lecture 5

7/21/2019 Lecture 5

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S. Turteltaub

Ae 2135-II - 2015

Topic 5

Generally-forced vibrations of

undamped and damped,

single degree-of-freedom system

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S. Turteltaub

Ae 2135-II - 2015

General (non-periodic) loading

Forced vibrations

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S. Turteltaub

Ae 2135-II - 2015

• In the previous lecture it was shown how to find the response

function of a system loaded with a simple harmonic function(sine and/or cosine). The method relied on a decomposition

(homogeneous + particular parts) and the approach of

undetermined coefficients to find the particular solution.

• It is possible to extend the method of undetermined

coefficients for more general loading conditions when the

loading function is periodic in time (i.e., when the load repeats

itself in time after one loading period)

• Due to time constraints the general periodic loading is not

covered in this course[optional: read section 3.3 of textbook if you want to learn how to use it]

Methods for arbitrary loading

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S. Turteltaub

Ae 2135-II - 2015

• For general (non-periodic) loading, two methods can be used:

– Method of convolutions (“superposition”) – Laplace transform

Note: the methods of convolutions and transformations are typically only

applicable for relatively simple loading. In general one has to use numerical

methods for the solution of problems.

Methods for arbitrary loading

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S. Turteltaub

Ae 2135-II - 2015

Convolutions: impulsive load

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S. Turteltaub

Ae 2135-II - 2015

Idea behind method:

Method of convolutions

• Find the response to a basic impulse loading (Dirac delta)

• Express general loading f as a “sum” of basic impulsive

loadings

• Use superposition to construct the general response x (in the form of a convolution integral)

• Solve the integral to obtain the response function x

Remark: from the basic impulsive loading, it is also possible to find the responsefunction to some useful “elementary” loadings (step loading, ramp loading, etc.)

In turn, using again linearity, these “elementary” loadings can be used to

construct other more complex loadings made out of sums of steps and/or ramps

(= a “super-superposition” approach)

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S. Turteltaub

Ae 2135-II - 2015

Consider a mass-damper-spring SDOF system at rest until an

impulsive load f i is applied at time t = t *:

Method of convolutions: impulsive load

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S. Turteltaub

Ae 2135-II - 2015

Method of convolutions: impulsive loadConsider a mass-damper-spring SDOF system at rest until an


The impulse load is obtained formally

as a limit for an infinitesimally smalltime interval:

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S. Turteltaub

Ae 2135-II - 2015

Method of convolutions: impulsive loadConsider a mass-damper-spring SDOF system at rest until an




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S. Turteltaub

Ae 2135-II - 2015

Method of convolutions: impulsive loadThe impulsive load can be expressed formally using Dirac’s delta



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S. Turteltaub

Ae 2135-II - 2015

Method of convolutions: impulsive loadThe impulsive load can be expressed formally using Dirac’s delta

Impulse I of load f i:

Remark: the impulse has units of Force x

Time hence the term P0 has also units of

Force x Time. Observe that the Dirac delta

has units inverse to the units of its

argument (in this case units of 1/time)

P0 is the impulse of the short pulse

Net impulse

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S. Turteltaub

Ae 2135-II - 2015

Method of convolutions: impulsive loadResponse of a system initially at rest subjected to a pulse

Equation of motion

Goal:

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Ae 2135-II - 2015

Solution:

• Analyze first the effect of the

impulse (key aspect of solution)

• Solve then the problem after the

pulse has been applied


Equation of motion

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S. Turteltaub

Ae 2135-II - 2015

Integrate equation of motion

from to


Equation of motion

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S. Turteltaub

Ae 2135-II - 2015

Transfer of impulse to linear momentum:

(cf. perfectly elastic central collision)


Equation of motion

x(t *-)= x(t *+)=0: Continuous integrands

Rest at t *-

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S. Turteltaub

Ae 2135-II - 2015

Effect of impulsive force: the impulse is

transferred to the mass as an (initial)

linear momentum at time

The (initial) velocity becomes:


Equation of motion

The position at instants and

remains unchanged:

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S. Turteltaub

Ae 2135-II - 2015

After the pulse has been applied, the

problem reduces to a free-vibration with

initial conditions at


Equation of motion

Example: underdamped case:

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S. Turteltaub

Ae 2135-II - 2015


h(t -t *): Unit impulseresponse function at t *

Under-

damped

case:

Exercise: find analogous functions for cases

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S. Turteltaub

Ae 2135-II - 2015

Method of convolutions: superpositionConsider an underdamped system subjected to several impulses:

The total loading can be expressed as a sum

of impulse loadings and the total response

as a sum of the corresponding responses:

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Ae 2135-II - 2015


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Ae 2135-II - 2015


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S. Turteltaub

Ae 2135-II - 2015

Method of convolutions: superposition

The total response is obtained as

a sum with the proper “shift"

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Ae 2135-II - 2015


Add all impulsive

responses up to the

“current” time t c

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Ae 2135-II - 2015


Duhamel integral (convolution)

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S. Turteltaub

Ae 2135-II - 2015

Method of convolutions: argument

Duhamel integral (convolution)

Useful property: swap arguments in integrand (use only if

integrand is zero up to t = 0)

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S. Turteltaub

Ae 2135-II - 2015

Heaviside (step) function:

Method of convolutions: Step loading

(derivative in the sense of distributions)

Step loading:

Properties:

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S. Turteltaub

Ae 2135-II - 2015

Find response in underdamped mass-damper-spring system


Use alternative form of Duhamel’s convolution

Recall that t is “fixed”, t > t * and the

variable τ ranges from 0 to t

The function H is equal to 1 if its

argument is positive and 0 otherwise:

if Relevant limits of integration

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S. Turteltaub

Ae 2135-II - 2015



Recall for the underdamped case that the response function to the

unit impulse is

Substitute in integral,

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S. Turteltaub

Ae 2135-II - 2015



Carry out the integration (use complex variable methods,

integration by parts, use Mathematica or consult a “table of

integrals” (as provided in the exam)

Use formula with

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S. Turteltaub

Ae 2135-II - 2015



Simplify expression

Evaluate expression and use

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S. Turteltaub

Ae 2135-II - 2015

Method of convolutions: Step loadingUnderdamped mass-damper-spring system

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S. TurteltaubAe 2135-II - 2015

A convenient way to express the response of an underdamped

mass-damper-spring system initially at rest subjected to a step

loading is as follows:


where hs is the response to a unit step loading (Heaviside

loading) at for times t > t *

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A convenient way to express the response of an underdamped

mass-damper-spring system initially at rest subjected to a step

loading is as follows:

Method of convolutions: SUPER-Position

Application: square pulse

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Laplace transform

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Laplace transform: overview

Original problem

(ODE)

(Real) time domain

Laplace transform

(Complex) frequency domain

Transformed problem

(Algebraic equation)

Simple

Solution of

transformed problem

Difficult

Solution of original

problem Inverse transform

Less difficult

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Laplace transformDefinition of Laplace transformation

In this course: tables for transformations of relevant functions will be provided

s: complex frequency

Inverse transformation

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Important (forward) transformations used to transform the

original ODE problem:

1. Transformations for derivatives of the unknown function x

2. Transformations for typical functions that describe the

loading f

The unknown function x is simply formally transformed as X and

left as an unknown in the new (transformed) problem

Laplace transform

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The transformation of a derivative of a given function f can be

computed using integration by parts assuming that the function

being transform does not grow to infinity in the limit as time

tends to infinity:

Laplace transform: derivatives

If limit of f is finite

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Second derivative: same procedure (apply integration by parts

twice)

Laplace transform: derivatives

Higher order derivatives are typically not required in vibrations(equation of motion is second-order)

Application to homogeneous problem:

Initial conditions are accounted

for in transformed problem

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The differential equation for x has been transformed into an

algebraic equation for the transformed unknown X

Laplace transform: homogeneous case

Solve for X :

Last step: inverse transform to find the actual response function

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Last step: inverse transform to find the actual response function

Application to non-homogeneous problem:

Laplace transform: non-homogeneous

Solve for X :

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Relevant functions and important relations

Laplace transform: common functions

A more complete table will be provided

(Step)

(Dirac)

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The method of convolutions (superposition) was used to solve

the problem of an underdamped system initially at rest loaded

with a square pulse. It can also be solved using the Laplace

transform

Laplace transform: square pulse revisited

Square pulse loading:

Transformed loading:

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Use in non-homogeneous problem


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Use in non-homogeneous problem


To compute the inverse transform x of a function X using a table

of transformations, it is often necessary to decompose the X in a

sum of terms that can be transformed more easily

The most commonly-used method is “partial fractions”

Example: underdamped system subjected to a square pulse

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Underdamped system subjected to a square pulse

Laplace transform: partial fractions

1. Separate exponential terms:

2. Factor denominator: Underdamped:

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1. Separate exponential terms:

2. Factor denominator: Underdamped:

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Underdamped:

3. Use partial fractions:

Find coefficients c0, c1, c2, d 1, d 2

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Underdamped:

3. Use partial fractions: (coefficients)

Note: It can be shown

that the coefficients

c1, c2 and d 1, d 2 are

complex conjugates

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3. Use partial fractions:

Underdamped:

(Coefficients computed in previous slide)

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4. Look-up inverse transforms and combine as required:

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Note: Coefficients c1, c2 and d 1, d 2 are complex conjugates hence x(t ) is real

Response function in terms of complex exponentials

f f

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l f

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Summary of method:

1. Transform all terms containing derivatives (this step will also

take care of the initial conditions)

2. Transform the loading term (crucial aspect: express the

loading in a convenient way! Dirac delta and/or step function

are quite useful at this stage)

3. Solve for the transformed response function and express in aconvenient way to transform back

4. Perform the inverse transform to get the solution (use tables,

a symbolic manipulator or compute it yourself)

Laplace transform

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