Modeling and analytical methods for a mounting system with double stage isolation

A calculation method for obtaining the displacements and rigid body modes of a Powertrain Mounting System (PMS) with double stage isolation is proposed in this paper. Firstly, the PMS with double stage isolation is modeled as a 12 Degree of Freedoms (DOFs) model, which includes six DOFs for the powertrain and the subframe respectively. The mounts are simplified as a three-dimensional spring along each axis of its Local Mount Coordinate System (LMCS), which takes the non-linear relation of the force versus the displacement of each spring into account. Secondly, the quasi-static equilibrium equation and the free vibration equation as well as the forced vibration equation of the proposed model are derived and the solutions of equations are presented. Then, the calculation and solution methods are validated by the simulation results. The differences of rigid body modes and displacements of the powertrain between single and double stage isolation are estimated, which demonstrates that the proposed model is more accurate, especially when powertrain mounts are stiff. Also, the effect of locations for powertrain mounts on car body is investigated, which shows that is beneficial for motion control of powertrain.

A Computational Design Synthesis Method for the Generation of Rigid Origami Crease Patterns

Today most origami crease patterns employed in technical applications are selected from a handful of well-known origami principles. Computational algorithms capable of generating novel crease patterns either target artistic origami, focus on quadrilateral creased paper, or do not incorporate direct knowledge for the purposeful design of crease patterns tailored to engineering applications. The lack of computational methods for the generative design of crease patterns for engineering applications arises from a multitude of geometric complexities intrinsic to origami, such as rigid foldability and rigid body modes, many of which have been addressed by recent work of the authors. Based on these findings, in this paper we introduce a Computational Design Synthesis method for the generative design of novel crease patterns to develop origami concepts for engineering applications. The proposed method first generates crease pattern graphs through a graph grammar that automatically builds the kinematic model of the underlying origami and introduces constraints for rigid foldability. Then, the method enumerates all design alternatives that arise from the assignment of different rigid body modes to the internal vertices. These design alternatives are then automatically optimized and checked for intersection to satisfy the given design task. The proposed method is generic and applied here to two design tasks that are a rigidly foldable gripper and a rigidly foldable robotic arm.

Influence of Structural Stiffness and Loss Factor on Railroad Vehicle Comfort

This paper presents a new formulation for analyzing a beam on elastic supports traveling on irregular profiles. The model is a first approximation of a passenger railway vehicle car body. The main difference with previous works is the use of a complex modulus to represent structural damping rather than relying on equivalent viscous terms. The formulation groups rigid body modes with flexible modes and proposes a matrix form that is easy to interpret and solve in the frequency domain. Comfort indexes are readily obtained from weighted response spectral densities. The model is used to assess the influence of structural damping and stiffness on comfort. It will be shown that the evolution of comfort with stiffness is non-monotonic and, therefore, comfort does not always improve as stiffness increases.

Flight Dynamics Simulation Modeling of a Large Flexible Tiltrotor Aircraft

A high-order tiltrotor mathematical model is developed and validated against flight-test data for the XV-15 and simulations of a large civil tiltrotor (LCTR) concept. Rigid body and inflow states, as well as flexible wing and blade states are used in the analysis. Wing flexibility is important when modeling large aircraft where structural modes effect the frequency range of interest for flight control, generally 1–20 rad/s. Details of the formulation of the mathematical model are given, including derivation of structural, aerodynamic, and inertial loads. A novel “quasi-multibody” approach, based on numerical kinematics but without equations of constraints, allows the modeling of complex, flexible aircraft configurations in an easy to set up, and computationally very efficient manner. Assessments of the effects of wing flexibility are given. Flexibility effects are also evaluated by looking at the nature of the couplings between rigid body modes and wing structural modes.

Hidden symmetries generate rigid folding mechanisms in periodic origami

We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell–Calladine index theorem we derive a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origami’s vertices. This supports the recent result by Tachi [T. Tachi,Origami6, 97–108 (2015)] which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero-energy deformations in the bulk that may be used to reconfigure the origami sheet.

Generalised assumed strain curved shell finite elements (CSFE-sh) with shifted-Lagrange and applications on N-T’s shells theory

We present a simple methodology to design curved shell finite elements based on Nzengwa-Tagne’s shell equations. The element has three degrees of freedom at each node. The displacements field of the element satisfies the exact requirement of rigid body modes in a ‘shifted-Lagrange’ polynomial basis. The element is based on independent strain assumption insofar as it is allowed by the compatibility equations. The element developed herein is first validated on analysis of benchmark problems involving a standard shell with simply supported edges. Examples illustrating the accuracy improvement are included in the analysis. It showed that reasonably accurate results were obtained even when using fewer elements compared to other shell elements. The element is then used to analyse spherical roof structures. The distribution of the various components of deflection is obtained. Furthermore, the effect of introducing concentrated load on a cylindrical clamped ends structure is investigated. It is found that the CSFE3-sh element considered is a very good candidate for the analysis of general shell structures in engineering practice in which the ratio h/R ranges between 1/1000 and 2/5.

Consistent frequency-matching calibration procedure for electromechanical shunt absorbers

Electromagnetic and piezoelectric shunt damping can be represented by an equivalent mechanical vibration absorber with a spring, dashpot, and inerter in series. A common calibration procedure for multiple electromechanical shunt absorbers is obtained by this equivalence, including a correction for the influence from residual vibration modes by additional modal terms, whose coefficients are consistently identified by frequency matching for the system with rigid absorber dashpots. The accuracy and robustness of the calibration procedure are verified numerically for an unsupported truss structure with rigid body modes.

Influence of Blade Flexibility on the Dynamic Response Simulation of a Turbomolecular Pump on Magnetic Bearings

This paper proposes the simulation of a complete mechanical model of a turbomolecular pump rotor, including rotor and blades flexibility, suspended by controlled active magnetic bearings. The mechanical model is composed of an eight stage blisk, attached to a shaft. Magnetic forces are linearized by the first-order Taylor expansion around a given point. Including blades and rotor flexibility makes the mechanical system asymmetric, so the equations of motion for the coupled system have periodic terms. A modal controller was designed to control rigid body modes, since the number of sensors is limited and no state observer is implemented. PID controllers are used for low frequency modes combined with the second order filters to damp high frequency modes. First of all, stability analysis was carried out for the axisymmetric case. Second, blades flexibility was included. Forced response of the whole system to an impulsive force was studied. Divergent responses for the system in rotation were obtained as a second order filter pole possibly interacting with blades modes. Taking the second order filters off the control loop allowed the system to be stable. These results show that the analysis method developed here is efficient to evaluate the performance of a controller in closed loop with the complete flexible system. This method may be used in industrial design processes as computation times for the complete system are very short.

Influence of Blade Flexibility on the Dynamic Response Simulation of a Turbomolecular Pump on Magnetic Bearings

This paper proposes the simulation of a complete mechanical model of a turbomolecular pump rotor, including rotor and blades flexibility, suspended by controlled active magnetic bearings. The mechanical model is composed of an eight stage blisk, attached to a shaft. Magnetic forces are linearized by first order Taylor expansion around a given point. Including blades and rotor flexibility makes the mechanical system asymmetric, so the equations of motion for the coupled system have periodic terms. A modal controller was designed to control rigid body modes, since the number of sensors is limited and no state observer is implemented. PID controllers are used for low frequency modes combined with second order filters to damp high frequency modes. First of all, stability analysis was carried out for the axisymmetric case. Secondly, blades flexibility was included. Forced response of the whole system to an impulsive force was studied. Divergent responses for the system in rotation were obtained as a second order filter pole possibly interacts with blades modes. Taking second order filters off the control loop allowed the system to be stable. These results show that the analysis method developed here is efficient to evaluate the performance of a controller in closed loop with the complete flexible system. This method may be used in industrial design processes as computation times for the complete system are very short.