Structural dynamic modification of circular disc using pre-stressed fields

Abstract

Dynamical properties of circular discs are investigated in this paper. One of the techniques of the disc modifications to achieve the required dynamic properties is to initiate pre-stress in disc plane. To obtain appropriate in-plane stress either roll-tensioning of disc surface or volume transformation of disc segment can be used. The role of in-plane stresses is assessed from the change in natural frequencies and modal shapes. The natural frequency characteristics for various rolling position and various rolling depth of the annulus are obtained by modal analysis using Finite Element Method (FEM). The fundamental considerations and derivation of equations of motion are based on Kirchhoff 's assumptions. The assumptions are valid only for thin circular disc. The field of displacements in the cylindrical coordinates r, φ, z, using Kirchhoff plate theory, can be written as˜ r ∂φwhere: u(r, φ), v(r, φ), w(r, φ) -displacements of point laying on neutral plane of the circular disc in coordinate directions.Generally, the stress-strain relations under consideration of initial stresses and initial strains are given bywhere: σ, ε -stress and strain vector, σ 0, ε 0 -initial stress and initial strain vector, D -elasticity matrix.Using the finite element formulation (ZIENKIE-WICZ, TAYLOR 2000), the equation of motion for a free vibration of in-plane stressed disc is described by expressionwhere: M -mass matrix, K -tiffness matrix, K σ -stiffness matrix resulting from stress distribution induced by rolling, ü, u -vector of nodal accelerations and vector of nodal displacements, respectively.We note, that the mass distribution of circular disc after rolling is not changed, but the bending stiffness is considerably changed.Equations (3) can be transformed to modal coordinates using the transformation equationswhere: φ -modal vector, q(t) -normal coordinate of the system.After applying the above transformation, the equation of motion (3) can be used to determination of the natural angular frequencies and mode shapes of the circular disc with roll-tensioning induced residual stress distribution. We obtain the following eigenvalue problemwhere: In order to calculate the modification of disc stiffness K σ after rolling, we must know the residual stress distribution in a disc plane. To determine the residual stress distribution, the method of thermal stress loading is used (KURATANI, YANO 2000). The thermal expansion induces a stress distribution, which is analogous to the stress distribution initiated by rolling. The dependence between temperature and depth of roll-tensioning is approximately described by equationwhere: µ -Poisson number, α -temperature expansion coefficient, h -disc thickness, ∆z -depth of roll-tensioning. 81The matrices M, K and additional matrix K σ , which follow from stress distribution arising from rolling (in this model analogy with thermal expansion is used), are calculated automatically by ANSYS (ANSYS 2001). Solution processes for determination of natural angular frequencies and modal shapes ...

Structural dynamic modification of an active suspended handle with a parallel coupled piezo stack actuator

“…For example, the response of a damped mechanical system was improved by shifting the natural frequencies of the structure to the desired value. 2 Parameters of modification included the mass, 3 the stiffness, 4,5 the damping, 5,6 the material 3 and the application of dynamic vibration absorber (DVA). 7 For example, the constraint damping layer and stiffeners have been added to reduce vehicle drum brake squeal 6 while DVA can be applied to reduce the vibration transmissibility of the grass trimmer handle by up to 67%.…”