Analysis of Vibration Transmission Path in Packaging System and Design of Teaching Experiment

. It is essential for realizing the most suitable product bufer packaging design to quantify the vibration transmission characteristics of the product packaging system. Te experiment system for the vibration transmission path of protective packaging is designed in this paper. Te practical system is used to analyze the vibration transfer path of the product packaging system and identify the critical transfer path. Te concepts of the cushions’ contribution rate and the cushions’ weighted contribution rate are introduced. Te product cushioning based on the weighted equal contribution rate of the cushions is proposed. It has been verifed by experiments that the system can accurately identify the transfer path with the weighted contribution rate of the cushions as a reference for the design of product bufer packaging, which improves the utilization rate of bufer packaging materials and reduces the cost of packaging materials. Te weighted equal contribution rates of bufer pads 1, 2, 3, and 4 are 40%, 27%, 22%, and 11%, respectively. For the needs of experiment teaching, the teaching content based on the protective packaging transfer path testing system is designed, which provides a reference for the practical education of the packaging specialty.


Introduction
With the rapid development of the logistics and transportation industry, the use of the foam to cushion and products protection is increasing with a series of economic and ecological problems.To respond to the call for plastic restriction [1], courier packaging reduction [2] has become an urgent problem in the packaging industry.At present, the cushioning design of products in transportation is usually based on the dynamic cushioning characteristics curve of (DCCC) cushioning materials and the transfer rate curve of material thickness and static stress.Te process includes calculating the area and thickness of cushioning materials required for the products, carrying out comprehensive cushioning or local cushioning, and testing the packaging prototypes to verify the cushioning vibration reduction efect.Tis method is based on the characteristics of cushioning materials based on empirical.Te cushioning area of each part is mostly uniformly distributed in the local cushioning, which needs to improve the utilization rate of cushioning materials.Te research points of the vibration transmission characteristics (VTC) of product packaging systems [3] under actual environmental excitation have more advantages compared to the DCCC method [4,5].Te VTC analyzes the critical vibration transmission path of the product and carries out essential bufer design for it.Te VTC carries out reduced bufer design for the nonkey vibration transmission path of the product, which makes full use of the bufer materials in each part to reduce the cost of bufer packaging.Te transfer path method [6,7] is an efective tool for VTC analysis.Te path can analyze the excitation energy of each excitation source or each transfer path into the target source of the proportion of the overall system vibration energy through the transfer [8].
A number of scholars have studied the dynamic characteristics of product packaging systems [9,10].Wang et al. [11] used the inverse analysis of the dynamical response of coupled product transport systems on the inverse substructuring method, which can optimize the design of cushioning packages.Li et al. [12] analyzed the contribution of each vibration path of the gearbox based on the velocity intervention loss to identify the main transmission path of the gearbox vibration.Te vibration test of the gearbox was carried out to verify the correctness of the transmission path model.Fan et al. [13] got the path contribution of fan noise in the converter and transformer noise sources to determine the main contributing paths by the transfer path analysis test.Te simulations were performed to verify the experimental results.
Among the transfer path methods, the advanced transfer path analysis (ATPA) method is widely used in the feld of noise and vibration control due to its experiment simplicity, no need for load identifcation, and theoretical decoupling between transfer paths.Magrans and Guasch [14] applied the ATPA methodology to identify the contribution of antennas and structural panels to the interior noise of the bus.Te modifcations were made to reduce the interior noise level at a given location based on the contribution from some structural elements of the bogie or the engine.Aragones et al. [15] used the ATPA method to the rectangular body model and developed a numerical model of the rectangular box based on fnite elements, which reconstructed the working condition signals through a direct transfer function.It presented a good agreement between measured and numerical results, which provided a reference for the numerical modeling study of the ATPA method.Wang and Zhu [16] further developed the predictive capabilities of the ATPA by extending 2 path-blocking techniques.It used easily measurable variables from the original system to predict the response of path-blocked systems to fnite element models of continuous systems, which can predict structural vibration transfer after path modifcation.Liao et al. [17] investigated the vibration transfer characteristics of coupled systems with Neumann series based on the higher transfer path method, which can further expand the application scope of the method.Malkoun et al. [18] developed a method for separating track and vehicle rolling noise based on ATPA, which obtained the separation of track and vehicle noise for a specifc rolling stock on a specifc track.It can realize the separation of track and vehicle noise for diferent types of rolling stock on diferent types of tracks.
Te establishment of the universally applicable experimental system for the transfer path of product packaging components is necessary for product packaging design.It is necessary to establish a universally applicable experiment system for product packaging transmission path for product packaging design.Tis paper designs a protective packaging transmission path experimental system, which can analyze the contribution of each vibration transmission path for any locally bufered product packaging system.Based on the design principle of weighted equal contribution rate of bufer pads, the experiment system provides a scientifc basis for the reduction design of bufer packaging.

Packaging Vibration Design Considerations and Requirements
Design of teaching experiment system design for transmission path and process of protective packaging of electrical products are based on the packaging design considerations which are listed as follows.
(1) Te attributes of the product itself: the allowable brittleness value, shape, size, weight, volume, center of gravity, and quantity of the product.(2) Environmental conditions during the circulation process, such as transportation intervals, transportation methods, loading and unloading times, equivalent drop height, impact direction, climatic conditions, and storage conditions (3) Characteristics of packaging materials (4) Te structure, shape, material, and strength of the outer packaging container (5) Characteristics of sealing materials (6) Packaging craftsmanship (7) Other packaging methods, such as moisture-proof, waterproof, rustproof, and dustproof.
Te requirements for packaging design are given in Figure 1.

Design of Teaching Experiment System Design for Transmission Path and Process of Protective Packaging
3.1.Experiment System Devices and Functions.Figure 2 shows the protective packaging transfer path experiment system components, which consist of the computer-side (data processing system), the data acquisition system, and the auxiliary test system.Te data processing system includes DASP V11 (data acquisition and signal processing) and MATLAB software.Te data acquisition system is made up of the data acquisition instruments, the force hammer, and the acceleration sensor.Te auxiliary test system consists of the hanger and the additional mass piece.Figure 3 shows the experiment system principle chart.Te hanger in the auxiliary test system lifts the product packaging system and puts it in a free suspension.Te additional mass piece is used to assist in signal acquisition.Te data acquisition system adopts a single-input multipleoutput test method, in which the excitation signal is given by the force hammer excitation and the acceleration response signal is collected by the acceleration sensor.Te outputs of both the force hammer and the acceleration sensor are ICP signals.Te INV data signal collector converts the excitation signal and the response signal.
Te converted signals were passed to the data processing system for analysis.Te computer-side software DASP V11 in the data processing system is used to  Shock and Vibration to obtain the weighted equal contribution rate of each cushion liner, which is used to guide the cushion packaging design.

Te System Work
Principles.Te experiment system is based on the ATPA of global transmissibility direct transmissibility function advanced transfer path method [19], which  Shock and Vibration 3 is one of the transfer path methods.Te ATPA transfer path method is divided into 2 steps [20].Te frst step is to test the overall transfer rate function of the system [21].Te second step is to obtain the direct transfer rate function by calculating the overall transfer rate function and then perform the vibration transfer path contribution analysis for each path [22].Te overall transfer rate function global transmissibility function (GTF) represents the ratio of the response signal of target point j to the response signal of subsystem i when the excitation is applied at the point i [22].Te direct transmissibility function (DTF) represents the ratio of the response signal of target point j caused by subsystem i only to the response signal of that subsystem i when excitation is applied at the point i.Te direct transfer function represents the direct transfer capability from subsystem i to subsystem j when the other subsystems are shielded.In the transportation process, the coupling excitation of road and vehicle is transferred to the product.Te vibration of key components is transmitted to the bottom plate of the car through the padding of the product packaging.Te protection of critical components is crucial, which are the most easily damaged components in a product.Te product packaging system is divided into a vital component subsystem and a cushion liner subsystem.Te advanced transfer path method is applied to study the vibration impact of each part of the cushion liner on the product.Te DTF is used to quantify the vibration transfer capability from a particular cushion liner subsystem to the critical component of the product.
For the product packaging system, the vibration response s o of the product critical component subsystem is equal to the vector superposition of the input cushion system excitation signal s i and the direct transfer function T D io multiplied from each cushion subsystem to the essential component subsystem.Te theoretical equation can be expressed as follows: Te product packaging system has n subsystems.It is assumed that the force hammer acts on the cushion liner subsystem i; the product key element subsystem is j (i ≠ j).Point e refers to the path point of a subsystem and the response at point j is equal to the sum of the response components transmitted directly to point j from all path points as given in the following equation: s D ej is the signal component passed directly from point e to point j.s D ej is equal to the work response signal at point e multiplied by the direct transfer rate function from point e to point j as listed in the following equation: According to the defnition of the overall transfer rate function, the overall transfer rate function from point i to point e is equal to the ratio of the response signal at point e to the response signal at end i as displayed in the following equation: T G ie is the overall transfer rate function from point i to point e.Substituting equations ( 2) and (3) into equation (1) yields the following equation: Equation ( 5) can be rewritten as follows: Substituting equations ( 6) into equation( 5) yields the following equation: Terefore, the overall transfer rate function matrix is related to the direct transfer rate matrix as given in the following equation: Equation ( 9) is the vibration contribution of each cushion liner to the critical components of the product s i o .
Te frequency point where the critical components of a product are most likely to be damaged is at the resonance peak.Te vibration contribution rate of the cushion pads at the resonance peak is signifcant.
Te concept of equal contribution rate of cushion pads (ECP) is introduced and defned as the ratio of the contribution of the i cushion pad to the resonance peak of the critical component s i op to the peak vibration response s op of the crucial component under the external environmental excitation as shown in the following equation: Tere are multiple resonance areas during vibration.Te analysis is performed at the most dominant resonance peak if the peak of each resonance area difers signifcantly.Te study is weighted if the height of multiple resonance areas does not vary greatly.
Te weighted equal contribution rate of cushion pads (WECP) is defned as the superposition of the cushion equal contribution rate of the i cushion pad at the j resonance peak multiplied by the ratio of the j resonance peak to the sum of all resonance peaks as shown in the following equation: Te cushion liner's weighted contribution rate is the cushion design evaluation basis.According to the weighted contribution rate of the cushion liner, the key or nonkey protection of the cushion liner is carried out at each position of the product.Te larger the weighted contribution rate of the cushion liner, the more protection is needed.Te smaller the weighted contribution rate of the cushion liner, the less protection is required.Te design of cushion liners to protect the product and minimize the cost of cushion packaging can be optimized.

Experiment System Operation
Procedure.Te experiment system operation process can be carried out in 6 steps as follows.
(1) Measure the response signals between all subsystems and then compare them to obtain the overall transfer rate function TG.Te function includes the overall transfer rate function between the bufer liner subsystems and the overall transfer rate function between the target subsystems.(2) Calculate the direct transfer rate matrix from the cushion liner subsystem to the target subsystem based on the relationship between the overall transfer rate function and the natural transfer rate (3) Measure the signals of the cushion liner subsystem and the target subsystem under the actual working conditions (4) Based on the direct transfer rate from the cushion liner subsystem to the crucial element subsystem and the excitation signal of the cushion liner subsystem under the actual working conditions, the response of the critical element subsystem is reconstructed.Te reconstructed response signal of the essential element subsystem is ftted with the measured movement of the subsystem to verify the correctness.( 5) Te equal contribution rate and weighted similar contribution rate of each cushion liner are calculated (6) Perform cushion packaging design analysis based on the weighted equal contribution rate of the cushion liner.
Figure 4 shows the experiment steps.

Experiment Subjects.
Te actual product packaging system was tested with the experiment system to verify efectiveness in order to verify the feasibility of the experimental system.Te computer mainframe is a typical electronic product with internal complexly connected components.Bufer protection is needed to avoid damage to its internal components in transport.Te computer mainframe is the test object and the computer motherboard is the critical protection component.Te main parts of the computer mainframe and selected key components are listed in Table 1.
Te dimensional parameters are given in Table 2. Te preliminary design of the bufer package of the product packaging system was carried out to ensure the rationality of the initial setup.
4 bufer corner pads of the same size were used to bufer and protect the computer mainframe, and the bufer material was foamed polyethylene with the density of 26 kg/m³.Te 4 cushion pads are numbered 1, 2, 3, and 4 in clockwise order from a corner near the computer motherboard, which is displayed in Figure 5. 3 lists the experiment equipment and apparatus.Te product packaging system is frst lifted with a hanger to make it free-hanging during the experiment.Te auxiliary mass piece is glued to the bottom of the cushion liner to keep the overall horizontal state.

Experiment Steps. Table
Figure 6 shows the computer mainframe components and the selected key components.Te acceleration sensor is installed on the auxiliary mass piece of each cushion liner after determining the location of the measurement points.Te acceleration sensor is installed on the computer motherboard.
Figure 7 shows the computer motherboard measurement points.Te force hammer is installed with the acceleration sensor to the channel of the data acquisition instrument.Te data acquisition instrument is connected with the computer.Ensure that the device is properly connected and then perform the test.Each measurement point was struck 7 times and sampled 7 times accordingly to reduce the error of the hammer.Te 7 samplings data were averaged as a set of data.Due to the interference of noise in the signal during the experiment, it is necessary to check the measurement data, such as poor coherence of the measurement points.Te experiment should continue until the measurement point of the frequency response function has better coherence and can be used.When the collected data are correct, the measured signals are averaged and windowed, which is input into the program prepared by MATLAB for the analysis of the vibration contribution of each cushion liner.Te road transportation is the primary mode of transportation for the computer mainframe.Te ASTM standard truck spectrum was selected as the simulated excitation signal for testing to obtain the measured vibration acceleration response of the computer mainframe under simulated excitation.Te movements of the 4 cushion pads input under simulated excitation were multiplied and superimposed with the direct transfer rate function to obtain the ftted response values of the computer motherboard.

Experiment Teaching Applications of the Laboratory.
Te experiment content revolves around cushioning reduction design for express packaging products.Te instructor teacher frst explains the experiment principles and Shock and Vibration data analysis and processing and then gives an introduction to the experiment system and experiment demonstration.Te students must master the empirical system principle, operation specifcation, and data processing method.On this basis, the students work in groups to select commonly used electrical products, analyze the current product packaging design, and put forward suggestions for reducing the amount of packaging in response to the shortcomings of the product design.
Te students apply the experiment system to analyze the vibration transmission characteristics of the product packaging system and calculate the cushion contribution rate and the weighted contribution rate of the cushion liner.Te helpful report will be written to compare the vibration response of critical components before and after the optimization and the cost of cushioning materials.In addition,     Shock and Vibration instructors encourage students to ask questions such as experiment system improvement design, data processing, and other suggestions to increase the bufer design's accuracy.Tese questions can deepen students' understanding of the practical system and stimulate students' interest in learning about the packaging profession.

Results and Discussion
In order to verify the accuracy of this experiment system, the measured acceleration response of the computer motherboard is compared with the ftted acceleration response in the frequency domain.Figure 8 shows the computer motherboard measured response and synthetic response amplitude frequency comparison.Te computer motherboard response ftted by the direct transfer rate tested by this experiment system which shows an overall agreement with the measured response with the frequency range from 0 to 100 Hz.Each prominent resonance peak corresponds to one another which indicated the accuracy of this practical test system.Te ftted response is slightly larger than the measured response, which made the experimental test results more conservative and reliable.Te areas with signifcant errors between the ftted response and the measured response are located in the second main resonance region, which is also a key consideration in bufer design.Te reasons for the errors are the nonlinearity of the actual system and the low coherence function values at individual frequencies.
Figure 9 shows vibration contribution of each cushion liner in the entire frequency band.Te excitation of products during transportation is mainly concentrated in  Shock and Vibration the low-frequency region.More attention is paid to the vibration response in the low-frequency region.In the full frequency range, the computer motherboard has two main resonance regions, namely, 0-10 Hz and 10−30 Hz.In the frequency range of 0-10 Hz from Figure 8, the contribution to the peak vibration response of the computer motherboard is 1, 3, 2, and 4 in descending order.In the frequency range of 10−30 Hz, the contribution to the peak vibration response of the computer motherboard is 2, 1, 3, and 4, respectively.
Figure 10 shows the vibration contribution of each cushion liner in the frst primary resonance region.Te resonance peak frequency of the frst main resonance region is 4 Hz.
Te vibration contribution of each cushion liner in the second primary resonance region is shown in Figure 11.Te resonance peak frequency of the second main resonance region is 25 Hz.Te vibration response of the product is most intense at the resonance peak.Te weighted equal contribution rates of bufer pads 1, 2, 3, and 4 are 40%, 27%, 22%, and 11%, respectively.
Te vibration contribution of the bufer liner in each main resonance region and the equal contribution rate are listed in Table 4. Te cushion packaging design is based on  Te cushion liner design focuses on cushion liners 1, 2, and 3 in order.Cushion liner 4 can be used as a nonfocused liner, which is more scientifc and reasonable than the previous way of all cushion liners of the same area.

Conclusion
Energy conservation is the theme of scientifc research [23][24][25][26] and packaging design also follows this principle.With the rapid development of the logistics and transportation industry [27,28], the reduction of the cost of transportation packaging [29,30] and product packaging puts forward higher requirements for product cushioning design [31,32].Te experiment system for the vibration transmission path of protective packaging is designed in this paper with the vibration transfer path of the product packaging system [33].
Te experiment system can be used in the teaching applications, which provides a good model for packaging experiment teaching [34].Te conclusions of this paper are as follows.
(1) Te concepts of the cushions' contribution rate and the cushions' weighted contribution rate are introduced.Te product cushioning based on the weighted equal contribution rate of the cushions is proposed.(2) Te feasibility of the experimental system for vibration transmission characteristics analysis of the product packaging system was verifed.Te weighted equal contribution rates of bufer pads 1, 2, 3, and 4 are 40%, 27%, 22%, and 11%, respectively.(3) For the needs of experiment teaching, the teaching content based on the protective packaging transfer path testing system is designed, which provides a reference for the practical education of the packaging specialty.
convert the time domain acceleration signals into frequency domain signals.Te frequency domain acceleration signals are processed and analyzed in a program prepared by MATLAB software

Figure 3 :
Figure 3: Te principle chart of the experiment system.
Measurement of response signals between all subsystems Measured signals of key component subsystems cushioning package design analysis Make analogies Global transfer function T G Calculation The direct transfer rate matrix from cushion liner subsystems to target subsystem Reconstruction signals of key component subsystem Measurement of signals from all subsystems under actual operation condition equivalent contribution rate and weighted equivalent contribution rate for each cushion liner Verification of consistency

Figure 6 :Figure 7 :
Figure 6: Computer mainframe components and selected key components.

Nomenclature n :
Subsystems s o : Vibration response s D ej : Signal component passed directly from point e to point j T G ie : Overall transfer rate function from point i to point e s i o : Vibration contribution of each cushion liner to the critical components.s op : Peak vibration response s i op : Contribution of the i cushion pad to the resonance peak of the critical component

Table 1 :
Computer mainframe components and selected key components.

Table 2 :
Dimensions of experimental objects.

Table 4 :
Vibration contribution of cushion liner in each main resonance area and equal contribution rate.