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.

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: Processof going from initial state to steady state.

Transient response is due to both the application of the force and the non zero initial conditions
Outline of this chapter

  1. Impulse response function
  2. Response to arbitrary excitation
  3. Shock spectrum
  4. Numerical calculation of response

  1. Impulse response function

Impulse function: idealization of short-duration force applied suddenly

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

Impulse responses of two systems with natural frequency 6.28 rad/sec and damping ratios 0.1 and 0.8.

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.

Case 2: Overdamped system, 1

Case 3: Critically damped system, =1

  1. 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.

Equivalentequation for response:

The above are called convolution integrals.

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
Step response

Consider underdamped systems only.

Response to unit step:

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.
  • 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)
  1. 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.

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 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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