C.D. Mote

Classical Vibration Analysis of Axially Moving Continua

The equations of motion are cast in a canonical state space form defined by one symmetric and one skew-symmetric differential operator. When an equation of motion is represented in this form, the eigenfunctions are orthogonal with respect to each operator. Following this formulation, a classical vibration theory, comprised of a modal analysis and a Greens function method, is derived for the class of axially moving continua

Comments on curve veering in eigenvalue problems

Abstract The dependence of eigenvalues on a system parameter is frequently illustrated by a family of loci. When two loci approach each other, they often cross or abruptly diverge. The latter case, called “curve veering”, has been observed in approximate solutions associated with discretized models. The influence of discretization in producing curve veering has raised doubt on the validity of many approximate solutions. The existence of curve veering in continuous models is illustrated by presenting the exact solution of an elementary eigenvalue problem. Veering is then examined in a general eigenvalue problem. Criteria are established to distinguish veerings from crossings in both continuous and discretized models. The application of the criteria is illustrated by examples.

Vibration Control of an Axially Moving String by Boundary Control

The stabilization of the transverse vibration ofan axially moving string is implemented using time-varying control of either the boundary transverse motion or the external boundary forces. The total mechanical energy of the translating string is a Lyapunov functional and boundary control laws are designed to dissipate the total vibration energy of the string at the left and/or right boundary. An optimal feedback gain determined by minimizing the energy reflected from the boundaries, is the ratio of tension to the propagation velocity of an incident wave to the boundary control. Also the maximum time required to stabilize all vibration energy of the system for any initial disturbance is the time required for a wave to propagate the span of the string before hitting boundary control. Asymptotic and exponential stability of the axially moving string under boundary control are verified analytically through the decay rate of the energy norm and the use of semigroup theory. Simulations are used to verify the theoretically predicted, optimal boundary control for the stabilization of the translating string.

Three-dimensional vibration of travelling elastic cables

A cable dynamics theory is derived for travelling cables having arbitrary initial sag and arbitrary support eyelet elevations. The theory incorporates the effect of cable elasticity through a finite strain model. Hamiltons principle provides three non-linear equations which describe the three-dimensional motion of the cable. The equations are linearized about a known equilibrium configuration. Natural frequencies, mode shapes and stability of equilibrium are predicted from the eigensolutions of the discretized model over a range of cable designs. Results of previous elastic cable theories are re-examined and new findings are presented.

Active Vibration Control of the Axially Moving String in the S Domain

A new method is presented for active vibration control of the axially moving string, one of the most common models of axially moving continua. The control is formulated in the Laplace transform domain. The transfer function of a closed-loop system, consisting of the plant, a feedback control law and the dynamics of the sensing and actuation devices, is derived. Analysis of the root loci of the closedloop system gives two stability criteria. Stabilizing controller design is carried out of both collocation and noncollocation of the sensor and actuator. It is found that all the modes of vibration can be stabilized and that in principle the spillover instability can be avoided. Also, the steady-state response of the stabilized string to periodic, external excitation is presented in closed form.

Vibration and parametric excitation in asymmetric circular plates under moving loads

Abstract The natural frequencies and modes of transverse vibration of circular plates containing small imperfections are determined through a perturbation method. Incision of equally spaced, equal-size radial slots at the rim of the plate creates asymmetry in some, but not all, of the vibration modes, and it causes the repeated natural frequencies of these modes in the symmetric plate to split into two distinct values. These vibration modes are called the split modes , and those associated with the repeated natural frequencies are called the repeated modes . A relationship identifying the split and repeated modes for any configuration of slots is presented. The vibration of a plate containing any number of thin slots cut into it at the rim and with any number of rotating linear springs is analyzed. Parametric instability can be excited in the split modes of the plate by the springs rotating below critical speed, but it cannot be excited in the repeated modes. The response of the plate in forms such as traveling or standing waves at parametric resonance is discussed. The theoretical predictions of split and repeated vibration modes and of the excitation of parametric instability are confirmed by experiments.

The maximum controlled follower force on a free-free beam carrying a concentrated mass

Abstract A uniform, free-free, Euler-Bernoulli beam, transporting a concentrated mass with rotary and transverse inertia, is driven by a follower force with controlled direction. A finite element model of the beam transverse motion in the plane is formulated through the extended Hamiltons principle. The stability of the model is investigated with respect to (i) the axial location and the inertia of the concentrated mass, (ii) the location of the follower force direction control sensor, (iii) the sensor gain, and (iv) the magnitude of the constant follower force. Both divergence and flutter instabilities can occur over the range of beam models examined. The analysis predicts the location and the magnitude of the additional mass, and the location and the gain of the follower force direction sensor that permits the follower force magnitude to be maximized for stable transverse motion of the beam.

Vibration coupling analysis of band/wheel mechanical systems

Measurements of vibration on continuous bands driven by rotating wheels show coupling occurs between vibration of the band spans and wheels. Significant error in the predicted vibration spectrum and the response of a span can occur if the coupling is neglected. An analytical model describing the vibration of the band/wheel system is presented. When the equilibrium curvatures of the band spans are finite, the transverse motion of the spans are linearly coupled to their longitudinal motions and to the oscillation of the wheels. The analytical predictions are validated by comparisons with experimental observations. The importance of the band tension, the band transport speed, the wheel inertias and radii, and the wheel support stiffnesses to the coupling are discussed. The model can be utilized in analysis of active and passive vibration control of band/wheel systems.

On the mechanisms of instability of a circular plate under a rotating spring-mass-dashpot system

Abstract The response of a stationary, undamped, asymmetric, classical, circular plate subjected to a rotating spring-mass-dashpot system is formulated in a co-ordinate system fixed to the plate. Use of an eigenfunction expansion reduces the equation of motion to a set of coupled Hills equations. Techniques for analytical solution of the Hills equations are illustrated on axisymmetric plates. It is found that a spring rotating at supercritical speed will parametrically excite the plate to single mode and combination resonances. A rotating mass will destabilize the plate above a speed greater than the critical speed at which the inertia force of the rotating mass exceeds the elastic restoring force of the plate. A viscous dashpot destabilizes the plate when it rotates at supercritical speed by negatively damping the backward-travelling wave components of the plate eigenmodes.

Moving‐load stability of a circular plate on a floating central collar

The eigenvalue problem and transverse response of a circular plate, that is free at the periphery and that slides freely along the axis of symmetry without bending rotation, are theoretically analyzed. The occurance of eigenvalues in the boundary conditions is accounted for with an extended operator definition in the equation of transverse motion. The stability of these plates under concentrated loads moving at uniform speed is analyzed for (i) harmonic transverse loading and (ii) loading proportional to transverse displacement and velocity. The harmonic loading case leads to a classical, critical‐speed analysis. The proportional loading case represents the excitation of the plate by transverse position guides. The number, orientation, and mechanical properties of the guides determine the transverse stability of the plate‐guide dynamic system.