Theoretical and Experimental Study on Improving the Dynamic Characteristics of Material Extrusion Additive Manufacturing Parts by Ultrasonic-Vibration Aided Machining

Abstract

Material extrusion (ME) is one of the most widely used additive manufacturing technologies, which can directly produce parts with complex geometries. However, since the working environment is increasingly complex, the built parts inevitably have to be subjected to dynamic loads, and it is urgent to study their dynamic performances so as to determine their reliability. In addition, the layer-by-layer forming process makes the ME parts inferior to those fabricated by traditional methods in terms of mechanical properties. It is important to find an effective method to improve the parts’ dynamic characteristics so that the reliability can be enhanced. To achieve these goals, the method of ultrasonic-vibration aided machining (UVAM) was proposed to improve the dynamic characteristics of polylactic acid ME parts in this paper, and the corresponding theoretical and experimental study was performed. Ordinary ME equipment was modified to an ultrasonic-vibrating one, by which the samples processed with and without UVAM utilized were prepared. With the modal test system, the dynamic characteristics of the samples were determined. Taking into account the influence of the porosity defects and ultrasonic vibration, the ME sample’s dynamic model was then established based on the finite element method. Through the comparison between the predicted and measured results, the proposed model was validated, and it can accurately predict the ME plates’ dynamic characteristics no matter processed with or without UVAM utilized. The results also showed that UVAM could significantly improve the ME plate’s dynamic characteristics. With the increasing vibration amplitude or frequency, the samples’ natural frequency and damping ratio would be increased, and the vibration response would be decreased. Compared to the three waveforms of ultrasonic vibration (sinusoidal, square, and triangle), the sinusoidal one has the best improvement effect.

Introduction

Additive manufacturing, integrating computer, material science, and circuit electronic technology,^1–3 can realize the forming process of industrial products from design to manufacturing. Due to the fast forming speed, simple process, low cost, and short manufacturing cycle,^4–6 material extrusion (ME) has become one of the most widely used addictive manufacturing technologies. With the increasingly complex working environment, ME parts have to suffer from dynamic loading or vibration. It is necessary to study their dynamic performances in order to determine their reliability. In addition, the layer-by-layer forming process inevitably causes built parts’ defects (such as pores, inclusions, and layer separation), resulting in the dynamic performance far inferior to those produced by traditional processing methods, which thereby hinders the development of the ME technique. Therefore, effectively improving the dynamic performance of ME products has become one of the key research issues for this technique.

Although a large number of scholars have studied the mechanical properties of ME products, such as tensile strength,^7–10 fracture toughness,^11–13 impact strength,^14–17 there is little information available in terms of built parts’ dynamic characteristics, and even less about the improvement study. Arivazhagan,¹⁸ used Dynamic Mechanical Analysis (DMA) equipment to perform a frequency sweep test (frequency range is 1–100 Hz) on ME parts to determine the modulus, damping, and viscosity values. The research showed that the strength of the product processed with solid built style was higher than others. The storage modulus increases with the increasing temperature, while the viscosity decreases. Wang,¹⁹ found that when the layer thickness decreases or the fill rate increases, the storage modulus, loss modulus, and loss factor of ME parts increase. While the storage modulus, loss modulus, and loss factor of the parts first increase and then decrease with the increase of the extrusion temperature. Mohamed,^20–23 studied the influence of different processing parameters on the dynamic mechanical performance of ME parts. The results illustrated that the layer thickness, air gap, and the number of contours have a great impact on the dynamic modulus and mechanical damping. Quintana,²⁴ measured the changes in the storage modulus, loss modulus, and loss tangent of ME parts through dynamic-loading experimental techniques. The results show that in the frequency range of 0.1–100 Hz, the storage modulus increases with the increase of frequency, while the loss modulus changes less. Generally, the above-mentioned studies were carried out based on the experimental tests and lacked a theoretical basis. Though the tests were performed under cyclic vibration loading conditions, only the material mechanical characteristics (such as storage modulus, and loss modulus) were analyzed. The information on dynamic characteristics (i.e., inherent characteristics and vibration responses) was little, and the corresponding improvement was not mentioned at all.

To improve ME parts’ mechanical properties, a few scholars have used ultrasonic-vibration aided machining (UVAM) in the manufacturing process. Gunduz,²⁵ indicated that introducing ultrasonic vibration onto the extrusion liquefier could drastically reduce the wall friction and flow stress of the inside melt. The viscosity was enhanced up to 14000 Pa·s. Tofangchi,²⁶ fixed the ultrasonic-vibration transducer onto the extrusion liquefier to use UVAM in the ME process. The results showed that the interlayer adhesion of the samples was increased by 10%. Li,²⁷ investigated the effect of ultrasonic vibration on the tensile strength of ME samples. It was found that the tensile strength of samples was greatly increased after ultrasonic vibration was applied. Wu,²⁸ and Qiao,²⁹ proved that ultrasonic strengthening technology could improve the flexural strength of ME samples. Ma,³⁰ investigated the effect of ultrasonic nano-crystal surface modification (UNSM) on the surface quality of AlSi10Mg alloy parts fabricated by direct metal laser sintering. It showed that UNSM could significantly improve the surface finish of the built parts. Although the method of UVAM has been used to improve the ME parts’ mechanical properties, the effect on dynamic characteristics was not taken into account.

In summary, most of the current literature was focused on the experimental static mechanical property of ME parts, the theoretical basis was lacking, and the dynamic characteristics were rarely studied. Although a few scholars have focused on the influence of UVAM on the mechanical properties of ME parts, the effect on the dynamic characteristics was barely mentioned. To cover these gaps, this paper uses UVAM in the manufacturing process. The dynamic characteristics of ME plates (i.e., natural frequency, mode shape, and vibration response) were both theoretically and experimentally investigated to accurately determine their reliability and to reveal the influencing mechanism of ultrasonic vibration machining. The research presents a novel method of utilizing ultrasonic vibration to enhance the dynamic characteristics of ME parts, contributing to the growth and implementation of ME technology. It serves as a reference for various other additive manufacturing technologies and supports the advancement of UVAM in the field of additive manufacturing.

Experimental Research

Ultrasonic vibrating ME equipment

To fabricate the parts with UVAM utilized, ordinary ME equipment (FLSUN) was modified to introduce ultrasonic vibration into the manufacturing process. It mainly includes a signal generator (VC2015H), amplifier (E2041), and ultrasonic vibrating piezoelectric ceramic, as shown in Figure 1. The piezoelectric ceramic is fixed at the extrusion liquefier. The voltage amplifier is connected to the signal generator to amplify the generated electrical signal, which provides a stable and high-resolution voltage to the piezoelectric ceramic. Due to the inverse piezoelectric effect, the ultrasonic vibrating piezoelectric ceramic can cause the vibration of the extrusion liquefier in the longitudinal direction. The actual vibration state is then determined by the acceleration sensor and data acquisition card.

FIG. 1.

The ultrasonic vibrating ME equipment. (a) The schematic diagram. (b) The actual equipment. ME, material extrusion.

Sample preparation

To determine the influence of UVAM on the ME plates’ dynamic characteristics, the samples processed without and with different ultrasonic vibrations were prepared with the equipment mentioned above. The samples’ material was polylactic acid (PLA), with its external dimensions shown in Figure 2. A total of 40 samples were manufactured, and they were divided into four types. The 1^st type was the ordinary samples (without UVAM utilized), which were represented by $R_{0_}^{0} i$ (i = 1–5); the 2^nd type was the samples processed with 2 g (acceleration) amplitude UVAM utilized at different frequencies, which were separately 20, 30, and 40 kHz. These samples were represented by $R_{20k_}^{2} i$ , $R_{30k_}^{2} i$ and $R_{40k_}^{2} i$ (i = 1–5); the 3^rd type was the samples processed at 40 kHz UVAM utilized with different amplitudes, which were 2, 4, and 6 g. They were individually represented by $R_{40k_}^{2} i$ , $R_{40k_}^{4} i$ and $R_{40k_}^{6} i$ (i = 1–5); the last type was the samples processed with the same amplitude and frequency UVAM utilized (2 g and 30 kHz), whose waveforms were different (sinusoidal, square, and triangle). These samples were represented by $R_{30k_}^{2} i$ , $S_{30k_}^{2} i$ and $T_{30k_}^{2} i$ (i = 1–5). It is noted that all the samples’ processing parameters were set the same except for the frequency, amplitude, or waveform of the UVAM utilized. Details are listed in Table 1.

FIG. 2.

Two-dimensional drawing of the ME sample. ME, material extrusion.

Table 1.

The Key Processing Parameter Settings of the ME Samples

Samples (i = 1–5)	$R_{0_}^{0} i$	$R_{20k_}^{2} i$	$R_{30k_}^{2} i$	$R_{40k_}^{2} i$	$R_{40k_}^{4} i$	$R_{40k_}^{6} i$	$T_{30k_}^{2} i$	$S_{30k_}^{2} i$
Vibration waveform	—	Sinusoidal					Triangle	Square
Vibration frequency (kHz)	0	20	30	40	40	40	30	30
Vibration amplitude (g)	0	2	2	2	4	6	2	2
Extrusion width (mm)	0.4
Printing direction	X direction (0°)
Layer height (mm)	0.2
Printing speed (mm/s)	60
Filling rate	100%
Extrusion temperature (°C)	200

The symbol was $R_{f}^{A}_i$ , where, $A$ is the vibration amplitude, $f$ is the vibration frequency, and $i$ is the serial number of each sample.

ME, material extrusion.

Dynamic characteristics test

To determine the dynamic characteristic parameters, i.e., the inherent characteristics and vibration response of the samples, a modal test system was set up (as shown in Fig. 3), which mainly includes a modal hammer (PCB-086C01, the sensitivity is 11.2 mV/N), data acquisition card (NI USB-4431, four analog input channels and one analog output channel) and accelerometer (PCB-352C33, 103.5 mV/g sensitivity). The fixture clamps the sample with a length of about 10 mm to maintain the cantilever state of the samples. The excitation point of the modal hammer is about 10 mm above the center of the clamping position of the thin plate. The acceleration sensor is fixed at the center of the top of the sample to obtain a larger vibration response and guarantee the reliability of the measurement results.

FIG. 3.

Schematics of the modal test system.

During the experiment, the modal hammer was used to give pulse excitation to the samples, and then the acceleration sensor would measure the vibration response caused by the pulse excitation. The sampling frequency of the data acquisition card was set to 2 times the frequency of the excitation signal to collect the vibration response signal in real-time, and thus to obtain the frequency response function of the thin plate. The multiple-input single-output method was then used to obtain the modal mode shape of the thin plate, that is, the accelerometer was fixed at the position where the vibration response was larger and the other measuring positions were excited separately. Finally, the dynamic parameters such as the inherent characteristics of the samples could be obtained by calculating the obtained data.

The frequency response function is commonly considered as the ratio of the output response of the structure to the input excitation. It is the inherent characteristic of the structure or system, which has nothing to do with external factors, such as the excitation or response. According to the half-power bandwidth method,^31,32 the damping ratio of the ME plate can be given by: $ζ = \frac{ω_{2} - ω_{1}}{2 ω_{0}}$ (1)where $ω_{0}$ is the natural frequency, $ω_{1}$ and $ω_{2}$ are the frequencies of half-power points.

Dynamic Model

Material property calculation

The extruded material filament has an elliptical cross-sectional shape mainly due to the nozzle extrusion and its gravity, and two bonding necks (i.e., horizontal and longitudinal) are generated between adjacent extrudates, as shown in Figure 4. During the forming process, the work done by the surface tension is equal to the work done by the viscous force, while the change of the extrudate’s length is ignored.

FIG. 4.

Schematic diagram of the bonding neck.

The long and short radii of the elliptical cross-section of the extruded material filament are $a'$ and $b'$ , respectively, and the length is l. Taking the longitudinal direction for example, in the forming process, the material flow is assumed to occur in a circle with the radius of the contact point being $r_{0}$ (the radius of curvature of the ellipse) at the initial time. At any time $t$ , the instantaneous radius, bonding neck length, and angle are separated $r_{1}$ , $2 y$ , and $2 θ$ . They have the following relationship: $y = r_{1} \sin θ$ (2)

According to the principle of volume conservation, the expression for the instantaneous radius $r_{1}$ at any time can be obtained as： $r_{1} = \frac{\sqrt{π} r_{0}}{\sqrt{π - θ + \sin θ \cos θ}}$ (3)where $r_{0} = \frac{{b'}^{2}}{a'}$

Under the influence of surface tension,³³ when material filaments are bonded, the work done by surface tension is: $W_{S} = - Γ \frac{d S}{d t}$ (4)where Γ is the surface tension coefficient.

When adjacent extruded filaments of length $l$ are bonded to each other, the net contact cross-sectional area at any time $t$ is: $S = \frac{4 l {b'}^{2} \sqrt{π}}{a'} \frac{π - θ}{\sqrt{π - θ + sin θ cos θ}}$ (5)

Substituting Equation (5) into Equation (4), the work done by surface tension can be expressed as: $W_{S} = \frac{4 \sqrt{π} l {b'}^{2} Γ}{a'} \frac{(π - θ) \cos^{2} θ + \sin θ \cos θ}{{[(π - θ) + \cos θ \sin θ]}^{3 / 2}} \dot{θ}$ (6)where $\dot{θ}$ is the instantaneous half-angle changing rate of the horizontal bonding neck, $\dot{θ} = \frac{d θ}{d t}$ .

Assuming that the molten mass is the Newtonian fluid,³⁴ when adjacent extrudate is bonded, the expression for the work done by viscous force of the Newtonian fluid is: $W_{V} = {\int\int\int}_{V} 3 η {\dot{ε}}^{2} d V$ (7)

Where η is the melt viscosity, which is an important factor dependent on the extrusion temperature and temperature gradient (difference between adjacent extrudate,^35–37) V is the bonded volume between extrudate, and $\dot{ε}$ is the fluid strain rate.

The expression of the strain rate is given by: $\dot{ε} = \frac{(θ -π) \sin θ}{(π - θ + \sin θ \cos θ)} \dot{θ}$ (8)

The instantaneous bonding volume $V_{V}$ between two adjacent extruded filaments is: $V_{V} = (π - θ + \sin θ \cos θ) r^{2} l$ (9)

With UVAM utilized in the ME process, the apparent viscosity of the extrudate can be expressed as: $\begin{array}{l} η = σ^{'} {(\frac{M_{0}}{1 + β^{'} t_{t} M_{0}})}^{α'} \times \exp (E_{E} {(R T_{0} + R \frac{ρ_{0} u^{*} A^{2} {ω'}^{2} π r_{1}^{2} t_{t} \sin δ'}{2 C' m_{0}})}^{- 1}) \times {K'}^{- \frac{n - 1}{n}} \cdot {(\frac{r_{1}}{2} \frac{Δ p}{L} + \frac{ρ_{0} r_{1}}{2} A ω' \sin ω' t)}^{\frac{n - 1}{n}} \end{array}$ (10)

Where M₀ is the initial molecular weight, $t_{t}$ is the relaxation time, $E_{E}$ is the viscous activation energy of the polymer melt, R is the gas constant, $ρ_{0}$ is the density of the melt, $u^{*}$ is the propagation speed of ultrasonic vibration in the polymer melt, $ω'$ is the ultrasonic-vibration angle frequency, $δ'$ is the energy conversion angle, $C'$ is the specific heat capacity, $m_{0}$ is the polymer melt mass, $K'$ is the fluidity coefficient, $Δ p$ is the pressure drop, A is the ultrasonic-vibration amplitude, $σ'$ is the empirical constant, which is related to the material of the polymer and the specific processing environment, $α'$ is the Mark–Houwink parameter, determined by the melt properties, $β'$ is the degradation constant and n is the power-law exponent.

Substituting Equations (8), (9), and (10) into (7) gives the work done by viscous force: $\begin{array}{l} W_{V} = \frac{6 π l {b'}^{4}}{{a'}^{2}} σ' {(\frac{M_{0}}{1 + β^{'} t_{t} M_{0}})}^{α'} \times \exp (E_{E} {(R T_{0} + R \frac{ρ_{0} u^{*} A^{2} {ω'}^{2} π r_{1}^{2} t_{t} \cdot \sin δ'}{2 C' m_{0}})}^{- 1}) \times {K^{'}}^{\frac{n - 1}{n}} \cdot {(\frac{r_{1}}{2} \frac{Δ p}{L} + \frac{ρ r_{1}}{2} A ω' \sin ω' t)}^{\frac{n - 1}{n}} \times \frac{{(π - θ)}^{2} \sin^{2} θ}{{(π - θ + \sin θ \cos θ)}^{2}} {\dot{θ}}^{2} \end{array}$ (11)

Equal the work done $W_{S}$ to $W_{V}$ , the expression of the instantaneous half-angle changing rate of the horizontal bonding neck can be given by: $\begin{array}{l} \dot{θ} = \frac{2 Γ a'}{3 \sqrt{π} {b'}^{2} σ'} \cdot {(\frac{M_{0}}{1 + β' t_{t} M_{0}})}^{- α'} \times \exp (E_{E}^{−1} (R T_{0} + R \frac{ρ_{0} u^{*} A^{2} {ω'}^{2} S t_{t} \sin δ'}{2 C^{'} m_{0}})) \times {K'}^{\frac{n - 1}{n}} \cdot {(\frac{r_{1}}{2} \frac{Δ p}{L} + \frac{ρ r}{2} A ω' \sin ω' t)}^{- \frac{n - 1}{n}} [\frac{[(π - θ) \cos^{2} θ + \sin θ \cos θ] {[π - θ + \sin θ \cos θ]}^{1 / 2}}{{(π - θ)}^{2} \sin^{2} θ}] \end{array}$ (12)

Solve Equation (12) with the initial condition of $θ (0) = 0$ , and substitute the result into Equations (1) and (2) to obtain the longitudinal bonding neck (2y). During the forming process, the bonding neck stops growing when the melt temperature of the extrudate drops to the critical temperature. Therefore, it is necessary to analyze the extrudate’s cooling time. Bellehumeur et al.³⁸ used the lumped capacity analysis for modeling the cooling process of the extrudate and thus proposed the cooling model as follows: $T = T_{\infty} + (T_{0} - T_{\infty}) . e^{- m x_{0}}$ (13)where, $m = (\sqrt{1 + 4 α β} - 1) / 2 α, x_{0} = v t, α = k' / ρ^{*} C' v, β = h_{1} P / ρ^{*} C' A' v,$ $T_{\infty}$ is the envelope temperature of the environment, $T_{0}$ is the melting temperature, T is the real-time temperature, $ρ^{*}$ is the density of the material filament, $k'$ is the thermal conductivity of the material, $h_{1}$ is the system convection coefficient, and the terms $C'$ , $A'$ , and P separately represent the specific heat capacity of the material, area, and perimeter of the elliptical section.

Due to the layer-by-layer forming process, it is inevitable for ME parts to have the defect of porosity, as shown in Figure 5. According to the literature,³⁹ the porosity can be given by: $ϕ^{*} = 2 \frac{h_{h}}{r_{r}} - \frac{1}{2} [β \frac{b_{L}}{r_{r}} - \frac{b_{l}}{r_{r}} \sin (β)] - 2 \frac{b_{l}}{b_{L}} \frac{h_{h}}{r_{r}}$ (14)

FIG. 5.

Schematic diagram of the cross-section porosity.

The ME thin plate studied in this paper is composed of multiple layers of PLA fiber material.⁴⁰ The following assumptions are made to study the dynamic characteristics: (1) the layers are firmly bonded, there is no slip, no relative displacement, and the interlayer coupling effect is not considered; (2) the laminate can be regarded as a whole structure. The bonding layer is very thin and has no deformation, that is, the deformation between the single-layer plates is continuous; (3) although the thin plate is formed by multiple layers, its total thickness still conforms to the assumption of a thin plate. Taking the middle plane of the plate before deformation as the xoy plane and establishing a coordinate system, the z axis perpendicular to the middle plane is the thickness direction, the x axis is the length direction, and the y axis is the width direction, as shown in Figure 6. The total length (a), width (b), and thickness (h) are separately 150, 50, and 2.4 mm.

FIG. 6.

Schematic diagram of ME plates’ structure. ME, material extrusion.

Based on the Kirchhoff hypothesis,⁴¹ in the case of small deformation, the displacement components of any point A (x, y, z) of the plate can be represented by the deflection w of the midplane as $\begin{matrix} u (x, y, z) = - z \frac{\partial w}{\partial x} \\ v (x, y, z) = - z \frac{\partial w}{\partial y} \\ w (x, y, z) \approx w (x, y, 0) = w (x, y) \end{matrix}$ (15)where u, v, and w represent the displacement components separately in the direction of x, y, and z. The three-direction strain components ( $ε_{x}, ε_{y}, γ_{x y}$ ) are expressed as $\begin{matrix} ε_{x} = \frac{\partial u}{\partial x} = - z \frac{\partial^{2} w}{\partial x^{2}} \\ ε_{y} = \frac{\partial v}{\partial y} = - z \frac{\partial^{2} w}{\partial y^{2}} \\ γ_{x y} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} = - 2 z \frac{\partial^{2} w}{\partial x \partial y} \end{matrix}$ (16)

Equation (16) can be further written as the relationship between the strain and curvature of the Z plane of the plate $ε = z χ$ (17)

In general, there are five stress components in different sections, which are $σ_{x}, σ_{y}, τ_{x y}, τ_{y z}$ and $τ_{x z}$ . Since the transverse shear stress $τ_{y z}$ and $τ_{x z}$ are much smaller than the in-plane stress $σ_{x}, σ_{y}$ and $τ_{x y}$ , they are not taken into account. Correspondingly, the bending moment of the thin plate is as follows $M = {\begin{cases} M_{x} \\ M_{y} \\ M_{z} \end{cases}} = {\begin{cases} \int_{- h / 2}^{h / 2} σ_{x} z d z \\ \int_{- h / 2}^{h / 2} σ_{y} z d z \\ \int_{- h / 2}^{h / 2} τ_{x y} z d z \end{cases}}$ (18)

Assuming that the elastic modulus of the ME thin plate in the X and Y direction is E₁ and E₂, the shear elastic modulus in the xoy plane is G₁₂, and the Poisson’s ratio in the X and Y direction is $υ_{12}$ and $υ_{21}$ , respectively. The stress-strain relationship of the thin plate can be written as ${\begin{cases} σ_{x} \\ σ_{y} \\ τ_{x y} \end{cases}} = [\begin{array}{l} D_{11} D_{12} 0 \\ D_{21} D_{22} 0 \\ 00 D_{33} \end{array}] {\begin{cases} ε_{x} \\ ε_{y} \\ ε_{z} \end{cases}}$ (19)where $\begin{matrix} D_{11} = \frac{E_{1}}{1 - υ_{12} υ_{21}}, D_{12} = \frac{υ_{12} E_{2}}{1 - υ_{12} υ_{21}} \\ D_{21} = D_{12}, D_{22} = \frac{E_{2}}{1 - υ_{12} υ_{21}} \\ D_{33} = G_{12}, υ_{21} = υ_{12} \frac{E_{2}}{E_{1}} \end{matrix}$ (20)

According to Rodríguez,^42,43 the elastic modulus, shear modulus, and Poisson’s ratio of the ME plate can be given by $\begin{array}{l} E_{1}^{*} = (1 - ϕ^{*}) E, E_{2}^{*} = (1 - \sqrt{ϕ^{*}}) \\ G_{12}^{*} = 2 G \frac{(1 - ϕ^{*}) (1 - \sqrt{ϕ^{*}})}{(1 - ϕ^{*}) + (1 - \sqrt{ϕ^{*}})} \\ υ_{12}^{*} = (1 - ϕ^{*}) υ \end{array}$ (21)where, E, G, and $υ$ are separately the elastic modulus, shear modulus, and Poisson’s ratio of the material filament.

Substituting Equation (17) into (19) can obtain the relationship between stress and curvature, and then substituting it into Equation (18) for integration, the relationship between the internal force and curvature can be obtained by $M = D_{b} χ$ (22)

The rectangular element with 4 nodes and 12 degrees of freedom is a common type of thin plate bending element, which has been widely used and recognized.⁴⁴ The degrees of freedom considered at each node shown in Figure 7 are separately the displacement w, the angles $θ_{x}$ , and $θ_{y}$ .

FIG. 7.

Rectangular plate element with four nodes.

The expression of a rectangular thin plate element shape is $\begin{matrix} N_{i} = \frac{1}{8} (1 + ξ_{i} ξ) (1 + η_{i} η) (2 + ξ_{i} ξ + η_{i} η - ξ^{2} - η^{2}) \\ N_{x i} = - \frac{1}{8} n η_{i} (1 + ξ_{i} ξ) (1 + η_{i} η) (1 - η^{2}) \\ N_{y i} = \frac{1}{8} m ξ_{i} (1 + ξ_{i} ξ) (1 + η_{i} η) (1 - ξ^{2}) \\ (i = 1, 2, 3, 4) \end{matrix}$ (23)

The relationship between node displacement and curvature can be obtained by $χ = [B_{1} B_{2} B_{3} B_{4}] δ_{e} = B δ_{e}$ (24)where the geometric submatrix is $B_{i} = - [\begin{array}{l} \frac{\partial^{2} N_{i}}{\partial x^{2}} \\ \frac{\partial^{2} N_{i}}{\partial y^{2}} \\ 2 \frac{\partial^{2} N_{i}}{\partial x \partial y} \end{array}] = - [\begin{matrix} \frac{1}{m^{2}} \cdot \frac{\partial^{2} N_{i}}{\partial ξ^{2}} \\ \frac{1}{n^{2}} \cdot \frac{\partial^{2} N_{i}}{\partial η^{2}} \\ \frac{2}{m n} \cdot \frac{\partial^{2} N i}{\partial ξ \partial η} \end{matrix}], (i = 1, 2, 3, 4)$ (25)

Substituting Equation (24) into (22) can obtain the relationship between bending moment and nodal displacement as $M = D_{b} χ = D_{b} B δ_{e}$ (26)

The strain energy can be expressed in terms of bending moment-torsion rate as $U_{e} = \frac{1}{2} \iint M^{T} χ dxdy = \frac{1}{2} δ_{e}^{T} k_{e} δ_{e}$ (27)

Substituting Equations (24) and (26) into (27) can obtain the element stiffness matrix as $k_{e} = \iint B^{T} D_{b} B dxdy = m n \int_{- 1}^{1} \int_{- 1}^{1} B^{T} D_{b} B d ξ d η$ (28)

Similarly, the element mass matrix can be obtained by $m_{e} = \int_{V} N^{T} ρ NdV = ρ m n \int_{- 1}^{1} \int_{- 1}^{1} N^{T} N d ξ d η$ (29)

Then the total stiffness matrix K and total mass matrix M_m can be obtained by summing and superimposing the stiffness matrix and mass matrix of the element, respectively.

The motion equation of the thin plate under stress is $M_{m} \overset{\cdot\cdot}{u} (t)+ + C \overset{\cdot}{u} (t)+ + K u (t)= F$ (30)where C is the overall damping matrix, F is the external load vector of the node, and u(t), u(t), u(t) individually represent the node displacement vector, velocity vector, and acceleration vector, $u = λ sin ϖ (t - t_{0})$ ( $λ$ is the n^th-order vector, $ϖ$ is the vibration circular frequency, and $t$ is the time variable).

Ignoring the damping and excitation force, the dynamic equation can be simplified as $M_{m} \overset{\cdot\cdot}{u} (t)+ + K u (t)= 0$ (31)

Solve the eigenvalues of the following equation can get the natural frequencies $[K - ϖ^{2} M_{m}] λ = 0$ (32)

Substituting the results into Equation (31) can obtain the mode shapes of the thin plate.

The frequency domain equation can be written as $[K + iω C - ω^{2} M_{m}] U_{0}^{*} = F_{0}$ (33)where $ω$ is the excitation frequency, $U_{0}^{*}$ and $F_{0}$ are the response amplitude and excitation force amplitude vector. The vibration response of the thin plate can be obtained by solving Equation (33).

The rectangular plate element is adopted to establish the finite element model of the ME plate and the meshing strategy is shown in Figure 8, which consists of 121 nodes and 100 elements. Each element has the same size in the established model, i.e., length, width, and thickness. The boundary conditions of the model are the cantilever boundary conditions (one short side is fixed and the other sides are unconstrained), and the positions of the excitation point and picking vibration point are the bottom and top of the sample, respectively.

FIG. 8.

Meshing strategy of the proposed model.

Results and Discussion

To validate the theoretical model and explore the influence of different ultrasonic vibrations on the dynamic characteristics of the samples, the theoretical and experimental dynamic characteristic parameters without and with UVAM of different amplitude or frequency were compared and analyzed, as well as the actual vibration state of the extrusion liquefier.

The actual vibration state of the extrusion liquefier

Figure 9 below shows the theoretical and actual vibration state of the extrusion liquefier with different UVAM utilized in the time and frequency domains, respectively. Generally, the theoretical vibration state agrees well with the actual one. The slight deviation is mainly because of the damping effect of the components attached to the extrusion liquefier (such as the cooling fan, and parallel arm), which not only weakens the vibration amplitude but also generates several sub-frequency signals. Compared to the sub-frequency signals with the main-frequency one, the amplitude is small to negligible. Therefore, the actual ultrasonic-vibration state of the extrusion liquefier can be considered as harmonic vibration, that is, shown as the theoretical curves.

FIG. 9.

Theoretical and actual ultrasonic-vibration state of the extrusion liquefier. (a) 30 kHz, 2 g. (b) 40 kHz, 2 g. (c) 40 kHz, 4 g. (d) 40 kHz, 6 g.

Effect of the UVAM with different frequency

Inherent characteristic

Taking the ME plate under the cantilever boundary condition as the research object, comparing and analyzing the predicted and measured inherent characteristic (i.e., natural frequency and mode shape) of the Sample $R_{0_}^{0}$ , $R_{20k_}^{2}$ , $R_{30k_}^{2}$ and $R_{40k_}^{2}$ , the dynamic model can be validated and the influencing rule of the UVAM with the same amplitude but different frequency can be determined. Details are listed in Table 2. It can be seen that the predicted results are in good agreement with the measured ones. The error between the predicted and measured natural frequencies is in the range of 5.29–7.77%. The mode shapes are almost the same. Therefore, the proposed theoretical model can reliably predict the inherent characteristics of ME plates processed no matter without or with UVAM utilized.

Table 2.

Measured and Predicted Inherent Characteristics of ME Plates Processed Without and with UVAM of the Same Amplitude but Different Frequencies

Category	Samples	Order
Category	Samples	1st	2th	3rd
Average measured natural frequency A (Hz)	$R_{0}^{0}$	18.00	134.43	417.57
	$R_{20k}^{2}$	18.91	142.47	435.42
	$R_{30k}^{2}$	19.65	149.30	453.63
	$R_{40k}^{2}$	20.03	154.70	468.86
Average predicted natural frequency B (Hz)	$R_{0}^{0}$	19.32	143.67	443.98
	$R_{20k}^{2}$	20.39	151.02	466.61
	$R_{30k}^{2}$	20.64	153.60	474.48
	$R_{40k}^{2}$	21.03	157.64	486.77
Error (%): \|B − A\|/A	$R_{0}^{0}$	7.33	6.87	6.32
	$R_{20k}^{2}$	7.83	6.00	7.16
	$R_{30k}^{2}$	5.04	2.88	4.60
	$R_{40k}^{2}$	4.99	1.90	3.82
Measured mode shape	—
Predicted mode shape	—

ME, material extrusion; UVAM, ultrasonic-vibration aided machining.

In addition, compared the natural frequency of the Sample $R_{20k_}^{2}$ , $R_{30k_}^{2}$ and $R_{40k_}^{2}$ with those of $R_{0_}^{0}$ , it can be found that UVAM can significantly increase the natural frequency of ME samples, which will be further increased with the increasing vibration frequency. This is because UVAM has the function of oscillating the molten material (melt PLA here), and it is beneficial to the material filament melting and reduction of pores and cracks of the extrudate, thereby improving the forming quality of ME samples. With the increasing frequency of UVAM, the improvement effect is more significant. Since the thin plate structural characteristics do not change, the corresponding mode shape remains unchanged, that is, the first three-order mode shapes are separately the first-order bending one, first-order twisting one, and second-order bending one.

Vibration response

Figure 10 compares and analyzes the experimental results of the vibration response of the Sample $R_{0_}^{0}$ , $R_{20k_}^{2}$ , $R_{30k_}^{2}$ and $R_{40k_}^{2}$ . It shows the influence of UVAM of the same amplitude but different frequencies on the samples’ vibration response. The specific data is shown in Table 3.

FIG. 10.

Measured vibration responses of ME plates processed without and with UVAM of the same amplitude but different frequency. (a) The 1st order. (b) The 2nd order. (c) The 3rd order. UVAM, ultrasonic-vibration aided machining.

Table 3.

Measured and Predicted Vibration Responses of the ME Plates Processed Without and with UVAM of the Same Amplitude but Different Frequency

Category	Samples	Order
Category	Samples	1st	2th	3rd
Average measured vibration response A (mm)	$R_{0}^{0}$	0.2581	0.0631	0.0345
	$R_{20k}^{2}$	0.2362	0.0564	0.0312
	$R_{30k}^{2}$	0.2173	0.0517	0.0285
	$R_{40k}^{2}$	0.2055	0.0489	0.0271
Average predicted vibration response B (mm)	$R_{0}^{0}$	0.2829	0.0673	0.0375
	$R_{20k}^{2}$	0.2501	0.0602	0.0335
	$R_{30k}^{2}$	0.2324	0.0568	0.0307
	$R_{40k}^{2}$	0.2257	0.0537	0.0297
Error (%): \|B − A\|/A	$R_{0}^{0}$	9.61	6.77	8.57
	$R_{20k}^{2}$	5.88	6.74	7.37
	$R_{30k}^{2}$	6.95	9.86	7.72
	$R_{40k}^{2}$	9.83	9.82	9.59

ME, material extrusion; UVAM, ultrasonic-vibration aided machining.

It can be seen that the vibration response is reduced when UVAM is utilized, and it will be further decreased as the vibration frequency increases. Details can be seen in Table 3, and the reason is similar as explained in Section 4.2.1. Moreover, the predicted vibration responses are close to the measured ones. The deviation is in the range of 5.88–9.86%. Therefore, the proposed model can also accurately predict the vibration responses of the ME plates no matter without or with UVAM utilized, which further verifies its correctness.

Damping ratio

Table 4 compares the experimental damping ratio of the ME sample (processing the vibration response curve by the half-power bandwidth method), further analyzing the influence of different UVAM on the dynamic characteristics. It can be seen that the damping ratio of the ME sample is significantly increased by UVAM, and it is further increased with the increasing vibration frequency. Therefore, UVAM can improve the dynamic characteristics of ME samples, i.e., anti-shock and vibration performance.

Table 4.

Measured Damping Ratio of ME Plates Built Without and with UVAM of the Same Amplitude but Different Frequency

Category	Samples	Order
Category	Samples	1^st	2^th	3^rd
Average measured damping ratio (%)	$R_{0}^{0}$	7.755	3.673	3.449
	$R_{20k}^{2}$	8.008	3.915	3.674
	$R_{30k}^{2}$	8.152	3.997	3.855
	$R_{40k}^{2}$	8.342	4.030	3.992

ME, material extrusion; UVAM, ultrasonic-vibration aided machining.

Effect of the UVAM with different amplitude

Inherent characteristics

Comparing and analyzing the inherent characteristics of Sample $R_{0_}^{0}$ , $R_{40k_}^{2}$ , $R_{40k_}^{4}$ and $R_{40k_}^{6}$ , it is able to further validate the proposed model and clarify the influencing rule of UVAM of the same frequency but different amplitude on the natural frequency and mode shape of ME plates, as detailed in Table 5. It can be seen that the deviation between the predicted and measured natural frequency is in the range of 1.90–5.47%, and the mode shapes are almost the same. Therefore, the proposed theoretical model can also reliably predict the inherent characteristics of ME plates processed with the UVAM of the same frequency but different amplitudes. With the UVAM utilized, the natural frequency is increased significantly, and it is further increased with the increasing vibration amplitude. While about the mode shape, they are still unchanged as the structural property does not change.

Table 5.

Measured and Predicted Inherent Characteristics of ME Plates Processed Without and with UVAM of the Same Frequency but Different Amplitude

Category	Samples	Order
Category	Samples	1st	2th	3rd
Average measured natural frequency A (Hz)	$R_{0}^{0}$	18.00	134.43	417.57
	$R_{40k}^{2}$	20.03	154.70	468.86
	$R_{40k}^{4}$	20.58	158.57	478.74
	$R_{40k}^{6}$	21.00	161.38	483.83
Average predicted natural frequency B (Hz)	$R_{0}^{0}$	19.32	143.67	443.98
	$R_{40k}^{2}$	21.03	157.64	486.77
	$R_{40k}^{4}$	21.55	163.51	504.63
	$R_{40k}^{6}$	21.95	165.27	510.30
Error (%): \|B − A\|/A	$R_{0}^{0}$	7.33	6.87	6.32
	$R_{40k}^{2}$	4.99	1.90	3.82
	$R_{40k}^{4}$	4.71	3.12	5.41
	$R_{40k}^{6}$	4.52	2.41	5.47
Measured mode shape	—
Predicted mode shape	—

ME, material extrusion; UVAM, ultrasonic-vibration aided machining.

Vibration response

Figure 11 compares and analyzes the measured average vibration response of the Sample $R_{0_}^{0}$ , $R_{40k_}^{2}$ , $R_{40k_}^{4}$ and $R_{40k_}^{6}$ . The vibration responses are reduced with UVAM utilized, and they will be further decreased when the vibration amplitude is increased, as detailed in Table 6. In addition, the deviation range between the theoretical and experimental vibration response is 7.05–10.75%, validating the proposed model again.

FIG. 11.

Measured vibration responses of ME plates processed without and with UVAM of the same frequency but different amplitude. (a) The 1st order. (b) The 2nd order. (c) The 3rd order. UVAM, ultrasonic-vibration aided machining; ME, material extrusion.

Table 6.

Measured and Predicted Vibration Responses of ME Plates Processed Without and with UVAM of the Same Frequency but Different Amplitude

Category	Samples	Order
Category	Samples	1st	2th	3rd
Average measured natural frequency A (Hz)	$R_{0}^{0}$	0.2581	0.0631	0.0345
	$R_{40k}^{2}$	0.2055	0.0489	0.0271
	$R_{40k}^{4}$	0.2006	0.0465	0.0261
	$R_{40k}^{6}$	0.1942	0.0453	0.0245
Average predicted natural frequency B (Hz)	$R_{0}^{0}$	0.2829	0.0673	0.0375
	$R_{40k}^{2}$	0.2257	0.0537	0.0297
	$R_{40k}^{4}$	0.2165	0.0515	0.0288
	$R_{40k}^{6}$	0.2079	0.0492	0.0267
Error (%): \|B − A\|/A	$R_{0}^{0}$	9.61	6.77	8.57
	$R_{40k}^{2}$	9.83	9.82	9.59
	$R_{40k}^{4}$	7.93	10.75	10.34
	$R_{40k}^{6}$	7.05	8.61	8.98

ME, material extrusion; UVAM, ultrasonic-vibration aided machining.

Damping ratio

Table 7 compares the experimental damping ratio of the Sample $R_{0_}^{0}$ , $R_{40k_}^{2}$ , $R_{40k_}^{4}$ and $R_{40k_}^{6}$ . As can be seen, UVAM can obviously increase the damping ratio of ME samples, which will be further enhanced as the vibration amplitude increases. Therefore, the dynamic characteristics of ME samples can be improved by UVAM, especially the anti-shock and vibration performance.

Table 7.

Measured Damping Ratio of ME Plates Built Without and with UVAM of the Same Frequency but Different Amplitude

Category	Samples	Order
Category	Samples	1st	2th	3rd
Average measured damping ratio (%)	$R_{0}^{0}$	7.755	3.673	3.449
	$R_{40k}^{2}$	8.342	4.030	3.992
	$R_{40k}^{4}$	8.369	4.077	4.035
	$R_{40k}^{6}$	8.400	4.103	4.062

UVAM, ultrasonic-vibration aided machining; ME, material extrusion.

Effect of the UVAM with different waveforms

Inherent characteristics

Homogeneously, the experimental influence of the UVAM with different waveforms on the inherent characteristics of ME samples was investigated. Compared the corresponding parameters of the Sample $R_{0_}^{0}$ , $R_{30k_}^{2}$ , $T_{30k_}^{2}$ and $S_{30k_}^{2}$ , it is able to clarify the influencing rule. Details are listed in Table 8. It can be seen that the UVAM of the sinusoidal waveform has the most significant effect, followed by that of the triangle and square waveform. About the mode shapes, they are still unchanged as the structural property does not change.

Table 8.

Measured Inherent Characteristics of ME Plates Processed Without and with UVAM of Different Waveform

Category	Samples	Order
Category	Samples	1st	2th	3rd
Average measured natural frequency (Hz)	$R_{0}^{0}$	18.00	134.43	417.57
	$R_{30k}^{2}$	19.65	149.30	453.63
	$T_{30k}^{2}$	19.14	146.93	448.73
	$S_{30k}^{2}$	18.88	140.82	443.27
Measured mode shape	—

ME, material extrusion; UVAM, ultrasonic-vibration aided machining.

Vibration response

About the vibration response, they are reduced most when the UVAM of the sinusoidal waveform is utilized, as shown in Figure 12. The UVAM of the square waveform has the minimum effect. Details can be seen in Table 9.

FIG. 12.

Measured vibration responses of ME plates processed without and with the UVAM of different waveform. (a) The 1st order. (b) The 2nd order. (c) The 3rd order. UVAM, ultrasonic-vibration aided machining; ME, material extrusion.

Table 9.

Measured Vibration Responses of ME Plates Processed Without and with the UVAM of Different Waveform

Category	Samples	Order
Category	Samples	1st	2th	3rd
Average measured vibration responses (mm)	$R_{0}^{0}$	0.2581	0.0631	0.0345
	$R_{30k}^{2}$	0.2173	0.0517	0.0285
	$T_{30k}^{2}$	0.2265	0.0526	0.0293
	$S_{30k}^{2}$	0.2389	0.0584	0.0326

ME, material extrusion; UVAM, ultrasonic-vibration aided machining.

Damping ratio

Correspondingly, the experimental effect of the UVAM with different waveforms on the damping ratio of ME samples was compared and analyzed in Table 10. As can be seen again, the damping ratio of ME samples increases obviously with the UVAM utilized, meaning that the anti-shock and vibration performance is improved. The sinusoidal waveform UVAM has the most significant effect, followed by triangle and square waveform one.

Table 10.

Measured Damping Ratio of ME Plates Built Without and with the UVAM of Different Waveform

Category	Samples	Order
Category	Samples	1st	2th	3rd
Average measured damping ratio (%)	$R_{0}^{0}$	7.755	3.673	3.449
	$R_{30k}^{2}$	8.152	3.997	3.855
	$T_{30k}^{2}$	8.092	3.957	3.834
	$S_{30k}^{2}$	7.894	3.871	3.614

ME, material extrusion; UVAM, ultrasonic-vibration aided machining.

Conclusion

This paper investigated the mechanism of using UVAM to improve the dynamic characteristics of ME parts, on which the influencing rule of different UVAM was clarified. The specific conclusions are as follows: (1)

A novel method of using UVAM to improve the dynamic characteristics of ME parts is proposed, that is, introducing ultrasonic vibration into the ME process by using ultrasonic piezoelectric ceramic.

(2)

Based on the finite element theory, the dynamic model of ME thin plates processed without and with UVAM is established and validated. It can give reliable predictions in terms of dynamic characteristic parameters, that is, natural frequency, mode shape, and vibration response of ME plates.

(3)

UVAM can obviously increase the ME plates’ natural frequency, decrease the vibration response, and increase the damping ratio, meaning that the ME plates’ anti-shock and vibration performance is significantly improved. As the vibration frequency or amplitude increases, the improvement effect is more obvious.

(4)

Compared to the effect of different-waveform UVAM on the dynamic characteristic of ME parts, the sinusoidal-waveform one has the most significant effect, followed by triangle and square-waveform one.

(5)

The modeling and experimental research methods proposed in this paper are also applicable to the study of other materials and other additive manufacturing techniques.

Footnotes

Authors’ Contributions

S.J.: Conceptualization, project administration, funding acquisition, supervision. C.C. and Y.Z.: Data curation, writing—original draft preparation. S.J., C.C., and Y.Z.: Formal analysis, validation. C.C., Y.Z., and C.Z.: Investigation. S.J. and C.C.: Methodology. S.J. and C.Z.: Resources, writing—review and editing. C.C.: Software.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This research is financially supported by national natural science foundation of China (52475093) and the Fundamental Research Funds for the Central Universities (N2303019, 2023-MSBA-028).

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