3. Effects of ultrasonic vibration on hot glass embossing process

### 3.1 Glass behavior under the application of ultrasonic vibration

In a dynamic experiment, if a sinusoidal strain with angular frequency ω and amplitude ε<sup>0</sup> is applied into a viscoelastic solid

$$
\varepsilon = \varepsilon\_0 \sin\left(\alpha \mathbf{t}\right) \tag{3}
$$

the resulting stress would be also sinusoidal with the same frequency which is lagging with a phase angle δ (Figure 9):

$$
\sigma = \sigma\_0 \sin\left(\alpha \mathbf{t} + \delta\right) \tag{4}
$$

Using complex notation

properly at a constant frequency. For thermal protection, a horn cooler is mounted outside the ultrasonic horn. O-rings placed between the ultrasonic horn and the horn cooler to form a water seal do not significantly affect the ability of the

Since the material properties of the horn would change in elevated temperature,

ffiffiffi E ρ

(1)

s

where E and ρ are Young's modulus and density, respectively. The wavelength is

where f is the resonance frequency of the ultrasonic vibration device. In longitudinal vibration mode, multiples of (λ=2) can be used as reference for the design of

As the temperature of a device whose geometry is fixed rises, its resonant frequency falls due to the decrease in Young's modulus [4], so such frequency will shift beyond the tracking range of the frequency generator. To increase the resonant frequency of the device at high temperature, its length must be reduced. With theoretical perspective, reducing the device length could increase the resonant frequency in longitudinal vibration mode. It means that the length reduction could compensate the frequency decrease caused by the increase in temperature. This trend has been verified by both finite element analysis and experiments as shown in

Resonant frequency of the ultrasonic vibration device with different horns and heating temperatures at

ffiffiffiffiffiffiffiffi E=ρ p

<sup>f</sup> (2)

its resonant frequency is shifted, and a mismatch with the frequency generator occurs. Hence, the ultrasonic device must be modified to ensure that it can operate correctly at high temperature. By simplifying theoretical equations, the speed of

cL¼

<sup>λ</sup> <sup>¼</sup> cL f ¼

a wave traveling along a one-dimensional medium is described by

ultrasonic device to vibrate.

Noise and Vibration Control - From Theory to Practice

the device length.

Figure 8.

Figure 8.

25°C [5].

20

$$
\sigma^\* = \mathfrak{e}\_0 \exp\left[i(\alpha \mathbf{t})\right]; \sigma^\* = \sigma\_0 \exp\left[i(\alpha \mathbf{t} + \boldsymbol{\delta})\right] \tag{5}
$$

the complex modulus G<sup>∗</sup> is then defined by the relation

$$\mathbf{G}^{\*} = \frac{\sigma^{\*}}{\mathbf{c}^{\*}} = \frac{\sigma\_{0}}{\mathbf{c}\_{0}} \exp\left(\dot{\mathbf{t}} \cdot \boldsymbol{\delta}\right) = \frac{\sigma\_{0}}{\mathbf{c}\_{0}} [\cos\boldsymbol{\delta} + \dot{\mathbf{t}} \cdot \sin\boldsymbol{\delta}] = \left(\mathbf{G}^{\prime} + \dot{\mathbf{t}} \cdot \mathbf{G}^{\prime}\right) \tag{6}$$

The first term on the right-hand side of Eq. (6) is in phase with the strain and is the real part of the complex modulus, often called the storage modulus:

$$\mathbf{G}' = \frac{\sigma\_0}{\mathbf{e}\_0} \cos \delta \tag{7}$$

The second term of Eq. (6) represents the imaginary part of the complex modulus, often called loss modulus:

$$\mathbf{G}'' = \frac{\sigma\_0}{\varepsilon\_0} \sin \delta \tag{8}$$

The ratio G}=G<sup>0</sup> ¼ tanδ, so-called loss factor, is widely used as a measure of the damping capacity of viscoelastic materials.

Figure 9. Oscillating strain ε, stress σ, and phase lag δ.
