Energy loss comparison between kinematically equivalent mechanisms: Slider-crank and eccentric cam

Figures & data

Figure 1. (a) Slider-crank mechanism; (b) Eccentric cam mechanism.

Figure 2. FBD for each link of a slider-crank mechanism.

Table 1. Dynamic variables and parameters for the slider-crank mechanism.

Parameter	Description
$T_{\mathrm{m}}$	Motor torque
$R_{x\mathrm{O}},$ $R_{y\mathrm{O}},$ $R_{x\mathrm{A}},$ $R_{y\mathrm{A}},$ $R_{x\mathrm{B}}$ and $R_{y\mathrm{B}}$	Internal reaction forces in the revolute pairs O, A and B, respectively
$N_3$ and $M_3$	Internal reaction normal force and moment in the prismatic pair, respectively
$T_{\mathrm{O}},$ $T_{\mathrm{A}}$ and $T_{\mathrm{B}}$	Friction torques in the revolute pairs O, A and B, respectively
$c_{\mathrm{O}},$ $c_{\mathrm{A}}$ and $c_{\mathrm{B}}$	Viscous friction coefficients in the revolute pairs O, A and B, respectively
$F_3$	Friction force in the prismatic pair
$c_3$	Viscous friction coefficient between the slider and the fixed guide
$F$	External force
$m_1,$ $m_2$ and $m_3$	Mass of the crank, the rod and the slider, respectively
$I_{{\mathrm{G}}_1}$	Crank moment of inertia along the z-axis referring to its center of inertia (${\mathrm{G}}_1$)
$I_{{\mathrm{G}}_2}$	Rod moment of inertia along the z-axis referring to its center of inertia (${\mathrm{G}}_2$)

Table 2. Geometric parameters for the eccentric cam mechanism.

Parameter	Description
${\phi }_1$	Eccentric cam angular displacement
${\phi }_{\mathrm{r}}$	Roller angular displacement
$d({\phi }_1)=x({\phi }_1)$	Linear displacement function imposed on the roller center
$\epsilon$	Positive offset between the follower’s motion axis and the rotational center of the eccentric cam
$l_1$	Eccentricity between the rotational center of the eccentric cam and the center of inertia (A = G1e); equivalent crank length
${\rho }_{\mathrm{c}}$	Eccentric cam curvature radius
$l_2$	Equivalent rod length
$r_{\mathrm{r}}$	Roller radius
$\varphi ={\phi }_2$	Pressure angle
$l_3$	Follower length
${\mathrm{I}}_{13}$	Instantaneous rotational center between the eccentric cam (link 1) and the follower (link 3), assuming no sliding at contact

Figure 3. Geometric parameters for an eccentric cam mechanism.

Figure 4. FBD for each link of an eccentric cam mechanism.

Table 3. Dynamic variables and parameters for the eccentric cam mechanism.

Parameter	Description
$T_{\mathrm{m}}$	Motor torque
$R_{x\mathrm{O}}$ and $R_{y\mathrm{O}},$ $R_{x\mathrm{B}}$ and $R_{y\mathrm{B}}$	Internal reaction forces in the revolute pairs O and B, respectively
$F_{\mathrm{n}}$	Internal normal force in the higher pair of contact point J
$F_{\mathrm{t}}$	Internal tangential force in the higher pair of contact point J, assuming no sliding
$N_3$ and $M_3$	Internal reaction normal force and moment in the prismatic pair, respectively
$T_{\mathrm{O}}$ and $T_{\mathrm{B}}$	Friction torques in the revolute pairs O and B, respectively
$c_{\mathrm{O}}$ and $c_{\mathrm{B}}$	Viscous friction coefficients in the revolute pairs O and B, respectively
$F_3$	Friction force in the prismatic pair
$c_3$	Viscous friction coefficient between the follower and the fixed guide
$F$	External force to avoid follower jump
$F_{\text{preload}}$	Preload force of the closure spring
$k_{\mathrm{s}}$	Linear stiffness of the closure spring
$m_{\text{1e}},$ $m_{\text{2r}}$ and $m_3$	Mass of the eccentric cam, the roller and the follower, respectively
$I_{\mathrm{O}}$	Eccentric cam moment of inertia along the z-axis referring to the rotational center O
$I_{{\mathrm{G}}_{2\mathrm{r}}}$	Roller moment of inertia along the z-axis referring to its center of inertia (${\mathrm{G}}_{2\mathrm{r}}$)

Table 4. Required parameters for the numerical examples.

Slider-crank mechanism		Eccentric cam. Small roller radius		Eccentric cam. Big roller radius
Parameter	Value	Parameter	Value	Parameter	Value
$l_1$ [mm]	25
$l_2$ [mm]	85	$r_{\mathrm{r}}$ [mm]	18.5	$r_{\mathrm{r}}$ [mm]	51
		${\rho }_{\mathrm{c}}$ [mm]	66.5	${\rho }_{\mathrm{c}}$ [mm]	34
$l_3$ [mm]	184.5
$\epsilon$ [mm]	0
${\dot{\phi }}_1$ [rad/s]	36.65
${\ddot{\phi }}_1$ [rad/s^2]	0
$m_1$ [kg]	0.031	$m_{1\mathrm{e}}$ [kg]	1.71	$m_{1\mathrm{e}}$ [kg]	0.41
$I_{{\mathrm{G}}_1}$ [kg·m^2]	9.5·10^−6	$I_{\mathrm{O}}$ [kg·m^2]	4.9·10^-3	$I_{\mathrm{O}}$ [kg·m^2]	4.9·10^-4
$m_2$ [kg]	0.094	$m_{2\mathrm{r}}$ [kg]	0.093	$m_{2\mathrm{r}}$ [kg]	0.53
$I_{{\mathrm{G}}_2}$ [kg·m^2]	9.2·10^−5	$I_{{\mathrm{G}}_{2\mathrm{r}}}$ [kg·m^2]	1.9·10^-5	$I_{{\mathrm{G}}_{2\mathrm{r}}}$ [kg·m^2]	8·10^-4
$m_3$ [kg]	1.3
$c_{\mathrm{O}}$ [N·m/(rad/s)]	0.17
$c_{\mathrm{A}}$ [N·m/(rad/s)]	0.026	No		No
$c_{\mathrm{B}}$ [N·m/(rad/s)]	0.026
$c_3$ [N/(m/s)]	24.40
$F_{\text{pre}}$ [N]	66.42
$k_{\mathrm{s}}$ [kN/m]	1.23

Figure 5. (a) Transmission efficiency of motion; (b) Linear acceleration of the slider and the follower.

Figure 6. Rollers and rod: (a) Angular velocities; (b) Angular accelerations.

Figure 7. Internal reactions for each pair: (a) Slider-crank; (b) Eccentric cam with small roller radius; (c) Eccentric cam with big roller radius; (d) Transmission efficiency of force.

Table 5. Required parameters for traction coefficient (Masjedi and Khonsari 2015).

Parameter	Value/specification
Contact length	0.013 m
$R,$ Equivalent contact radius	Calculated with eccentric cam radius and roller radius from Table 4
$E^{\prime },$ Effective Young’s modulus	228 GPa
Thermal conductivity of contact solids	60.5 W/(m·K)
Specific heat of contact solids	434 J/(kg·K)
Vickers hardness	2.35 GPa
$R_{\mathrm{a}},$ Surface roughness	0.2 µm
Limiting shear stress coefficient	0.0434
Asperity friction coefficient	0.12
SRR	0.05
$v_{\text{slid}},$ Mean sliding velocity big roller	1.25 m/s·SRR = 0.063 m/s
$v_{\text{slid}},$ Mean sliding velocity small roller	2.4 m/s·SRR = 0.12 m/s
Oil density	0.87 g/ml
Oil kinematic viscosity	20.5 mm^2/s at 40 °C
Inlet lubricant temperature	315 K
Viscosity temperature coefficient	0.027 K^-1
Viscosity pressure coefficient	1.5·10^-8 m^2/N

Figure 8. Energy loss per cycle.

Figure 9. Energy loss per cycle workspaces.

Figure 10. Relative differences in energy loss per cycle with regard to the most efficient case.

Figure 11. Test stand indicating the shared common elements.

Table 6. Test stand specifications of the shared components.

Element	Specification
DC Motor	DC shunt motor. P = 232 W; I = 1.8 A; U = 220 V
Gear reducer	2 stage reduction: worm gear and spur gears i = (31·38)/(5·51) = 4.619
Main shaft	$I_{{\mathrm{G}}_{\text{ms}}}=$5.4·10^-4kg·m^2
Flywheel	$I_{{\mathrm{G}}_{\mathrm{f}}}=$7.2·10^-2kg·m^2
Final shaft coupling	$I_{{\mathrm{G}}_{\text{fsc}}}=$1.8·10^-3kg·m^2

Figure 12. (a) Slider-crank; (b) Eccentric cam with small roller radius; (c) Eccentric cam with big roller radius.

Figure 13. Test results: (a) Eccentric cam with small roller radius; (b) Eccentric cam with big roller radius; (c) Slider-crank.

Figure 14. Inertia reduced to ${\phi }_1.$

Figure 15. Experimental comparison of energy loss per cycle.

Figure A1. Bushing oscillating system: (a) Base input test; (b) Schematic representation.

Figure A2. (a) Pendulum system; (b) Schematic representation.

Masjedi, M., and M. M. Khonsari. 2015. “An Engineering Approach for Rapid Evaluation of Traction Coefficient and Wear in Mixed EHL.” Tribology International 92: 184–190. https://doi.org/10.1016/j.triboint.2015.05.013

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