Response of linear SDOF to arbitrary excitation

Objective: Learn how to find the response of

a linear SDOF system to a given input

(excitation)

Preliminary definitions:

Response = natural response + forced response

Natural response: solution of equation of

motion of the system when the excitation is

zero. The expression for natural response

contains constants.

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Forced response: any solution of equation of

motion of the system for non zero excitation.

If the natural response tends to zero when

time tends to infinity and the limit of the

forced response as time goes to infinity

exists and is bounded (not infinite), then the

limit is called steady state response.

Transient response: Process of going from

initial state to steady state.

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Transient response is due to both the

application of the force and the non zero

initial conditions

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Transient responseSteady state

Outline of this chapter

1.Impulse response function

2.Response to arbitrary excitation

3.Shock spectrum

4.Numerical calculation of response

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1. Impulse response function

Impulse function: idealization of short-

duration force applied suddenly

Force

time

Force

time

)(ˆ tFArea=

Idealization

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Impulse response

Find response of single degree of freedom system to a unit impulse:

Case 1: <1

If impulse was applied at time, τ, then

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m

kc

h(t)δ(t)

Impulse responses of two systems with natural frequency 6.28 rad/sec and damping ratios 0.1

and 0.8.

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Slope is 1/m here

Observations:

Transient response dies out faster when

damping increases.

Displacement overshoot decreases with

damping.

Maximum displacement does not occur

exactly at time equal to one fourth of a

period, unless damping is zero.

Slope just after impulse has been

applied (i.e. at t=0+) is 1/m.

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Case 2: Overdamped system, >1

Case 3: Critically damped system, =1

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2. Response to arbitrary excitation

First, assume that system is at rest at t=0.

Idea: Use superposition principle. Split

excitation to sum of impulses. Find

response to each impulse and sum up the

responses.

Equivalent equation for response:

The above are called convolution integrals.

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If system is not at rest at t=0, then the

response is the free vibration response due

the non zero initial conditions plus the above

convolution integral

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Step response

Consider underdamped systems only.

Response to unit step:

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t1

Unit step function

Observations:

Response oscillates about quasi static

response with frequency, ωd.

Response converges to quasi static

response as time tends to infinity. This

response is the steady state.

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Steady state (quasi static response)

Overshoot

Slope is zero here

Time to peak, tp=half damped natural period

At time t=0, the velocity is zero. (note

difference with slope at time zero of

impulse response)

Time to peak is equal to half period.

(note difference with impulse response)

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3. Shock spectrum

Shock: Sudden application of force resulting in transient response.

Shock spectrum: maximum response vs. normalized frequency

Usually normalized frequency is the ratio of the shock duration divided by the natural period of the system.

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4. Numerical simulation of response

It is often difficult to calculate the

convolution integral or solve the differential

equation of motion. We could use

numerical simulation in this case. There are

two approaches for numerical simulation:

1) Solve the differential equation of motion

2) Calculate numerically the convolution integral

1) Numerical solution of differential equation of motion:

Most computer programs can only solve first

order differential equations. We can convert

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the equation of motion to a system of two

first order differential equations as follows.

Start with the original equation:

Let . Then the above equation of

motion becomes:

We can solve the above equations numerically using Mathematica, Mathcad or Matlab.

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