Subharmonic Resonance Cascades in a Class of Coupled Resonators

We consider a chain of N nonlinear resonators with natural frequency ratios of approximately 2:1 along the chain and weak nonlinear coupling of a form that allows energy to flow between resonators. Specifically, the coupling is such that the response of one resonator parametrically excites the next resonator in the chain, and also creates a resonant backaction on the previous resonator in the chain. This class of systems, which is being proposed for micro-electro-mechanical frequency dividers, is shown to have rich dynamical behavior. Of particular interest is the case when the high frequency end of the chain is resonantly excited, and coupling results in the potential for a cascade of sub-harmonic bifurcations down the chain. When the entire chain is activated, that is, when all N resonators have non-zero amplitudes, if the input frequency on the first resonator is Ω, then the terminal resonator responds with frequency Ω/2N. The details of the activation depend on the strength and frequency of the input, the level of resonator dissipation, and the mistuning in the chain. In this paper we present analytical results, based on perturbation methods, which provide useful predictions about these responses in terms of system and input parameters. Parameter conditions for activation of the entire chain are derived, along with results about other phenomena, such as bistability and partial activation of the chain. We demonstrate the utility of the predictive results by direct comparison with simulations of the equations of motion, and we also present samples of mechanical and electromechanical systems that realize the desired properties. These results will be useful for the design and operation of mechanical frequency dividers based on subharmonic resonances.

Advanced System Identification of Plates Using a Higher-Order-Spectral Approach: Theory and Experiment

A nonlinear system identification technique exploiting the dynamic response features of fully nonlinear physics-based plate models extracted by Higher-Order Spectral (HOS) analysis tools is developed. The changes induced by an imperfection in the dynamics through the structural nonlinearities are used as key detection mechanism. The differences in dynamic response of a baseline and a modified/imperfect structure are enhanced by the local nonlinearities induced by the structural modification which thus represent the specific objective of identification. The validation of the procedure and the developed algorithms is carried out through extensive experimental testing employing various plates, including isotropic and composite lay-ups, and excitation sources, including White Gaussian Noise and a train of impulses.

Synchronization of Mechanical Oscillators: An Experimental Study

In this paper we investigate synchronization of oscillators. We use mechanical metronomes that are coupled through a mechanical medium. In passive coupling of two oscillators, the coupling medium is a one degree of freedom passive mechanical basis. The analysis of the system is shown and supported by simulations of the proposed model and experimental results. We show how the oscillators synchronize and discuss the affecting parameters in synchronization. In another case, the oscillators are forced by an external input while that input is also affected by the oscillators. This feedback loop introduces dynamics to the whole system. For this case, we place the mechanical metronomes on a one degree of freedom moving base. The movements of the base are a function of a feedback from the phases of the metronomes. We study the space of possibilities for the movements of the base and consider impacts of the base movement on the synchronization of metronomes. We also show how such a system evolves in time when we introduce an adjusting parameter that changes over time and updates based on feedbacks from the system.

Trajectory Tracking of Underactuated Multibody Systems

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

Efficient Sensitivity Analysis for Multibody Dynamics Systems Using an Iterative Steps Method With Application in Topology Optimization

Efficient and reliable sensitivity analyses are critical for topology optimization, especially for multibody dynamics systems, because of the large number of design variables and the complexities and expense in solving the state equations. This research addresses a general and efficient sensitivity analysis method for topology optimization with design objectives associated with time dependent dynamics responses of multibody dynamics systems that include nonlinear geometric effects associated with large translational and rotational motions. An iterative sensitivity analysis relation is proposed, based on typical finite difference methods for the differential algebraic equations (DAEs). These iterative equations can be simplified for specific cases to obtain more efficient sensitivity analysis methods. Since finite difference methods are general and widely used, the iterative sensitivity analysis is also applicable to various numerical solution approaches. The proposed sensitivity analysis method is demonstrated using a truss structure topology optimization problem with consideration of the dynamic response including large translational and rotational motions. The topology optimization problem of the general truss structure is formulated using the SIMP (Simply Isotropic Material with Penalization) assumption for the design variables associated with each truss member. It is shown that the proposed iterative steps sensitivity analysis method is both reliable and efficient.

Simulation of Differentials in Four-Wheel Drive Vehicles Using Multibody Dynamics

The dynamic performance of vehicle drivetrains is significantly influenced by differentials which are subjected to complex phenomena. In this paper, detailed models of TORSEN differentials are presented using a flexible multibody simulation approach, based on the nonlinear finite element method. A central and a front TORSEN differential have been studied and the numerical results have been compared with experimental data obtained on test bench. The models are composed of several rigid and flexible bodies mainly constrainted by flexible gear pair joints and contact conditions. The three differentials of a four wheel drive vehicle have been assembled in a full drivetrain in a simplified vehicle model with modeling of driveshafts and tires. These simulations enable to observe the four working modes of the differentials with a good accuracy.

On the Application of the Lu-Gre Model to Simulate Joint Friction in Multi-Body Systems

Advances in the application of multi-body simulation technology to real world problems have led to an ever increasing demand for higher fidelity modeling techniques. Of these, accurate modeling of friction is of strategic interest in applications such as control system design, automotive suspension analysis, robotics etc. Joints (sometimes referred to as constraints) play an important role in defining the dynamics of a multi-body system. Hence, it is imperative to accurately account for friction forces arising at these joints due to the underlying interface dynamics. In this paper, we discuss the application of LuGre, a dynamic friction model to simulate joint friction. We choose the LuGre model due to its ability to capture important effects such as the Stribeck effect and the Dahl effect. The native 1-d LuGre model is extended to formulate friction computations for non-trivial joint geometries and dynamics in 2 and 3 dimensions. It is also extended to work in the quasi-static regime. Specific applications to revolute, cylindrical and spherical joints in multi-body systems are discussed. Finally, an engineering case study on the effects of joint friction in automotive suspension analysis is presented.

Comfort and Handling of a Commercial Vehicle With Individual Front Suspension

The main focus in the presented work is on the sensitivity analysis of the comfort and handling characteristics of a commercial vehicle with Individual Front Suspension (IFS). For the sensitivity analysis, two simulations in frequency domain are conducted: road input and steering input frequency responses. Employing the model of the tractor semitrailer combination for the above mentioned analyses, this study evaluates the influence of five damping coefficients — cab, front and rear axle shock absorbers — on our objectives. The results that are provided in Pareto fronts clearly show the great influence of the studied parameters, particularly the front axle and cab lateral dampers, on the vehicle comfort and handling for the performed simulations. Finally, the model is updated with the chassis and cab lateral damper coefficients to run a simulation on random roads for more detailed comfort examinations. This analysis confirms the obtained improvements in the outcome of the sensitivity study.

State Dependent Riccati Equation (SDRE) Control Method Applied in Cancellation of a Parametric Resonance in a Magnetically Levitated Body

Here, a simplified dynamical model of a magnetically levitated body is considered. The origin of an inertial Cartesian reference frame is set at the pivot point of the pendulum on the levitated body in its static equilibrium state (ie, the gap between the magnet on the base and the magnet on the body, in this state). The governing equations of motion has been derived and the characteristic feature of the strategy is the exploitation of the nonlinear effect of the inertial force associated, with the motion of a pendulum-type vibration absorber driven, by an appropriate control torque [4]. In the present paper, we analyzed the nonlinear dynamics of problem, discussed the energy transfer between the main system and the pendulum in time, and developed State Dependent Riccati Equation (SDRE) control design to reducing the unstable oscillatory movement of the magnetically levitated body to a stable fixed point. The simulations results showed the effectiveness of the (SDRE) control design.

The Selection of the Linearized Natural Frequency for the Second-Order Normal Form Method

The normal form technique is an established method for analysing weakly nonlinear vibrating systems. It involves applying a simplifying nonlinear transform to the first-order representation of the equations of motion. In this paper we consider the normal form technique applied to a forced nonlinear system with the equations of motion expressed in second-order form. Specifically we consider the selection of the linearised natural frequencies on the accuracy of the normal form prediction of sub- and superharmonic responses. Using the second-order formulation offers specific advantages in terms of modeling lightly damped nonlinear dynamic response. In the second-order version of the normal form, one of the approximations made during the process is to assume that the linear natural frequency for each mode may be replaced with the response frequencies. Here we will show that this step, far from reducing the accuracy of the technique, does not affect the accuracy of the predicted response at the forcing frequency and actually improves the predictions of sub and superharmonic responses. To gain insight into why this is the case, we consider the Duffing oscillator. The results show that the second-order approach gives an intuitive model of the nonlinear dynamic response which can be applied to engineering applications with weakly nonlinear characteristics.