4 review response arbitrary excitation
TRANSCRIPT
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.
1
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.
2
Transient response is due to both the
application of the force and the non zero
initial conditions
3
Transient responseSteady state
Outline of this chapter
1.Impulse response function
2.Response to arbitrary excitation
3.Shock spectrum
4.Numerical calculation of response
4
1. Impulse response function
Impulse function: idealization of short-
duration force applied suddenly
Force
time
Force
time
)(ˆ tFArea=
Idealization
5
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
6
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.
7
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.
8
Case 2: Overdamped system, >1
Case 3: Critically damped system, =1
9
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.
10
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
11
Step response
Consider underdamped systems only.
Response to unit step:
12
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.
13
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)
14
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.
15
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
16
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.
17