Complex Mode Theory Research Articles

Seismic demand estimation of isolated buildings with nonlinear bearings subjected to linear response spectra

Response spectrum analysis is a commonly used method for the seismic design of linear structures with fixed bases since the seismic input is normally provided in terms of the response spectrum in many codes. This study proposes a more generalized response spectrum method for the seismic design of three-dimensional isolated structures. Firstly, the theory basis of this method is derived in the framework of the random vibration theory. The equivalent parameters of each bearing are expressed in terms of the statistics of the displacement amplitude. Then, the complex mode theory is introduced into this method to consider the effect of the nonclassical damping resulting from the isolation layer. Finally, an iterative analysis procedure is presented, which combines the equivalent linearization with the complex mode theory. Moreover, the convergence and applicability of this method are discussed in detail. The parameter analyses show that the relative error of the base shear of the superstructure estimated by using this method is within 10% compared with that of the nonlinear time history analysis. In addition, a benchmark base-isolated structure is used as a numerical example to examine the effectiveness of the proposed method. The results show that the iteration is convergent and computationally efficient. The results also show that the maximum relative errors of the predicted isolation layer displacement, the inter-story drift of the bottom floor and the absolute acceleration of the top floor are 9.5%, 3.9% and 18.9%, respectively, by comparing them with the nonlinear time history analysis, while these responses are underestimated by 9.8%, 25.9% and 40.2%, respectively, when using the classical complete quadratic combination (CQC) method that neglects the nonclassical damping effect. Although in this study the proposed approach mainly focuses on isolated structures, in essence, this approach can also be applied to inelastic structures or structures with nonlinear dampers.

Frequency response of spinning cylindrical shells with discontinuous boundary conditions: A semi-analytical method

This paper develops a semi-analytical method for solving the frequency response of spinning cylindrical shells. The dynamic model of the spinning cylindrical shells with discontinuous boundary conditions is established. In the modeling, Sanders’ shell theory is employed and Chebyshev polynomials are adopted as the admissible displacement functions. By using the Lagrange equation, the equation of motion is obtained. Then, the equation is transformed into a state-space form, wherein the orthogonality of complex eigenvectors is derived. Further, the equation of motion is decoupled on the basis of the complex mode theory. Utilizing the reduced complex eigenvector, a semi-analytical method for solving the frequency response of the spinning shells is finally developed, wherein the point and distributed harmonic loads are all considered. The influences of the damping coefficient, the spinning speed, and the boundary spring stiffness on the frequency response of the shells are analyzed. The proposed method can conveniently predict the vibration characteristics of spinning cylindrical shells through a reduced-order dynamic model, which is much more efficient than numerical methods.

Parameters optimization and performance evaluation of multiple tuned mass dampers to mitigate the vortex-induced vibration of a long-span bridge

Vortex-induced vibration(VIV) is one of the primary dynamic problems of long-span bridges, which affects the safety and serviceability of bridges. The control and mitigation of VIV is a hotspot to researchers. Multiple tuned mass dampers(MTMD) consist of a set of tuned mass dampers(TMD) with varied frequencies and damping ratios, which is a common method for vibration control of structures. In this study, a concept of complex effective mass is proposed, which can entirely represent the dynamic effect of MTMD on bridges, and a derivative-based MTMD optimization process for VIV control of long-span bridges is presented. The VIV mitigation of a six-span continuous steel box-girder bridge is used to analyze the performance and mechanism of MTMD from the aspects of frequency domain and time domain. Optimal parameters of MTMD for suppressing VIV are obtained by the proposed algorithm. The results indicate that MTMD can significantly reduce the displacement amplitude of the beam due to VIV, and the performance of MTMD is better than that of single tuned mass damper(STMD) with the same mass ratio. The optimal performance of MTMD is independent of the mass distribution of the sub-TMDs. Moreover, the damping ratio of MTMD is much lower than that of STMD, but it can still help the vibration of the primary structure to decay faster. Complex mode theory and transfer function theory are used to explain this phenomenon. It indicates that the performance of MTMD does not depend on its damping ratio but is due to the phase interference between complex modes, and a one-way energy barrier between the bridge deck and MTMD can be formed, which prevents the transfer of vibration energy from MTMD to the primary structure after unloading. Finally, the stroke of MTMD in the control of long-span bridge VIV is analyzed, and a simplified formula is proposed.

Ultrasensitive temperature sensor based on a urethane acrylate-coated off-axis spiral long period fiber grating

We propose and experimentally demonstrate an ultra-high sensitivity fiber-optic sensor to measure temperature changes in order to establish a high-precision temperature model in the bio-fermentation process. The proposed sensor based on off-axis spiral long-period fiber grating (OAS-LPFG) is fabricated in a single mode fiber (SMF) by arc discharge technology. This kind of grating is distinguished in the refractive index (RI) modulation, geometry changes, and shear stress of a spiral structure. Because the RI of the urethane acrylate is close to that of the SMF cladding and has a high thermo-optic coefficient, the resonant wavelength of the OAS-LPFG dip has a high sensitivity to the temperature measurement. The response of the OAS-LPFG to environmental RI is analyzed by full vector complex coupling mode theory. In the experiment, the temperature sensitivity of the proposed sensor can be reached ∼-108.69 nm/°C over a range of 25.0 °C to 25.5 °C. Compared with other all-fiber-based temperature sensors, the sensor has higher sensitivity.

Structural characteristics of a propulsion mechanical system considering aeromechanical excitations

This paper studies the structural dynamic characteristics of an essential mechanical vector device on airship, which is described as a capsule-wing-rotor system. The root flexibility of the capsule and the gyro-effect from the tip rotor are considered to be distinct factors relative to the previous traditional situations, and the inherent structural geometries are also incorporated for multiple parametric analyses. The governing equation is established by Hamilton’s principle, and the analytical solution and the numerical solution are derived by virtue of complex mode theory and Rayleigh-Ritz method, respectively. In case studies, the validity of the numerical solution is verified via a special error test technology. It is demonstrated that the boundary capsule flexibility, the gyro-effect and the system parameters all have great influences on the structural performance. The analyses indicate that the root flexibility tends to decrease the overall frequencies of the structure, and the gyro-effect strengthens the dimensional coupling of the structure intensely, leading to the occurrence of complicated complex-mode vibration phenomenon. The computations of critical boundaries under known aeromechanical excitations show that the design parameters used to stabilize the system are affected by the gyro-effect severely, demonstrating further optimization work for stability improvement is requisite.

Structural and aeroelastic analyses of a wing with tip rotor

Based on parametric studies, this paper studies the structural and aerodynamic properties of a typical wing-rotor system by considering additional coupling effects from the tip rotor. The governing equations are established by virtue of Yamanaka’s theory and Hamilton’s principle in an Euler–Bernoulli precision. In the derivation process, complex mode theory and the Rayleigh–Ritz approach are employed to obtain the analytical solution and the numerical solution, respectively, and a set of trial functions are generated for the sake of reliability. Two aeroelastic models are generated via steady and unsteady aerodynamic models to investigate the divergence and flutter performance, respectively, where the computational p-k method is adopted because of the gyros matrix. In case studies, the effectiveness of the numerical solution is validated through comparative analysis; thereafter, the aeroelastic solutions are further verified. The results demonstrate that the gyro effect and the thrust effect have significant influences on the fundamental structural and aeroelastic performances, and parametric designs have great potential to postpone aeroelastic instabilities.

An adaptive higher order finite element model and modal energy for the vibration of a traveling string

A nonlinear equation describing the transverse vibration of an axially traveling string with constant and time-varying length is obtained by developing a new finite element model described by quadratic shape functions. A novel nonlinear coordinate transform is introduced with regard to its nonlinear terms. Subsequently, a new hybrid Newmark-beta/time varying degree of freedom method, which can adjust the element number automatically according to the change of string length, is proposed to improve accuracy. The proposed method as well as normal numerical methods are compared with an analytical solution. Results show that the proposed method is in good agreement with the Newmark-beta method for the case of small variations in string length, whilst it is superior in accuracy to the latter in the case of large length variations. Complex mode theory is adopted firstly to obtain the modal components as well as subsequently the modal energy for a traveling string. A phenomenon is observed where the free vibration energy leaks from one mode to the others in a traveling string. The higher the speed of translation and the modal order, the more energy that is leaked into the modes close to the initially excited mode.

Mechanical Parameters of Standing Body and Applications in Human–Structure Interaction

In this paper, the dynamic interaction of human body and structure is studied The shaking table experiment with a person standing on a rigid table supported by springs is firstly carried out to determine the dynamic characteristics of the coupled system. It is shown that the body mainly contributes only one degree of freedom to the human-structure coupled system. Then, the two-degree-of-freedom (TDOF) coupled model of the human-structure system is developed through the energy variation by considering the standing human body as an elastic bar of two segments with distributed mass, stiffness and damping. Based on the experiment data, the dynamic parameters of the TDOF coupled system are determined by using the least square method (LSM). The mechanical parameters such as the damping ratio and the distributions of mass and stiffness of the human body model of two segments are identified by adopting the inversing technique Finally, the determined body model is applied to analyze the free vibration of beams and plates occupied by standing persons. The governing differential equations of the human-beam system and the human-plate system are, respectively, derived out. The dynamic characteristics of the human-structure interaction are obtained by the use of the complex mode theory. The results are compared with the experimental ones and those from the finite element simulations. Good agreement is observed for all cases.

Analysis on Aircraft Brake Squeal Problem Based on Finite Element Method

Brake squeal phenomenon is a problem that has been long studied using multiple methods and theories. Finite Element Method (FEM) has been applied to the study of brake squeal problem. First, a disc brake model has been established. Complex mode theory has been applied to the mode analysis and unstable vibration modes can be extracted subsequently. The form of unstable vibration mode has been studied. Then, transient dynamic simulation using explicit dynamic method has been performed. Response in both time and frequency domain has been analyzed. Two methods have been compared, considering accuracy and calculation consumption. Then, the effect of different parameters such as coefficient of friction, stiffness, and brake force fluctuation frequency on squeal phenomenon has been analyzed. It can be found that coefficient of friction and the brake stiffness have a positive correlation with the extent of brake squeal phenomenon, while the frequency of brake force fluctuation should remain as low as possible. Afterwards, a ring-shaped layer of viscoelastic damping material is constrained to outer margin of the stator to restrain the unstable modal. This method can change the vibration nature and improve the brake squeal problem.

Dynamic response of a front end accessory drive system and parameter optimization for vibration reduction via a genetic algorithm

A typical engine front end accessory drive system (FEADS) is mathematically modeled through Hamilton’s principle and Newton’s second law. In this model, the belt’s flexural rigidity and pulley’s eccentricity are considered. Eccentricities of the pulleys are introduced into governing motion equations of the belt spans through the boundary conditions and then transformed to external forces acting on the belt spans. Vibration modes and natural frequencies of the FEADS are calculated by the state-space technique of the complex mode theory. Dynamic responses of the FEADS at different rotational rates of the crankshaft are calculated by solving the spatially discretized governing equations obtained by Galerkin method. The modeling and solution methods are formulated and programmed in a general purposed code. The study shows that the typical resonance and beat phenomenon happen in a certain portion of the belt spans at a certain rotational rate by the excitations of the pulley’s eccentricity. According to the modal analysis and dynamic response analysis, an optimization method based on a genetic algorithm is proposed. By comparing the vibration amplitudes of belt spans before and after optimization at different rotational rates, this optimization method is verified to be effective in reducing transverse vibrations of the belt spans.

Improved complex mode theory and truncation and acceleration of complex mode superposition

The traditional complex mode theory is substantially improved. By means of the conjugacy of the complex characteristic value and vector, the formulae to compute the accuracy solution to the time-domain dynamic response for a symmetrical system with generalized linear viscous damping are established. Based on the formulation constructed, the truncation of the complex mode superposition is investigated. By employing the complex mode expansion of the system flexibility matrix discovered in this work, an accelerating method for the complex mode superposition after truncation is proposed. According to such improved complex mode theory, the system dynamical parameters, for example, the natural frequency and the complex vibration shape, are all the functions with respect to the mass, stiffness, and damping. However, the influence of the damping on the natural frequency may always be relatively slight. The numerical simulating investigation shows that the precise dynamic response can be obtained simply and conveniently in line with the formulae established. Meanwhile, the truncating and accelerating methods are effective for both the narrow or wide band external excitation and can provide the approximate solution to the dynamic response with better precision even when the frequency of the excitation is higher. Under certain conditions, the accelerating approach can be used to boost the accuracy of the truncated response.

Active Vibration Control of Macroscopically Smart Mechanisms

This paper presents an investigation on the active vibration control of flexible linkage mechanisms featuring piezoceramic actuators and strain gauge sensors. The dynamic equation of the macro-scopically smart mechanism is decoupled by means of the complex mode theory. The state-space expression of the controlled system is developed, which includes the system noise and the observation noise. Moreover, a discrete linear quadratic Gaussian (LQG) state feedback controller and a discrete Kalman filter are designed separately. Finally, the proposed method is applied to the on-line vibration control of a macroscopically smart mechanism. The experimental results reveal that the strain amplitude of the flexible link is suppressed by 80% and the dynamic performance of mechanism has been ameliorated significantly.

Smart Material Based Complex Mode Active Control of Flexible Manipulators with Kinematic Redundancy

An active control methodology is presented for suppressing the vibratory response of flexible redundant manipulators with bonded piezoceramic actuators and strain gage sensors. Firstly, the dynamic equation of the manipulator is decoupled by means of the complex mode theory and the state-space expression of the controlled system is developed. Secondly, a continuous linear quadratic regulator (LQR) state feedback controller is designed based on the minimum principle. Thirdly, a full-order Luenberger state observer featuring an assigned degree of stability is determined via the duality between control and estimation. Finally, a numerical simulation is carried out on a planar 3R flexible redundant manipulator. The simulation results reveal that the dynamic performance of the system is improved rapidly and significantly.

Active vibration controller design and comparison study of flexible linkage mechanism systems

Three active vibration controller design methodologies include the reduced modal controller, the classical and the robust H ∞ controller for the high-speed flexible linkage mechanism systems with piezoelectric actuators and sensors are investigated. Firstly, the state space form control model is formulated based on the complex mode theory. Secondly, the reduced modal controller, the classical and the robust H ∞ controller are designed. Finally, numerical simulations are carried out on a 4-bar linkage mechanism. The simulation results show that the vibration of the system is significantly suppressed with permitted actuator input voltages by each of the three controllers. The reduced modal control makes it more convenient to increase the modal damping of the system than that of the H ∞ control does, while the robust H ∞ control can avoid the spillover due to mode truncation.

Complex Mode Dynamic Analysis of Flexible Mechanism Systems with Piezoelectric Sensors and Actuators

A method has been developed for analyzing the elastodynamic response ofhigh-speed flexible linkage mechanisms with piezoelectric sensors andactuators. The finite element analysis model is obtained by using amixed variational approach with Hamilton's principle. An efficientsolution method is developed on the basis of complex mode theory. Afour-bar linkage mechanism with all flexible links is employed as anexample to demonstrate the capability of the proposed method. Theinvestigation provides a theoretical basis for complex mode activevibration control of high-speed flexible mechanism systems.

COMPLEX MODE ACTIVE VIBRATION CONTROL OF HIGH-SPEED FLEXIBLE LINKAGE MECHANISMS

A methodology for suppressing the elastodynamic response of high-speed flexible linkage mechanisms with piezoelectric elements is developed. The finite element active vibration control model is given by a mixed variational approach with Hamilton's principle. On the basis of complex mode theory, a hybrid independent modal controller is developed, which is composed of state feedback and disturbance feedforward control laws. As an illustrative example, the strategy is used to control the elastic vibration response of a four-bar linkage mechanism. The simulation results show that the vibration is efficiently suppressed.

Free Vibration Analysis Of Linear Vibrating Deficient Systems With Arbitrary Damping

The "deficient" system that exists when the geometric multiplicity of an eigenvalue of a linear vibrating system is less than its algebraic multiplicity is defined and constructed. A complex mode theory is developed for the deficient system. The normal complex modes and generalized complex modes are defined. The free vibration characteristics of the deficient system are discussed. It is shown that the state equations of motion of the deficient system cannot be decoupled completely, and that the free vibration of the deficient system will decay more slowly than a damped normal system with the same damping ratio. A three-degree-of-freedom deficient system of springs and masses is used to illustrate the utility and correctness of the results.

Modelling and identification of offshore towers with general damping

Abstract The modelling problem of offshore towers is considered. A model is proposed which includes viscous and hydrodynamic drag damping and takes into account the stochastic nature of the problem. The model is decoupled into a set of modal equations by using the complex mode theory. A technique is developed to identify the modal parameters and the complex mode shapes by using the noisy excitation and response measurements. This technique is computationally efficient and accurate in the sense of minimizing a criterion function of the estimates variance. A simulation example is presented to demonstrate the results.

First-order perturbation solution to the complex eigenvalues

The matrix perturbation method is extended to discrete linear nonconservative system with unsymmetrical matrices in this article. By introducing the concept of the adjoint complex eigenvector and by making use of the orthogonality relationship in the complex mode theory, the first-order perturbation solution to the complex eigenvalues is derived. Numerical example shows that this method is efficient and practicable.