Kinematic Analysis-Sewing Machine

Figure 1:  Sewing Machine Schematic

Figure 2: Position of Rocker-Crank Mechanism

Figure 3: Angular Velocity of Rocker-Crank Mechanism

Figure 4: Mechanical Advantage

This is the kinematic analysis of the Singer Treadle Sewing Machine power transfer mechanism from the input rocker to the upper wheel of the pulley.

The power transfer system consists of two fourbar mechanisms:

• a rocker-crank mechanism and
• a belt-pulley system.

Rocker-Crank

The first fourbar mechanism consists of a rocker-coupler-crank configuration. (See Schematic, Figure 1.) The ground of the input (O₂) is the fulcrum of the treadle and the ground of the output (O₄) is the center of the large wheel. These are vertically aligned, defining the x-axis and simplifying the analysis.

The limits of the range of motion of the rocker were solved by calculating the position of the crank in line with the coupler, see Figure 5 (ϴ₂ = 83°) and the position of the crank overlapping the coupler, see Figure 6 (ϴ₂ = 58°). These angles were calculated with the law of cosines.

Thus, the treadle is the input rocker of length, a, with a range of motion of 58.6-83.0°. It is connected to the output crank by a coupler (b) of 11.5 in. length. The first output crank (c) is 2.5 inches in diameter with a 360° range of motion.

The motion of the rocker-crank mechanism is affected by the to two toggle points which coincide with the instant when the treadle reverses direction. The linkage will sometimes start the motion in the negative (CW) direction or reverse direction at the toggle point. The design accounts for this challenge with added mass in the rotating elements. The coupler shaft is only 0.25 inches in diameter but has a much thicker (Ø 0.50 in) ring on the end (Point B) and the output crank is an offset shaft of the same diameter. The additional mass provides the momentum to carry the crank through the full revolution.

The position of the fourbar mechanism through the range of motion is calculated with the equations for the analytical solution of the position of a fourbar mechanism provided in the textbook¹, section 4.5. See Figure 2 for ϴ₂ vs. ϴ₃ and ϴ₄. Note that the coupler only rotates about 13 degrees from its initial position, defined at point A.

The velocity of the fourbar mechanism is calculated with the analytical solution of the velocity of a fourbar mechanism provided in the textbook¹, section 6.7. ω2 is assumed as 1 rev/s. Of interest in the velocity analysis is how the crank reverses direction halfway through the cycle. The output velocity (ω4) is fairly constant at 4.5-6.0 rev/sec.

Belt-Pulley

The second fourbar mechanism is the pulley system. The output crank (c) of the rocker-crank mechanism is fixed to the input wheel (e) of the pulley mechanism, which has a 12-inch diameter. The output wheel of the pulley (g) mechanism has a 3-inch diameter. The pulley functions like a set of gears, although both rotate in the same direction, with the resulting gear ration of ¼. The velocity of the upper wheel of the pulley (ω7) is 4-times that of the lower wheel (ω4), approximately 20 rev/sec. The angular velocity is subject to surge at the toggle points.

Mechanical Advantage

The result of this system is a Mechanical Advantage less than 0.25.

The mechanisms that transfer the force from the upper wheel of the pulley to the sewing mechanism is outside the scope of this analysis. However, the motion is transferred to a horizontal shaft which connects to a cam located directly above the needle. The cam and associated linkage serve to convert the rotary motion of the shaft to a reciprocating, vertical motion. The force from the pulley is also transferred to a vertical rocker that transmits the motion to the bottom of the machine and then over to the bobbin casing. The bobbin casing rotates back and forth with every stroke of the needle.

The point of force application is the needle. The force produced at the needle is approximately one-fourth of the force introduced at the treadle. This results in increased speed and control at the point of application. For every cycle of the treadle, the upper wheel turns 4 times, making 4 precise stitches.

Figure 5: Toggle Position 1 (ϴ₂ = 83.0°)

Figure 6: Toggle Position 2 (ϴ₂ = 58.6°)