Mechanical Vibrations Chapter 8

Transcription

1 Mechanical Vibrations Chapter 8 Peter Avitabile Mechanical Engineering Department University of Massachusetts Lowell 1 Dr. Peter Avitabile

2 Numerical Methods - Direct Integration The equation of motion is integrated using numerical step-by-step procedure for a number of t steps. The term 'direct' means that the equations of motion are formulated in a physical space at Ndof without any transformation to another space (ie,modal space). The equation of motion is to be satisfied not at any time t but rather an equivalent 'static' equilibrium is sought at discrete time intervals t apart. Note that from this statement, the static solution techniques will be employed for the system at many different t time steps. In essence, 'effective' loads are computed from the manipulation of the velocity and acceleration terms of the equation of motion to reduce the problem to a simple-to-solve static equivalent problem. 2 Dr. Peter Avitabile

3 Numerical Methods - Direct Integration Basically, numerical integration is a process of marching along in time where response parameters (acceleration, velocity and displacement) at time t are evaluated from their known historic values. Typically, three values are needed for three unknowns. Two of these values are derived from assumptions regarding the manner in which response parameters vary during a time step. The third equation is the equation of motion written at a selected point. An important aspect of numerical integration is the selection of the time step used in the integration process. If the time step is selected to be too large then the computed response will suffer from the effects of 'numerical damping' that will distort the results even the scheme is unconditionally stable. 3 Dr. Peter Avitabile

4 Numerical Methods - Schemes Explicit Schemes scheme is explicit when the equation of motion is written at time t (which is the current time) computationally efficient when compared to implicit schemes these techniques are only conditionally stable typically there is no factorization of [K] or [M] needed for most cases Implicit Schemes scheme is implicit when the equation of motion is written at next time step (which is t + t) requires more computation when compared to explicit schemes these techniques are usually unconditionally stable usually factorization of [K] or [M] is needed for most cases 4 Dr. Peter Avitabile

5 Numerical Methods - Schemes Some common integration techniques Central diff Explicit Houbolt Implicit Wilson θ Implicit Newmark Implicit γ=0, β=0 (constant acceleration) Newmark Implicit γ=1/2, β=1/4 (average acceleration) Newmark Implicit γ=1/2, β=1/6 (linear acceleration) 5 Dr. Peter Avitabile

6 Numerical Methods - Central Difference Source: UMASS Lowell Structural Dynamic Modeling Techniques Seminar Notes 6 Dr. Peter Avitabile

7 Numerical Methods - Central & Houbolt Source: UMASS Lowell Structural Dynamic Modeling Techniques Seminar Notes 7 Dr. Peter Avitabile

8 Numerical Methods - Wilson Source: UMASS Lowell Structural Dynamic Modeling Techniques Seminar Notes 8 Dr. Peter Avitabile

9 Numerical Methods - Comparison Source: Bathe - Numerical Methods in Finite Element Analysis 9 Dr. Peter Avitabile