Investigation of Damping of Vibrations in a System With Two‐Disc Inseparable Clutch

This paper presents the results of tests on free and forced harmonic torsional vibrations in a system with a two‐disc inseparable clutch, with structural friction taken into account. Nonlinear histeresis loop describing the frictional‐elastic properties of the system was introduced into the model. The mathematical model of the vibrating system containing two disks inseparable clutch was built. During free vibrations of the system, its damping characteristics were tested by a digital simulation method. The vibration damping decrement as a function of amplitude torsional displacement was determined. When vibrations were harmonically forced, the amplitude ‐ frequency characteristics of the system were determined numerically. The system was used as a nonlinear torsional vibration damper in a linear system with a harmonic force. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

集中阻尼弦本征解的性质

利用Dirac δ函数,在全域建立并求解集中阻尼弦的动力学方程,导出其本征方程组、频率方程和本征函数的一般形式,推导了单项阻尼下本征函数的具体形式,并分析了中点阻尼对本征解的影响.同时,讨论了混合动力学系统在频率 阻尼关系、衰减率和完全抑制振动的最优阻尼3个方面既不同于连续系统,又不同于离散系统的特性:1）系统频率与其阻尼无关;2）各阶本征函数在单位时间内的衰减率都相同,衰减率与本征值的阶次无关;3）当阻尼取2时,系统衰减率趋于无穷大,系统不能发生任何有阻尼振动.     

Damping of Vibrations in a SystemWith Two Beams by Structural Friction

This paper presents the results of tests on free and forced harmonic vibrations in a system with two beams with structural friction taken into account. The beams are clamped together with uniform unitary pressure. The hysteresis loop describing the frictional-elastic properties of the system has a form of a parallelogram. The autor created a mathematical model of the vibrating system with two beams. During free vibrations of the system, its damping characteristics were tested by a digital simulation method. The vibration damping decrement as a function of amplitude displacement was determined. When vibrations were harmonically forced, the amplitude - frequency characteristics of the system were determined numerically. The system was used as a nonlinear vibration damper in a linear system with a harmonic force. The equations of motion of the nonlinear two-degree of freedom system were solved by means of a digital simulation method. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Damping Of Pedestrian‐Induced Bridge Vibrations By Tuned Liquid Column Dampers

The excessive lateral vibrations of Londons Millenium Bridge and the Toda Park Bridge in Japan due to a large number of crossing pedestrians have raised an unexpected problem in footbridge constructions. Secondary tuned structures, like the conventional tuned mass damper (TMD) or the tuned liquid damper (TLD) were installed to the bridge in order to suppress these vibrations. In the present investigation it is proposed to apply the more efficient and more economic tuned liquid column damper (TLCD), which relies on the motion of a liquid mass in a sealed tube to counteract the external motion, while a built‐in orifice plate induces turbulent damping forces that dissipate kinetic energy. For optimal tuning of TLCDs the natural frequency and equivalent linear damping coefficient have to be chosen suitable, likewise to the conventional TMDs, as given in Den Hartog [1]. The advantages of TLCDs are: simple tuning of natural frequency and damping, low cost of design and maintenance and a simple construction. A mathematical model of a three degree‐of‐freedom (DOF) bridge coupled with an optimal tuned TLCD is derived and analyzed numerically. Furthermore, a small scale experimental model set‐up has been constructed in the laboratory of the TU‐Insitute. The experimental results are in good agreement with the theoretical predictions and indicate that TLCDs are effective damping devices for the undesired pedestrian induced footbridge vibrations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibration suppression of non-linear system via non-linear absorber

In this paper, the coupled non-linear differential equations of the non-linear dynamical two-degree-of-freedom vibrating system including quadratic and cubic non-linearities are studied. The system consists of the main system and the absorber. The absorber is used to control the main system vibrations when subjected to multi-external excitation forces at simultaneous primary and internal resonance. This system represents many applications in machine tools, ultrasonic cutting process, etc. The method of multiple scales perturbation technique (MSPT) is applied throughout to determine the solution up to third order approximations. The different resonance cases are reported and studied numerically. Stability is studied applying frequency response functions. The effects of different parameters of the system are studied numerically. Optimum working conditions for the absorber where obtained at internal resonance ratio 1:3. This means smaller mass for the absorber which solves the problem of space limitation. A comparison is made with the available published work.     

Mathematical model for coupled roll and yaw motions of a floating body in regular waves under resonant and non-resonant conditions

The prediction of resonance is very important with respect to the vessels stability in the early stages of design. In this paper, an efficient modeling approach is presented to determine coupled roll and yaw motions of a symmetric and slender floating body when the influences of small amplitude regular waves are dominant. The angular motions described in time domain by considering all internal and external forces are transformed into frequency domain to obtain motion characteristics. We adopt a semi-analytical treatment to obtain roll and yaw motions and derive system instability due to roll resonance. To compute hydrodynamic forces, we employ strip theory method where frequency dependent sectional added-mass, damping and restoring coefficients are derived from the Frank’s close-fit curve. Numerical experiments carried out for a vessel of mass 19,190 ton under the action of wave of frequencies 0.56 and 0.76 rad/s with zero and non-zero initial conditions are reported and the effect of various parameters on system stability is investigated. Model results indicate that damping factor (ς) plays a pivotal role when wave encountering frequency (ω) and undamped natural frequency (β) are nearly equal. The essence of this study lies in the efficient modeling technique to evaluate damping factor and critical encountering frequency regime for a given ship particulars when experimentally derived resonance zone is absent.     

An inverse eigenvalue method for frequency isolation in spring–mass systems

The action of external vibrating forces on mechanical structures can cause severe damages when resonance occurs. The removal of natural frequencies of the structure from resonance bands is therefore of great importance. This problem is called frequency isolation problem and is the subject of this paper. A new inverse eigenvalue method is proposed and applied to spring–mass systems, which have generated much interest in the literature as prototypes of vibrating structures. The novelty of the method lies in using the zeros of the frequency response function at the last mass as control variables in an optimization problem to minimize the impact of redesign. Numerically accurate algorithms for computing the sensitivity with respect to the control variables are presented, which form the basis of an efficient multidimensional search strategy to solve the frequency isolation problem. Copyright © 2001 by John Wiley & Sons, Ltd.     

Effects of Damping Mechanisms in Free and Forced Vibrations of Some Structural Members

For continuous vibrating systems, such as bars and beams, end-mounted in the environment, knowledge about the mass, damping and stiffness properties of the resonating environment is important for analyzing free and forced vibrations of such structural members which are also damped themselves. To finally get an identification of the clamping parameters, examinations of both vibrating structural members for various restraint conditions and dynamic interaction with viscoelastic halfspaces, etc., are required. As a first step, longitudinal bar vibrations are studied in detail. The method of separation of variables combined with the reformatted orthogonality relation, and numerical integration is applied for calculating the free and forced oscillations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forced vibrations of a viscoelastic damping layer

Forced vibrations of a viscoelastic damping layer are investigated. The equations obtained are reduced to normal modes, and the solutions are determined for nonresonance and resonance cases of vibrations.Moscow Institute of Electronic Machine Construction. Translated from Mekhanika Polimerov, No. 1, pp. 124–132, January–February, 1972.     

The vibrational behaviour of bladed disks with multiple coupling devices

In turbomachinery applications turbine blades are subjected to high static and dynamic loads. Static loads are due to centrifugal stresses and thermal strains. Especially the dynamic excitation caused by fluctuating gas forces results in high vibration amplitudes which can lead to high cycle fatigue failures (HCF). Therefore, in practical applications, coupling devices like underplatform dampers, lacing wires and tip shrouds are installed to the structure. In case of blade vibrations the relative displacements between these coupling devices and the blades generate friction forces. The resulting energy dissipation provides additional damping to the structure. Furthermore, coupling devices, in particular tip shrouds, snubbers and lacing wires, increase the stiffness of the structure. Hence, they lead to a shift of the resonance frequencies. So far, only effects of single coupling devices and the influencing properties have been examined. Within this paper the effect of multiple couplings is determined and compared with single couplings. The forced response of turbine bladings with multiple couplings is calculated under consideration of geometrical and mechanical parameters of the blading and contacts, respectively. The results are compared with the single coupled blading. Furthermore, a multiple coupled device with under-platform damper and connecting pin is compared with respect to his effectiveness. Especially the influence on the resonance frequency and the achievable damping is analysed. The results of the simulation are verified by measurements at a two-blade non-rotating test rig with an underplatform damper and connecting pin. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eigen‐frequencies in thin elastic 3‐D domains and Reissner–Mindlin plate models

The eigen‐frequencies of elastic three‐dimensional thin plates are addressed and compared to the eigen‐frequencies of two‐dimensional Reissner–Mindlin plate models obtained by dimension reduction. The qualitative mathematical analysis is supported by quantitative numerical data obtained by the p‐version finite element method. The mathematical analysis establishes an asymptotic expansion for the eigen‐frequencies in power series of the thickness parameter. Such results are new for orthotropic materials and for the Reissner–Mindlin model. The 3‐D and R–M asymptotics have a common first term but differ in their second terms. Numerical experiments for clamped plates show that for isotropic materials and relatively thin plates the Reissner–Mindlin eigen‐frequencies provide a good approximation to the three‐dimensional eigen‐frequencies. However, for some anisotropic materials this is no longer the case, and relative errors of the order of 30 per cent are obtained even for relatively thin plates. Moreover, we showed that no shear correction factor is known to be optimal in the sense that it provides the best approximation of the R–M eigen‐frequencies to their 3‐D counterparts uniformly (for all relevant thicknesses range). Copyright © 2002 John Wiley & Sons, Ltd.     

Resonance methods for determining the complex shear moduli of orthotropic composites

A survey of various methods for determining the complex elasticity and shear moduli from the resonant frequencies of flexural and torsional vibrations of rectangular rods cut out from a plate of an orthotropic composite is presented. The errors in the computed values of dynamic shear moduli caused by inaccuracies in the experimental determination of resonance frequencies are examined. A new variant of the resonance method is developed, which permits one to find three complex shear moduli of a composite from the resonant frequencies and the damping of torsional vibrations of three rods oriented along three symmetry axes of the material. For computing the moduli in the case of an overdetermined system, an algorithm of nonlinear optimization based on the least-squares method is recommended. From the results obtained it follows that, for determining the interlaminar shear moduli with a necessary accuracy, the rods must be sufficiently thick. It is shown that a good agreement alone between calculated and experimental frequencies of flexural and torsional vibrations of rods does not ensure a reliable determination of the moduli of interlaminar shear if experiments are carried out on wide test specimens cut out from a thin plate. Recommendations are given for the choice of geometrical sizes of test specimens for resonance experiments. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 721–744, November–December, 2007.     

Impurity modes in a curved Fermi–Pasta–Ulam chain

The spatial localization properties of nonlinear excitations/modes supported by a curved Fermi–Pasta–Ulam (FPU) lattice chain in presence of an isolated impurity of mass lighter or heavier than the host mass, is investigated. The impurity modes oscillate locally at and around the impurity site. It is examined that a light-mass impurity mode fulfills non-resonance with the linear (or phonon) spectrum because its frequency is located above the phonon band whereas frequency of a heavy-mass impurity mode drops into the phonon band. The phenomenon of resonance of impurities with plane waves explains the lifetimes of localized impurity modes in the nonlinear system.     

Non-linear free spatial vibrations of combined suspension systems

Free vibrations of a combined suspension system with a beam with stiffness which has one axis of symmetry are studied. Unlike /1/ the case of internal resonance in the system is analysed in detail. It is shown that both in this case and in the case where the frequencies of the vertical modes and the bending and twisting modes are close to each other /1/ there are two possible forms of vibrations, namely, those which are subject to phase and amplitude modulation and are accompanied by energy transmission, and those subject to phase modulation only, which allow no energy transmission.     

隔水套管波流联合作用下非线性动力响应

考虑流及波流联合作用,研究了深水套管的涡激非线性振动．将套管简化为梁模型,计及Morison非线性流体动力和涡激荷载,建立套管的涡激振动方程．采用Korolov函数求解套管的固有频率和模态,提出了计算涡激非线性动力响应的Galerkin方法,计算了160 m水深中170 m长套管的固有频率和模态,研究了流引起的主共振和波流联合引起的组合共振．计算结果表明波流联合作用下套管的动力响应明显增大,结果也揭示了波流联合激励下套管复杂的动力响应特性．     

Mechatronic system with shunted piezoelectric damper modelled with structural damping

Paper presents analysis of an one-dimension flexural vibrating mechatronic system. The considered system is a cantilever beam with a piezoelectric transducer bonded to the beam's surface. An external electric circuit is adjoined to the transducer's clamps in order to damp vibrations. System was analyzed on the basis of an approximate Galerkin method. Verification and assumptions of the approximate method were described in the previous papers where analysis of the mechatronic system with piezoelectric shunt damper was presented. Structural damping of all system's components was being taken into consideration. Rheological properties were introduced using Kelvin-Voigt model of materials. Influences of component's structural damping coefficients values on the system's dynamic flexibility were defined. Obtained results were presented on 3D graphs as dynamic flexibility dependence on the structural damping coefficient and frequency of an external force that was applied to the system. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Features of the construction of correction functions in the analysis of the free vibrations of mechanical systems by Bubnov''s method

Free vibrations of mechanical systems are investigated by Bubnov's method, using coordinate functions that do not satisfy all the boundary conditions of the problem. To eliminate discrepancies in the boundary conditions, correction functions are introduced, orthogonal to all coordinate functions involved in the analysis of the natural vibrations of mechanical systems. This method of constructing correction functions reveals the existence of differences between nearly equal numbers in the frequency equation and in the expressions for the vibrational modes. Moreover, for a given number N of coordinate functions, the method guarantees increased precision in computing all the natural frequencies of the mechanical system up to and including the (N + 1)th partial frequency as the number of terms of the power series in the frequency parameter representing the correction functions is increased. In the limit, as the number of terms of the power series tends to infinity, one is assured of obtaining exact values of the frequencies and vibrational modes in the indicated frequency range.     

Stability study and control of helicopter blade flapping vibrations

The response of a two-degree-of-freedom, controlled, autoparametric system to harmonic excitations is studied and solved. The objective of this research is to investigate the effect of linear absorber on the vibrating system and the saturation control of a linear absorber to reduce vibrations due to rotor blade flapping motion. The method of multiple scale perturbation technique is applied to obtain the periodic response equation near the primary resonance in the presence of internal resonance of the system. The stability of the obtained numerical solution is investigated using both phase plane methods and frequency response equations. Variation of some parameters leads to the bending of the frequency response curves and hence to the jump phenomenon occurrence. The reported results are compared to the available published work.     

On the effect of damping on the stabilization of mechanical systems via parametric excitation

The effect of damping on the re-stabilization of statically unstable linear Hamiltonian systems, performed via parametric excitation, is studied. A general multi-degree-of-freedom mechanical system is considered, close to a divergence point, at which a mode is incipiently stable and n ? 1 modes are (marginally) stable. The asymptotic dynamics of system is studied via the Multiple Scale Method, which supplies amplitude modulation equations ruling the slow flow. Several resonances between the excitation and the natural frequencies, of direct 1:1, 1:2, 2:1, or sum and difference combination types, are studied. The algorithm calls for using integer or fractional asymptotic power expansions and performing nonstandard steps. It is found that a slight damping is able to increase the performances of the control system, but only far from resonance. Results relevant to a sample system are compared with numerical findings based on the Floquet theory.     

An equivalent moving force model for consideration of human-structure interaction

To predict the vibration response of footbridges, many codes of practice use a deterministic moving force (MF) model. This approach may not be well suited for the design of slender, lightweight, low-damping, and low-frequency footbridges because it ignores the pedestrian interaction with the vibrating footbridge. On the other hand, a spring-mass-damper (SMD) model is able to incorporate human mass, stiffness, and damping into the vibration response prediction. However, the SMD model is computationally demanding and not commonly available in engineering practice. To address this shortfall, a framework is proposed to derive a computationally-efficient equivalent MF-structure system to the reference SMD-structure system such that both systems give a similar vibration response metric. Analytical and numerical approaches to the equivalent MF (EMF) system are described in detail and applied to bridges with approximately simply-supported mode shapes. A sensitivity analysis is carried out to show the effects of different pedestrian parameters on the equivalent damping of the EMF system. The effects of pedestrian damping, frequency, and weight are found to be pronounced, while those of dynamic load factors and pedestrian step length are insignificant. Finally, empirical expressions are proposed in a probabilistic framework to determine the equivalent damping for simply-supported low-frequency footbridges as a function of bridge frequency. This work should find use in the serviceability assessment of low-frequency footbridges in engineering practice.