Transient Vibration of SDOF Systems

Contents

Introduction

Impulse excitation

Arbitrary excitation

Laplace transform formulation

Pulse excitation and rise time

Shock response spectrum

Finite difference numerical computation

Introduction

Transient vibration is defined as a temporarily sustained vibration of a mechanical system. It may consist of forced or free vibrations, or both.

Examples:-Forging hammer, bump on road

Impulse excitation

Impulse is the time integral of force.

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Now, a sudden change in velocity is equal to the actual applied input divided by the force.

Response to unit impulse

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Similary for damping case,

Arbitrary excitation

Using the unit impulse response function, the response due to arbitrary loadings can be determined.Arbitrary force is considered as a series of impulses.

This integral is called Convolution integral and also called as superposition integral

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Q. Use the convolution integral to determine the response of an undamped 1DOF system of natural frequency and mass m when subject to a constant force of magnitude F0. The system is at rest in equillibrium at t=0

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Q. Same problem but with damping factor. Find the response.Q. Same problem without damping and force is

Laplace Transfrom Formulation

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Pulse Excitation and Rise Time

If a force F0 is not applied untill t0, it can be expressed using a delayed unit step function:-

Eg.Use the unit step function to write a unified mathematical expression for each of the forces

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Q. Find the response for the triangular pulse as shown:-

F0

t0/2 t0

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Q. Develop a mathematical expression for the excitation using unit step function

Shock Response Spectrum

Plot of maximum response(s,v,a) of the system against the natural frequency of the system is called Shock Response Spectrum for that particular disturbance.Useful for the designer to know the maximum response of a structure as a function of system parameters.

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Q. Determine the Shock Spectrum of a rectangular pulse.

Shock Isolation

For shock isolation, peak response must be less than unity.

Finite Difference Numerical Computation

Runge Kutta method:- A method to solve ODE. Most commonly used method is 4th order approximation Runge Kutta method.

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End of Chapter