keywords

Grass trimmer handle vibration vibration absorber imposing node vibration isolation

Introduction

The petrol driven grass trimmers are generally used in maintenance of parks and areas with bushes. The operator of trimmer may subjected to large magnitude of hand-arm vibration and may cause complex vascular and neurological and musculoskeletal disorder, collectively named as hand-arm vibration syndrome. There are number of ways by which vibrations of the grass trimmer can be minimized. The first approach is to eliminate or reduce the source of vibration, that is, engine vibrations. However, for I.C engines, vibrations cannot be eliminated completely. Second technique is to use of vibration isolator,^1–3 in which the elastic member, that is, isolator is inserted between the grass trimmer handle and the source of vibration. The limitation of this technique is that in order to accomplish the vibration isolations, the excitation frequency has to be greater than $\sqrt{2}$ times the natural frequency of the system. Alternatively, the excessive vibrations of the grass trimmer are suppressed by attaching a tunable vibration absorber (TVA) to the system.⁴ The TVA has certain advantages over other techniques of vibration reduction. It is external to the structure; therefore, re-installation of the device is not required. Nevertheless, a TVA can be designed and tested before installation.

Research has shown that the vibration of a desired location of the beam is suppressed by the proper selection of the TVA parameters, that is, mass and stiffness.^5–10 Hao and Ripin¹¹ applied the imposing node technique to reduce handle vibration of grass trimmer. The engine is mounted on operator back and the MATLAB routine global search was utilized to obtain the solution of equation. However, the damping in the absorber which affects its performance in the vibration reduction is not considered while modeling. Also the absorber attachment location is arbitrarily selected.

In the present research, imposing node technique is applied to reduce handle vibration of grass trimmer where the engine is mounted on the grass trimmer pipe. The algorithm developed by the present authors¹⁰ is modified to find resonance frequency considering damping in the absorber. Additionally, in present work, the plots are generated which are useful in selecting the absorber attachment location.

Grass trimmer system

A small petrol engine grass trimmer as shown in Figure 1 is considered in the present study. It is powered by an internal combustion engine mounted at one end of hollow circular pipe and rotating cutter head at the other end. The nylon string attached to the cutter head cuts the grass. The specifications of the grass trimmer used for numerical and experimental tests are listed in Table 1.

Figure 1.

Grass trimmer system.

Table 1.

Specifications of grass trimmer.

Engine	TB 43 Mitsubishi Japan
Displacement	42.7 cm³
Power output	1.3 H.P/6500 r/min
Fuel tank capacity	0.9 L
Handle	Double (W)
Pipe diameter	28 mm
Gear ratio	14:19
Size (L × W × H)	1810 mm × 610 mm × 480 mm
Weight	8.3 kg

The grass used for study is well developed and dense for which the speed of grass trimmer was around 7000–7500 r/min. For laboratory and field test, the speed of grass trimmer is selected as 7250 r/min. The hand-transmitted vibration is measured by means of the frequency-weighted root-mean-square (RMS) acceleration as per ISO 5349-1:2001.¹² Figure 2(a) and (b) shows the axis system for the trimmer pipe and handle, respectively. As the single-axis accelerometer is available, the sequential measurement (measure in one direction at a time) of vibration is carried using lightweight mounting block which is attached to vibrating surface as depicted in Figure 2(c). The accelerometer is attached to the block for the measurement of x-, y-, and z-direction and all the operating conditions remain same for three-axis measurements ISO 5349-2:2001.¹³

Figure 2.

Axis systems on (a) trimmer pipe, (b) trimmer handle, and (c) mounting block for vibration measurement.

Frequency analysis of grass trimmer

To find the primary source of vibration, the frequency spectrums of grass trimmer near engine support location and handle location for cutting condition are studied, as shown in Figure 3. The acceleration spectrum of grass trimmer measured in x-, y-, and z-axes are illustrated in Figure 4 for engine location. Only frequencies in the operating speed range of 0–200 Hz are shown. Table 2 lists the magnitudes of the acceleration in x-, y-, and z-axes at engine and handle location of the grass trimmer.

Figure 3.

Frequency spectrum measurement of trimmer during cutting operation.

Figure 4.

Input spectrum of trimmer in x-direction (a), y-direction (b), and z-direction (c) near engine.

Table 2.

Magnitude of acceleration in “g” at engine and handle location of grass trimmer.

	x-axis	y-axis	z-axis
Engine location	7.8	0.72	0.18
Handle location	4.7	0.77	0.52

Grass trimmer employed in this study had highest peak in the x-axes of magnitude 7.8g at 121 Hz. This frequency is correlated with the speed of the engine of 7250 r/min, which is the primary source of vibration excitation. The vibration magnitude at 121 Hz in the y-axis was 0.72g and peak in z-axis of magnitude 0.18g. There is a second peak in x, y, and z-axis with 90 Hz which is due to rotating cutter head speed of 5400 r/min (gear ratio 14:19). The magnitude of this source of excitation is small compared with engine excitation, hence neglected. Meanwhile, near handle location the magnitude of vibration at 121 Hz in the x-axis was 4.7g, y-axis was 0.77g, and in z-axis was 0.52g. These results provide a guideline for the vibration suppression of grass trimmer used in the present study.

Mathematical modeling and algorithm development

The grass trimmer modeled as free-free beam with effective mass of cutter head, handle, and engine as three lumped masses as shown in Figure 5. The grass trimmer system is subjected to harmonic excitation force of forcing amplitude $F$ with frequency $ω_{e}$ , $f (t) = F e^{j ω_{e} t}$ , acting at $x_{f} = 0.98 L$ due to engine vibration. The absorber is modeled as single-degree-of-freedom spring–mass–damper system having mass $m_{1}$ , variable stiffness $k_{1}$ , and damping coefficient $c_{1}$ of the absorber. The system parameters and material properties used in the numerical and experimental tests are listed in Table 3. The weight of each end mass of the absorber is selected as 200 g, so as to limit the percentage increase in the total weight of the trimmer system. Note that the structural damping of the absorbers used is considered as equivalent viscous damping as discussed in Dayou and Brennan.¹⁴

Figure 5.

Model of grass trimmer with structurally damped vibration absorber.

Table 3.

The system parameters and material properties used in the numerical and experimental test.

Length of the trimmer rod	L	1.5 m
Mass of rod	m	0.9 kg
Density of the trimmer rod material	$ρ_{m}$	2800 kg/m³
Young’s modulus of the trimmer rod	E	80 × 10⁹ N/m²
Mass per unit length of trimmer rod	$ρ$	0.6 kg/m
Area moment of inertia of trimmer rod	I	1.2 × 10⁻⁸ m⁴
Parameter $E I / L^{3}$ used to nondimensionalized stiffness		329.15 N/m
Parameter $ρ L$ used to nondimensionalized mass of the trimmer and absorber		0.9 kg
Weight of cutter head	$m l_{1}$	0.895 kg = 0.955ρL
Weight of handle	$m l_{2}$	0.833 kg = 0.925ρL
Weight of engine	$m l_{3}$	5.21 kg = 5.78ρL
Excitation force location	$x_{f}$	0.985L
Cutter head location	$x_{a}^{1}$	0.0180L
Handle location	$x_{a}^{2}$	0.68L
Engine location	$x_{a}^{3}$	0.985L
Total weight of end masses of absorber	$m_{1}$	0.2 kg = 0.22ρL
Damping ratio of absorber	$ζ_{1}$	0.001
Damping ratio of trimmer rod	$ζ_{b}$	0.08
Parameter $\sqrt{E I / (ρ L^{4})}$ used to nondimensionalized frequencies		18.30 N/m/kg
Parameter $F / (E I / L^{3})$ used to nondimensionalized vibration amplitude of beam and absorber masses		2 × 10⁻⁶ m

Using the assumed-modes method, the deflection of the beam (grass trimmer pipe) at any point $x$ along the length is given by

w (x, t) = \sum_{i = 1}^{N} ϕ_{i} (x) η_{i} (t)

(1)

where $N$ is the number of modes used in the assumed-modes expansion, $ϕ_{i} (x)$ are the eigenfunctions of the undamped pipe, and $η_{i} (t)$ are corresponding generalized co-ordinates.

Applying Lagrange’s equations and assuming simple harmonic motion with same response frequency as the excitation frequency, the following equations of motion are obtained

(\begin{array}{l} - ω_{e}^{2} [\begin{matrix} [M] & 0 \\ 0^{T} & [m] \end{matrix}] \\ + j ω_{e} [\begin{matrix} [C] & [R_{c}] \\ {[R_{c}]}^{T} & [c] \end{matrix}] \\ + [\begin{matrix} [K] & [R_{k}] \\ {[R_{k}]}^{T} & [k] \end{matrix}] \end{array}) [\begin{matrix} \underline{\bar{η}} \\ \underline{\bar{z}} \end{matrix}] = [\begin{matrix} F \underline{ϕ} (x_{f}) \\ 0 \end{matrix}]

(2)

where $\bar{η} = {[{\bar{η}}_{1} {\bar{η}}_{2} \dots {\bar{η}}_{N}]}^{T}$ , $\bar{z} = {[{\bar{z}}_{1}]}^{T}$ , $m = [m_{1}]$ , $c = [c_{1}]$ , $k = [k_{1}]$ .

The $N \times N$ $[M]$ , $[C]$ , and $[K]$ matrices of equation (2) are

\begin{array}{l} [M] = [M^{d}] + \sum_{i = 1}^{3} m l_{i} \underline{ϕ} (x_{a}^{i}) {\underline{ϕ}}^{T} (x_{a}^{i}) \\ [C] = [C^{d}] + c_{1} \underline{ϕ} (x_{p}) {\underline{ϕ}}^{T} (x_{p}) \\ [K] = [K^{d}] + k_{1} \underline{ϕ} (x_{p}) {\underline{ϕ}}^{T} (x_{p}) \end{array}

(3)

where $[M^{d}]$ , $[C^{d}]$ , and $[K^{d}]$ are diagonal matrices whose ith elements are $M_{j}$ , $C_{j}$ , and $K_{j}$ are the generalized masses, damping, and stiffnesses of beam.

Vector of the eigenfunctions of the beam

\begin{array}{l} \underline{ϕ} (x) = {[\begin{matrix} ϕ_{1} (x) & ϕ_{2} (x) \dots & ϕ_{N} (x) \end{matrix}]}^{T} \\ \underline{ϕ} (x_{a}^{i}) = {[\begin{matrix} ϕ_{1} (x_{a}^{i}) & ϕ_{2} (x_{a}^{i}) \dots & ϕ_{N} (x_{a}^{i}) \end{matrix}]}^{T} \\ ϕ (x_{p}) = [ϕ (x_{p})] \\ \underline{ϕ} (x_{f}) = {[\begin{matrix} ϕ_{1} (x_{f}) & ϕ_{2} (x_{f}) \dots & ϕ_{N} (x_{f}) \end{matrix}]}^{T} \end{array}

(4)

The matrices in equation (2) $[R_{c}]$ and $[R_{k}]$ are given by

[R_{c}] = [- c_{1} ϕ (x_{p})] [R_{k}] = [- k_{1} ϕ (x_{p})]

(5)

The damping matrix of the beam in nondimensional form is given by

[C^{d}] = 2 ζ_{b j} ω_{b j} [I] j = 1, \dots, N

(6)

In above equation, the experimental approach can be used,¹⁵ to measure the damping in a beam using experimental modal analysis which determines the modal damping ratios $ζ_{b j}$ and natural frequencies $ω_{b j}$ .

Using second equation of equation (2), the ${\bar{z}}_{1}$ is found to be

{\bar{z}}_{1} = \frac{ω_{1}^{2} + j ω_{e} c_{1} / m_{1}}{ω_{1}^{2} - ω_{e}^{2} + j ω_{e} c_{1} / m_{1}} ϕ_{i} \bar{η}

(7)

In equation (7), resonance frequency $ω_{1}$ and damping coefficient $c_{1}$ of the absorber are given by

ω_{1} = \sqrt{k_{1} / m_{1}} c_{1} = 2 ζ_{1} ω_{1}

(8)

where $ζ_{1}$ is the damping ratio of the absorber.

Equation (7) is substituted into the first equation of equation (2) and then solving for $\bar{η}$ to obtain

\bar{η} = {\begin{cases} - ω_{e}^{2} [M] + j ω_{e} [C^{d}] + [K^{d}] \\ + σ_{1} ϕ (x_{1}) ϕ^{T} (x_{1}) \end{cases}}^{- 1} F ϕ (x_{f})

(9)

where

σ_{1} = \frac{(m_{1} ω_{1}^{2} + j c_{1} ω_{e}) ω_{e}^{2}}{ω_{e}^{2} - ω_{1}^{2} - j c_{1} / m_{1} ω_{e}}

(10)

Substituting equation (9) into equation (2) to induce node at desired locations, $x_{q}$ along the beam requires that

\begin{array}{l} W (x_{q}) = \\ {\underline{ϕ}}^{T} (x_{q}) {\begin{cases} - ω_{e}^{2} [M] + j ω_{e} [C^{d}] + [K^{d}] + \\ \frac{(m_{1} ω_{1}^{2} + j c_{1} ω_{e}) ω_{e}^{2}}{ω_{e}^{2} - ω_{1}^{2} - j c_{1} / m_{1} ω_{e}} \underline{ϕ} (x_{p}) {\underline{ϕ}}^{T} (x_{p}) \end{cases}}^{- 1} \\ F \underline{ϕ} (x_{f}) = 0 \end{array}

(11)

Equation (11) is used to find the resonance frequency of the absorber $ω_{1}$ for absorber mass $m_{1}$ and damping coefficient $c_{1}$ at which the value of the displacement of beam $W (x_{q})$ becomes zero to impose nodes at $x_{q}$ . The algorithm based on iterative procedure developed in Patil and Awasare¹⁰ is modified to accommodate the damping in the absorber and the procedure is as given as follows.

The algorithm procedure to find the frequency of absorber $ω$ to impose a node:

Set the initial frequency of the absorber $ω < ω_{e}$ .

Compute $W (x_{q})$ using equation (11).

Increase the frequency of absorber $ω$ in steps up to $ω > ω_{e}$ and find $| W (x_{q}) |$ for each increment.

Select the frequency of absorber $ω_{1}$ at which $| W (x_{q}) |$ is minimum.

It should be noted that if the method does not converge to zero value of $| W (x_{q}) |$ or does not give the feasible value of $ω$ (the value of $ω$ return by iteration is very high), this implies that the node cannot be enforced at the selected location for the given value of mass $m$ . In such cases, the new value of mass $m$ is selected, to get a feasible value of $ω$ and to $| W (x_{q}) |,$ converge to zero. In addition, absorber attachment location $x_{p}$ can also be changed if no feasible solution is found.

It is important to mention that for damped beam and absorber, the amplitudes, that is, $W (x_{q})$ and ${\bar{z}}_{1}$ , are complex quantities. Hence, absolute values are used for comparison, for plots, and for list in the tables.

Numerical simulations

The code is developed in MATLAB using algorithm developed in the previous section to perform the numerical simulations for imposing nodes at the handle support location on trimmer pipe using vibrations absorber. The eigenfunctions $ϕ_{i} (x)$ , generalized masses $M_{i}$ , and generalized stiffnesses $K_{i}$ for grass trimmer which is modeled as free-free beam are given by

\begin{array}{l} ϕ_{i} (x) = \frac{1}{\sqrt{ρ L}} \\ (\begin{array}{l} \sinh β_{i} x + \sin β_{i} x + \\ \frac{\sin β_{i} L - \sinh β_{i} L}{\cosh β_{i} L - \cos β_{i} L} (\cos β_{i} x + \cosh β_{i} x) \end{array}) \end{array}

(12)

M_{i} = 1 and K_{i} = {(β_{i} L)}^{4} E I / (ρ L^{4})

(13)

where $β_{i} L$ satisfies the following transcendental equation

\cos β_{i} L \cosh β_{i} L = 1

(14)

In the following simulation results, the masses, frequencies, and vibration amplitudes are nondimensionalized by dividing by $ρ L$ , $\sqrt{E I / (ρ L^{4})}$ , and $F / (E I / L^{3})$ , respectively. The plots are generated as shown in Figure 6(a)–(c), for absorber resonance frequency, vibration amplitude of the beam at the node location, and absorber mass amplitude for different absorber attachment locations for given absorber mass. The plot shows that the trimmer pipe displacement at node location is minimum for absorber attachment location $x_{p} = 0.655 L$ . Choosing this absorber attachment location, resonance frequency required to impose node obtained from the plot is $41.74 \sqrt{E I / (ρ L^{4})}$ , that is, 121.56 Hz. The absolute steady-state response of the trimmer pipe with and without absorber is shown in Figure 7. Responses show that vibrations are suppressed to a great extent at node locations $x_{q} = 0.68 L,$ therefore isolating trimmer handle form vibration.

Figure 6.

The plots of absorber frequency (a), displacement of trimmer pipe at handle location (b), and absorber mass amplitude (c), versus trimmer pipe length, for the given absorber mass.

Figure 7.

The absolute steady-state response of the trimmer pipe with and without absorber.

Vibration reduction of grass trimmer handle due to variations in speed

The operating speed of grass trimmer varies during its operation. In order to evaluate the effectiveness of absorber with variations in operating speed, simulation is conducted to find the response of trimmer due to change in excitation frequency. Figure 8 shows the absolute steady-state response of the trimmer pipe with and without absorber with the variation in speed from 7200 to 7320 r/min. The dotted line in figure represents the response of trimmer pipe without absorber. It is observed that the node position gets shifted from handle location with change in speed till the displacement at handle location remains small compared with the displacement without absorber.

Figure 8.

The absolute steady-state response of the trimmer pipe with and without absorber subjected to variation in the speed.

The petrol engine grass trimmer is operated at the speed which suited the task. The speed of the trimmer for medium dense grass is around 6000 r/min whereas for well-developed and dense grass, the cutting speed is around 7500 r/min. The total operating range for the trimmer is around 1500 r/min. In the present study, test is conducted for cutting dense grass at speed around 7250 r/min and simulation results show that absorber is effective for the speed range of $\pm 60 r / \min$ . In order to apply imposing node method for medium dense grass, the absorber resonance frequency need to be change with the change in speed of the grass trimmer and this is the limitation of the present absorber.

Experimental testing of trimmer with and without absorber

In first step, a tuned vibration absorber was designed and constructed as illustrated in Figure 9(a). The dimensions and material properties of the absorber are listed in Table 4. The experimental modal analysis of the absorber was carried out to determine the resonance frequencies of absorber for different mass positions on the rods. Figure 9(b) describes the relationship between the mass position and the resonance frequencies of the absorber. The weight of the absorber is 480 g and the percentage increase in the weight of grass trimmer system due to absorber is 0.06%. Next, Figure 10 shows the experimental setup for vibration measurement of grass trimmer with and without absorber. The grass trimmer suspended horizontally in a “free-free” orientation approximated by two bungee cords, as shown in Figure 10. The vibration amplitudes are measured at thirty points on the trimmer surface by the accelerometer and recorded by vibration analyzer to plot experimental steady-state response of trimmer pipe rotating at 7250 r/min (i.e. $ω_{e} \approx 41.5 \sqrt{E I / (ρ L^{4})}$ or $f_{e} = 121 Hz$ ).

Figure 9.

(a) Experimental modal analysis of absorber and (b) variation of resonance frequency of absorber with mass position from base support.

Table 4.

Dimensions and material properties of absorber.

Density of rod and end mass	$ρ_{m}$	7830 kg/m³
Modulus of elasticity	$E$	2.1 × 10¹¹ N/m²
Length of rod	$l$	0.12 m
Diameter of rod	$d_{b}$	0.006 m
Diameter of end mass	$d$	0.0380 m
Length of end mass	$l_{a}$	0.012 m
Mass of the absorber end mass	$m_{a}$	0.1 kg

Figure 10.

Grass trimmer with vibration absorber.

To find the effectiveness of the absorber the x-direction, the absorber is mounted on the grass trimmer pipe at location 980 mm, that is, $x_{p} = 0.655 L$ and it is tuned by moving masses in or out such that the vibration at handle mount location of pipe is reduced to the minimum level. The vibration amplitudes were measured on the trimmer surface by the accelerometer and recorded by vibration analyzer to plot experimental steady-state response of trimmer pipe with absorber. Response in Figure 11(a) shows that vibrations are suppressed to a great extent at node locations, therefore isolating trimmer handle form vibration. Comparing the steady-state deformed shapes of trimmer pipe obtained from numerical experiments and experimental test, that is, Figure 7 with Figure 11(b), there is good agreement between numerical and experimental results.

Figure 11.

Experimental (a) measured and (b) nondimensional steady-state deformed shape of trimmer pipe.

It is evident from experimental steady-state response of trimmer pipe shown in Figure 11(a) that the amplitude of vibration is suppressed at handle location, that is, at 1020 mm from 101 to 25 µm. The percentage reduction in the amplitude at handle location is 75%. The amplitude at handle location can be further reduced by increasing the absorber mass but it increases the total weight of the system. Note that the displacement at node location cannot reduce to zero due to presence of the damping in the absorber and trimmer pipe.

Frequency-weighted RMS acceleration of handle vibration

The measurement of acceleration of handle vibration near hand grip position is carried out with and without absorber. The measured values of acceleration are listed in Table 5. The handle vibrations are measured by means of the frequency-weighted RMS acceleration, expressed in meter per second squared as per ISO 5349-1:2001. Table 6 illustrates the frequency weighting RMS acceleration ( $x_{h}$ , $y_{h}$ , and $z_{h}$ axes) and vibration total value measured at handle of grass trimmer during no cutting (laboratory testing) and cutting conditions (field testing). The reduction of vibration was calculated by subtracting the vibration acceleration of the grass trimmer when operating without the TVAs from the condition with the TVAs installed.

Table 5.

Acceleration (RMS) of grass trimmer handle.

Condition	Absorber	x-direction, m/s²	y-direction, m/s²	z-direction, m/s²	Total vibration, m/s²
No cutting	Without absorber	25.4	35.1	46	63.20
	With absorber	13.5	9.4	33	36.88
	Reduction	11.9	25.7	13	24.02
Cutting	Without absorber	39.2	33.4	48.2	70.54
	With absorber	14.1	10.2	35.1	39.17
	Reduction	25.1	23.2	13.1	31.37

RMS: root mean square.

Table 6.

Frequency-weighted RMS acceleration of grass trimmer handle.

Condition	Absorber	x-direction, m/s²	y-direction, m/s²	z-direction, m/s²	Total vibration, m/s²
No cutting	Without absorber	3.30	4.56	5.98	8.21
	With absorber	1.75	1.22	4.29	4.79
	Reduction	1.55	3.34	1.69	3.42
	% Reduction	47%	73%	28%	42%
Cutting	Without absorber	5.096	4.34	6.26	9.16
	With absorber	1.84	1.32	4.56	5.09
	Reduction	3.25	3.02	1.7	4.07
	% Reduction	63%	70%	27%	44%

RMS: root mean square.

The reduction of frequency-weighted RMS accelerations during no cutting condition was 1.55 m/s² in $x_{h}$ , 3.34 m/s² in $y_{h}$ , 1.69 m/s² in z_h-axis and total vibration by 3.42 m/s², yield the percentage reduction in the x_h-axis was 47%, 73% in the y_h-axis, 28% in the $z_{h}$ and 42% for the vibration total value. It was observed that reduction in frequency-weighted RMS during the cutting operation was highest in the $y_{h}$ axis, which is 70%. The vibration reduction for the $x_{h}$ axes and $z_{h}$ axes is 63% and 27%, respectively, and total vibration reduction was 44%.

Conclusion

The variable stiffness vibration absorber is used to suppress the handle vibration of petrol engine grass trimmer by imposing node at handle support location. The absorber resonance frequency required to impose node at desired location on beam is determined by the algorithm developed in the present investigation. The design plots are generated to assist the selection of the absorber attachment location on grass trimmer beam. The numerical simulation shows that the vibrations at handle support location on the trimmer pipe are suppressed by the absorber and it is effective for the speed range of $\pm 60 r / \min$ . The result shows that the total vibrations are reduced by 42% for noncutting condition and by 44% for cutting operation.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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