Dynamics Primer Lectures Dermot O’Dwyer

Objectives Need some theory to inderstand general dynamics Need more theory understand the implementation of some of the analytical approaches Theory can answer some fundamental questions Try to make the presentation very applied

Topics Simple Harmonic Motion Multi-degree of freedom systems Free vibration and eigen values and vectors Modal Analysis Time domain analyses Frequency domain

Simple Harmonic Motion - I Typical of many structural systems Force is linear function of displacement Therefore,

Simple Harmonic Motion - II Which has the form Where

Simple Harmonic Motion - III

Simple Harmonic Motion - IV The basic homogeneous undamped differential equation Has the following closed form solution Where is the initial displacement and Is the initial velocity

Multi-Degree of Freedom Systems Topic II

Multi-Degree of Freedom Systems - I Most dynamic systems require more than a single degree of freedom to describe them The general equation that describes the motion of a multi-degreee of freedom system is

Multi-Degree of Freedom Systems - II Consider the term within the fundamental differential equation - Stiffness Matrix - Mass Matrix - Damping Matrix - Forcing function - Displacment vector

Multi-Degree of Freedom Systems - III Homogeneous equation – free response Calculate free response to an impulse Calculate natural frequencies Calculate eigen vectors

Natural Response Topic III

Free Vibration I The response of a dynamic system in the absence of external forces is very important because the free response of a system gives very useful information about the system’s characteristics –Natural Frequencies & Resonance –Eigen Vectors – modal analysis

Free Vibration II Finding the Natural Frequencies of a dynamic system Solve the undamped equation This equates to finding the values for omega for which the determinate of the above equation is zero

Resonance

Free Vibration III Once the natural frequencies of the system have been calculated, the eigen values can be found by substituting the known values of omega back into the equation

Eigen Vectors - 3 DOF

Modal Analysis Topic IV

Orthogonality of Eigen Vectors The Eigen vectors of a system are orthogonal with respect to the stiffness and mass matrices Where

Modal Analysis - I The displaced shape of a system can be described in terms of its Eigen vectors (natrual modes) Thus, where

Modal Analysis - II In a similar manner the dynamic response of a multi degree of freedom system can be represented in terms of the mode shapes Advantages - Significant problem reduction - Identify potential resonance - Calculate peak effects Disadvantages - None unless modes are removed

Time Stepping Topic V

Calculating the forced response of a dynamic system There are a number of techniques –In limited cases a closed form solution is available Fryba’s solution Response of a single degree of freedom system to a harmonic forcing function –Numerical Integration schemes Time-stepping algorithms Du Hammel’s Integral

Time Stepping - I If the position, velocity and acceleration of a dynamic system is known at t, then the state of the system an instant later, at, can be calculated

Time Stepping - II In the general multi-degree-of-freedom case the fundamental equation can be rearranged into the following form

Time Stepping - III Need figure of acceleration versus Note potential problems i.e. the difficulty with long time intervals Advantages include ability to deal with non-linear systems There are numerous time-stepping algorithms –Runge Kutta –Wilson Theta –Newmark Beta Stable non-stable

Runge-Kutta - I The Runge-Kutta algorithm proceeds by introducing a dummy variable If the value of x i and y i are known then a Taylor’s expansion can be used to calculate their values a short time later

Runge-Kutta - II In practice the Taylor series is truncated after the first term And an aveage value for the first term is used, thus The average values are calculate using Simpson’s rule

Runge-Kutta – III Integration table

Potential Time-Stepping Problems Duration of the time-step –Should be less than one tenth of period Enormous quantities of Data