Introduction

The objective of this paper was to investigate possible vibration isolator solutions for a 4D entertainment theater. The approach was to conduct a material selection search within the elastomers family to resolve a vibration leakage issue claimed by customers of Company A. This company manufactures a vibratory system that actuates vibrations transmitted from an amplifier in order to provide a 4D experience while watching movies. The device manufactured by Company A operates as a vertical tactile motion actuator that transforms electrical signals into mechanical vibration motion.

The device is placed underneath each leg of a couch. The vibratory system including the vibration actuators responds to specific scenery in the movie and creates vibration responses to mimic the scenery in the seating environment. However, the device is in direct contact with the floor where it exposes vibratory energy that disturbs the surroundings. The vibration losses transfer into the floor and are not damped. Therefore, the issue is vibration leakage into floor. The specification requirements required by Company A for two different geometries are listed in Tables 1 and 2.

Table 1 Specification list by Company A for square pad
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Table 2 Specification list by Company A for circular rubber feet
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Background

The main idea behind this study was to select a material that is high in damping. Highly damped materials have the capability to absorb vibrational energy and damp it thoroughly. The best known vibration isolation materials are elastomers. Those materials have a significant mechanical loss factor coefficient, η, and yet have the least Young’s modulus values [1].

The Ashby’s charts [2] were used in narrowing down the materials that have high mechanical loss factor coefficient, η, and low Young’s modulus, E. The system being investigated represents an active application that aims to reduce the vibration from the machine to the ground shown in Fig. 1.

Fig. 1
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Active application

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The natural frequency for the system is shown in the following equation:

$$ f_{o} = \frac{1}{2\pi }\left( {\frac{K}{M}} \right)^{{\frac{1}{2}}} $$
(1)

where K represents the dynamic spring stiffness, and M represents the weight (applied load). The natural frequency range is provided from the specification list. The excitation frequency of the vibration actuator, specifically for the device manufactured by Company A, had a range of 0–600 Hz. The vibrational transmissibility is the ratio of the energy going out of the system to the energy coming into the system. The lower transmissibility means the lower damage to sensitive components. Transmissibility analytical approach is shown in “Appendix” and calculated by the following equation:

$$ T = \frac{F}{{F_{0} }} = \left| {\frac{1}{{1 - \left| {\frac{N}{{f_{0} }}} \right|^{2} }}} \right| $$
(2)

where T represents transmissibility, F is the transmitted force, F 0 is the machine’s excitation force, N is the interference frequency, and f o is the natural frequency. The vibrational transmissibility was compared to the frequency ratio of the system studied. The key is to ensure that the material selected was under the area of isolation where magnification factor is equal to or greater than 1.4. The magnification factor is defined as the point at intersection between an underdamped system to a critical damped system, which is always constant having a value of \( \sqrt 2 \). The dynamic stiffness can also be determined using the natural frequency chart by indicating the natural frequency and the nominal loading values [3]. The analytical calculation for frequency ratio is shown in “Appendix” and calculated by using the following equation:

$$ {\text{Frequency}}\,{\text{ratio}} = \frac{N}{{f_{0} }} $$
(3)

The frequency ratio and the transmissibility were plotted to ensure that the material selected lies under the isolation area, Fig. 2. The system has to impose a frequency ratio that is above the value of magnification factor.

Fig. 2
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Magnification factor

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Materials and Methods

The methodology was to use CES [2] in order to examine the relevant properties of the materials such as the mechanical loss factor coefficient, η, and Young’s modulus, E, and updated market price. Figure 3 shows multiple selection options varying from metals having lowest mechanical loss factor coefficient, η, to elastomers having the best η.

Fig. 3
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Ashby’s chart for mechanical loss coefficient vs. Young’s modulus for all materials [2]

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In Fig. 3, four different materials polyurethane foam, polypropylene foam, melamine foam and butyl/halobutyl rubber were compared. The first objective was to have a high mechanical loss coefficient and a low Young’s modulus E. The process of selecting the best material starts by drawing a diagonal line in Ashby’s chart. The materials that lie above the diagonal line will be considered and further examined. Since the polyurethane lies above the diagonal line, it has the highest mechanical loss coefficient and lowest Young’s modulus; then, it will be selected as the best material for the study. The price is the second objective for the study. Company A indicated that the price should not be higher than five dollars per piece. Table 3 shows that polypropylene, butyl/halobutyl rubber and polypropylene meet the objective, but melamine does not meet the objective due to its high cost. Accordingly, melamine was eliminated.

Table 3 Various elastomer prices and mechanical properties [2]
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In Figs. 3 and 4, the elastomer bubbles on top far left have highest η. In Figs. 5 and 6, the prices for elastomers selected versus their mechanical loss coefficient are shown.

Fig. 4
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Ashby’s chart (η vs. E) for all materials [2]

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Fig. 5
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Ashby’s chart (price vs. η) for all materials [2]

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Fig. 6
figure 6

Ashby’s chart (price vs. η) for all materials [2]

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The materials selected from CES [2] as shown in Figs. 4, 5, and 6 were all polyurethanes having differences in their microstructure. The prices given in Table 4 for the selected materials were relatively economical.

Table 4 Materials selected based on price and mechanical loss coefficient from CES [2]
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The polyurethanes shown in Table 4 illustrate the best possible solutions for the selection. An initial selection has been made on the polyurethane foam (elastomeric, open cell, 0.024) based on its relatively low Young’s modulus and low cost compared to other polyurethanes shown in Table 4. Hence, the 0.024 in the polyurethane name means the relative density value of the material. Once polyurethane foam (elastomeric, open cell, 0.024) was selected, the selection was suggested to Company A to find the raw material in the market, to create a mold design and to find a supplier. However, Company A was unable to find a supplier for the raw materials as well as a mold design creation. Therefore, Company A requested to research other possible sources besides Ashby’s charts in order to find a supplier willing to provide a rubber molded by their unique design. A research online has been conducted to find a rubber that is specifically suitable for vibration isolation applications. The research resulted in finding out that there was a modified polyurethane composition owned by Sorbothane Company, which carries the same name, but shows a better performance than unmodified polyurethane in controlling mechanical vibrations and is suitable for vibration damping operations. Sorbothane is a viscoelastic polymer (polyurethane) that has low transmissibility verifying its damping superiority over other elastomers [3].

Results

Table 5 represents the properties for Sorbothane provided by Sorbothane Company. The durometer shore 00 indicates the hardness scale for the rubber. The durometer values at which the selection is made have to be stiff appropriately that it does not affect the 4D experience. The rubber durometer has to be stiff enough that it is statically fixed at normal conditions. That is, shore 00 offers durometer 30, 50, 70, where the less durometer value implies less stiffness. At shore 00 grades 30 and 50, they fall under the extra soft rubber criteria, but shore 00 grade 70 falls under medium soft rubber, which is slightly stiffer than grades 30 and 50.

Table 5 Sorbothane material properties [3]
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The mechanical loss coefficient for Sorbothane is represented in Table 6. At given resonant frequencies, the Sorbothane responds with different loss factor coefficients. The less the durometer, the more the mechanical loss coefficient. However, accounting for an appropriate stiffness for the application, if the least durometer is selected, even though it has the highest mechanical loss coefficient, the rubber will create undesired motion itself under the loadings. This will affect the desired mimicking motion created by the vibration actuator. Ultimately, the 4D experience will not be pleasurable.

Table 6 Mechanical loss coefficient (tan delta) for Sorbothane [3]
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In Figs. 7 and 8, Sorbothane was being compared to various types of common vibrational isolators used such as natural rubber and neoprene. Sorbothane was compared to natural rubber and neoprene to verify its performance in damping effectiveness. In Fig. 7, Sorbothane shows its capability to damp vibrations with a faster response by having smaller oscillations at given impact forces (G-force). Also, Sorbothane exhibited very low rebound when compared to other materials [5]. In Fig. 8, Sorbothane has the least transmissibility while maintaining a relatively similar frequency ratio to neoprene and natural rubber [5]. Low transmissibility means less damage to sensitive components [5].

Fig. 7
figure 7

Controlling shock [5]

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Fig. 8
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Controlling vibration [5]

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Table 7 shows transmissibility analyzed for the vibrational system using Eq 2. The natural frequency is given by the range specified by Tables 1 and 2 for Company A. The driving frequency is averaged for the vibrational actuator, and the frequency ratio is calculated using Eq 3. The interference frequency is for the vibration actuator.

Table 7 Data analysis
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Figure 9 represents a square rubber pad that will be placed underneath the couch isolating it from the floor ground. Figure 10 represents circular rubber feet that will be placed underneath each chair leg.

Fig. 9
figure 9

Square rubber pad [6]

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Fig. 10
figure 10

Circular rubber feet [7]

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Tables 8 and 9 show the general properties of the Sorbothane, which have been conducted by Sorbothane. The prices and the loading capability of the product are the market off-shelf price.

Table 8 Sorbothane prices [6,7,8,9]
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Table 9 Sorbothane loading capability [8, 9]
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Discussion

Based on the results found, the selected polyurethane rubbers from Ashby’s charts met the criteria of selection by evaluating their mechanical loss coefficient and price initially. Yet, the best possible solution was to consider far more factors than the material properties themselves. Company A is aiming to find a supplier that can supply rubber material in an off-shelf product or a unique mold designed by Company A. In addition, the company is aiming to establish a professional contract with the supplier for future business. Sorbothane as a vibration isolation solution company has a unique composition polyurethane rubber that is made specifically for vibration isolation tasks. Also, Sorbothane has a stock rubber pad that matches the exact dimensions given by the specification lists with ability to withstand excessive nominal loadings ranging between 900 and 1200 lbs (408–544 kg) for shore 00 durometer 30 and 1200–2400 lbs (544–1089 kg) for shore 00 durometer 50 [10]. Moreover, Sorbothane has circular rubber feet shape that has a similar performance and price of $2.5 for each. The mechanical loss coefficient and price were the most valuable parameters in the study. Sorbothane had a relatively high mechanical loss coefficient. The transmissibility analyzed for the viboratory system showed that the rubber lied under the isolation area having 2.86% transmissibility and a frequency ratio of 6 at occasional loadings. Also, the price if ordered by minimum order quantity is managed to be compromised if ordered in large quantities.

Summary

  • Material selection for a suitable vibration isolation application was found to be an elastomer.

  • Ashby’s charts were utilized in searching for elastomer based on three main parameters: mechanical loss coefficient, Young’s modulus and price.

  • Transmissibility of the vibrational system is calculated to be 2.86%, whereas the frequency ratio was determined to be 6, yielding a rubber material that lies under isolation area.

  • Sorbothane is selected based on its rapid performance in controlling shock and vibration compared to other elastomers.

  • For pad selection, Sorbothane with shore 00 durometer 50 was available at Sorbothane with dimensions matching the specification list of 0.127 × 0.127 m (5″ × 5″) and 0.0127 m (0.5″) thickness for a price of $12 for a singular pad; total savings are 19%.

  • For circular rubber feet selection, Sorbothane offers them with shore 00 durometer 50 with a price of $2.50 for each, saving up to 50% for specification list maximum price.