Comparison of the Vibration Response of a Rotary Dobby with Cam-Link and Cam-Slider Modulators

2.1 Motion principle of modulators

Schematic drawings of the cam-link (CL) and cam-slider (CS) modulators are presented in Figure 1(a) and (b), respectively. The CL modulator, shown in Figure 1(a), consists of a gear, conjugate cam (including main cam and auxiliary cam (aux cam)), roller, swing arms, links, and main shaft. The swing arms are fixed to the gear. When the loom is running, the swing arms are hinged to the gear, and motion is transmitted to the cam rollers. The rotational speed of the gear is uniform. The conjugate cam is fixed to the dobby body, such that the four rollers move along the main and auxiliary cam contours of the conjugate cam. Clockwise rotation of the follower rollers causes the links to move, which drives the main shaft of the dobby.

Figure 1

Schematic illustration of the motion principles of the dobby modulator. (a) Cam-link modulator and (b) Cam-slider modulator.

Similarly, the CS modulator, shown in Figure 1(b), consists of a gear, conjugate cam (including main cam and auxiliary cam), rollers, swing arms, sliders, and the main shaft. The swing arms are fixed on the large gear. When the loom is running, the slider is embedded in the groove of the slide rack. As the gear continuously rotates, the slider rack and main shaft rotate synchronously. The groove direction ensures the swing arms always travel toward the center of the main shaft, and the groove direction can be altered by the conjugate cam. The swing arms are equipped with four rollers. The groove direction of the swing arms can be changed by rotating the gear according to certain rules, resulting in rotation of the dobby main shaft.

2.2 Acquiring the cam profile

The cam profile is an important aspect of cam-follower systems and is typically designed to move the follower in the desired manner [24,25,26]. Given the wide range of applications, there is a wide variety of possible cam geometries, ranging from cycloidal cams to cam motion designed using spline functions to accommodate discontinuities in the acceleration of the follower. In the present study, first we assume that the main shaft of the dobby will have the same kinematic curves for both modulators. Then, cam profiles of the CL and CS modulator can be obtained. Figure 2 illustrates the kinematic motion characteristics of the main shaft of the modulators.

Figure 2

Kinematic motion characteristics of modulators, which are the same for both the cam-link and cam-slider modulators. (a) Angular displacement and velocity and (b) angular acceleration.

2.2.1 The CL modulator

Figure 3 presents a schematic diagram of the CL modulator. The angle φ of link OB is a known parameter and φ is the main shaft angle of the modulator.

Figure 3

Sketch of the cam-link modulator.

In Figure 3, ω is the angular velocity of the gear. The starting position of the cam swing arm coincides with the hinge point C of the gear. The gear rotation angle θ in the Cartesian coordinate (X, Y) system can be expressed as θ = ωt.

From the geometric relationship:

(1) {(xA−xB)2+(yA−yB)2=lAB2(xA−xC)2+(yA−yC)2=lAC2

where x_B = l_OB cos φ, y_B = l_OB sin φ, x_c = l_OC cos θ, and y_C = l_OC sin θ.

From Eq. (1), the point A (x_A,y_A) can be obtained.

The distance between the cam actual profile and the cam pitch curve is equal to the roller radius r. To obtain the coordinates of the corresponding point T (x_T, y_T) on the actual main cam profile, distance r is taken as the distance from the main cam profile along the direction normal to any point A.

The slope of the line n − n – normal to the main cam pitch curve A is

(2) tanγ=dxAdyA=−dxA/dθdyA/dθ=sinγcosγ

where γ is the angle between the normal line n – n and the x -axis; (x_A, y_A) are coordinates of any point A on the main cam pitch curve.

Eq. (2) can be rewritten as

(3) {sinγ=(dxA/dθ)/(dxA/dθ)2+(dyA/dθ)2cosγ=−(dyA/dθ)/(dxA/dθ)2+(dyA/dθ)2

According to Eq. (1)–(3), the coordinates of point T (x_T, y_T) of the actual main cam profile can be determined as follows:

(4) {xT=xA±rcosγyT=yA±rsinγ

Then, the analytical solution of the cam profile equation can be determined using Eq. (1)–(4).

To obtain the coordinates of the corresponding point M (x_M, y_M) on the actual auxiliary cam profile, the auxiliary cam profile along the direction normal to any point D is taken as distance r.

Based on the law of cosines:

(5) {lAD2=lAC2+lCD2−2lAClCDcosα(xC2−xD2)+(yC2−yD2)=lCD2

Using Eq. (5), point D (x_D, y_D) can be obtained.

From the geometric relationship:

(6) {sinη=(dxD/dθ)/(dxD/dθ)2+(dyD/dθ)2cosη=−(dyD/dθ)/(dxD/dθ)2+(dyD/dθ)2

The coordinates of point M (x_M, y_M) of the actual auxiliary cam profile can be determined, as follows:

(7) {xM=xD±rcosηyM=yD±rsinη

2.2.2 The CS modulator

Figure 4 presents the structure of the CS modulator. The radius of the roller is r₁; point C₁ on the gear is centered at point O₁; and the theoretical cam profile is the trajectory formed by point A₁.

Figure 4

Sketch of cam-slider modulator.

Point A₁ (x_A1, y_A1) satisfies

(8) lA1C12=(xA1−xC1)2+(yA1−yC1)2

Point C₁ (x_C1, y_C1) satisfies

(9) {xC1=lO1C1cosεyC1=lO1C1sinε

According to the law of cosines:

(10) cosβ=(lA1C12+lB1C12−lA1C12)/2lA1C1lB1C1

The geometric relationship is

(11) lB1C1=(xB1−xC1)2+(yB1−yC1)2

(12) lA1B1=(xA1−xB1)2+(yA1−yB1)2

(13) cosφ=xB1yB1

where angle β, l_{A1 C1}, l_O1C1, and angle φ are known parameters.

From Eq. (8)–(13), the point A₁ (x_A1, y_A1) can be obtained.

Similar to the principle of the CL modulator mechanism:

(14) {sinδ=(dxA1/dθ)/(dxA1/dθ)2+(dyA1/dθ)2cosδ=−(dyA1/dθ)/(dxA1/dθ)2+(dyA1/dθ)2

The coordinates of point T₁ (x_T1, y_T1) of the actual main cam profile can be determined as

(15) {xT1=xA1±r1cosδyT1=yA1±r1sinδ

From the geometric relationship:

(16) {lA1D12=lA1C12+lC1D12−2lA1C1lC1D1cos2β(xC12−xD12)+(yC12−yD12)=lC1D12

From Eq. (16), point D₁ (x_D1, y_D1) can be obtained.

The coordinates of point M₁ (x_M1, y_M1) of the actual auxiliary cam profile can be determined as

(17) {xM1=xD1±r1cosσyM1=yD1±r1sinσ

where

(18) {sinσ=(dxD1/dε)/(dxD1/dε)2+(dyD1/dε)2cosσ=−(dyD1/dε)/(dxD1/dε)2+(dyD1/dε)2

3.1 Dynamic model

The dobby modulator was modeled as two masses, three springs, and three dampers. Figure 1(a) and (b) shows models of the CL and CS modulators, respectively. When the gear rotates at low speeds, the cam surface and the contact points of rollers 1 and 2 never separate. Roller 1 meets the desired displacement characteristic s₁ (θ) and roller 2 meets characteristic s₂ (θ). Assuming no assembly errors, when the two rollers are in close contact with the conjugate cam profile, then s₂ (θ) = s₁ (θ). Figure 5(a) shows the dynamic models of the CL and CS modulators. In the CL modulator dynamic model, m₁ (follower 1-CL) is the concentrated mass of cam roller 1, swing arm 1, and half of the link. In the ideal case, displacement y₁ is the expected displacement and m₂ (follower 2-CL) is the conc entrated mass of cam roller 2, swing arm 2, and half of swing arm 3.

Figure 5

Schematic representations and dynamic models of cam-link (CL) and cam-slider (CS) modulators. (a) Dynamic models, (b) equivalent dynamics model, and (c) force analysis.

Moreover, in the CS modulator dynamic model, m₁ (follower 1-CS) is the concentrated mass of cam roller 1, swing arm 1, link 1, and half of the main shaft link; m₂ (follower 2-CS) is the concentrated mass of cam roller 2 and swing arm 2. During high-speed motion, the intermediate link may experience irregular vibrations, which cause noise and friction, abrasion of moving parts, and reduce the service life of the system. Therefore, displacement y₂ of the follower 2-CS should also be taken seriously (y₁ and y₂ are absolute displacements).

The additional spring and damping elements are simplified as spring coefficients k₁, k₂, and k₃ and viscous damping coefficients b₁, b₂, and b₃ connecting masses m₁ and m₂ with the contact points, as shown in Figure 5(b).

The detached bodies of follower 1-CL and follower 2-CL are considered for force analysis. In the vertical direction, follower 1-CL and follower 2-CL are subjected to an elastic restoring force and a damping force. The force analysis is presented in Figure 5(c). The CL and CS modulators have the same dynamic model.

The dynamic equations can be written as [27]

Eq. (19) can then be reorganized as

3.2 Dynamic model response

The nonlinear system of equations, presented in Eq. (20), was implemented in Simulink using the parameters listed in Table 2. The Runge-Kutta method was used to calculate y₁ and y₂ of the two mechanisms. Then, the dynamics of the cam-follower system were investigated, paying particular attention to the impact behavior following the detachment of the follower from the cam.

Figure 6 shows the value of s₁ (θ) for the two types of modulators when the rotational speed of the gear is very low. Owing to the different mechanical structures of the two modulators, the cam contours must also differ to achieve the same motion characteristics of the spindle output. Therefore, the s₁ (θ) curves produced by each modulator are different.

Figure 6

Comparison of s₁ (θ) curves of cam-link and cam-slider modulators.

4.1 Vibration response of CL modulator

To understand the vibration response of follower 1-CL, a bifurcation diagram was derived for different values of ω, as shown in Figure 7. The evolution of gear angle θ with rotational speed is presented in Figure 8(a)–(d).

Figure 7

Bifurcation diagram of follower 1-CL system with changes in gear rotational speed ω.

Figure 8

Gear angle θ. Evolution of the system response for different values of ω. (a) ω = 700 rpm, (b) ω = 1,400 rpm, (c) ω = 1,500 rpm, and (d) ω = 1,800 rpm.

Figure 7 shows the vibration response of the follower 1-CL at about 700 rpm, and the corresponding vibration response to the gear angle θ is shown in Figure 8(a). For a large range of ω ∈ [0, 700], the vibration of the system is relatively stable. Severe vibration suddenly occurs at about 1,500 rpm, which rapidly evolves into a chaotic attractor, as illustrated in Figure 8(c) and (d).

In addition, the vibration responses of follower 2-CL at different values of ω were analyzed. Trends were similar for follower 2-CL and follower 1-CL. Taking ω = 1,000 rpm as an example, the vibration response of follower 1-CL and follower 2-CL are compared in Figure 9.

Figure 9

Comparison of follower-CL displacements when ω = 1,000 rpm.

According to the above analysis, the maximum value of ω cannot theoretically exceed 1,400 rpm in the CL modulator system and should ideally operate under 700 rpm.

4.2 Vibration of CS modulator

The bifurcation diagram of follower 1-CS position of CS modulator is presented in Figure 10. As an example, trajectories at different instances in time are shown in Figure 11 for two gear cycles only.

Figure 10

Experimental bifurcation diagram of follower 1-CS position of CS modulator versus gear rotational speed ω.

Figure 11

Evolution of gear angle θ of the system for different values of ω. (a) ω = 500 rpm, (b) ω = 1,000 rpm, (c) ω = 1,100 rpm, and (d) ω = 1,800 rpm.

Figure 10 shows the vibration response of follower 1-CS at about 500 rpm, and the corresponding vibration response to the gear angle θ is shown in Figure 11(a). For large ω ∈ [0, 500], the vibration response of the CS modulator system is stable. However, severe vibration suddenly occurs at about 1,100 rpm. The behavior is similar to that of the CL modulator, and again, quickly evolves into a chaotic attractor, as shown in Figure 11(c)–(d).

Similar to the analysis of follower 2-CL, vibration characteristics were also calculated for follower 2-CS at different values of ω. It can be concluded that the vibration conditions of follower 1-CS and follower 2-CS are approximately the same. Figure 12 shows a comparison of the follower-CS displacements when ω = 1,000 rpm.

Figure 12

Comparison of follower-CS displacements when ω = 1,000 rpm.

Theoretically, the maximum value of ω should not exceed 1,000 rpm, and ideally, the CS modulator system should operate under 500 rpm.

4.3 Comparison of vibration results

Bifurcation diagrams of the vibration/displacement of follower 1-CL and follower 1-CS are presented in Figure 13. In Figure 13(a), the vibration/displacement is equal to the vibration of follower 1-CL/S_l (θ) of the CL modulator. In Figure 13(b), the vibration/displacement is equal to the vibration of follower 1-CS/S₁(θ) of the CS modulator.

Figure 13

Vibration/displacement of follower 1-CL and follower 1-CS. (a) Vibration/displacement of follower 1-CL and (b) vibration/displacement of follower 1-CS.

From Figure 13, it can be observed that, overall, the vibration/displacement of follower 1-CS is smaller than that of follower 1-CL. The ideal speed of the CL modulator is under 700 rpm, however, theoretically, the modulator can be used at speed of up to 1,400 rpm. On the other hand, the ideal speed of the CS modulator is under 500 rpm, and theoretically, the modulator can be used at speed of up to 1,000 rpm. Based on the analysis of vibration/displacement, it can be concluded that the CS modulator is suitable for gear speeds ω under 500 rpm, and the CL modulator is suitable for high speeds.