Driver Model Based on Mechanical Traveling Wave Principle

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In the figure, the motion trail of the particle of the elastic layer is elliptical ^{[4, 5, 6, 7, 17, 18]}. When traveling wave moves toward the right, the movement direction of the particles is counter clock wise. When the particles arrive at the wave peak, the direction of the vibration velocity is toward the left. Under the condition of preload between elastic layer and output layer, the horizontal movement of the particles causes friction force on the output layer, driving the output layer movement towards the left and vice versa.

The shape of the motion trail for the elastic particles is an ellipse (shown in Figure 1). Point P₀ is the initial position of the particle before bend deformation of the elastic layer and point P is another position of that point after bending deformation, is the bend angle between P and P₀ and is the vibration displacement along x direction. his the thickness of un-deformed elastic layer and a rectangular coordinate system can be built up along neutral surface of elastic layer. The equation of traveling wave for neutral surface is as follows ^{[8, 9, 11, 12]}:

(1)

where is the wavelength of traveling wave for elastic layer

Then, the vibration w(x,t) displacement of particle from point P₀ to P is as follows:

(2)

Through a method of partial differential equations, the vibration velocity(v_x) for particles of the elastic layer is as follows:

(3)

A negative sign means that the motion direction of the particle is opposite to that of the motion direction for the traveling wave. If we do not consider the sliding effect between elastic layer and output layer, the output layer velocity(v_F) equals the vibration velocity of the elastic particle at the bending wave crest location.

(4)

Figure 2. Velocity curve of output layer

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Velocity curve of output layer is shown in Figure 2. The figure illustrates that only a half cycle of the velocity is useful for the movement of output layer. Without taking the influence of relative sliding into account, by calculating the dashed area of velocity curve shown in the figure. The mean velocity of output layer is given by:

(5)

3. Structure of Mechanical Traveling Wave Driver

Based on the theory of mechanical traveling wave drive, a mechanical traveling wave driver of a rotary type was designed. Figure 3 shows the structure of the driver, which includes such components as input shaft, output shaft and traveling wave producing roller etc. In the driver, a series of cylindrical balls are distributed equally along the input shaft. Mechanical traveling wave would be produced on the condition of cylindrical rollers rotating around input shaft, and the output shaft rotates correspondingly based on a traveling wave principle. The key part of the driver is the traveling wave producer, which includes an elastic layer and cylindrical rollers uniformly distributed along the input shaft (shown in Figure 3a). Traveling wave producer can be designed as of several shapes such as circular conical shape, torus shape and disk shape. Cross-section shape of traveling wave producer is shown in Figure 3 (b).

Figure 3. Structure of mechanism traveling wave driver

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4. Modeling for Mechanical Traveling Wave Producer

4.1. Drive Ratio of Mechanical Traveling Wave Producer

Figure 4. several parameters of driver

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Figure 4 shows the relationship between parameters for input shaft, traveling wave producer and output shaft. R₁, R₂, r, r₁ and h represent distance between input shaft center and elastic layer center, input shaft radius, output shaft radius, curve radius between elastic layer and cylinder ball, radius of cylinder roller and the thickness of elastic layer respectively. Consider that the deformation of the elastic layer is small and that it can be ignored. Then the wave length of mechanical travelling wave driver is as follows:

while T (wave period) is:

We have: . Combined with Equation(5), the average velocity of output shaft is obtained as follows:

(6)

Evidently, the biggest squeeze deformation of elasticlayer is the amplitude of the elastic layer, i.e. r₁=A. The velocity of center part for elastic layer is obtained as:

(7)

Setting total preload between input shaft and output shaft as F, the number of rollers is n, and the preload for single roller is:

The contact area between roller and the elastic layer is as follows:

(8)

where, r₁(r₁=A) is the deformation depth of the elastic layer, E is elastic modulus of the elastic layer and B is length of the cylindrical roller.

Consider that the value of the deformation depth of elastic layer(r₁) is much smaller than that of the cylinder roller(r), the stress of the elastic layer is as follows:

(9)

In a traveling wave length period, the strain of elastic layer is given by ^{[8, 9]}:

where, 2L is the contact length in a wavelength between input shaft and output shaft. Consider that the strain is produced in half of the wavelength period. The average value of the valid strain is as follows:

(10)

Based on stress-strain relation formula: , the relationship between squeeze deformation and preload is as follows:

Because r₁<<r, by simplifying the formula, the relationship between r₁ and F (preload) is given by:

(11)

Then, the total drive ratio is as follows:

(12)

It is obvious that the total drive ratio is influenced by such parameters as thickness of elastic layer, elastic layer materials, preload, radius of input (output) shaft, number of cylindrical rollers, radius of cylindrical rollers and amplitude etc.

4.2. Output Power for Mechanical Traveling Wave Producer

Figure 5. The contact model between the elastic layer and friction layer considering shear deformation [10-16]

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Friction layer between the elastic layer and output shaft is the key part to drive the output shaft. Considering the stiffness of elastic layer is larger than that of friction layer, it is supposed that friction layer hasdeformed while elastic layer has no deformation, and the elastic layer has the waveform shape. Considering theory of shearing deformation and Coulomb friction law, a contact model between the elastic layer and friction layer has been built, which is shown in Figure 5. In the figure, the interval [-x_r,-x_r1] is the viscous region, point (-x_r1) is the demarcation point of viscous region to sliding friction region . Considering the influence of the viscous region, the effective area between the input layer and output layer decreases, and the output torque correspondingly decreases. Driving force from between input shaft and output shaft is given by ^{[9, 10]}:

(13)

where, B is the contact width of roller between input shaft and output shaft, f₁ is the horizontal force between input shaft and output shaft, and , is the dynamic friction coefficient and is the preload.

It is difficult to accurately measure some parameters such as because the viscous region and sliding region are variable with materials and preload change. So, the equation (13) can be simplified as follows:

(14)

where, is constant, which reflects the friction value. The average output torque of the producer is as follows:

(15)

Then, the output power is given by:

(16)

5. Testing Device Experiment

A testing device with traveling wave principle was manufactured, which is shown in Figure 6. In the device, the kinematic chain consists of the electromotor, input shaft, mechanical traveling wave producer and output shaft. Rotational speed is measured by a speed sensor and displacement was measured by a linear scale sensor. Here, a straightedge was used as the output link instead of the output shaft. When the mechanical traveling wave was produced on the elastic layer, the straightedge would move toward a certain direction, and the displacement value was measured by a linear scale sensor. A feed screw nut device was used as the preload adjuster, and the preload can be adjusted by rotating the nut. The material used for the elastic layer is important, as it is a key part of the mechanical traveling wave producer. Here, PU (Polycarbamate) was selected as the initial material used for the elastic layer, and the thicknesses of the elastic layers was 3mm, 5mm, 7mm and 9mm, respectively.

Figure 6. Mechanical traveling wave testing device

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According to Equation (12), we know that the total velocity ratio value is dependent on such parameters as thickness of elastic layer, material of elastic layer, amplitude of preload, radius of input and output shaft, number of rollers, radius of roller and amplitude etc. In the experiment, several parameters such as material of elastic layer, radius of input and output shaft and radius of roller were set as constants, while such parameters as thickness of elastic layer, number of rollers and preload were set as variables. In the testing device, the radius of the input shaft was 20mm and the angular velocity was 136r/min; the radius of the cylindrical roller was 1.8mm and the length of roller was 9mm, which were equally distributed along input shaft length, the diameter of the elastic layer is 48.16mm. Here, experiments on total drive ratio for the device by adjusting such parameters as thickness of elastic layer, number of cylinder rollers and preload were done.

5.1. Experiment through Adjusting the Thickness of the Elastic Layer

To test the influence of the thickness of the elastic layer vs. total drive ratio, several elastic layers with a thickness of 3mm, 5mm, 7mm and 9mm were studied. The preload between the elastic layer and the straightedge was changed by rotating the screw nut, and four displacement curves are shown in Figure 7. The velocity of the straightedge is almost constant, and the larger drive ratio is about 401.7. Secondly, the displacement value of the straightedge is from 189mm to 150mm. With the increase of thickness for the elastic layer, it was revealed that the thickness has an unobvious effect on the total drive ratio.

Figure 7. The displacement of output straightedge on condition of different elastic layer thicknesses (a-3mm, b-5mm, c-7mm, d-9mm)

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5.2. Experiment through Adjusting Preload between Elastic and Straightedge

In the experiment, the preload between elastic and straightedge was adjusted by rotating the nut of the screw rod, and the adjusted distance was 0.5mm. Here, two elastic layers with thicknesses of 3mm and 5mm were used in the experiment.

The curve of the drive ratio VS the deformation of the elastic layer is shown in Figure 8. we know that with the increase of preload (i.e. the increase of deformation for elastic layer) between the elastic layer and the straightedge, the total drive ratio decreased in a degree correspondingly, and the total drive ratio decreased with the thickness of elastic layer increase, which corresponds to the theoretical model of the total drive ratio.

Figure 8. The curve of drive ratio VS deformation of the elastic layer

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5.3. Experiment through Adjusting Roller Number

The number of rollers is one of the key factors in a mechanical traveling wave producer. In the experiment, the numbers of rollers were set to 6,8,12 and 16, and the preload was constant. The curve of total drive ratio VS the rollers number is shown in Figure 9, which has exponential profile approximately. This illustrates that the total drive ratio decreases with the increase of roller numbers, which is consistent with the theoretical model.

Figure 9. The curve of drive ratio VS the roller number

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