Impacts of PU Foam Stand-Off Layer on the Vibration Damping Performance of Stand-Off Free Layer Damping Cantilever Beams

Stand-off free layer damping is a vibration reduction method based on the traditional free layer damping. In this paper, a stand-off free layer damping cantilever beam is prepared with the steel plate as the base layer, rigid polyurethane (PU) foam as the stand-off layer, and rubber as the damping layer, and the motion equation of the cantilever beam is derived. ,e dynamic mechanical properties of damping rubber and PU foam are tested and analyzed. ,rough hammering tests, we have studied the effect of the density and thickness of the PU foam layer on the amplitude-frequency curves, modal frequencies, and loss factors of the cantilever beams. ,e results show that the rubber damping material is a major font of energy dissipation of the cantilever beam, and PU foam acts mainly to expand the deformation of the damping layer and plays a role in energy consumption. By increasing the density and thickness of PU foam within a certain range, the vibration peaks of the first five modes of the cantilever beam decreases gradually, the loss factors rise, and the damping performance is improved. Meanwhile, increased density and thickness enhances the overall stiffness of the beam, making the modal frequencies get higher.


Introduction
e vibration and noise generated during the navigation of a ship not only seriously affects the comfort of the ride but also affects the rest of the crew. It will also reduce the technical performance of the ship, causing fatigue damage to the mechanical structure and abnormal instrumentation. Due to the special requirements of the production and development of the ship manufacturing industry, trying to reduce the vibration and noise of the ship is a focus of research [1,2].
ere are generally two technical measures to reduce vibration and noise: one is to try to reduce the intensity of the noise source itself and the other is to use damping and absorbing materials to dissipate the vibration and noise energy. e use of damping materials to control vibration and noise is more convenient to implement, and it is a commonly used means of vibration and noise reduction [3,4].
Damping materials on ships often exist in two structural forms: free layer damping structure and constrained layer damping structure [5,6]. In 1959, Yellin et al. [7] first proposed adding a stand-off layer between the base layer and the damping layer to expand the energy dissipation deformation of the damping layer and to improve the vibration damping performance. By introducing a stand-off layer into the traditional damping structure, a stand-off structure can be formed, including stand-off free layer damping and constrained layer damping. An ideal stand-off layer has a bending stiffness close to zero, a shear stiffness approaching infinity, and incompressibility [8]. Stand-off constrained layer damping is an efficient damping treatment, which has been studied extensively and deeply. Yellin et al. [9,10], based on the Euler-Bernoulli beam, analyzed the frequency response of the stand-off damping and compared it with the traditional constrained layer damping. e results show that the standoff layer can significantly increase the energy dissipation level of the damping layer. Research by Yan et al. [11][12][13] showed that when the elastic modulus of the stand-off layer is 100 times that of the damping layer, the tube-shaped stand-off damping model can achieve better vibration reduction performance without changing the natural frequency of the structure. Kumar [14] claimed that increasing the thickness and modulus of the constrained layer can improve the damping performance. Yi et al. [15,16] demonstrated that the particle swarm optimization algorithm can better solve the dynamic optimization problem of stand-off damping structures by using the cosimulation method of ANSYS and MATLAB. Garrison et al. [17][18][19] conducted a targeted analysis of the constrained damping with the local stand-off layer and believed that the locally stand-off damping treatment has a higher damping efficiency though the damping area is reduced. ey made optimal design of its vibration damping performance. Rao [20] pointed out that the slotted stand-off layer reduces the bending stiffness and overall mass of the structure and thus can effectively control the vibration and noise in aircraft cabins. Meng [21] showed that the introduction of the slotted stand-off layer is very helpful in improving the damping effect of the structure. Zhao et al. [22][23][24] discovered, through numerical simulations and experimental tests, that the noise of constrained mute rails with slotted stand-off is reduced by more than 6-9 dB under vertical and lateral excitation.
Despite extensive research on and applications of standoff constrained layer damping, few studies are available on stand-off free layer damping structures [25][26][27]. us, this paper focuses on the stand-off free layer damping structure. We use lightweight, easy-to-form, and economical rigid PU foam as the stand-off layer and rubber as the damping layer material, which are combined with steel plates to prepare stand-off free layer damping cantilever beams. Based on the modal superposition method and the Lagrange equation, the motion equation of the cantilever beam is derived. e dynamic mechanical properties of PU foam and rubber damping materials are tested and analyzed. rough hammering tests in which the damping layer remains constant, the influence of the density and thickness of PU foam on the vibration damping performance of the stand-off free layer damping cantilever beam is analyzed from the perspectives of the amplitude-frequency curve loss factor and modal frequency.

Derivation of Motion Equation
Motion equation is a mathematical expression for describing the dynamic displacement of a structural system. ere are different ways to establish the motion equation of the vibration system. Using the Hamilton principle to establish the equation of motion can avoid vector operations, which is a common method [28][29][30][31][32][33][34][35]. e stand-off free layer damping cantilever beam is shown in Figure 1. It consists of a base layer, a stand-off layer, and a damping layer from bottom to top. e stand-off layer plays a leverage role, which can expand the tension and compression deformation of the damping layer, so the structural damping increases accordingly. In addition, the stand-off layer has the effect of broadening the damping temperature range. Based on the model, a rectangular coordinate system is established, where the length of the cantilever beam is l, the width is b, and the thickness of the base layer, the stand-off layer, and the damping layer is h b , h s , and h v , respectively.
Reference [36] derived the motion equation of the free layer damping cantilever beam with the shear deformation of the damping layer taken into consideration. In this paper, the nonlinear vibration and damping of the cantilever beam are not considered. e following assumptions are made in analyzing the deformation of the stand-off free layer damping cantilever beam. (1) e overall structure produces small deflection deformation, and the resonance remain in the elastic region. (2) e shear deformation of each layer is not considered, and the moment of inertia of the structure is ignored. (3) ere is no relative slip between the layers, and the lateral displacement of the layers is equal. (4) e cantilever beam conforms to the plane assumption. Figure 2 shows the local deformation relation of the stand-off free layer damping cantilever beam. e axial displacement of the base layer, the stand-off layer, and the damping layer on the neutral plane is u b , u s , and u v , respectively. e lateral displacement of each layer is the same, which is w.
According to the assumption, we have e distance between the neutral axis of the cantilever beam and neutral axis of the base is [37] where E b , E s , and E v are the elastic modulus of the base layer, the stand-off layer, and the damping layer, respectively, and d 1 and d 2 are the distances from the neutral axis of the stand-off layer and the damping layer to the neutral axis of the base layer, respectively. e total potential energy of the cantilever beam includes deformation potential energy of the base layer, the stand-off layer, and the damping layer. e potential energy of the base layer is where T b , T s , and T v are the kinetic energy of the base layer, the stand-off layer, and the damping layer, respectively, and ρ b , ρ s , and ρ v are the material density of the three layers, respectively. e total potential energy of the cantilever beam structure is e total kinetic energy is We expand the displacement of the cantilever beam structure according to the assumed mode: where n w and n b are the numbers of half waves according to the accuracy. Let L be the Lagrangian function of the beam, and we have Substituting equation (12) into the first equation and the third equation in equation (13), we obtain where en, the motion equation of the stand-off free layer damping cantilever beam is as follows: where We make en, equation (16) is reduced to Let the axial and lateral vibrations (both simple harmonic oscillations) be ω. We make € q(t) � q e iωt , and then, motion equation (19) of the stand-off free layer damping cantilever beam can be reduced to an eigenvalue equation: By solving the equation, we can obtain the modal frequency (ω i ) of the cantilever beam.
A stand-off free layer damping cantilever beam model is established by means of experiment. e model materials and dimensions are shown in Table 1 (sample 3#). e modal frequencies and errors of the first five order bending vibration of the sample are shown in Table 2.
e modal frequencies of the first five order bending vibration obtained by experimental measurement and motion equation solution are very close (with errors ranging from 2.03% to 5.09%), which proves the validity of the motion equation.

Material Preparation.
e rigid PU foam used in the experiment is manufactured by Qingdao Yongde Polyurethane Co., Ltd. Four types of foam with different densities (40 kg/m 3 , 80 kg/m 3 , 175 kg/m 3 , and 260 kg/m 3 ) are selected as the stand-off layer. e damping layer is a D-803-Z rubber damping material provided by Tianjin Rubber Industry Research Institute Co., Ltd., with a thickness of 6 mm, a density of 1420 kg/m 3 , and an elastic modulus of 8.3 × 10 7 N/ m 2 . e interlayer adhesive is Qtech-113 from Qingdao Shamu Advanced Material Co., Ltd. e base layer of the cantilever beam is the Q235 steel plate with a thickness of 3.5 mm, a density of 7800 kg/m 3 , and an elastic modulus of 2.06 × 10 11 N/m 2 .

Dynamic Mechanics Test of Materials.
e dynamic mechanical analysis is an important means to study the mechanical properties of viscoelastic materials. Its main purpose is to measure the stiffness and damping of materials under certain conditions and obtain the characteristic parameters related to the structure and molecular motions of materials. e dynamic mechanical analyzer (Netzsch DMA242) was used to analyze the mechanical properties of rubber and PU foam (ρ � 175 kg/m 3 ). e temperature range was −80∼100°C, the heating rate was 3°C/min, and the test frequency was 5, 25, and 100 Hz.

Sample Preparation and Hammering Test.
e rigid layer on the upper and lower surfaces of the PU foam board was removed, and the middle layer with uniform foaming was taken as the material of the stand-off layer. e Qtech-113 adhesive was evenly coated on the surfaces of PU foam and the rubber damping plate to bond them together. en, they were compacted gently to drive out the bubbles, thus forming PU-rubber composite damping material. e composite material was bonded with the 3.5 mm Q235 steel plate by adhesive to prepare the stand-off free layer damping cantilever beams. e beams were placed in an oven of 40°C for curing for 24 hours and then taken out. After being stored for 24 hours at room temperature (20～25°C), they were ready for the hammering test.
e cantilever beams are numbered as in Table 1. Beams 1# to 4# have the same parameters except the density of the stand-off layer, and beams 4# to 7# have the same parameters except the thickness of the stand-off layer.
e test and analysis system includes cantilever beams, fixture, force hammer, accelerometer, signal acquisition device, control, and analysis system (includes data analysis software), as shown in Figure 3. e detailed information of the test and analysis system is shown in Table 3.
e excitation point of the force hammer and the vibration pick-up point of the accelerometer are located in the middle of the steel plate, 25 mm from the fixed end and the free end of the beam, respectively. In order to ensure the resonance of the cantilever beam in the elastic region, the hammering tests adopt small exciting force to avoid large deformation of the cantilever. e data of three repeated hammer strikes were taken for the final analysis. e analysis frequency was1250 Hz, and the experimental temperature was 23.5°C. rough transfer function analysis of the collected data, the transfer function curves of the beams were obtained, and the modal frequency and loss factor were obtained by the INV method.

Dynamic Mechanical Analysis of Materials.
e damping capacity of viscoelastic material is characterized by its dynamic mechanical properties, and the basic parameters are dynamic modulus M * (storage modulus M ′ and loss modulus M ′ ) and loss factor β, as shown in the following equations, where i is the imaginary unit: e dynamic modulus (storage modulus and loss modulus) of D-803-Z and PU foam (175 kg/m 3 ) at different frequencies is shown in Figures 4 and 5, respectively, and the variation of loss factor is shown in Figures 6 and 7.
As can be seen from Figure 4, when the temperature is between −80°C and −40°C, D-803-Z is in the glassy state with a large storage modulus. e internal molecular chains are "frozen," and it is difficult to produce relative sliding. With the rise of temperature, the storage modulus decreases slowly, while the loss modulus continues to increase. When the temperature rises from −40°C to 20°C, the storage modulus decreases rapidly, and the loss modulus reaches a peak in this temperature domain. It indicates that the material has been in a glass transition region and that the internal chains start to change from a "frozen" state to an "unfrozen" state. At this time, although the molecular chains are capable of moving, the movement process needs to overcome the large internal friction, and mechanical energy   Figure 3: Test and analysis system.  is converted into heat energy through internal friction between molecules. After the external force is removed, the chains cannot return to the original state completely, resulting in permanent deformation. erefore, the movement is irreversible, and the material exhibits viscous behavior. e portion of mechanical energy applied to the viscous component by the external force cannot return to outside and is dissipated as heat, thus reducing vibration by damping. When the temperature is higher than 20°C, the storage modulus and loss modulus continue to decrease and then gradually stabilize.
As shown in Figure 5, when the temperature rises from −80°C to 60°C, the storage modulus of PU foam (175 kg/m 3 ) shows a decreasing trend, and the loss modulus decreases before increasing, displaying a "U" change. When the temperature exceeds 60°C, the storage modulus of the material decreases faster, while the loss modulus increases gradually, and the material begins to enter into a glass transition region. In addition, when the testing frequency is increased to 100 Hz, the storage modulus does not change significantly, while the loss modulus is significantly improved, which shows that the loss modulus of polyurethane foam has obvious frequency dependence. Compared with low-frequency vibration, polyurethane foam has a more obvious damping effect on high-frequency vibration.
As shown in Figure 6(a), the loss factor of D-803-Z exceeds 0.3 at 15∼90°C, with a wide damping temperature domain, and two peaks occur at 20∼40°C displaying a "saddle" shape. is is because the D-803-Z rubber damping material is a kind of the semicompatible blend system with macroscopic homogeneous and microscopic heterogeneous phases. us, a double glass transition phenomenon occurs, resulting in two loss factor peaks overlapping each other [4]. Mixing rubber with other polymers with high glass  transition temperature (T g ) is an effective method to improve the damping properties of rubber materials. After blending, the T g of polymer (such as plastic) will move and the damping peak will decrease. e addition of high T g viscoelastic polymer increases T g of the rubber system. By adjusting the blending ratio of each component, the effective damping temperature range of the blend can be extended beyond the glass transition region of each homopolymer, so as to expand the effective damping temperature range. e loss factor of PU foam is in the range of −80∼100°C (as shown in Figure 6(b), which tends to increase with the rise of temperature, especially after 50°C. At around 20°C, the loss factor of PU foam is about an order of magnitude smaller than that of rubber; so, the main energy consumer of the cantilever beam structure is rubber. e PU foam mainly plays the role of expanding the deformation of the damping layer and auxiliary energy dissipation. In addition, the loss modulus and loss factor of rubber and foam material at 100 Hz are improved considerably, as compared with 5 Hz and 25 Hz. It indicates that the damping performance of the material is frequency dependent to a great extent.

Effect of Foam Density on Vibration Performance.
e PU foam density of cantilever beams 1#∼4# are 40 kg/m 3 , 80 kg/m 3 , 175 kg/m 3 , and 260 kg/m 3 , respectively, all other parameters being the same. e first five transfer function curves are shown in Figure 7. When the density of the PU foam stand-off layer increases from 40 kg/m 3 to 80 kg/m 3 , 175 kg/m 3 , and 260 kg/m 3 in sequence, the vibration response peaks of the first five modes all show a decreasing trend. Low-density PU foam has a large porosity and a low modulus and is apt to deform, which is not conducive to the transfer of the deformation generated by the vibration (especially the high frequency vibration with small deformation) of the base layer to the damping layer. As the density of PU foam increases, the modulus ratio of foam to rubber increases, which helps the vibration of the substrate to be transferred more efficiently to the damping layer for energy dissipation. us, the structure exhibits better damping performance, in agreement with the results in [14].
Moreover, when the density of foam increased from 40 kg/m 3 to 80 kg/m 3 and 175 kg/m 3 , the vibration peaks of the first five orders decreased sharply. When the density was further increased to 260 kg/m 3 , the peak decline of most modes was reduced obviously. Take the third order and the fourth order for examples; when the foam density changed from 40 kg/m 3 to 175 kg/m 3 , the peak value decreased by 42% and 52%, respectively. When it was increased to 260 kg/m 3 , the peak value decreased by 47% and 59%, respectively. e change of the vibration peak value was relatively small, as compared with the foam composite structure of 175 kg/m 3 . It can be seen that increasing the density of the foam is helpful to improve the vibration damping performance of cantilever beam. When the foam density increases to a certain level, the change of modal vibration peak tends to be gentle, and the vibration suppression effect tends to be stable. e modal frequencies and loss factors of the first five orders of 1#∼4# beams are shown in Table 4. e increase of foam density leads to increased elastic modulus, which significantly increases the overall stiffness of the cantilever beam, thus causing its modal frequency to move towards high frequency. Meanwhile, the loss factor rises, and the damping performance is improved.
In addition, compared with 1#, the modal frequency of the first five orders of 4# increased by 6.40%, 2.68%, 8.42%, 12.16%, and 15.99%, respectively, and the loss factor increased by 79.74%, 36.71%, 113.19%, 147.61%, and 172.78%, respectively. With the increase of the modal order, the change rate of its modal frequency and loss factor increased significantly (except for the second order), and the same is true for 2# and 3#. is shows that a higher density of the foam layer has more an obvious influence on the higher order mode of the cantilever beam because rubber and PU foam are strongly frequency dependent. e higher the frequency, the greater the storage modulus and the loss factor.
In summary, given the same thickness (6 mm) of the foam layer, increasing its density is helpful to improve the vibration reduction performance of the stand-off free layer damping cantilever beam. In this study, the foam with a density of 260 kg/m 3 has the best effect.

Effect of Foam
ickness on Vibration Performance. Based on the above results, PU foam with a density of 260 kg/ m 3 was selected as the stand-off layer to obtain stand-off free layer damping cantilever beams with different thicknesses of PU foam. For example, for 4#∼7#, the thicknesses of the foam stand-off layer are 6 mm, 9 mm, 12 mm, and 15 mm, respectively, all other parameters of the cantilever beam being the same. e first five transfer function curves are shown in Figure 8.
It can be seen that the peak value of the first five order shows a downward trend with the increase of the thickness of the stand-off layer. e metal itself consumes little energy, and the vibration energy of the substrate is all transmitted to the stand-off layer. A small portion of the energy is dissipated by the stand-off layer, while most of it is transmitted to the damping layer to be dissipated. According to equation (2), the distance between the neutral axis of the damping layer and the neutral axis of the cantilever beam is at is, With the increase of the thickness (h s ) of the stand-off layer, the distance (p) processed by the damping layer increases.
erefore, for the same deformation of the base layer, increased deformation of the damping layer leads to the increase of the relative slip of the molecular chain segments and increase of the energy consumption of the Shock and Vibration    damping layer. erefore, the vibration damping performance is improved, and the vibration response is reduced. Moreover, PU foam itself consumes some energy; so, increasing the thickness of the stand-off layer not only extends the transmission distance of vibration but also acts to increase the thickness of the damping layer to some extent, which is conducive to improving the damping performance of the structure.
As the thickness of the stand-off layer increases, another vibration peak gradually emerges near the second-order resonance peak. e reason might be that the increase in the overall thickness of the cantilever beam and the upward movement of the neutral axis cause the structure to be excited into a torsional mode near the second-order bending mode, which partially overlaps with the resonance peak of the bending mode. Moreover, the thicker the stand-off layer is and the farther the neutral axis is from the base layer, the more likely the torsional mode is to be excited and the more obvious it is. As a result, the modal frequency and loss factor of the second mode are different from those of other modes. e modal frequencies and loss factors of the first five orders of cantilever beams 4#∼7# are shown in Table 5.
Compared with 1#, the modal frequency of the first five orders of 4# increased by 15.77%, 1.04%, 25.15%, 21.85%, and 17.65%, respectively, and the loss factor of the first four orders increased by 58.36%, 3.10%, 52.37%, and 62.46%, respectively. With the increase of the thickness of the standoff layer, the modal frequencies move towards high frequency, which due to the bending modal frequency of the cantilever beam is [4] When the thickness of the stand-off layer changes, the parameter that determines its natural vibration frequency is I/S. us, we have With the increase of the thickness of the stand-off layer, its natural vibration frequency increases, thus causing the natural vibration rate of the structure to rise. e increase of foam layer thickness significantly increases the loss factor of the cantilever beam and improves the damping performance. e results are in good agreement with the previous analysis results.
To sum up, within the scope of this study, increasing the thickness of the foam layer can further improve the damping

Conclusions
(1) e motion equation of the stand-off free layer damping cantilever beam is derived by means of the modal superposition principle and the Lagrange equation. A comparison with the experiment results proves the accuracy of the motion equation. (2) D-803-Z is a blend rubber with double glass transition temperature, which has a high loss factor and good damping temperature domain, and is a major energy consumer in the whole composite structure. e loss factor of PU foam is smaller than that of damping rubber, which mainly plays the role of expanding the deformation of the damping layer and auxiliary energy dissipation. Moreover, the dynamic mechanical properties of the two materials show obvious frequency dependence.
(3) Increasing the density and thickness of the PU foam stand-off layer within a certain range can improve the vibration suppression performance of the standoff free layer damping cantilever beam and enhance the overall stiffness of the cantilever beam in the meanwhile, making the modal frequency get higher. erefore, without considering the additional mass and thickness of the structure, we can increase the density and thickness of PU foam appropriately to improve the vibration damping performance of the stand-off free layer damping cantilever beam.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.