Kinematic Modeling and Simulation of Dual-Sided Shaper Machine: A Newton-Euler and Lagrangian Approach

GamachisRagasaGutata1

GalanaAbayKebedee1

GurmessaHorataAbbera2

GalanaAbayKebede1✉Emailgalana.abay@hu.edu.et

Department of Mechanical EngineeringDambi Dollo UniversityDambi DolloEthiopia

2Department of Mechanical EngineeringHawassa Institute of TechnologyHawassaEthiopia

Gamachis Ragasa Gutata¹, Galana Abay Kebedee² and Gurmessa Horata Abbera³

¹Department of Mechanical Engineering, Dambi Dollo University, Dambi Dollo, Ethiopia

^,2,3 Department of Mechanical Engineering, Hawassa Institute of Technology, Hawassa, Ethiopia,

Keywords:

Equation of motion

Lagrange approach

MATLAB

Newton-Euler approach

Scotch yoke mechanism

Correspondence to

Galana Abay Kebede /galana.abay@hu.edu.et

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract

Most industries have various types of reciprocating machines for performing machine operations on small work pieces. Usually, the onside shaper machine is conventional and removes material from one job only at forward stroke thus, more machining time is needed to complete the production. To overcome this problem, a small dual shaper machine operating via a scotch yoke mechanism is developed for machining two work pieces at the same time to reduce the machining time and increase the production rate. Additionally, it has fewer moving parts than the conventional shaper does. For smooth operation of this machine it is important to develop an equation of motion for the whole system. However, most papers focus on mechanically designing the system for strength rather than its kinematic analysis. This project focuses on solving kinematic problems, and the equation of motion was developed via both the Newton–Euler approach and the Lagrange approach for proper motion of the whole system. The results were simulated via the MATLAB code.

1. Introduction

A shaper is a type of machine tool that uses linear relative motion between work piece and a single-point cutting tool. It is an important machine tool available in almost all industry workshops and is analogous to a lathe, except that it is linear instead of helical. Samuel Bentham developed a shaper between 1971 and 1793 [1], [2]. However, Reo (1916) credits James Nasmyth with the invention of the shaper in 1836 [3], [4]. The shaper was very common in industrial production from the mid19th century through the mid-20th century. A typical shaper machine operates via a principle of a quick return mechanism in which materials are processed at one end and the other end remains idle. However, in a dual side shaper machine, materials are processed at both ends, which becomes advantageous compared with the usual shaper[2], [5].

Currently, industries are under increasing pressure to achieve high production rates while minimizing time, operational costs, and energy consumption[6]. In this context, the adoption of a dual side shaper machine presents a significant advancement over conventional single side shaper machines[7], [8], [9]. Unlike traditional shaping machines, which operate with a single tool cutting in one direction, the dual side shaper incorporates two cutting tools positioned on opposite sides of the workpiece. These tools operate simultaneously and reciprocate in opposite directions, effectively doubling the material removal rate within the same cycle time[10],[11].

The primary advantage of this configuration is a substantial reduction in machining time and overall production cost. This improved efficiency directly contributes to increased productivity and throughput in manufacturing processes, especially in high-volume production environments such as automotive, heavy machinery, and fabrication industries[12], [13].

Additionally, the dual side shaper is designed with fewer moving components compared to its single-side counterpart. The reduction in mechanical complexity not only decreases the likelihood of mechanical failure but also leads to lower maintenance requirements, improved reliability, and extended machine life[14], [15].

This system typically uses a single power source—such as an electric motor that drives a central crank or cam mechanism. Power is transmitted through a set of gears and linkages, which are engineered to convert rotary motion into the linear reciprocating motion required by the shaping tools. The gear system can be configured to adjust cutting speed as needed, allowing operators to tailor the process for different materials and desired surface finishes. This adaptability improves machining flexibility without requiring separate power systems for each tool head [16].

Moreover, the machine can be integrated with automation and control systems, allowing for programmable shaping cycles, improved precision, and real-time monitoring. Such enhancements support smart manufacturing practices, aligning with Industry 4.0 goals. The dual side shaper machine offers a robust and efficient solution for modern manufacturing demands. By minimizing downtime, optimizing resource usage, and increasing cutting efficiency, it significantly enhances the overall productivity and cost-effectiveness of shaping operations [15], [16].

Fig. 1

Conventional one-sided shaper machine[19]

To solve such problems, it is important to design a dual-sided shaper machine using scotch yoke mechanism, as currently, industries are trying to achieve high production rates at minimal amounts of time, cost, etc. In this paper, we describe a dual-sided shaper machine using a scotch yoke mechanism that can be used in industries for the cutting process[20], [21]. Most studies have focused on the strength of the system rather than dynamical analysis, and less research has been done on kinematic analysis. Therefore, this paper focuses on kinematic analysis to develop an equation of motion analytically determines the position, velocity and acceleration of points on the system and writes a math lab code[19],[22].

The aim of this work to provide kinematic ally well-designed dual shaper machine with fewer moving components and increasing production rates. Specifically, governing the equation of motion via Newton’s Eulers and the Lagrangian approach, the same results of the two approaches are checked mathematically, geometrically and via 3D modeling of the dual shaper machine via SOLIDWORK.

The motion of the dual side shaper machine is simulated via MATLAB programming software, the numerical solution for the dual side shaper machine is obtained via MATLAB, and displacement, velocity and acceleration graphs for the motion of the dual side shaper machine are drawn via MATLAB.

2. Materials and Methods

Dual sided shaper machine has different components to accomplish its objectives[23], [6]. Therefore, the 3D model of the machine and its components are shown in Fig. 2 below. The components of the dual-sided shaper machine by the scotch yoke mechanism are as follows: -frame, crank, connecting rode, slotted bar, shaft, Plummer block, cutting tool, bolts and nuts, pulleys and v-belts and feed table[10], [24].

Fig. 2

3D Model of a machine by solid work.

For this dual–sided shaping machine purpose, the Scotch Yoke mechanism is used for reciprocating the cutting tool, and to accomplish this objective, two connecting rods are welded to the slotted link. It is a reciprocating mechanism that converts rotational motion to reciprocating motion or vice versa. This mechanism is commonly used to control valve actuators in high-pressure gas and oil pipelines. The reason is that the scotch yoke mechanism advantages of higher torque output, fewer moving parts and hence a smoother operation can be used to perform various operations such as cutting and slotting. This process can be automated, and rotary motion can be directly converted into reciprocating motion.

Fig. 3

Scotch yoke mechanism [25].

Dual-sided shaper designed for cutting two flat pieces by a tool at one time. The operation of machine is simplified to few simple operations involving a motor and tool head arrangement (i.e., pulley and belt), [24]. The smaller pulley is driven by a motor, and by the use of V-belts, the speed of the motor is reduced and transmitted to the larger pulley. The larger pulley transmits the rotation to the crank that rotates about its axis by increasing the speed to some extent, so the yoke slides inside the slot of the slotted plate[26], [27]. As the crank rotates, the slotted bar reciprocates due to the sliding of the yoke (crank pin). The connecting rod attached to the slotted plate on both sides reciprocates as the crank rotates. The cutting tool is attached at both sides of the connecting rod to carry out the cutting operation, and the feed of the work piece is given for the feed table[26],. Since the cutting tool is placed on both sides, operations can be performed on both sides of the machine, i.e., the return stroke at one end is converted into cutting stroke at the other end, thereby reducing the production time and increasing the metal removal rate[27], [30].

2.1 Mathematical Modeling

The length of the cutting stroke is the main important parameter for the shaper machine. In a dual-sided shaper machine using a scotch yoke mechanism, this length is dependent mainly on the circumference of the crank. Considering these facts, the following specifications are intended for our analysis.

Radius of the crank, r = 300 mm

Length of cutting stroke, L = 2πr/2 = πr ≅ 942mm

The height of cutting tool is h = 50mm

Total length from the center of the crank to the end edge of the cutting tool at maximum at maximum extension of the connecting rod,

$\:{L}_{tot}$

=1320 mm.

Fig. 4

Free body diagram of the machine at the max. and min. positions.

Fig. 5

Position representation in 2D coordinate system.

The position of any point on a system can be represented by a 2D Cartesian coordinate system (x, y). The position of the yoke pined to the crank at radial distance r can be represented by point P. The linear position of the cutting tool is represented by

$\:{x}_{Q}$

, and the length of the belt from the datum to the larger pulley is represented by Sb.

Therefore,

a) The position of yoke

$\:{r}_{p}={x}_{p}\:i+{y}_{p}\:j\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(1\right)$

$\:\text{W}\text{h}\text{e}\text{r}\text{e},{X}_{P}=r\text{cos}\theta\:\:\text{a}\text{n}\text{d}\:{Y}_{P}=r\text{sin}\theta\:$

$\:{r}_{p}=r\text{cos}\theta\:i+r\text{sin}\theta\:j\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(2\right)$

b) Linear position of the cutting tool

$\:{x}_{Q}=r\left(\text{cos}\theta\:\right)+a\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(3\right)$

Where, 𝑎 is the length between the link slot and the cutting tool.

c) Position analysis of the belt

$\:{l}_{b}=2{S}_{b}+{r}_{1}\beta\:+{r}_{2}\theta\:\:\:\:\:\:\:\:\:\:\:\:\:\left(4\right)$

Where,

$\:{l}_{b}$

-total length of the belt

$\:{S}_{b}$

-length of the belt between two pulleys

$\:\beta\:$

-angular velocity of the smaller pulley

$\:\theta\:\:$

-angular velocity of the larger pulley and crank

d. Velocity of the yoke

The velocity of a certain rigid body is obtained by the time derivative of its position.

Note

the velocity of the crank is equal to the velocity of the yoke

$\:{v}_{p}=\frac{d{r}_{p}}{dt}=-r\dot{\theta\:}\text{sin}\theta\:i+r\dot{\theta\:}\text{cos}\theta\:j\:\:\:\:\:\:\:\:\:\:\:\left(5\right)$

Assuming that there is no slip between the pulley and the belt, we can relate the angular velocity of the smaller pulley 𝛽 and the angular velocity of the larger pulley 𝜃.

$\:\dot{\beta\:}=\frac{{r}_{2}}{{r}_{1}}\dot{\theta\:}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(6\right)$

The angular velocity of the larger pulley is equal to the angular velocity of the crank 𝜃.

e) Velocity of the cutting tool

$\:{v}_{Q}=\frac{d{x}_{Q}}{dt}\:=\:\frac{d(rcos\theta\:+a]}{dt}\:=\:-\text{r}\dot{\theta\:}\text{sin}\theta\:i\:\:\:\:\:\:\:\:\:\:\:\left(7\right)$

f) Velocity of the belt

Derivations of Eq. (4

$\:\:)$

with respect to time yield the velocity of thebelt,

$\:{,v}_{b}\:$

From Eq. (6

$\:\:)$

$\:{v}_{b}\:=\:\frac{-1}{2}({r}_{1}\dot{\beta\:}+{r}_{2}\dot{\theta\:}$

)

The acceleration of a certain body is determined by derivation of its velocity with respect to time. (i.e.

$\:a=\frac{dv}{dt}$

g) Acceleration of yoke

$\:{a}_{p}\:=\:\frac{d{v}_{p}}{dt}\:=-\left(r\ddot{\theta\:}\text{sin}\theta\:+r{\dot{\theta\:}}^{2}\text{cos}\theta\:\right)i+\left(r\ddot{\theta\:}\text{cos}\theta\:-r{\dot{\theta\:}}^{2}\text{sin}\theta\:\right)j\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(9\right)$

h) Acceleration of the cutting tool

$\:{a}_{Q}\:=\:\frac{d{v}_{Q}}{dt}\:=-(r\ddot{\theta\:}\text{sin}\theta\:+r{\dot{\theta\:}}^{2}\text{cos}\theta\:\left)\:i\:\:\:\:\:\:\:\:\:\right(10\:)$

k) Acceleration of the belt

$\:{a}_{b}\:=\:\frac{d{v}_{b}}{dt}\:=-{r}_{2}\ddot{\theta\:}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(11\right)$

2.2 Newton–Euler method formulation of equation of motion

For rotational motion, the work done equal to change in rotational kinetic energy [28]

Newton second law,

$\:\:\sum\:_{\varvec{i}=1}^{\varvec{n}}{\varvec{F}}_{\varvec{i}}$

$\:\sum\:_{\varvec{i}=1}^{\varvec{n}}{\varvec{m}}_{\varvec{i}}{\varvec{a}}_{\varvec{i}}$

for linear motion, where

$\:\:{,M}_{i}$

-Inertial moment

$\:\sum\:_{\varvec{i}=1}^{\varvec{n}}{\varvec{\tau\:}}_{\varvec{i}}$

$\:\sum\:_{\varvec{i}=1}^{\varvec{n}}{\varvec{M}}_{\varvec{i}}{\varvec{\alpha\:}}_{\varvec{i}}$

for rotational motion,

$\:{a}_{i}$

-Angular acceleration for rotation

$\:\sum\:_{\varvec{i}=1}^{\varvec{n}}{\varvec{\tau\:}}_{\varvec{i}}$

-the resultant torque of the system and its zero for kinematic analysis, T-total kinetic energy of the system

$\:{\varvec{M}}_{\varvec{i}}\dot{\theta\:=\frac{dT}{dt}}=\sum\:_{\varvec{i}=1}^{\varvec{n}}{\varvec{\tau\:}}_{\varvec{i}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(12\right)\:$

Since we need kinematic analysis (motion without considering the force causing the motion), it is possible to assume that no external forces or force due to weight is negligible. Therefore, the resultant torque becomes zero. Minimizes equation [24],:

$\:\dot{\frac{dT}{dt}}=\sum\:_{\varvec{i}=1}^{\varvec{n}}{\varvec{\tau\:}}_{\varvec{i}}=0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(13\:\right)$

Therefore, it is important to determine the kinetic energy of the system. We can represent the system with the following terms.

$\:{m}_{p}=\text{m}\text{a}\text{s}\text{s}\:\text{o}\text{f}\:\text{y}\text{o}\text{k}\text{e}\:\:\:\:\:$

$\:{m}_{Q}$

=mass of the cutting tool

$\:{m}_{b}=\text{m}\text{a}\text{s}\text{s}\:\text{o}\text{f}\:\text{t}\text{h}\text{e}\text{b}\text{e}\text{l}\text{t}\:$

$\:\:{m}_{s}$

=mass of the smaller pulley

$\:{m}_{r}$

=mass of the crank

$\:{m}_{l}$

=mass of larger pulley

$\:\:{m}_{c}$

=mass of connecting rode and yoke about its center of mass Note: Each mass (

$\:{m}_{i}$

), velocity (

$\:{v}_{i}$

) and acceleration (

$\:{a}_{i}$

) is about the center of mass.

$\:{v}_{l}$

=velocity of the larger pulley about its c.m.=

$\:{r}_{2}\dot{\theta\:}$

$\:{v}_{s}$

=velocity of the smaller pulley at c.m=

$\:{r}_{1}\dot{\beta\:}$

$\:{r}_{1}\frac{{{r}_{2}}^{2}}{{r}_{1}}\dot{\theta\:}$

$\:{r}_{2}\dot{\theta\:}$

$\:{v}_{c}$

Represents the velocity of the connected rod and slotted link welded together at approximately c.m. and is equal to the velocity of the cutting tool

$\:{v}_{Q}$

(i.e.,

$\:{v}_{c}{=v}_{Q}$

$\:{v}_{r}\:-r\dot{\theta\:}\text{sin}\theta\:i+r\dot{\theta\:}\text{cos}\theta\:j$

Where,

$\:{v}_{r}$

=Velocity of the crank=velocity of the yoke

$\:{Y}_{p}$

= position yoke from datum (fixed point O) = L+

$\:{y}_{p}$

=L+

$\:\:r\text{sin}\theta\:$

$\:{Y}_{Q}$

= position cutting tool from datum (fixed point O) = L+

$\:{l}_{1}$

= constant

$\:{Y}_{c}$

=position connecting rode and slotted link at c.m. from the datum (fixed point O) position cutting tool from the datum (fixed point O), [i.e.,

$\:{\:Y}_{Q}$

$\:{Y}_{c}=$

L+

$\:{l}_{1}$

= constant]

= position larger pulley from datum (fixed point O) = position crank from datum

$\:{,Y}_{r}$

(fixed point O) [i.e.,

$\:{Y}_{l}$

$\:{,Y}_{r}$

=L]

$\:{I}_{r,}{I}_{l}{,I}_{s}=$

Axial inertia of the crank, large pulley and smaller pulley, respectively=

$\:\frac{1}{3}{m}_{i}{{r}_{i}}^{2}$

$\:\text{T}=\sum\:_{i=1}^{n}\frac{1}{2}{m}_{i}{{v}_{i}}^{2}+\sum\:_{i=1}^{n}\frac{1}{2}{I}_{i}{\dot{\theta\:}}^{2}+{I}_{s}\left(\frac{{{r}_{2}}^{2}}{{{r}_{1}}^{2}}\right){\dot{\theta\:}}^{2}$

Before determining the kinetic energy, it is better to write the square of the velocities to reduce complexity.

Velocity of the yoke

From equation [5],

$\:{{v}_{p}}^{2}\:={\left(-r\dot{\theta\:}\text{sin}\theta\:i+r\dot{\theta\:}\text{cos}\theta\:j\right)}^{2}={r}^{2}{\dot{\theta\:}}^{2}{sin}^{2}\theta\:+{r}^{2}{\dot{\theta\:}}^{2}{cos}^{2}\theta\:={r}^{2}{\dot{\theta\:}}^{2}\:\:\:\:\:\:\:\:\:\left(16\right)$

Velocity of cutting tool

From equation [7],

$\:{{v}_{Q}}^{2}={r}^{2}{\dot{\theta\:}}^{2}{sin}^{2}\theta\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(17\right)$

Velocity of the belt

From equation [8],

$\:{{v}_{b}}^{2}=\:{\:{r}_{2}}^{2}{\dot{\theta\:}}^{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(18\right)$

Velocity of the remaining body about their cm

From equation [14],

$\:{{v}_{l}}^{2}={\:{r}_{2}}^{2}{\dot{\theta\:}}^{2}$

$\:{{v}_{s}}^{2}=\:{\:{r}_{2}}^{2}{\dot{\theta\:}}^{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(19\right)$

$\:{{v}_{c}}^{2}={r}^{2}{\dot{\theta\:}}^{2}{sin}^{2}\theta\:$

$\:{{v}_{r}}^{2}={r}^{2}{\dot{\theta\:}}^{2}$

Now, it is easy to calculate the kinetics. Using equation [15] and neglecting the inertial effect of the belt

$\:\:\frac{1}{2}{\dot{\theta\:}}^{2}\left[{I}_{r}+{I}_{l}+{I}_{s}\left(\frac{{{r}_{2}}^{2}}{{{r}_{1}}^{2}}\right)+\left({m}_{b}+{m}_{l}+{m}_{s}\right){\:{r}_{2}}^{2}+\left({m}_{p}+{m}_{r}\right){r}^{2}+\:\left({m}_{Q}+{m}_{c}\right){r}^{2}{sin}^{2}\theta\:\right]\:\:\:\:\:\:\:\:\:\:\:(20$

)

Derivation of equation [20] with respect to time; then, equation [13] can be written as

$\:\left[{I}_{r}+{I}_{l}+{I}_{s}\left(\frac{{{r}_{2}}^{2}}{{{r}_{1}}^{2}}\right)+\left({m}_{b}+{m}_{l}+{m}_{s}\right){\:{r}_{2}}^{2}+\left({m}_{p}+{m}_{r}\right){r}^{2}+\:\left({m}_{Q}+{m}_{c}\right){r}^{2}{sin}^{2}\theta\:\right]\ddot{\theta\:}\:+\:\left[\:\frac{1}{2}\left({m}_{Q}+{m}_{c}\right){r}^{2}\text{sin}2\theta\:\right]{\dot{\theta\:}}^{2}=0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(21\right)$

This equation of motion of a system does not consider acceleration due to gravity (or because effect of weight is neglected), assuming that no slip between the pulleys and the belt and friction between the connecting rod and its supports are small and negligible. The equation is a second-order ordinary differential equation of form a

$\:\ddot{x}$

$\:\dot{x}$

If we consider only the masses of crank, yoke, connecting rode and cutting tool and neglect the remaining masses, we can obtain the equation of motion for scotch yoke mechanism rather than the system[32].

$\:\:\left[{I}_{r}+\left({m}_{p}+{m}_{r}\right){r}^{2}+\:\left({m}_{Q}+{m}_{c}\right){r}^{2}{sin}^{2}\theta\:\right]\ddot{\theta\:}$

$\:\:\left[\:\frac{1}{2}\left({m}_{Q}+{m}_{c}\right){r}^{2}\text{sin}2\theta\:\right]{\dot{\theta\:}}^{2}=$

0 (22)

2.3 Lagrange method formulation of equation of motion

We can describe the motion of the whole system via generalized independent coordinates

$\:\theta\:$

and apply Lagrange approach to determine the governing equation of motion.

Assumptions

• There is no slip between the pulleys and the belt.

The friction between the connecting rod and its supports is small and negligible.

No external force on the system or force due to weight is negligible. Therefore, the resultant torque can be neglected.

The system is conservative.

The potential energy of the belt is small and negligible.

Neglecting the inertial effect of the belt

L = T-V (23)

Where, T= =

$\:\sum\:_{i=1}^{n}\frac{1}{2}{m}_{i}{{v}_{i}}^{2}+\sum\:_{i=1}^{n}\frac{1}{2}{I}_{i}{\dot{\theta\:}}^{2}+{I}_{s}\left(\frac{{{r}_{2}}^{2}}{{{r}_{1}}^{2}}\right){\dot{\theta\:}}^{2}$

= kinetic energy of the system

$\:\sum\:_{i=1}^{n}{m}_{i}g{y}_{i}$

= Potential energy of the system

$\:\sum\:_{i=1}^{n}{m}_{i}g{y}_{i}$

$\:{m}_{p}g{Y}_{p}$

$\:{m}_{Q}g{Y}_{Q}$

$\:{m}_{l}g{Y}_{l}$

$\:{m}_{c}g{Y}_{c}$

$\:{m}_{r}g{Y}_{r}$

(24)

$\:\frac{d}{dt}\left(\frac{\partial\:L}{\partial\:\dot{q}}\right)-\frac{\partial\:L}{\partial\:q}=0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(25\right)$

From equations [14] and [24],

$\:\text{V}=\left({m}_{p}+{m}_{Q}+{m}_{l}+{m}_{c}+{m}_{r}\right)gL\:+\:\left({m}_{Q}+{m}_{c}\right)g{l}_{1}\:+{m}_{p}g\:r\text{sin}\theta\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(26\right)$

Substitution of equations (24) and (25) into Eq. (16) results Lagrangian equation as given below.

$\:\:L=\:\frac{1}{2}{\dot{\theta\:}}^{2}\left[{I}_{r}+{I}_{l}+{I}_{s}\left(\frac{{{r}_{2}}^{2}}{{{r}_{1}}^{2}}\right)+\left({m}_{b}+{m}_{l}+{m}_{s}\right){\:{r}_{2}}^{2}+\left({m}_{p}+{m}_{r}\right){r}^{2}+\:\left({m}_{Q}+{m}_{c}\right){r}^{2}{sin}^{2}\theta\:\right]-\left[\left({m}_{p}+{m}_{Q}+{m}_{l}+{m}_{c}+{m}_{r}\right)gL\:+\:\left({m}_{Q}+{m}_{c}\right)g{l}_{1}\:+\:{m}_{p}g\:r\text{sin}\theta\:\right]$

We can define the motion of the whole system by independent generalized coordinate

$\:\theta\:$

Equation of motion (

$\:\varvec{f}\varvec{o}\varvec{r}\:\varvec{\theta\:}$

)

From equation [25],

$\:\frac{d}{dt}\left(\frac{\partial\:L}{\partial\:\dot{q}}\right)-\frac{\partial\:L}{\partial\:q}=0$

$\:\frac{\partial\:L}{\partial\:\theta\:}=\left({m}_{Q}+{m}_{c}\right){\dot{\theta\:}}^{2}{r}^{2}\text{sin}\theta\:\text{cos}\theta\:-{m}_{p}g\:r\text{cos}\theta\:\:\:\:\:\left(28\right)$

$\:\frac{\partial\:L}{\partial\:\dot{\theta\:}}=\dot{\theta\:}\left[{I}_{r}+{I}_{l}+{I}_{s}\left(\frac{{{r}_{2}}^{2}}{{{r}_{1}}^{2}}\right)+\left({m}_{b}+{m}_{l}+{m}_{s}\right){\:{r}_{2}}^{2}+\left({m}_{p}+{m}_{r}\right){r}^{2}+\:\left({m}_{Q}+{m}_{c}\right){r}^{2}{sin}^{2}\theta\:\right]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(29\right)$

$\:\frac{d}{dt}\left(\frac{\partial\:L}{\partial\:\dot{\theta\:}}\right)=\ddot{\theta\:}\left[{I}_{r}+{I}_{l}+{I}_{s}\left(\frac{{{r}_{2}}^{2}}{{{r}_{1}}^{2}}\right)+\left({m}_{b}+{m}_{l}+{m}_{s}\right){\:{r}_{2}}^{2}+\left({m}_{p}+{m}_{r}\right){r}^{2}+\:\left({m}_{Q}+{m}_{c}\right){r}^{2}{sin}^{2}\theta\:\right]+2\left({m}_{Q}+{m}_{c}\right){\dot{\theta\:}}^{2}{r}^{2}\text{sin}\theta\:\text{cos}\theta\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(30\right)$

By subtracting equation [24] from [26], equation [16] yields

This equation is a second-order ordinary differential equation of the form a

$\:\ddot{x}$

$\:\dot{x}+x$

=0, which is the governing equation of motion for the system of machines.

If we consider only the masses of the crank, yoke, connecting rode and cutting tool and neglect the remaining masses, we can obtain the equation of motion for scotch yoke mechanism rather than the system.

$\:\:\left[{I}_{r}+\left({m}_{p}+{m}_{r}\right){r}^{2}+\:\left({m}_{Q}+{m}_{c}\right){r}^{2}{sin}^{2}\theta\:\right]\ddot{\theta\:}+\:\left[\:\frac{1}{2}\left({m}_{Q}+{m}_{c}\right){r}^{2}\text{sin}2\theta\:\right]{\dot{\theta\:}}^{2}+\:{m}_{p}g\:r\text{cos}\theta\:=0\:\left(32\right)$

If we neglect the gravitational effect (potential energy), the Lagrangian equation motion of equations (31) and (32) is the same as the Newton Euler equation of equations (21) and (22).

2.4 MATLAB simulation for a dual-sided shaper machine

The simulation of the system is performed on the basis of the design specification stated in section 2.1. The MATLAB simulation code is shown below.

a. MATLAB code, which simulates motion:

%%% MACHINE DYANMICS MINI PROJECT ON DUAL SIDED SHAPER MACHINE BY SCOTCH YOKE MECHANISM

%%% TITTLE: - Determination of Equation of motion for Dual sided shaper machine by using scotch yoke

%%% mechanism

%%% Name:-GALANA ABAY ID NO GSR/9971/12

% MATLAB simulation result BY MATLAB R2018a

% all dimension given below are in millimeter(mm)

r = 300; % the radius of crank its dimension is by mm

a = 1000; % length of connecting rod holding a cutting tool on the positive x-axis

la=-1000;% length of connecting rod holding a cutting tool on the negative x-axis

ls = 942; % the length of slot in which the yoke slides through it

h = 50; % the height of the cutting tool

lt = 1320; % total length of the mechanism from the crank center to the end of the cutting tool

O=[0 0];% the position of point O or the origin crank

for teta = 0:0.3:20.5*pi% rotation of the crank with a 0.3 value gap to 20.5*pi

Xp = r*cos(teta); % x coordinate of point p

Yp = r*sin(teta); % Y coordinate of point p

p=[r*cos(teta) r*sin(teta)]; % x‒y coordinate of point p

% Distance from the end of the cutting tool to the minimum reciprocating distance

N = lt-r*cos(teta);

R = r*sin(teta);

X=[O(1) Xp];

Q=[O(1) Xp];

Y=[O(2) Yp];

T=[O(2) Yp];

% Ploting the line of crank by making the size of line 3 with color of black including points

plot(X,Y,'-ko','LineWidth',3,'MarkerEdgeColor','k','MarkerSize',2)

% ploting the line of crank making the size of line 3 with color of black including points

plot(Q,T,'-co','LineWidth',3','MarkerEdgeColor','k','MarkerSize',2)

% waiting till to plot any other lines or not to cancel the plot before

hold on

% The line of the crank in which it rotates

B_traj = viscircles([0 0],r,'linestyle','--');

% ploting the line of connecting rod making the size of line 3

plot(X,Y,'-bs','LineWidth',3,'MarkerEdgeColor','g', 'MarkerSize',5)

%ploting the line of connecting rode making the size of line 3

plot(Q,T,'-cs','LineWidth',3,'MarkerEdgeColor','g', 'MarkerSize',5)

XE = lt-N;

QE = lt-N;

YE = ls/3;

TE = ls/3;

XF = lt-N;

QF = lt-N;

YF=-ls/3;

TF=-ls/3;

X=[QE QF];

Y=[TE TF];

% ploting the line of slot link in which the yoke slides by making the size of line 3

plot(X,Y,'-ms','LineWidth',3,'MarkerEdgeColor','b', 'MarkerSize',2)

% ploting the line of slot link in which the yoke slides making the size of line 3

plot(Q,T,'-ms','LineWidth',3,'MarkerEdgeColor','b', 'MarkerSize',2)

XD = lt-N;

QD = lt-N;

YD = 0;

TD = 0;

XG = lt + a-N;

QG = lt + la-N;

YG = 0;

TG = 0;

X=[XD XG];

Q=[QD QG];

Y=[YD YG];

T=[TD TG];

% plot the line of cutting tool hold rode by making the size of line 3

plot(X,Y,'-bo','LineWidth',3,'MarkerEdgeColor','y', 'MarkerSize',2)

% ploting the line of cutting tool hold rode making the size of line 3

plot(Q,T,'-bo','LineWidth',3,'MarkerEdgeColor','y', 'MarkerSize',2)

H=[lt + a-N h/2];

V=[lt + la-N h/2];

J=[lt + a-N -h/2];

U=[lt + la-N -h/2];

X=[H(1) J(1)];

Q=[V(1) U(1)];

Y=[H(2) J(2)];

T=[V(2) U(2)];

% plotting the line of cutting tool rode by making the size of line 3

plot(X,Y,'-ks','LineWidth',3,'MarkerEdgeColor','r', 'MarkerSize',2)

% plotting the line of cutting tool rode making the size of line 3

plot(Q,T,'-ks','LineWidth',3,'MarkerEdgeColor','r', 'MarkerSize',2)

X=[lt-1200 lt-1000];% the

Q=[lt-1200 lt-1000];

Y=[30 30];

T=[30 30];

plot(X,Y,Q,T)

X=[lt-1200 lt-1000];

Q=[lt-1200 lt-1000];

Y=[-30 -30];

T=[-30 -30];

% The dimension of the window in which the simulation plot

axis([-2000 3000 − 2000 2000])

grid on

hold off

pause(0.005)

end

b) Motion analysis (position, velocity and acceleration)

As we have seen in the simulation code above, all links and points from the crank to the cutting tool are programmed and coded to have relative motion specifically to perform the shaping operation. The following MATLAB code shows the position, velocity and acceleration of selected points for the crank (point p) and cutting tool (point Q).

%MATLAB CODE FOR MOTION ANALYSIS CODED BY MATLAB R2018a

% The program intended to show the position, velocity and acceleration of the crank and cutting tool.

% All dimensions are in mm

% Note:-All representations of symbols are written in simulation code

r = 300;

a = 1000;

ls = 942;

h = 50;

lt = 1320;

teta = linspace(0,2*pi,50);

t = 0:0.03:10;

teta = 2*pi*t/3;

omega = 2*pi/3;

ang_teta = omega./t;

% Position of crank P, which is only under rotation.

Xp = r*cos(teta);

Yp = r*sin(teta);

r_p = sqrt(Xp.^2 + Yp.^2);

% Position of cutting tool Q, which is under translation only.

XQ = r*cos(teta) + a;

YQ = 0;

r_Q = sqrt(XQ.^2 + YQ.^2);

% The resultant velocity equation of crank p

Vp_x=-r*omega. *sin(teta);

Vp_y = r*omega. *cos(teta);

velo = sqrt(Vp_x.^2 + Vp_y.^2);

Vp = sqrt((-r*omega.*sin(teta)).^2+(r*omega. *cos(teta)).^2);

%Resultant velocity equation of cutting tool Q

VQ_x=-r*omega. *sin(teta);

VQ_y = 0;

velo2 = sqrt(VQ_x.^2 + VQ_y.^2);

VQ = sqrt((-r*omega. *sin(teta)).^2 + 0);

%%%%% The resultant acceleration equation of the crank

ap_x=-r.*ang_teta.*sin(teta)-r.*omega.^2. *cos(teta);

ap_y = r.*ang_teta.*cos(teta)-r.*omega.^2. *sin(teta);

acc = sqrt(ap_x.^2 + ap_y.^2);

ap = sqrt((-r.*ang_teta.*sin(teta)-r.*omega.^2.*cos(teta)).^2+(r.*ang_teta.*cos(teta)-r.*omega.^2. *sin(teta)).^2);

% the resultant acceleration equation of cutting tool Q

aQ_x=-r.*(ang_teta.*sin(teta) + omega.^2. *cos(teta));

aQ_y = 0;

acc2 = sqrt(aQ_x.^2 + aQ_y.^2);

aQ = sqrt((-r.*(ang_teta.*sin(teta) + omega.^2. *cos(teta))).^2 + 0);

plot(t,r_p,t,r_Q,t,Vp,t,VQ,t, ap,t,aQ)

title('Graphical Representation for Motion of crank and cutting tool','FontSize',12)

legend('Position of crank or rp','Position of cutting tool or xQ','Velocity of crank or Vp','Velocity of cutting tool or VQ','Acceleration of crank or ap','Acceleration of cutting tool or aQ')

xlabel('time(s)')

ylabel('motion in (mm, mm/s, mm/s^2)')

xlim([0 10])

ylim([-1 2000])

3. Results and Discussions

The Newton–Euler and Lagrange approaches yield the same results for the governing equation of motion, as shown in sections 2.2 and 2.3. The MATLAB simulation results are shown in the figure below.

Fig. 6

a) MATLAB simulation result picture, b) MATLAB simulation result at an angle, θ c) maximum positive and d) negative x- values

Point’s p and Q represent the positions of the yoke and cutting tool, and the lengths “a” and “la” are equal and represent the lengths of the connecting rods on the positive and negative x-axes. The remaining points shown in figure are selected to define distance for simulation, and their descriptions are written in the MATLAB code shown below. The MATLAB software shows the motions of all points, which represent the output results from the machines. MATLAB simulation is held by the software with the corresponding results of position, velocity and acceleration of the yoke, in which the connecting rode is assembled and the rotational and translational motion of the yoke results in translational motion (reciprocating motion) of the cutting tool attached to the end of a connecting rode at both sides of the slotted link to shape two workpieces at one time. These results are highly consistent with the manually expected results from the motion of each point on a body.

Fig. 7

Motion analysis of the crank and cutting tool

As we can see from the graph, the position of the cutting tool generates sinusoidal, with the time having an amplitude equal to the length of the cutting stroke, and its velocity is a cosine function. The results are cyclic as well as consistent throughout the rotation of the system which is consistent with the real calculations manually; for example, the position of the crank and the point p is constant if the rotation of the crank is constant. Additionally, its velocity is cyclic, which repeats itself after one revolution rotation. In general, the results of equation motion obtained from Lagrange and Newton–Euler methods are simulated via MATLAB.

4. Conclusions

This study successfully derived the governing equations of motion for a mechanical system using both the Newton–Euler and Lagrange methods, demonstrating their equivalence in predicting system dynamics. The MATLAB simulations validated the theoretical results, showing consistent cyclic behavior in the position, velocity, and acceleration of the yoke and cutting tool. The simulation outputs, including sinusoidal position profiles and cosine velocity functions, align precisely with manual calculations, confirming the accuracy of the derived models. The motion analysis revealed that the cutting tool undergoes reciprocating motion with a stroke length equal to the amplitude of its displacement, while the crank maintains a constant rotational speed, producing periodic velocity profiles. These results not only verify the theoretical formulations but also provide practical insights into the system's kinematic behavior, essential for optimizing machine performance in real-world applications.

Overall, this work bridges theoretical mechanics and computational simulation, offering a robust framework for analyzing similar mechanical systems. Future research could explore dynamic responses under varying loads or incorporate control strategies to enhance system efficiency. The consistency between analytical and simulated results underscores the reliability of the proposed approach, making it suitable for engineering applications requiring precise motion control.

Acknowledgments

This work was not supported by any Research Program.

Nomenclature-

-----------------------------------------------------------------

$\:{m}_{p}$

Mass of yoke

$\:{m}_{Q}$

: Mass of the cutting tool

$\:{m}_{b}$

: Mass of the belt

$\:{m}_{s}$

: Mass of the smaller pulley

$\:{m}_{r}$

: Mass of the crank

$\:{m}_{l}$

: Mass of larger pulley

$\:{m}_{c}$

: Mass of connecting rode and yoke

$\:T$

: Kinetic energy

Declarations

Conflict of Interest

• We have no competing financial interests or personal relationships that could have appeared to influence our work reported in this paper

Funding

The Research receives No fund

Data Availability

All data analyzed during the current study is available from corresponding author: Galana Abay Kebede, galana.abay@hu.edu.et

Author Contribution

Authors ContributionsGamachis Ragasa Gutata: Formal analysis, Writing - original draft, Writing - review & editing, Conceptualization, MethodologyGalana Abay Kebede: Investigation, Methodology, Writing - review & editing. Gurmessa Horata Abbera: Visualization, Writing - review & editing

Data Availability

Declaration of interest’sAll data analyzed during the current study is available from corresponding author: Galana Abay Kebede, galana.abay@hu.edu.et [Gamachis Ragasa Gutata]Researcher and LecturerDambi Dollo University, Ethiopia[gamachisragasa@dadu.edu.et ]

References

Pranav, B. A Dual Cutting Edge for A Single Tool Shaper Machine. Int. J. Eng. Res. V9, 362–364. 10.17577/ijertv9is070233 (2020).

Gagnol, V., Le, T. P. & Ray, P. Modal identification of spindle-tool unit in high-speed machining. Mech. Syst. Signal. Process. 25, 2388–2398. 10.1016/j.ymssp.2011.02.019 (2011).

Hulke, A., Moon, V., Dome, P., Shrirame, A. & Kakde, P. A Dual Side Shaping Machine for Industrial Applications. Int. J. Mod. Stud. Mech. Eng. 3 (3), 102–113. 10.20431/2454-9711.0303005 (2017).

Arakelian, V., Le Baron, J. P. & Mkrtchyan, M. Design of Scotch yoke mechanisms with improved driving dynamics, Proc. Inst. Mech. Eng. Part K J. Multi-body Dyn., vol. 230, no. 4, pp. 379–386, (2016). 10.1177/1464419315614431

Berthold, J., Kolouch, M., Wittstock, V. & Putz, M. Identification of modal parameters of machine tools during cutting by operational modal analysis. Procedia CIRP. 77, 473–476. 10.1016/j.procir.2018.08.268 (2018).

Talamucci, F. Nonlinear nonholonomic systems: a simple approach and various examples. Meccanica 59 (3), 333–362. 10.1007/s11012-024-01755-9 (2024).

Cheng, C., Yuan, X., Li, Y. & Liu, J. Comparisons of Inverse Dynamics Formulations in a Spatial Redundantly Actuated Parallel Mechanism Constrained by Two Point Contact Higher Kinematic Pairs. Biomimetics 9 (9). 10.3390/biomimetics9090564 (2024).

Talamucci, F. A simple approach to nonlinear nonholonomic systems with several examples, pp. 1–26, [Online]. (2023). Available: http://arxiv.org/abs/2305.17136

Gardie Damtie, E. & Dubale, H. Finite element analysis of crank and slotted lever quick return mechanism for shaper machine application. Results Mater. 14, 100288. 10.1016/j.rinma.2022.100288 (2022).

10.

Bharath, R. M. L. & Patil, S. Shaper Automation Using Electro-Pneumatic Devices And Plcs, Int. J. Eng. Res. Dev. e-ISSN 2278-067X, p-ISSN 2278-800X, www.ijerd.comVol. 8, Issue 2 (August PP. 49–56 Shap., vol. 8, no. 2, pp. 49–56, 2013. (2013).

11.

Law, M., Altintas, Y. & Srikantha Phani, A. Rapid evaluation and optimization of machine tools with position-dependent stability. Int. J. Mach. Tools Manuf. 68, 81–90. 10.1016/j.ijmachtools.2013.02.003 (2013).

12.

Ataei, M., Khajepour, A. & Jeon, S. Development of a novel general reconfigurable vehicle dynamics model. Mech. Mach. Theory. 156, 104147. 10.1016/j.mechmachtheory.2020.104147 (2021).

13.

Marques, F., Roupa, I., Silva, M. T., Flores, P. & Lankarani, H. M. Examination and comparison of different methods to model closed loop kinematic chains using Lagrangian formulation with cut joint, clearance joint constraint and elastic joint approaches. Mech. Mach. Theory. 160, 104294. 10.1016/j.mechmachtheory.2021.104294 (2021).

14.

Villano, E., Lenzo, B. & Sakhnevych, A. Cross-combined UKF for vehicle sideslip angle estimation with a modified Dugoff tire model: design and experimental results. Meccanica 56 (11), 2653–2668. 10.1007/s11012-021-01403-6 (2021).

15.

Zulaika, J. J., Campa, F. J., Lopez, L. N. & De Lacalle An integrated processmachine approach for designing productive and lightweight milling machines. Int. J. Mach. Tools Manuf. 51, 7–8. 10.1016/j.ijmachtools.2011.04.003 (2011).

16.

Li, C. et al. Analysis of dynamic characteristics for machine tools based on dynamic stiffness sensitivity. Processes 9 (12). 10.3390/pr9122260 (2021).

17.

Wan, M., Feng, J., Ma, Y. C. & Zhang, W. H. Identification of milling process damping using operational modal analysis. Int. J. Mach. Tools Manuf. 122, 120–131. 10.1016/j.ijmachtools.2017.06.006 (2017).

18.

Deng, C., Liu, Y., Zhao, J., Wei, B. & Yin, G. Analysis of the machine tool dynamic characteristics in manufacturing space based on the generalized dynamic response model. Int. J. Adv. Manuf. Technol. 92, 1–4. 10.1007/s00170-017-0201-9 (2017).

19.

Bin Zhou, L. & Chen, H. G. Kinematics Simulation of Shaper Quick-Return Mechanism. Appl. Mech. Mater. 741, 607–610. 10.4028/www.scientific.net/amm.741.607 (2015).

20.

Luo, B. et al. A method to predict position-dependent structural natural frequencies of machine tool. Int. J. Mach. Tools Manuf. 92, 72–84. 10.1016/j.ijmachtools.2015.02.009 (2015).

21.

Deng, C., Miao, J., Wei, B., Feng, Y. & Zhao, Y. Evaluation of machine tools with position-dependent milling stability based on Kriging model. Int. J. Mach. Tools Manuf. 124, 33–42. 10.1016/j.ijmachtools.2017.09.004 (2018).

22.

Hung, J. P., Lai, Y. L., Luo, T. L. & Su, H. C. Analysis of the machining stability of a milling machine considering the effect of machine frame structure and spindle bearings: Experimental and finite element approaches. Int. J. Adv. Manuf. Technol. 68, 9–12. 10.1007/s00170-013-4848-6 (2013).

23.

Al-Shuka, H. F. N., Corves, B. J. & Zhu, W. H. Dynamic Modeling of Biped Robot using Lagrangian and Recursive Newton-Euler Formulations. Int. J. Comput. Appl. 101 (3), 1–8. 10.5120/17664-8485 (2014).

24.

Zhao, H., Zhen, S. & Chen, Y. H. Dynamic modeling and simulation of multi-body systems using the Udwadia-Kalaba theory. Chin. J. Mech. Eng. (English Ed. 26 (5), 839–850. 10.3901/CJME.2013.05.839 (2013).

25.

Buradkar, M. et al. Design of Dual Slider Shaper Machine. Int. J. Anal. Exp. Finite Elem. Anal. 8 (3), 108–112. 10.26706/ijaefea.3.8.20210712 (2021).

26.

Bascetta, L., Ferretti, G. & Scaglioni, B. Closed form Newton-Euler dynamic model of flexible manipulators. Robotica 35 (5), 1006–1030. 10.1017/S0263574715000934 (2017).

27.

Wang, X., Guo, Y. & Chen, T. Measurement research of motorized spindle dynamic stiffness under high speed rotating, Shock Vib., vol. 2015, (2015). 10.1155/2015/284126

28.

Schmitz, T. L. Predicting high-speed machining dynamics by substructure analysis. CIRP Ann. - Manuf. Technol. 49 (1), 303–308. 10.1016/S0007-8506(07)62951-5 (2000).

29.

Belrzaeg, M. et al. Vehicle dynamics and tire models: An overview. World J. Adv. Res. Rev. 12 (1), 331–348. 10.30574/wjarr.2021.12.1.0524 (2021).

30.

Peng, Y. et al. Partition of the workspace for machine tool based on position-dependent modal energy distribution and clustering algorithm. Int. J. Adv. Manuf. Technol. 108 (3), 943–955. 10.1007/s00170-020-05487-4 (2020).

31.

Srang, S., Ath, S. & Yamakita, M. Newton-euler based dynamic modeling and control simulation for dual-axis parallel mechanism solar tracker. Adv. Sci. Technol. Eng. Syst. 5 (5), 709–716. 10.25046/AJ050587 (2020).

32.

Zhou, G., Jin, S., Wang, Y., Zhou, Z. & Cao, S. Dynamic Analysis of a Planar Suspension Mechanism Based on Kinestatic Relations. Electron 11 (12), 1–14. 10.3390/electronics11121856 (2022).

Yes