Soft Vibrations Research Articles

Approximate analytical solutions of nonlinear vibration and bifurcation of dielectric elastomer balloon

Dielectric elastomers (DE), a novel class of actuators, are increasingly employed in fields such as soft robotics and vibration control. Most existing studies investigating the dynamic characteristics of DE structures neglect the viscoelasticity and strain-stiffening effects of materials, with only a few offering approximate analytical solutions. This paper establishes the governing equation for the dynamics of a DE balloon, utilizing the principle of virtual work and considering material viscoelasticity and strain-stiffening effects through linear damping and the Gent hyperelastic model, respectively. The nonlinear governing equations are simplified using the Chebyshev polynomial approximation. The multiple scales method is applied to derive approximate solutions for the structure's free vibration under static voltage excitation and parametrically excited vibration under harmonic voltage excitation. The accuracy of these approximate analytical solutions is validated by comparison with numerical solutions. Additionally, the paper analyzes stability and bifurcation phenomena through time-domain response curves, amplitude-frequency curves, and phase trajectories. Furthermore, the influence of the stretching limit, voltage, and damping on the nonlinear vibration of the balloon structure is studied, providing a valuable theoretical foundation for the future design and utilization of dielectric elastomers.

Room-Temperature Synthesis and Low Thermal Conductivity in Nanocrystalline Ag3CuS2.

Noble-metal-based chalcogenide materials recently gained massive attention in the field of thermoelectrics. In most cases, materials are synthesized using (i) high-temperature solid-state reactions or (ii) soft chemical methods where temperature requirements are lower than those of solid-state reactions (generally below 400 °C). Herein, we present a simple, surfactant-free, room-temperature, and energy-efficient synthesis of Ag3CuS2 nanocrystals. The present synthesis technique is scalable and capable of gram-scale production. A spark plasma sintering (SPS) pressed sample exhibits ultralow thermal conductivity (∼0.31 W/mK at room temperature). We found that Ag3CuS2 exhibits low sound velocity, as well as a non-Debye-like behavior based on a low-temperature heat capacity measurement. A high degree of anharmonicity of bonding, soft vibrations modes, and nanoscale grain boundary scattering in Ag3CuS2 lead to ultralow thermal conductivity, which can be important for thermoelectrics, optoelectronics, and thermal barrier coating applications.

Large vortices from soft vibrations

Steady streaming is a peculiar flow, induced by the oscillation of an object with respect to the fluid in which it is embedded. While the oscillation amplitude may be extremely small, the nonlinearity due to fluid inertia gives rise to this steady flow. Large and powerful vortices develop around the object. Usually, the streaming is scrutinised in the case of homogeneous objects. Here, Cui et al. (J. Fluid Mech., vol. 979, 2024, A7) derive the streaming around a soft elastic sphere containing a solid core. This elaborate derivation opens a new description of streaming in biological soft environments and robotics.

Impact of palatopharyngeal sizes changing on pharyngeal airflow fluctuation and airway vibration in a pediatric airway

Snoring is common in children and is associated with many adverse consequences. One must study the relationships between pharyngeal morphology and snoring physics to understand snoring progression. Although some model studies have provided fluid–structure interaction dynamic descriptions for the correlation between airway size and snoring physics, the descriptions still need to be further investigated in patient-specific airway models. Fluid-structure interaction studies using patient-specific airway structures complement the above model studies. Based on reported cephalometric measurement methods, this study quantified and preset the size of the palatopharynx airway in a patient-specific airway and investigated how the palatopharynx size affects the pharyngeal airflow fluctuation, soft palate vibration, and glossopharynx vibration with the help of a verified FSI method. The results showed that the stenosis anterior airway of the soft palate increased airway resistance and airway resistance fluctuations, which can lead to increased sleep effort and frequent snoring. Widening of the anterior airway can reduce airflow resistance and avoid obstructing the anterior airway by the soft palate vibration. The pharyngeal airflow resistance, mouth inflow proportion, and soft palate apex displacement have components at the same frequencies in all airway models, and the glossopharynx vibration and instantaneous inflow rate have components at the same frequencies, too. The mechanism of this same frequency fluctuation phenomenon can be explained by the fluid–structure interaction dynamics of an ideal coupled model consisting of a flexible plate model and a collapsible tube model. The results of this study demonstrate the potential of FSI in studying snoring physics and clarify to some degree the mechanism of airway morphology affecting airway vibration physics.

Structure searches and superconductor discovery in XB2 (X = Sc, Ti, V, Cr, and Tc).

With extensive structure searches for XB2 (X = Sc, Ti, V, Cr, and Tc) under pressures up to 100 GPa, we uncovered that the crystal structures of these compounds with the lowest enthalpy have the same space group (P6/mmm) as MgB2 at ambient pressure. Among them, ScB2, TiB2 and VB2 are dynamically stable at ambient pressure, but they do not superconduct. CrB2 becomes dynamically stable at 108 GPa and shows superconductivity with a transition temperature (Tc) of 26.0 K. TcB2 is not dynamically stable until 9 GPa. At 20 GPa, it has a Tc of 23.5 K. Further calculations indicate that CrB2 and TcB2 are also thermodynamically stable, suggesting that it is highly likely that they can be synthesized successfully in the laboratory. We found that transition metal atoms (Cr/Tc) dominate soft phonon vibrations and make significant contributions to the electron-phonon coupling (EPC) and superconductivity in CrB2/TcB2, which is in strong contrast to the case of MgB2, where high-frequency B vibrations dominate the EPC and superconductivity. Our work enriches the understanding of superconductivity in transition metal borides.

The local atomic distribution in tetragonal PZT.

Expanding on the comprehensive research conducted by previous scholars, herein, we aim to elucidate the intrinsic piezoelectricity of tetragonal Pb(Zr1-xTix)O3 (PZT), by focusing on the local atomic distribution which was neglected for a long time, through the supercell approach based on colour symmetry. Density functional theory (DFT) was employed to perform first-principles calculations on the electronic, phononic structures and piezoelectricity of various tetragonal PZT supercells. Building upon the evaluation of the piezoelectric properties of 22 distinct distributions, classical Monte Carlo methods were utilized to explore the statistical macroscopic properties at the morphotropic phase boundary (MPB). The results reveal that at x = 0.5 and x = 0.55, the d33 reached 957 pm V-1 and 893 pm V-1, respectively. The analysis of phonon vibration modes exposed significant disparity between different colour symmetries. The supercells of lower symmetry contain B-site atoms in asymmetric positions, and they exhibit softer vibrational frequencies in the phonon spectrum. These soft phonon vibration modes resulting from colour symmetry breaking were previously unknown. The weakening and reorientation of the covalent bonds between the O 2p and d orbitals were found in electronic structures. The free energy flattening in the polarization rotation path is the origin of the high piezoelectricity of this type of supercell. The analysis of the electronic structure is consistent with the experimental observations. Finally, colour symmetry proved to be an effective and accurate way to describe the local atomic distribution in supercells. It will also bring new perspectives to understanding the structure of domain walls, phase boundaries, etc.

Potential predictive effect of mechanical properties of the plantar skin and superficial soft tissue, and vibration perception on plantar loading during gait in individuals with diabetes

BackgroundThis exploratory study aimed to investigate the extent to which mechanical properties of the plantar skin and superficial soft tissue (hardness, stiffness, and thickness) and vibration perception thresholds (VPTs) predict plantar pressure loading during gait in people with diabetes compared to healthy controls.MethodsMechanical properties, VPTs, and plantar loadings during gait at the heel and first metatarsal head (MTH) of 20 subjects with diabetes, 13 with DPN, and 33 healthy controls were acquired. Multiple regression analyses were used to predict plantar pressure peaks and pressure-time integrals at both locations based on the mechanical properties of the skin and superficial soft tissues and VPTs.ResultsIn the diabetes group at the MTH, skin hardness associated with 30-Hz (R2 = 0.343) and 200-Hz (R2 = 0.314) VPTs predicted peak pressure at the forefoot. In the controls at the heel, peak pressure was predicted by the skin thickness, hardness, and stiffness associated with 30-Hz (R2 = 0.269, 0.268, and 0.267, respectively) and 200-Hz (R2 = 0.214, 0.247, and 0.265, respectively) VPTs.ConclusionThe forefoot loading of people with diabetes can be predicted by the hardness of the skin when combined with loss of vibration perception at low (30-Hz) and high (200-Hz) frequencies. Further data from larger sample sizes are needed to confirm the current findings.

Dorsal Column Bedside Examination Test: Tips for the Neurosurgical Resident.

The dorsal column medial lemniscus (DCML) system is a sensory pathway of the central nervous system; it carries sensations of soft touch, vibration, proprioception, two-point discrimination, and pressure from the skin and joints. The clinical signs of the DCML pathway lesions include loss of soft touch, vibratory sense, proprioception, discrimination sense, and a positive Rhomberg test. Diseases that affect this pathway are usually degenerative, for example, spinal cord degeneration due to vitamin B12 deficiency; it can also be affected by trauma or infarction of the posterior spinal artery causing posterior cord syndrome. This video manuscript provides a step-by-step examination technique of the dorsal column examination, specially catered for Malaysian medical students and trainees. A series of videos show the techniques for soft touch sensation examination, examination of the vibratory sense, examination of the joint position sense, examination of two-point discrimination and the Rhomberg test. We hope that students can adhere to these techniques and apply them in their daily neurological assessments.

Enhanced Schiff and magnetic quadrupole moments in deformed nuclei and their connection to the search for axion dark matter

Deformed nuclei possess enhanced moments violating time reversal invariance ($T$) and parity ($P$). Collective magnetic quadrupole moments (MQM) appear in nuclei with a quadrupole deformation (which have ordinary $T$,$P$-conserving collective electric quadrupole moments). Nuclei with an octupole deformation have a collective electric octupole moment, electric dipole moment (EDM), Schiff moment and MQM in the intrinsic frame which rotates with the nucleus. In a state with definite angular momentum in the laboratory frame, these moments are forbidden by $T$ and $P$ conservation, meaning their expectation values vanish due to nuclear rotation. However, nuclei with an octupole deformation have doublets of close opposite parity rotational states with the same spin, which are mixed by $T$,$P$-violating nuclear forces. This mixing polarises the orientation of the nuclear axis along the nuclear spin, and all moments existing in the intrinsic frame appear in the laboratory frame (provided the nuclear spin $I$ is sufficiently large to allow such a moment). Such a mechanism produces enhanced $T$,$P$-violating nuclear moments. This enhancement also takes place in nuclei with a soft octupole vibration mode. In this paper we present updated estimates for the enhanced Schiff moment in isotopes of Eu, Sm, Gd, Dy, Er, Fr, Rn, Ac, Ra, Th, Pa, U, Np and Pu in terms of the CP-violating $\pi$-meson--nucleon interaction constants $\bar{g}_{0},\bar{g}_{1}$ and $\bar{g}_{2}$, the QCD parameter $\bar{\theta}$ and the quark chromo-EDMs. The implications of the enhanced $T$,$P$-violating moments to the search for axion dark matter in solid state experiments are also discussed, with potential alternative candidate compounds in which we may expect enhanced effects suggested.

Nonlinear vibration analysis for bolted CFRC plates based on displacement-dependent surface spring-damping model of bolted joint

Under the effect of external excitation, bolted structures are prone to stiffness weakening and damping energy dissipation at the bolted joint, which makes the structures show nonlinear vibration behavior. In this paper, the displacement-dependent surface springs and tangential surface damping expressed by the S-curve model are proposed to simulate the nonlinear mechanical properties of the bolted joint. The non-uniform distribution of the mechanical parameters for the bolted joint is given to simulate the actual joint interface pressure distribution. Based on these, a nonlinear semi-analytical model for bolted carbon fiber reinforced composites (CFRC) plates considering the nonlinear mechanical characteristics of the bolted joint is established. Combined with the experimental results for a bolted CFRC beam, the specific S-curve expressions for the relevant mechanical parameters of the bolted joint are obtained by the inverse identification technology and numerical fitting method. After that, the bolted joint model is transplanted to the dynamic model of a bolted CFRC plate for numerical calculation. The comparison with the corresponding measured results shows that the proposed modeling method can well reproduce the soft nonlinear vibration of the actual bolted CFRC plate, which verifies the correctness and applicability of the proposed modeling method.

Sound absorption in glasses

The paper presents a description of the sound wave absorption in glasses, from the lowest temperatures up to the glass transition, in terms of three compatible phenomenological models. Resonant tunneling, the rise of the relaxational tunneling to the tunneling plateau and the crossover to classical relaxation are universal features of glasses and are well described by the tunneling model and its extension to include soft vibrations and low barrier relaxations, the soft potential model. Its further extension to non-universal features at higher temperatures is the very flexible Gilroy-Phillips model, which allows to determine the barrier density of the energy landscape of the specific glass from the frequency and temperature dependence of the sound wave absorption in the classical relaxation domain. To apply it properly at elevated temperatures, one needs its formulation in terms of the shear compliance. As one approaches the glass transition, universality sets in again with an exponential rise of the barrier density reflecting the frozen fast Kohlrausch t β -tail (in time t , with β close to 1/2) of the viscous flow at the glass temperature. The validity of the scheme is checked for literature data of several glasses and polymers with and without secondary relaxation peaks. The frozen Kohlrausch tail of the mechanical relaxation shows no indication of the strongly temperature-dependent barrier density observed in dielectric data of molecular glasses with hydrogen bonds. Instead, the mechanical relaxation data indicate an energy landscape describable with a frozen temperature-independent barrier density for any glass.

Nonlinear chirped interferometry for frequency-shift measurement and χ(3) spectroscopy

Four-wave mixing processes are ubiquitous in ultrafast optics and the determination of the coefficients of the χ(3) tensor is thus essential. We introduce a novel time-resolved ultrafast spectroscopic method to characterize the third-order nonlinearity on the femtosecond time-scale. This approach, coined as “nonlinear chirped interferometry,” makes use of the variation of the optical group delay of a transmitted probe under the effect of an intense pump pulse in the nonlinear medium of interest. The observable is the spectral interference between the probe and a reference pulse sampled upstream and the metric is the transient swing of the probe group delay. We show that the detected signal is enhanced when the pulses are weakly chirped, and that, although interferometric, the method is intrinsically less sensitive to environmental phase fluctuations and drifts. By chirping adequately the reference pulse, the transient frequency shift of the probe pulses is also detected in the time domain and the detected nonlinear signal is enhanced. Nonlinear phase shifts as low as 10 mrad, corresponding to a frequency shift of 30 GHz, i.e., 0.01% of the carrier frequency, are detected without heterodyne detection or active phase-stabilization. The diagonal and/or non-diagonal terms of reference glasses (SiO2) and crystals (Al2O3, BaF2, CaF2) are characterized. The method is finally applied to measure the soft vibration mode of KTiOAsO4 (KTA).

Origins of the diffuse shared proton vibrational signatures in proton-coupled electron transfer model dyad complexes.

Phenol-benzimidazole and phenol-pyridine dyad complexes have served as popular model systems for the study of proton-coupled electron transfer (PCET) kinetics in solution-phase experiments. Interpretation of measured PCET rates in terms of key structural parameters, such as the H-bond donor-acceptor distance, however, remains challenging. Herein, we report vibrational spectra in the electronic ground state for a series of phenol-benzimidazole and phenol-pyridine complexes isolated and cryogenically cooled in an ion trap. The four models studied each display highly red-shifted and broadened OH stretching transitions that arise from strong H-bonding interactions between the phenol OH group and the basic N site on benzimidazole/pyridine rings. The OH stretch transition in each model displays relatively strong absorption onsets near 2500cm-1 with broad shoulders that extend asymmetrically to higher frequencies over hundreds of wavenumbers. In contrast, the deuterated isotopologues yield much weaker OD stretch transitions that appear symmetrically broadened. The spectral breadth and shape of the OD stretch transitions are ascribed to variations in OD stretch frequencies that arise from zero-point distributions in the proton donor-acceptor low-frequency soft mode vibration. The asymmetric structure of the OH stretch transitions is attributed to a set of combination bands between the OH stretch and a series of low-frequency H-bond soft modes. The spectra and modeling highlight the importance of OH stretch-soft mode couplings, which are thought to play important roles in PCET and proton transfer dynamics.

A fully soft, self-powered vibration sensor by laser direct writing

Soft electronics are comprised of stretchable and user-friendly components that mechanically conform to complex surfaces and epidermis, extending their applicability over commercial electronics. Vibration sensors that detect operation conditions of common motions generally rely on rigid materials or tedious fabrications. Here, a fully soft self-powered vibration sensor (SSVS) engineered by laser-assisted fabrication is proposed. Different from its rigid counterparts, the device is constructed with entirely ductile materials including a liquid metal kernel and an elastic shell. Furthermore, laser direct writing elaborates intricate patterns and functional surfaces of SSVS in a rapid and mask-free manner. A ubiquitous IR laser combined with an origami-inspired transfer creates arbitrary out-of-plane electrodes on soft media, while a pulsed UV laser judiciously tunes nonstick microstructures to the liquid metal droplet. To demonstrate the capability on curved surfaces, the SSVS is interfaced with an exhaust silencer of an air compressor and detects failures of screw connection. Additionally, a cap reinforced by SSVS with wireless transmission is devised for motion monitoring, which effectively distinguishes slip from walking, jumping and running. This work affords promising routes toward monitoring complex machinery and remote guardianship for the vulnerable populations.

Fluid–structure-coupled Koopman mode analysis of free oscillating twin-cylinders

Flow-induced vibration (FIV) of twin square cylinders in a tandem arrangement was numerically investigated at Reynolds numbers 200 and gap L/D = 2.0, 4.0, and 6.0 (D is the side length of the cylinders). Fluid-structure-coupled Koopman mode analysis method was developed to synchronously identify the coherence flow and structural modes. Then, the energy transfer between cylinders and Koopman modes was analyzed to uncover the underlying mechanism of FIV. The results showed that at L/D = 2.0 and 4.0, only soft lock-in vortex-induced vibration (VIV) was observed. The oscillating amplitude for L/D = 4.0 was much higher than that of L/D = 2.0, due to the interference effects induced by fully developed gap vortices. As L/D = 6.0, VIV and galloping coexisted. For the coherence mode, the primary flow mode induced by the vortex shedding dominated the flow field at L/D = 2.0 and 4.0. The direct mode energy dominated the energy transfer process. The upstream cylinder (UC) contributed to the negative work done and thus tended to stabilize the vibration; in contrast, the downstream cylinder (DC) exhibited the opposite behavior. In the galloping branch at L/D = 6.0, both the flow field and structural response contained three main modes: one vortex-shedding-induced mode and two vibration-induced modes. For the direct mode energy, owing to the interference effects, DC contributed to more positive work done than UC by the vibration-induced modes. The vortex-induced mode was governed by DC and afforded negative work done. Moreover, all the coupled mode energy was almost equal to zero.

Glassy thermal conductivity in Cs3Bi2I6Cl3 single crystal

As the periodic atomic arrangement of a crystal is made to a disorder or glassy-amorphous system by destroying the long-range order, lattice thermal conductivity, κL, decreases, and its fundamental characteristics changes. The realization of ultralow and unusual glass-like κL in a crystalline material is challenging but crucial to many applications like thermoelectrics and thermal barrier coatings. Herein, we demonstrate an ultralow (~0.20 W/m·K at room temperature) and glass-like temperature dependence (2–400 K) of κL in a single crystal of layered halide perovskite, Cs3Bi2I6Cl3. Acoustic phonons with low cut-off frequency (20 cm−1) are responsible for the low sound velocity in Cs3Bi2I6Cl3 and make the structure elastically soft. While a strong anharmonicity originates from the low energy and localized rattling-like vibration of Cs atoms, synchrotron X-ray pair-distribution function evidence a local structural distortion in the Bi-halide octahedra and Cl vacancy. The hierarchical chemical bonding and soft vibrations from selective sublattice leading to low κL is intriguing from lattice dynamical perspective as well as have potential applications.

Bidirectional Locomotion of Soft Inchworm Crawler Using Dynamic Gaits.

Inchworm-styled locomotion is one of the simplest gaits for mobile robots, which enables easy actuation, effective movement, and strong adaptation in nature. However, an agile inchworm-like robot that realizes versatile locomotion usually requires effective friction force manipulation with a complicated actuation structure and control algorithm. In this study, we embody a friction force controller based on the deformation of the robot body, to realize bidirectional locomotion. Two kinds of differential friction forces are integrated into a beam-like soft robot body, and along with the cyclical actuation of the robot body, two locomotion gaits with opposite locomotion directions can be generated and controlled by the deformation process of the robot body, that is, the dynamic gaits. Based on these dynamic gaits, two kinds of locomotion control schemes, the amplitude-based control and the frequency-based control, are proposed, analyzed, and validated with both theoretical simulations and prototype experiments. The soft inchworm crawler achieves the versatile locomotion result via a simple system configuration and minimalist actuation input. This work is an example of using soft structure vibrations for challenging robotic tasks.

Enhanced magnetic quadrupole moments in nuclei with octupole deformation and their CP -violating effects in molecules

Nuclei with an octupole deformation have a non-zero electric octupole moment, electric dipole moment (EDM), Schiff moment and magnetic quadrupole moment (MQM) in the intrinsic frame which rotates with the nucleus. In a state with definite angular momentum in the laboratory frame, these moments are forbidden by parity (P) and time reversal invariance (T) conservation, meaning their expectation values vanish due to nuclear rotation. However, nuclei with an octupole deformation have doublets of close opposite parity rotational states with the same spin, which are mixed by T,P-odd nuclear forces. This mixing produces the orientation of the nuclear axis along nuclear spin and all moments existing in the intrinsic frame appear in the laboratory frame (provided the nuclear spin I is sufficiently large to allow such moment). Such a mechanism produces enhanced T,P-violating nuclear moments. This enhancement also takes place in nuclei with a soft octupole vibration mode. Schiff moments in such nuclei have been calculated in previous works. In the present paper we consider the magnetic quadrupole moment which appears in isotopes with nuclear spin $I \geq 1$. Magnetic interaction between the nuclear MQM and electrons produces an atomic EDM and T,P-violating nuclear spin - molecular axis interaction constants for molecules in electronic states with non-zero electron angular momentum. Measurements of these constants may be used to test CP-violation theories and search for axion dark matter in atomic, molecular and solid state experiments. Potential candidate nuclei include 153Eu, 161Dy, 221Fr, 223Fr, 223Ra, 223Rn, 225Ac, 227Ac, 229Th, 229Pa, 233U and 235U. We subsequently consider molecules containing these nuclei (EuO, EuN+, RaF, AcO, AcN+, AcF+, ThO and ThF+).

Deformation behavior of fluid-filled porous elastomers: Analytical estimates and validation

Fluid-filled elastomers are used in soft body wearable electronics devices, optics, soft pressure transducers, vibration damping and impact protection. Deformation behavior of these materials is complex and would depend on elastomer properties, fluid properties, loading rate, porosity, pore size, pore shape, and pore end conditions. In this work, closed form analytical estimates are derived for effective behavior of a special class of fluid-filled elastomers composed of incompressible, hyperelastic matrix with mono-disperse circular pores, distributed in a square grid, filled with an incompressible, Newtonian fluid. The analytical estimates are validated using three dimensional numerical simulations modeled in COMSOL multiphysics software. The analytical estimates for pressure and velocity fields in the fluid, deformation of the matrix, viscous dissipation in fluid, and stored energy in the solid are seen to match well with that from numerical simulations. For small deformations and small porosities, while storage modulus is seen to be linearly proportional to shear modulus of elastomer and inversely proportional to the porosity, the loss modulus is seen to be linearly proportional to the product of loading frequency and fluid viscosity and inversely proportional to the product of porosity and square of ratio of pore radius to pore length. The analytical estimates presented here can be used to design fluid-filled elastomers with desired stiffness and damping by selecting appropriate elastomer stiffness, fluid viscosity, porosity, pore size and pore aspect ratio.

Nonlinear vibration modeling and analysis of bolted thin plate based on non-uniformly distributed complex spring elements

Due to the existence of bolt influence zone, the interface pressure of bolted joint presents a non-uniform distribution. The nonlinear mechanical behaviors of variable stiffness and variable damping will also occur in bolted joints under varying degrees of forced vibration. In this paper, a nonlinear finite element analysis model of bolted thin plate is established by introducing displacement-dependent nonlinear complex spring elements with non-uniformly distributed parameters. Sinusoidal, parabolic and linearly distributed complex spring elements are used to simulate the non-uniform pressure distribution in the bolt influence zone, respectively. The stiffness and damping displacement-dependent parameters of the complex spring elements are described by higher-order polynomials, and a method to obtain the above parameters by inverse identification is proposed. In the case study, the non-linear finite element model created is used to calculate the resonance frequency and frequency domain response of bolted thin plate under five excitation levels. The maximum differences between the results and the measured values are 2.7% and 0.66%, respectively. The simulation results also reproduce the soft nonlinear vibration phenomenon of the bolted thin plate. Furthermore, by comparing the vibration analysis results of the linear model and nonlinear model for bolted thin plate, the reason for the soft nonlinear phenomenon produced is explained.