Elastodynamics of Planar Mechanisms Using Planar Actual Finite Line Elements, Lumped Mass Systems, Matrix-Exponential Method, and the Method of “Critical-Geometry-Kineto-Elasto-Statics” (CGKES)

Abstract

The article presents a general method for the elastodynamic analysis of planar mechanisms. It uses planar actual finite line elements (regular and irregular elements given in a companion article) and lumped mass systems to formulate the equations of motion of a mechanism. Damping coefficient matrix can incorporate time dependent viscous or coulomb damping coefficients in addition to the coefficients of velocity dependent internal damping. The forcing vector can incorporate any externally applied time dependent force or torque, inertial forces and inertial torques, any nonlinear viscous or Coulomb damping forces and torques. The matrix exponential method is introduced for the numerical solution of the equations of motion. Matrix displacement method of determining dynamic stresses using the generalized coordinate displacements is given. Elastodynamic analysis of a plane four-bar mechanism is performed for several cycles of kinematic motion, and the dynamic stresses are compared with those obtained by experiments. The method of “Critical-Geometry-Kineto-Elasto-Statics” (CGKES) is proposed for the computation of dynamic stress magnitudes making use of the critical geometry of the mechanism. It requires the analysis of a mechanism at the critical geometry position of the mechanism which is defined by the lowest fundamental frequency of the mechanism. The results predicted by the method of CGKES compares within two percent with the experimental results.

Analysis method of dynamics and structure optimization for sealing mechanism under reducing vibration condition

“…In this paper, Newmark method is used to solve the dynamic equation of sealing system. The detailed calculation process is as follows: 26 Obtain the related coefficient matrix ( M , C, K ) of system kinetic equation and initial state vectors of system Ut=t0, U·t=t0, and U··t=t0 through analyzing the shear mechanism. Select a proper time step Δt and solving parameters μ 1 , μ 2 , and calculate the following solving parameters: a 0 = 1/( μ 2 Δ t 2 ), a 1 = μ 1 /( μ 2 Δ t ), a 2 =1/( μ 2 Δ t ), a 3 = 1/(2 μ 2 ) − 1, a 4 = μ 1 / μ 2 − 1, a 5 = μ 1 Δ t /(2 μ 2 ) − Δ t , a 6 = Δ t (1 − μ 1 ), and a 7 = μ 1 Δ t . Calculate the effective rigid matrix of the system: K··=K+a0a1C. Calculate effective load of the system at t + Δt Calculate the elastic motion parameters of the sealing mechanism system …”

“…Select a proper time step Δt and solving parameters μ 1 , μ 2 , and calculate the following solving parameters: a 0 = 1/( μ 2 Δ t 2 ), a 1 = μ 1 /( μ 2 Δ t ), a 2 =1/( μ 2 Δ t ), a 3 = 1/(2 μ 2 ) − 1, a 4 = μ 1 / μ 2 − 1, a 5 = μ 1 Δ t /(2 μ 2 ) − Δ t , a 6 = Δ t (1 − μ 1 ), and a 7 = μ 1 Δ t .…”

“…A and B refer to unit number. 1, (2), and 3refer to component number. Sealing mechanism element finite element model is expressed in Figure 6.…”

“…According to the characteristics of the object, the finite element method is used to analyze the mechanical structure dynamics. In the early eighties, Bagci and Kalaycioglu 2 used the finite element method, centralized quality method, the matrix exponential method, and the key geometric motion elasticity method to analyze the elastic dynamics of the plane mechanism, but they did not further explore the high-speed elastic link mechanism in the course of the dynamic stress changes. Midha et al.…”

Two-Dimensional Equivalent Models in the Analysis of a Multibody Elastic System Using the Finite Element Analysis

“…Classical methods applied in the MBS are described in [26,27]. A high-precision formulation for a 3D beam element is presented in [28], and the mathematical methods used to study such problems are studied in [29,30].…”