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... According to the equivalent principles of mass, damping, and stiffness in mechanical vibration [33,34], the equivalent model of blade testing system can be established as shown in Figure 7. According to dynamic analysis, the following equations can be deduced: ...

... Under the action of a constantly changing external force, the position of the blade has experienced 1-2-3-4-5-6-7-8-1 movement. According to the equivalent principles of mass, damping, and stiffness in mechanical vibration [33,35], the equivalent model of blade testing system can be established as shown in Figure 9. According to the dynamic analysis, it can be deduced that: ′ ″ Figure 8. Movement process of blades in a single cycle when using hybrid biaxial fatigue testing device. ...

... a is hydraulic loader of flapwise, b is cross section of blade and fixture, c is hydraulic rod and vibration mass, d is push rod, d is push rod, e is rocker arm, f is foundation, g is hydraulic loader of edgewise. According to the equivalent principles of mass, damping, and stiffness in mechanical vibration [33,35], the equivalent model of blade testing system can be established as shown in Figure 9. According to the dynamic analysis, it can be deduced that: ...

Wind power utilization is attracting worldwide attention in the renewable energy field, and as wind power develops from land to sea, the size of the blades is becoming incredibly larger. The fatigue test, especially the biaxial synchronous fatigue test for the blades, is becoming an indispensable step to ensure the blade’s quality before mass production, which means the biaxial independent test presently used may have difficulty reproducing the real damage for large-sized blades that oscillate simultaneously in flap-wise and edgewise directions in service conditions. The main point of the fatigue test is to carry out accelerated and reinforced oscillations on blades in the experimental plan. The target moments of critical blade sections are reached or not during the test are treated as one significant evaluation criterion. For independent tests, it is not hard to realize moment matching using additional masses fixed on certain critical blade sections, which may be not easy to put into effect for biaxial synchronous tests, since the mechanical properties and target moments in the flap-wise and edgewise directions are widely varied. To realize the mechanical decoupling for loading force or additional mass inertia force in two directions is becoming one of the key issues for blade biaxial synchronous fatigue testing. For this problem, the present paper proposed one mechanical decoupling design concept after a related literature review. After that, the blade moment design and target matching approach are also proposed, using the Transfer Matrix Method (TMM) for moment quick calculation and Particle Swarm Optimization (PSO) for case optimization.

... According to the modal analysis, any stimulated mechanical system vibrates on one or a combination of its natural frequencies. The cantilever beam system has a natural frequency that depends on the length, material property, mass, and boundary condition of the system (45,46). An elastic or non-rigid boundary condition reduces the fixation and stability, and consequently, reduces the natural frequency (47). ...

... The recorded accelerometer response was then analyzed, and the damped natural frequency (ω d ) value was determined. According to equations 1-3, decaying oscillation amplitude (δ), the system's damp ratio (ζ), and the natural frequency (ω n ) were calculated, respectively (46). A(x) and A(x+T) are the amplitudes of two sequential periods at the accelerometer time response curve ...

... natural frequency by damped ratio indicates decay time (46). The modal testing for each sample was repeated four times. ...

Background:
Pedicle screw fixation devices are the predominant stabilization systems adopted for a wide variety of spinal defects. Accordingly, both pedicle screw design and bone quality are known as the main parameters affecting the fixation strength as measured by the pull-out force and insertion torque. The pull-out test method, which is recommended by the standards of the American Society for Testing and Materials (ASTM), is destructive. A non-destructive test method was proposed to evaluate the mechanical stability of the pedicle screw using modal analysis. Natural frequency (ωn) extracted from the modal analysis was then correlated with peak pull-out force (PPF) and peak insertion torque (PIT).
Methods:
Cylindrical pedicle screws with a conical core were inserted into two different polyurethane (PU) foams with densities of 0.16 and 0.32 g/cm3. The PIT and PPF were measured according to the well-established ASTM-F543 standard at three different insertion depths of 10, 20, and 30 mm. Modal analysis was carried out through recording time response of an accelerometer attached to the head of the screw impacted by a shock hammer. The effect of the insertion depth and foam density on the insertion torque, natural frequency, and pull-out force were quantified.
Results:
The maximum values of ωn, PIT, and PPT were obtained at 2,186 Hz, 123.75 N.cm, and 981.50 N, respectively, when the screw was inserted into the high-density foam at the depth of 30 mm. The minimum values were estimated at 332 Hz, 16 N.cm, and 127 N, respectively, within the low-density PU at the depth of 10 mm. The higher value of ωn was originated from higher bone screw stability and thus more fixation strength. According to the regression analysis outcomes, the natural frequency (ωn) was linearly dependent on the PIT (ωn=14 PIT) and also on the PPF (ωn=1.7 PPF). Coefficients of variation as the results of the modal analysis were significantly less than those in conventional methods (i.e. pull-out and insertion torque).
Conclusion:
The modal analysis was found to be a reliable, repeatable, and non-destructive method, which could be considered a prospective alternative to the destructive pull-out test that is limited to the in-vitro application only. The modal analysis could be applied to assess the stability of implantable screws, such as orthopedic and spinal screws.

... a . time-signal inspection (various) [2] CONCERTO [6] higher order statistics/FRFs [2] associated linear equations: SDOF [7], MDOF [8] nonlinear normal modes: FANS [9], NLRD [10], experimental (Shaw-Pierre) [11] restoring force surfaces: SDOF [2,12,13] non-stationary analysis: [14][15][16][17][18][19][20][21][22] characterization CONCERTO [6] FRF distortion methods (various) [2,3] associated linear equations: SDOF [7], MDOF [8] time-signal inspection (various) [2] inverse harmonic balance [23] higher order statistics/FRFs [2] restoring force surfaces: SDOF [2,12,13], MDOF (direct parameter estimation) [24] proper orthogonal modes [1] reverse path method: standard [25] (and NIFO [26]), conditioned [27], orthogonal [28] nonlinear structural model updating [1] restoring force surfaces: modal space [3] nonlinear PCA [29] non-stationary analysis [14][15][16][17][18][19][20][21][22] nonlinear normal modes: experimental (Shaw-Pierre) [11] . location nonlinear output FRFs [30] n . ...

... a . time-signal inspection (various) [2] CONCERTO [6] higher order statistics/FRFs [2] associated linear equations: SDOF [7], MDOF [8] nonlinear normal modes: FANS [9], NLRD [10], experimental (Shaw-Pierre) [11] restoring force surfaces: SDOF [2,12,13] non-stationary analysis: [14][15][16][17][18][19][20][21][22] characterization CONCERTO [6] FRF distortion methods (various) [2,3] associated linear equations: SDOF [7], MDOF [8] time-signal inspection (various) [2] inverse harmonic balance [23] higher order statistics/FRFs [2] restoring force surfaces: SDOF [2,12,13], MDOF (direct parameter estimation) [24] proper orthogonal modes [1] reverse path method: standard [25] (and NIFO [26]), conditioned [27], orthogonal [28] nonlinear structural model updating [1] restoring force surfaces: modal space [3] nonlinear PCA [29] non-stationary analysis [14][15][16][17][18][19][20][21][22] nonlinear normal modes: experimental (Shaw-Pierre) [11] . location nonlinear output FRFs [30] n . ...

... inverse harmonic balance [23] restoring force surfaces: MDOF (direct parameter estimation) [24] associated linear equations: MDOF [8] reverse path method: standard [25] (and NIFO [26]), conditioned [27], orthogonal [28] nonlinear structural model updating [1] . quantification CONCERTO [6] restoring force surfaces: modal space [3] associated linear equations: SDOF [7], MDOF [8] nonlinear normal modes: NLRD [10], experimental (Shaw-Pierre) [11] inverse harmonic balance [23] restoring force surfaces: SDOF [2,12,13], MDOF (direct parameter estimation) [24] reverse path method: standard [25] (and NIFO [26]), conditioned [27], orthogonal [28] nonlinear structural model updating [1] nonlinear subspace identification [31] non-stationary analysis [14][15][16][17][18][19][20][21][22] . ...

Model validation using data from modal tests is now widely practiced in many industries for advanced structural dynamic design analysis, especially where structural integrity is a primary requirement. These industries tend to demand highly efficient designs for their critical structures which, as a result, are increasingly operating in regimes where traditional linearity assumptions are no longer adequate. In particular, many modern structures are found to contain localized areas, often around joints or boundaries, where the actual mechanical behaviour is far from linear. Such structures need to have appropriate representation of these nonlinear features incorporated into the otherwise largely linear models that are used for design and operation. This paper proposes an approach to this task which is an extension of existing linear techniques, especially in the testing phase, involving only just as much nonlinear analysis as is necessary to construct a model which is good enough, or 'valid': i.e. capable of predicting the nonlinear response behaviour of the structure under all in-service operating and test conditions with a prescribed accuracy. A short-list of methods described in the recent literature categorized using our framework is given, which identifies those areas in which further development is most urgently required.
© 2015 The Authors.

... To derive an energy balance equation analogous to SEA, initially an assumption will be made that the nonlinear term on the right-hand side of equation (2.4) is an external force acting on a linear system. Linear vibration theory [17] may then be used to describe the response of the linear structure consisting of the oscillators a j to a random external force F j . For a linear structure with randomly distributed natural frequencies ω j , undergoing random excitation, it is common on December 25, 2015 http://rsta.royalsocietypublishing.org/ Downloaded from [18] to approximate the zero mean, random process of each oscillator a j as being statistically uncorrelated, defined as E[a i a j ] = 0, where E indicates the ensemble average across the different realizations of the random structure. ...

... where S k is the spectral response of the kth oscillator and H j is the linear transfer function between force and displacement for the jth oscillator if all the nonlinear parameters C in equation (2.4) were zero. Solving for the mean-square response of the jth oscillator σ 2 j by integrating over ω and assuming that the forcing is flat near the resonant peak of each oscillator so a white noise approximation is valid for S FF [17] gives ...

Nonlinearities in practical systems can arise in contacts between components, possibly from friction or impacts. However, it is also known that quadratic and cubic nonlinearity can occur in the stiffness of structural elements undergoing large amplitude vibration, without the need for local contacts. Nonlinearity due purely to large amplitude vibration can then result in significant energy being found in frequency bands other than those being driven by external forces. To analyse this phenomenon, a method is developed here in which the response of the structure in the frequency domain is divided into frequency bands, and the energy flow between the frequency bands is calculated. The frequency bands are assigned an energy variable to describe the mean response and the nonlinear coupling between bands is described in terms of weighted summations of the convolutions of linear modal transfer functions. This represents a nonlinear extension to an established linear theory known as statistical energy analysis (SEA). The nonlinear extension to SEA theory is presented for the case of a plate structure with quadratic and cubic nonlinearity.
© 2015 The Author(s).

... Two additional families of missing mode shapes are identified and described by the sum and the difference of beam natural modes, respectively. The Rayleigh quotient is used not only for the first but also for the higher modes, [21]. In such a way, a complete and denser spectrum of natural frequencies is obtained. ...

... , where m and n are the numbers of beam vibration nodes, then the plate mode shape function has to manifest m + n cross straight lines of zero displacement, Fig 3.Such a mesh can be noticed only in the modes 1, 8, and 17, and roughly in the modes 4, 5, 6, 7, 11, 12, 15, 16, 20, 21, and 24 in Fig. 2. As for all other remaining modes, it is necessary to find out another type of the plate mode shape functions. An Analytical Solution to I. Senjanović, M. Tomić, Free Rectangular Plate Natural Vibrations N. Vladimir, N. Hadžić by Beam Modes – Ordinary and Missing Plate Modes An Analytical Solution to N. Vladimir, N. Hadžić Free Rectangular Plate Natural Vibrations by Beam Modes – Ordinary and Missing Plate Modes ...

Relatively simple analytical procedures for the estimation of natural frequencies of free thin rectangular plates, based on the Rayleigh quotient and the Rayleigh-Ritz method, are presented. First, natural modes are assumed in the usual form as products of beam natural modes in the longitudinal and transverse directions, satisfying the grillage boundary conditions. Based on a detailed FEM analysis, the missing of some natural modes, defined as a sum and a difference of the cross products of beam modes, is noted. The frequencies of these modes are very similar and identical in some special cases, manifesting in such a way a double frequency phenomenon. These families of natural mode shapes form a complete natural frequency spectrum of a free rectangular plate as a novelty. The reliable approximation of natural modes enables the application of the Rayleigh quotient for the estimation of higher natural frequencies. The application of the developed procedure is illustrated by the case of a free thin square plate. The obtained results are compared with those determined by FEM and also with rigorous ones from the relevant literature based on the Rayleigh-Ritz method. The achieved accuracy is acceptable from the engineering point of view. Furthermore, the same problem is solved by the Rayleigh-Ritz method using approximate natural modes as mathematical ones. Direct and iterative procedures are presented. A small number of mathematical modes and iteration steps are sufficient to achieve reliable results.

... The nonlinear terms cause a peak-frequency shift with vibration-amplitude increases. The real time frequency can be deduced as [45]: ...

... Combining Equations (13), (45) and (46) with Equation (47), we can deduce the excess phase caused by the A-S effect: ...

In this paper, a detailed analysis of the influence of phase noise on the micro-electro-mechanical system (MEMS) disk resonator gyroscope (DRG) is presented. Firstly, a new time-varying phase noise model for the gyroscope is established, which explains how the drive loop circuit noise converts into phase noise. Different from previous works, the time-varying phase noise model in this paper is established in mechanical domain, which gain more physical insight into the origin of the phase noise in gyroscope. Furthermore, the impact of phase noise on DRG is derived, which shows how the phase noise affects angular velocity measurement. The analysis shows that, in MEMS DRG, the phase noise, together with other non-ideal factors such as direct excitation of secondary resonator, may cause a low frequency noise in the output of the gyroscope system and affect the bias stability of the gyroscope. Finally, numerical simulations and experiment tests are designed to prove the theories above.

... In accordance to the modal theory, two different structures with similar dimensional characteristics, similar weight distributions and same materials (similar rigidities) should have very similar modal behaviors (modal spectrums) (De Silva, 2000;Newland, 2006;Robert Bosch GmbH, 2007). ...

... As it was expected based on the specialty literature (De Silva, 2000;Newland, 2006), the obtained spectrums present multiple similarities, in fact that their general aspect is very similar. Nevertheless, modal peaks among which there are no significant differences (1) and modal peaks that present significant differences (2) can be seen. ...

The analysis of structures with finite elements models represents one of the most utilized an accepted technique in the modern industry. For the analysis of large tubular structures similar to buses and coaches upper structures, beam type elements are utilized and recommended due to the fact that these elements provide satisfactory results at relatively reduced computational performances. However, the beam type elements have a main disadvantage determined by the fact that the modeled joints have an infinite rigid behavior, this shortcoming determines a stiffer behavior of the modeled structures which translates into error sources for the structural simulations (up to 60%).
By modeling tubular junctions with shell and volume elements, more realistic models can be obtained, because the topological characteristics of the junction at the joint level can be reproduced more accurately. This way, the shortcoming that the beam type elements present can be solved. Despite this fact, modeling large tubular structures with shell or volume type elements represents an unattractive alternative due to the complexity of the modeling process and the large number of elements that result which imply the necessity of vast computational performances. The main objective of the research presented in this thesis was to develop a new beam type element that would be able to provide more accurate estimations for the local behavior of the modeled junctions at the same time maintaining the simplicity of the modeling process the regular beam type elements have. In order to reach the established objectives of the research activities, a series of different methodologies and investigations have been necessary.
From these investigations an alternative beam T-junction model was obtained, in which a total of six elastic elements at the joint level were introduced, the elastic elements allowed us to adapt the local behavior of the modeled junctions. Additionally, for the estimation of the stiffness values corresponding to the elastic elements two methodologies were developed, one based on the T-junction’s static behavior and a second one based on the T-junction’s dynamic behavior by means of modal analysis. The improvements achieved throughout the implementation of this alternative T-junction model were analyzed though mechanical validation in a complex tubular structures that had a representative configuration for buses and coaches upper structures.
From the comparative analyses of the finite element modeled T-junctions and mechanical experimental analysis, was determined that the beam type modeled T-junctions have a stiffer behavior compared to equivalent shell and volume modeled T-junctions with average differences ranging from 5-60% based on the profile configurations. It was also determined that the shell and volume models have a stiffer behavior compared to real T-junctions varying from 5 to 10% depending on the profile configurations.
Based on the analysis of the complex tubular structure, significant improvements were obtained by the implementation of the alternative beam T-junction model, the model estimations were improved from a 49% to approximately 14%

... Eq. (27) and (35) indicate that matrix has the same non-zero elements with . By substituting and into Eq. ...

... The controllers for two SISO systems can be designed separately. As to the MIMO system, the modal superposition method is adopted to make them in decoupling form [27]. The kinetic equation of the − direction is: ...

Piezoelectric actuators (PEA) act an important role in active vibration control area due to the advantages of fast response, high output force, small size and light weight. A 6-axis orthogonal vibration isolation platform based on PEAs is designed, which satisfies the demands of heavy payload, small installation space and multi degree of freedom vibration isolation. The dynamic model of the six-axis orthogonal vibration isolation platform with PEAs is established using Newton-Euler method. With the layout of six PEAs around the axis of symmetry, the dynamic equations could be decoupled into two single-input-single-output (SISO) subsystems and two multi-input-multi-output (MIMO) subsystems. Based on the modal superposition method, the two MIMO subsystems are further decoupled. The control strategy for each SISO system is developed with LQR control method. To evaluate the effectiveness of the control method, the simulation and verification experiment are conducted. The simulation result and experimental data indicate that the decoupling control of the proposed six-axis orthogonal vibration isolation platform with piezoelectric actuators effectively reduces the vibration response of payload within the target frequency range of 20 Hz to 200 Hz.

... Shafting torsional vibration lumped parameter model is usually simplified as straight branched-chain type. Straight branched-chain type vibration model (undamped [6,7] or damped [8,9]) is usually calculated by Holzer method [10,11]. With the development of shipbuilding technology, the propulsion system of ship power plant is becoming more and more complex, and the lumped parameter model that describes the torsional vibration of the shaft system is in the process of simplification. ...

... To sum up, according to the diesel engine exciting force equation (9), motor electromagnetic harmonic exciting force caused by (16) and the forced vibration equation (6) of hybrid propulsion shafting vibration response is calculated. As the phase angle of twin-engine parallel operation changes in the range of 0 degrees to 720 degrees, only the calculation result of the vibration amplitude of the transmission shaft at = 120 ∘ is shown in Figure 9. From Figure 9, we can see that the resonance amplitude of the transmission shaft is ...

This study analyzes the characteristics of hybrid propulsion shafting and builds mathematical models and vibration equations of shafting using the lumped parameter method. Main focus is on the asymmetric double diesel propulsion shafting operation process and the impact of the phase angle and motor excitation on torsional vibration of shafting. Model result is validated by testing results conducted on double diesel propulsion shafting bench. Mathematical model and model-building methods of shafting are correct.

... The HPT, LPT, and generator rotor were modelled as discs connected by massless springs. Figure 11 illustrates the simplified model of the single cylinder monoblock rotor as a vibrating system with three (3) DOF [29]. Figure 11. ...

... The calculated shaftline properties for torsional dynamic analysis are shown in Table 5. Table 4. Torsional parameter formulae [29]. ...

The AM600 represents the conceptual design and layout of a Nuclear Power Plant Turbine Island intended to address challenges associated with emerging markets interested in nuclear power. When coupled with a medium sized nuclear reactor plant, the AM600 is designed with a unit capacity that aligns with constraints where grid interconnections and load flows are limiting. Through design simplification, the baseline turbine-generator shaftline employs a single low-pressure turbine cylinder, a design which to date has not been offered commercially at this capacity. Though the use of a ‘stiffer’ design, this configuration is intended to withstand, with a margin, the damage potential of torsional excitation from the grid-machine interface, specifically due to transient disturbances and negative sequence currents. To demonstrate the robust nature of the design, torsional rotordynamic analysis is performed for the prototype shaftline using three dimensional finite element modelling with ANSYS® software. The intent is to demonstrate large separation of the shaftline natural frequencies from the dominant frequencies for excitation. The analysis examined both welded drum and monoblock type Low Pressure Turbine rotors for single cylinder and double cylinder configurations. For each, the first seven (7) torsional natural frequencies (ranging from zero–190 Hz) were extracted and evaluated against the frequency exclusion range (i.e., avoidance of 1× and 2× grid frequency). Results indicate that the prototype design of AM600 shaftline has adequate separation from the dominant excitation frequencies. For verification of the ANSYS® modelling of the shaftline, a simplified lumped mass calculation of the natural frequencies was performed with results matching the finite element analysis values.

... Both algorithms and parallel computing techniques should be therefore considered. Modal analysis takes an important role in dynamic computation, which is also the base of many other vibration analysis types [1, 2]. For large scale vibration analysis problems, the mode superposition method [3] is usually a feasible and effective way to obtain the dynamic response of structures. ...

In the development of large and complex equipment, a large-scale finite element analysis (FEA) with high efficiency is often strongly required. This paper provides some progress on parallel solution of large-scale modal and vibration FE problems. Some predominant algorithms for modal and vibration analysis are firstly reviewed and studied. Based on the newly developed JAUMIN framework, the corresponding procedures are developed and integrated to form a parallel modal and vibration solution system; the details of parallel implementation are given. Numerical experiments are carried out to evaluate the parallel scalability of our procedures, and the results show that the maximum solution scale attains ninety million degrees of freedom (DOFs) and the maximum parallel CPU processors attain 8192 with favorable computing efficiency.

... is no antiresonance frequency in Yt there is some measure of ~t at all frequencies in this response [70]. ...

p>An active mechanism, the cochlear amplifier, enhances the response of the cochlea to low-level stimuli and is assumed to be controlled by the action of the outer hair cells (OHCs) located within the organ of Corti.
Classical models of the cochlea use an array of isolated lumped parameter systems along its length, coupled through the cochlear fluid. These models employ active feedback loops between the basilar and tectorial membrane (TM) for the mechanism of the cochlear amplifier. Several such models are reviewed and their underlying dynamic behaviour examined, in order to compare the predicted response with recent measurements of the relative motion within the organ of Corti. Their stability is also tested to establish reliability of calculated frequency responses. The models are conditionally stable and operate close to instability to achieve high sensitivity of the cochlear amplifier. Compressive nonlinearity is also included in one of the classical models using a quasi-linear approach.
It has recently been suggested that wave motion within the organ of Corti may also play a role in the cochlear amplifier. The behaviour of two possible types of wave between the reticular lamina and the TM is examined, one in which the TM is assumed to behave as a plate in bending and another in which it is assumed to behave like an elastic half-space. The propagation speed is very low for both waves and incorporation of the losses induced by viscosity causes the waves to decay significantly within a wavelength. Feedback from the OHCs coupled into these waves outcomes the effects of viscosity and enhances waves’ resonant supporting this form of amplification in the cochlea.</p

... If the system damping is not proportional then Eq. (10) no longer applies; in this case the response of the system can be represented in terms of complex modes [21] and the resulting expression again has exactly the form of Eq. (9). For generality Eq. (9) rather than Eq. ...

In nuclear physics it is known that under broad restrictions a random scattering matrix element H satisfies a condition known as the analyticity-ergodicity (AE) requirement, which states that E[f(H)] = f(E[H])], where f is some function of H, and E[] represents the ensemble average. A scattering matrix element is directly analogous to a vibrational frequency response function, and it is of significant interest to consider whether the AE requirement is also applicable to random engineering systems. The proof of the AE condition in nuclear physics rests on the assumptions that H is causal and ergodic: causality implies that a Lorentzian frequency average satisfies the AE equation, and ergodicity implies that Lorentzian frequency averages are equal to ensemble averages. In vibrational systems it is readily shown that a typical frequency response function is non-stationary and non-ergodic, so that the Lorentzian and ensemble averages can differ significantly, and this means that the standard proof of the AE requirement breaks down. The question then arises as to whether the AE requirement might nonetheless apply to vibrational systems. It is shown in the present paper that the requirement does apply providing that the random point process representing the system natural frequencies is at least locally stationary (which is a much weaker condition than local stationarity of the frequency response function), and a number of the implications of this result are explored.

... For any stationary and ergodic random variable x(t), the spectral density function S xx (ω) is given by (7) with (8) where R xx (τ) is the autocorrelation of x(t), or in other words, it is the expected value E[x(t)x(t + τ)], i.e., (9) and, under certain conditions, the Fourier transform is valid to be applied over R xx (τ). ...

... But technical limitations held back the widespread use of the methods until recently when many of these limitations have been overcome [4,5], following which there has been a resurgence due to the development of new and efficient algorithms for the various aspects of the analysis. For a full technical background, the reader is referred to, for example, [6,7,8,9]. ...

... A low pass filter was used to remove the high frequency noise. Fast Fourier Transform (FFT) was performed to obtain the natural frequency of the composite foots and to calculate the damping ratio using the logarithmic decrement method [23]. The natural frequency and damping ratio of the composite foot structures with respect to the stacking sequences were measured and calculated as shown in Table 5. ...

Although aluminum structures are generally used for robot structures due to their high specific strength, aluminum feet for fast running biped robots are vulnerable to fatigue failure due to the low fatigue limit and low vibration damping of aluminum structures under repeated impact loadings on the feet. On the other hand, carbon/epoxy composites not only have a much higher specific fatigue limit but also have a higher material damping than that of aluminum.In this study, a carbon/epoxy composite foot structure of a biped robot was developed. The composite foot structure was designed for optimum performances such as weight saving, natural frequency, damping, and compliance for vibration isolation. Then its performances were analytically and experimentally obtained and compared with those of an aluminum foot structure. Finally, an optimum configuration of the composite foot structure was suggested for the reliable dynamic performance of the biped robot.

... R. Assuming that a damping force acts on the beam, we describe the evolution of the system by the following equations (see [24,26,41,48]) ...

In this paper, a dynamic contact problem between a Timoshenko beam and two rigid obstacles is considered. Thermal effects are also taken into account and the contact is modeled using the classical Signorini condition. The global existence in time of solutions is found by considering related penalized problems, proving some a priori estimates and passing to the limit. An exponential decay property is also showed.

... Equation (8) is reviewed in the state-space. The state-space method is based on transforming the N second-order coupled equations into a set of 2N first-order coupled equations [30,31]. Equations of dynamic system motion can be recast as: ...

There are many reasons for rehabilitation of existing buildings. Adding stories is one of the most common reasons. When a steel building is retrofitted by concrete jacketing for adding stories, this system contains several structural systems. These systems are composite concrete and steel systems in initial stories, welded steel system in middle stories and cold-formed steel frames in upper stories. Dynamic analysis of hybrid structures is usually a complex procedure due to various dynamic characteristics of each part, i.e. stiffness, mass and especially damping. Availability of different damping factors causes a higher degree of complication for evaluating seismic responses of hybrid systems. Due to using several structural systems, an existing building is changed to hybrid system. Damping matrix of these structures is non-classical. Also, the nonlinear software is not able to analyze these structures precisely. In this study, a method and graphs have been proposed to determine the equivalent modal damping ratios for rehabilitated existing steel buildings for adding stories.

... Besides, (•) T denotes the transpose of the vector. Since the system focus in this research is asymmetric, the right and left eigenvectors are different and they can be calculated using the first-order state-space method [15]. Eqn. ...

Generally, a mechanical system always has symmetric system matrices. Nevertheless, when some non-conservative forces are included, such as friction and aerodynamic force, the symmetry of the stiffness matrix or damping matrix or both violated. Moreover, such an asymmetric system is prone to dynamic instability. Distinct from the eigenvalue assignment for symmetric systems to reassign their natural frequencies, the main purpose of eigenvalue assignment for asymmetric systems is to shift the unstable eigenvalues to the stable region. In this research, only the unstable eigenvalues and eigenvalues which are close to the imaginary axis of the complex eigenvalue plane are assigned due to their predominant influence on the response of the system. The remaining eigenvalues remain unchanged. The state-feedback control gains are obtained by solving the constrained linear least-squares problems in which the linear system matrices are deduced based on the receptance method and the constraint is derived from the unobservability condition. The numerical simulation results demonstrate that the proposed method is capable of partially assigning those targeted eigenvalues of the system for stabilisation.

... Clutch hysteresis is also taken into account as equivalent viscous damping; 3. Transmission is considered as a single inertia and reflected stiffness and damping at its input; 4. The drive shafts and half shafts are considered as a single inertia located in the end of the shaft (Author and Ungar 1992); 5. The torsional stiffness is computed using the finite element method with quadratic tetrahedron elements. ...

The driveline torsional vibration issue is one of the most significant Noise, Vibration and Harshness (NVH) problems, especially in rear-wheel drive vehicles with manual transmission. In this article, a new driveline and rear axle coupled torsional vibration model (DRCTVM) is developed that considers the relationship between the driveline and the rear axle. The experiments show that the DRCTVM can provide much better results than the traditional model. In addition, for the first time, uncertainty theory is introduced to the analysis and optimization of driveline torsional vibration based on the DRCTVM. A truncated normal distribution is used to describe the uncertainty of DRCTVM, which considers both the probability distribution and the bounds of uncertain variables. Furthermore, robustness of the driveline torsional vibration was analysed using the Monte Carlo (MC) process and optimized using the Multi-Island Genetic Algorithm. The optimization results show that the proposed model and method are effective and improve the robustness of driveline torsional vibration performance.

... These vibrations have significant effects on the physical and mental health [6,7]. It has been observed that the stresses imposed by vibration have produced changes in the normal functions of the human body [8,9]. The Whole body vibration can cause severe motion sickness which depends on the mechanical properties of the human body [10]. ...

The aim of this work is to study the effect of vibration on the amputee’s Comfort experimentally. The vibration transmitted from the artificial limb to the human body by free and forced vibration is taken into consideration. An experimental rig was constructed for this purpose and it consists of a shaking table equipped with a cam mechanism, AC motor, PLC speed variator (inventor) and suitable springs. A measurement device is used to collect the displacement, velocity, and acceleration spectrum with computer interface in different points along the prosthetic limb and the healthy limb. The experiments were carried out on an amputee whose amputation is in his leg during the Iraqi-Iranian war in the eighteens of the last century. This work is planned to serve the increasing number of amputee to supply them with more comfortable artificial limbs.

... This is particularly true for machines with bearings. Therefore, in traditional PHM, vibrations of rotat- ing parts have played an important role (Dalpiaz & Rivola, 1997;Goyal & Pabla, 2016;Jardine, Lin & Banjevic, 2006;Newland, 1989;Scheffer & Girdhar, 2004). To the oppos- ite, vibrations caused by feedback loops, as it seems to be the case here, have been far less explored for condition monit- oring. ...

Data-driven Prognostic and Health Management is gaining more and more popularity, to the point where it is often seen as a potential solution for any maintenance problem. To the opposite, it is known that, when the physic of a system is known and tractable, it is usually the most efficient way to infer relevant indicators for detecting system behaviour changes. And sometimes, when the physic of a system is only partially known, data-driven approaches can highlight critical information that would help to make the right physical assumptions, which in turn, can be used for elaborating the right health indicators. This paper presents a real-life example of such process. With focus on the estimation of the degradation of a measurement system head, a data-driven approach, with its limitations, helped to identify the right signal processing tools with which to analyse the data. With the results obtained from Spectral Analysis with Windowed Fourier Transform, the physics of the system could be hypothesized, and then used for deriving the most relevant indicators for monitoring the degradation. These indicators have been implemented online and so far used successfully on several machines encountering degradation of their measurement head. This successful ``reversed'' case study of degradation monitoring is a strong reminder of the benefits of field symbiosis: data-driven approach made possible the understanding of the physics, which in turn could be used for data monitoring.

... As shown in Fig. 4(i), when using the elastic coefficient parameter of silicon material, the impulse response signal of the supercapacitor shows several high-frequency oscillation peaks after the main voltage peak, which significantly increases the difficulty of recognizing the number of successive impacts. This is because the rubber skeleton has a lower intrinsic frequency of vibration than a metal material of high rigidity, and thus, the response signal contains fewer high-frequency oscillating components [27]. ...

... Introducing the regular coordinates q r (t), based on the formation superposition method, the solution d(x, t) for a given boundary condition can be expressed as [41]: ...

Due to its excellent strength, plasticity, and fracture toughness, titanium alloy has been widely used in the aerospace field. The specificity of its application environment places high demands on the surface quality of titanium alloy. In this paper, we study the effects of different lubrication methods on the microscopic topography, surface roughness, and microhardness of titanium alloy TC21 during the milling process. The lubrication methods include dry, high-pressure air cooling, and minimum quantity lubrication (MQL). Compared with dry milling and high-pressure air cooling conditions, the MQL environment can effectively suppress plastic deformation and surface defects of titanium alloy TC21. MQL is significant for improving the milling process of titanium alloy TC21. In addition, a surface roughness model considering milling vibration is developed. According to the results of orthogonal experiments, the prediction accuracy of the surface roughness model is acceptable, and the prediction errors are all below 20%.

... The mooring point is the center of mass, so that the elastic coefficient is 0 when only the pitch is considered. Figure 1 shows a dynamic model of a pitch-float wave energy converter [37], where K1 and K2 are both 0 and C2 = CPTO. In Figure 1, the coupling motion of the pitching float, I1, and the wave converts the kinetic and potential energy of the wave into the mechanical energy of the float. ...

This study analyzes the hydrodynamic performance and application of a pitching float-type wave energy conversion device under complex sea conditions in the South China Sea. Potential flow theory and ANSYS-AQWA software are used to establish a method for analyzing hydrodynamic performance in both time and frequency domains, as well as the various factors that influence hydrodynamic performance. The frequency domain characteristics of the conversion device are explored, as well as the time-domain characteristics when exposed to regular and irregular waves. The results show that the frequency domain of hydrodynamic performance conforms to the requirements of an offshore mobile platform. A mooring point that is closer to the center of mass leads to improved stability of the conversion device. The angle arrangement of the anchor-chain mooring method fully conforms to safety requirements. When the wave direction is 45°, the conversion device is highly stressed and its movement is the most strenuous; however, the device can operate safely and stably under all working conditions. These results provide a significant reference for expanding the wave-energy capture range and the hydrodynamic performance of floating wave-energy conversion devices.

... Mechanical systems driven by random excitations have been discussed by Lin [49], Nigam [53], Bolotin [54], Roberts and Spanos [55] and Newland [56] within the scope of random vibration theory. To obtain the samples of the random response quantities such as the displacement of the mass y(t) and the voltage v(t), one needs to solve the coupled stochastic differential equations (27) and (28) or (27) and (51). ...

... For the sake of simplicity, we neglect the effect of damping forces, as they do not change the mode shapes of vibration (considering weakly damped systems) and exert only a marginal influence on the eigenvalues-the treatment of damped systems is comprehensively discussed in many references such as Refs. [10,[41][42][43]. The homogeneous equations of motion excluding damping effects are given as ...

An efficient and robust finite element-based transient analysis of structures is important in many engineering applications. In this context, a diagonal or lumped mass matrix is an essential prerequisite. In the last decades, several methods to construct such a mass matrix have been proposed and therefore a comprehensive and unbiased evaluation of these approaches in needed. In the present article, we consequently investigate established mass lumping schemes, such as the row-sum method and the diagonal scaling method, as well as the recently proposed manifold-based method. Furthermore, the latter approach is rigorously extended to higher order serendipity finite elements in two and three dimensions. Note that the manifold-based method and the diagonal scaling method are very general approaches, in the sense that they are applicable to arbitrary nodal-based shape functions with arbitrary polynomial orders, while guaranteeing the positive-definiteness of the lumped mass matrix. With respect to the row-sum method the positivity is not ensured and negative or zero diagonal components might be computed. Consequently, a detailed analysis of the influence and performance of these mass lumping schemes on the numerical results is conducted. To this end, several dynamic benchmark models (including modal, harmonic, and transient analyses) are selected, showing that only suboptimal rates of convergence are attained if serendipity finite elements based on a diagonal mass matrix are employed.

Two important issues of the automotive turbochargers in the rotordynamics are resonance and instability of the rotor. Both have a common harmful effect that causes damages of the turbochargers during the operation. However, there is a big difference between the resonance and instability if we take a close look on them.

Exhaust gas turbochargers used in the automobiles, such as passenger, on-road vehicles, and off-road engines have some discrepancies to the heavy turbomachines applied to the power plants and chemical industries. The first ones are much smaller and work at high rotor speeds in various operating conditions, such as variable rotor speeds, pressures, temperatures, and as well as mass flow rates. Contrary to the automotive turbochargers, the industrial turbomachines are bigger, heavier and mostly operate at a stationary working condition. Due to their large sizes of compressor and turbine wheels, the turbomachines only operate at low rotor speeds between 3,000 to 20,000 rpm. The maximum circumferential velocities of the turbine and compressor wheels used in the automotive turbochargers are approximately 530 and 560 m/s, respectively. The maximum circumferential velocities of the compressor and turbine wheels are determined by the durability of materials at various driving cycles. Their thermo-mechanical characteristics and lifetime depend on the using material, producing method, and as well as driving cycles.

The block-diagonalization method for systems with geometric symmetries is introduced based on the framework of group-representation theory. This method is put to use in the numerical analysis of the systems with regular polygonal symmetries, that is, Dn-invariant systems. The method is superior to the direct method in that the former can enhance the numerical efficiency and, at the same time, reduce numerical errors.

Harvesting energy from the wind to replace the conventional electrochemical batteries has gained considerable interest. An axially retractable bracket-shaped piezoelectric vibrator excited by magnetic force is proposed to provide a practical solution to the low reliability and narrow bandwidth of the traditional turbine-based piezoelectric wind energy harvesters (PWEHs). To prove the feasibility and explore the influence of the structure parameters of the bracket-shaped vibrator on the power generation performance, the theoretical analysis, simulation, fabrication and experimental tests were performed for the turbine-based PWEH. The results showed that the bending angle, the excitation distance of magnets, the number ratio of exciting magnets and the stiffness ratio of springs brought significant effects on the performance in terms of electric output, cut-in wind speed, cut-out wind speed, and wind speed bandwidth. The maximal peak output voltage was significantly affected by the magnet spacing and the number of exciting magnets. The cut-in wind speed decreased with the increasing magnet spacing and meanwhile it was hardly influenced by both the number ratio of magnets and the stiffness ratio of springs. The cut-out wind speed was also decreased with the increase of magnet spacing and there was an optimal number ratio of magnets and an optimal stiffness ratio of spring to maximize the cut-out wind speed. Besides, there was an optimal excitation distance, number ratio of magnets and stiffness ratio of springs respectively, where the wind speed could reach the maximum value. A maximum power of 2.13 mW could be achieved with the magnet spacing of 7 mm, the magnet number ratio of 0.83 and stiffness ratio of 10.08, respectively. It was expected this study could provide a useful reference for improving the reliability and experimental adaptability of PWEHs.

The purpose of this study is to facilitate engineering applications and allied researches by presenting a detailed comparative picture based on dynamic analysis of Jeffcott rotor, diameter 32 mm for three materials, namely low carbon steel, SAE1045, and SAE5120 employing finite element analysis simulation software, ANSYS. The authors covered modal, harmonic response, and transient analyses along with natural frequencies and whirling speeds at different modes. Resonance occurrence speeds are also analyzed using the Campbell diagram. The harmonic response is investigated, with added material (0.1 kg) on the disc, to reveal the stresses and deformation. Transient analysis is performed with time-varying loads to work out stresses. FEA is carried out to estimate critical speeds for these three materials. The result highlights that SAE5120 exhibits the least resonance with a high value of natural frequency. Similarly, Campbell diagrams reveal no instabilities with a majority of critical speeds at the backward whirl, thus modal-based analysis confirms the superiority of SAE5120. Stress analysis involving harmonic responses also supports SAE5120 as the stress values are very high for both SAE1045 and SAE5120. The transient analysis involving equivalent, principal, and shear stresses supports the above results. The study compares and establishes the overall usefulness of SAE5120 for rotor applications among three materials.

Rotational machinery such as horizontal axis wind turbines exhibits complex and nonlinear dynamics (e.g. precession and Coriolis effects, torsional coupling) and is subjected to nonlinear constrained conditions (i.e. aeroelastic interaction). For those reasons, aeroelastic and computer-aided models reproduced under controlled conditions may fail to predict the correct non-stationary loading and resistance patterns of wind turbines in actual operation. Operational techniques for extracting modal properties under actual non-stationary loadings are needed in order to improve computer-aided elasto-aerodynamic models to better characterize the actual behavior of horizontal axis wind turbines in operational scenarios, monitor and diagnose the system for integrity and damage through time, and optimize control systems. For structural health monitoring applications, model updating of stochastic aerodynamic problems has gained interest over the past decades. A probability theory framework is employed in this study to update a horizontal axis wind turbine model using such a stochastic global optimization approach. Structural identification is addressed under regular wind turbine operation conditions for non-stationary, unmeasured, and uncontrolled excitations by means of stochastic subspace identification techniques. This numerical framework is then coupled with an adaptive simulated annealing numerical engine for solving the problem of model updating. Numerical results are presented for an experimental deployment of a small horizontal axis wind turbine structure.

The existing systems in operation, for instance in mechanical, civil, transport, electrical engineering, etc., are usually considered as having only spatial r and fast time t-dynamic coordinates and properties. This description is sufficient if only the design properties are considered, optimized, and the dynamic behaviour is simulated. But when the mechanical properties of real systems in operation are considered, which means taking into account imperfections and damage that have appeared during the manufacturing stage, and when their evolution in operation is considered, one should introduce the concept of life with the lifetime θ, where the damage evolution can proceed and can be measured. The life properties depend on the environmental conditions as well as on the operational ones. Both can introduce faults and damage into the system, which, if they exist, will have an instantaneous effect or will affect the mechanical system under operation in an evolutionary manner to a certain extent. In consequence, a holistic approach is needed in order to model the total dynamics of the system in operation.

Um den Kontakt zwischen einem rotierenden und einem ruhenden Maschinenteil zu untersuchen, war es von Interesse, definierte Anfangsbedingungen in einer rotierenden, axisymmetrischen Struktur zu erzeugen.

This chapter contains two topics which extend the stability and vibration analysis of the arches. They are the dynamical stability of arch and dynamic action of the moving load on the arched structure.

This chapter contains information about the reasons for nonlinearity and the general properties of nonlinear vibration, and discusses the basics of the harmonic linearization method. Applications of this method for the analysis of free and forced vibrations of systems with one degree of freedom are presented. Different types of nonlinearities are considered. These include Duffing’s rigidity characteristic, a combination of nonlinear stiffness with viscous resistance, linear stiffness with dry friction. A nonlinear dynamic absorber is considered.

Modern theory of automatic control of dynamical systems contains in its arsenal an extremely valuable tool. We are talking about the structural representation of an arbitrary dynamical system. Such representation allows us to divert attention from the physical nature of a process (thermal, vibrational, diffusion, etc) to the physical nature of the elements (mechanical, pneumatic, etc). In the context of structural representation of a mechanical system, we can explore diverse aspects of dynamic processes (controllability, invariance, stability, etc.) [1–3]. The theory of vibration protection is a very attractive application area of structural theory for several reasons. First, many fundamental aspects and concepts of control theory in general and the theory of vibration protection coincide; these include input–output concepts, transfer function, etc. Second, a vibration protection system consists of pronounced blocks and can be represented in symbolic form by a functional block diagram. Successful attempts that consider the problems of vibration protection in terms of the structural theory have been performed by Kolovsky [4, 5], Eliseev [6], and Bozhko et al. [7]. Systematic exposition of the structural theory to systems with distributed parameters was presented by Butkovsky [8]. Structural representation of the system in conjunction with the vibration protection device is a common way of describing complex dynamical systems with lumped and distributed parameters. Structural theory allows us to easily introduce changes into a vibration protection system of the object and find a relationship between any coordinates of a system, while the differential equation of the system assumes a fixed input–output. The Simulink (MATLAB) package has a full set of blocks that allows us to implement just about any structural model.

This chapter deals with fundamental functions of linear dynamical systems including the transfer function, Green’s function, Duhamel’s integral, and standardizing function. We show their application to different problems dealing with dynamical systems.

This chapter presents the theory of dynamic suppression of vibration of systems with lumped parameters. First, in the example of the simplest dynamic absorber, we consider the idea of suppressing of vibrations. Then we discuss the different types of absorbers have been considered by Babicky [1, Chap. 14, 2], Haxton and Barr [3], and Karamyshkin [4]. These include impact absorbers, gyroscopic vibration suppressors, and autoparametric vibration absorbers. Such devices can also be effectively used for reducing vibrations of systems with distributed parameters [5].

An analysis of steady-state vibration of linear dynamical systems subjected to harmonic force and/or kinematic exposure can be reduced to analysis of mechanical two-term networks (M2TN), also known as replacement schemes, which are equivalent to the original scheme. The two representations of the system are equivalent in the sense that both representations can be described by the same differential equations. The theory of analogy [1–4] is what makes such an interchange possible. The advantage of representing a dynamical system as a replacement scheme is that its construction for multi-element dynamical systems is fairly simple and consists in analyzing M2TN by algebraic methods [5], whereas analysis of the original design diagram must be performed by solutions to differential equations. Another advantage of representing systems through M2TN is that theorems often used to analyze electrical circuits (Kirchhoff’s rule, Thevenin and Norton’s theorem, principle of superposition, etc.) can also be applied to replacement schemes.

This chapter is devoted to the analysis of one degree of freedom systems subjected to shock excitation [1, 2, Chap. 9, 3], etc. Some important concepts are discussed. among which are types of shock excitation and different approaches to the shock problem. Fourier transformation of aperiodic functions and corresponding concepts are considered and are then applied to the shock phenomenon. The spectral shock theory method and the concepts of residual and primary shock spectrums are discussed [4]. The transient vibration caused by different force and kinematic shock excitation (Heaviside step excitation, step excitation of finite duration, impulse excitation) are considered. Dynamic and transmissibility coefficients are derived and discussed in detail.

Dynamic analysis of multi-mesh planetary gearings is very important for reduction of noise and vibration. Splitting of force flow into several planet wings is the main advantage of planetary gearings. As it can be devaluated by unequal load sharing on individual planet stages, the floating sun gear and flexible pins of planet gears are applied. This paper shows that dynamic model of such a gearing box is very complicated with many multiple eigenfrequencies. The gained frequency spectrum with multiple eigenvalues is derived and analyzed.

از جمله مهمترین عوامل ایجاد صدای ماشینهای لباسشویی، ارتعاشات بدنة خارجی آنهاست. وظیفة سیستم تعلیق ماشین لباسشویی، که اتصالدهندة ماشین لباسشویی (مخزن ثابت در لباسشویی) به بدنه میباشد، تلاش برای عدم انتقال ارتعاشات مخزن به بدنه است. بهطور کلی سه نوع طراحی برای سیستم تعلیق ماشین لباسشویی وجود دارد: سسیتم تعلیق در بالای مخزن، سسیتم تعلیق در پایین مخزن و نهایتاً سیستم تعلیق ترکیبی از دو مورد پیشین. در این مقاله از مدل سوم استفاده شده است. در این رهگذر، سیستم تعلیق ماشین لباسشویی خودکار با محور دوران افقی مورد ارزیابی قرار گرفته و سعی شده است تا مدل تحلیلی دقیقی از دینامیک سیستم تعلیق آن ارائه گردد. معادلات دینامیکی برای حرکت دوبعدی مخزن یک ماشین لباسشویی خودکار در صفحه استخراج شده و بهصورت عددی مورد ارزیابی قرار گرفته است. همچنین آزمونهای عملی روی این ماشین لباسشویی - که در آزمایشگاه واحد تحقیق و توسعة گروه صنعتی انتخاب انجام شده - با نتایج تئوری مقایسه شده است. نتیجة مقایسه بیانگر تطابق مناسب نتایج تئوری و تجربی است. از مدل بهدست آمده میتوان بهمنظور بهینهسازی اولیة طراحی ماشینهای لباسشویی خودکار استفاده کرد.

p>Waves, winds and currents can cause specific environmental effects that a marine structure has to withstand. Amongst these, wave action is the fundamental source of load on the marine structure. In order to ensure safety, operability, economy and design-life duration of a marine structure, theoretical estimates of wave loads and structural response play an increasingly important role in the overall design process. The interaction between a structure and a fluid medium is of great concern in numerous engineering problems, e.g., slamming of ships in rough seas, vibration of water retaining structures under earthquake loading etc. All these dynamic problems include the interaction, which takes place between the structure and surrounding fluid. It is of practical importance to estimate the effect of the induced fluid loading on the dynamic state of the vibrating structure. If the vibration takes place in a relatively low-density fluid, such as air, in comparison with the structural material, in most situations, the loading will have a comparatively small influence on the vibration. However, when the vibrating structure is in contact with a fluid which has a comparable density, such as water, the fluid loading which depends on the structural surface motions will significantly alter the dynamic state of the structure from that of the in vacuo vibration. In other words, the equations of structural and fluid motions are inexorably linked. Therefore, development, improvement and application of numerical techniques for analyzing such an interaction become one of the most important activities of naval architecture researchers. The following document is about the interaction mentioned above and particularly studied on the slamming issue and its main characteristic, transient excitation and response. A dry analysis is presented on simple beams, idealized SWATH ship as a preamble to a future wet deck slamming analysis and plates (unstiffened and stiffened). As the basis of subsequent harmonic and transient analyses, modal characteristics of each system is studied and in conjunction with the results obtained from these, responses on frequency and time domain are calculated in this document. In the following part of the thesis beams and plates are analysed under transient excitation, since this is the basis for modelling the excitation and response induced by slamming. Results are produced and compared both using theoretically established convolution method and ANSYS (transient analysis with full and mode superposition methods). Realistic stiffened plates and their equivalent flat plates are also studied and analysed in the subsequent sections. Difficulties encountered during the structural modelling (finite element modelling) are briefly outlined, with particular emphasis to the importance of the selection of appropriate finite elements.</p

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