examples of pulse loading
- impact of cars
- explosion loading
- impulsive types of earthquake loading
impulsive force
very large force that acts for a very small time
unit impulse has
- start at time t = tau
- duration e
- force p(t) = 1 / e
damping and impulse
neglect effect of restoring and damping forces due to small displacement and velocity relative to acceleration, caused by infinitesimal duration of impulse
Duhamel’s integral
provides a general result for evaluating the response of a LINEAR SDOF system to an arbitrary force
pulse-like loading
important class of excitations fundamentally different from harmonic loading
- essentially consist of a single pulse
- can occur from blasts and explosions, or rapid impact
fundamental difference between pulse load and harmonic load
- pulse loads never reach steady state conditions therefore effect of initial conditions must be considered
two phases of pulse displacement solution
a) forced vibration phase
b) free vibration phase
forced vibration phase
t
free vibration phase
t > tp
displacement solution for t
the same regardless of pulse duration
effect of duration of pulse on dynamic amplitude
the larger the duration of the pulse, the larger the dynamic amplitude ( more energy imparted )
maximum response (for a short pulse)
max response occurs in free vibration phase
I
impulse
I = p(0) t(p)
when is approximate pulse analysis considered accurate?
when tp / Tn
effect of damping on pulse loads
effect of damping on amplitude of response is very small for pulse loads
- reason for lack of sensitivity is that max response occurs on first displacement cycle, after which time very little energy has been damped out
- in contrast by the time steady state for harmonic loading has been reached, many more displacement cycles have occured and so a significant amount of energy has been damped out
two major drawbacks of Duhamel integral
- only applies for linear systems
- when p(tau) is not a ‘simple’ function, Duhamel integral must be solved numerically (particularly inefficient to solve)
Most common numerical method used
Newmark-Beta Method
two assumptions of Newmark-B method
- average constant acceleration
- linear acceleration
Newmark-B derivation
- draw accn assumption diagram
- derive formula for accn
- integrate to find velocity and disp
- evaluate at tau = t(i+1)
- Define incremental terms
- apply these to eqautions from (4)
- rearrange to find deltaaccn
- substitute delta accn to find delta velocity
- now have incremental formula so substitute into equation of motion
Stability of Newmark-B method
- constant accn assumption unconditionally stable
- linear accn stable if time step is less than 0.551 times the shortest period of vibration in the system
Accuracy of Newmark-B method
for sufficient accuracy, timestep
preffered acceleration assumption
constant average acceleration - due to unconditional stability
Effect of damping on Response Spectrum
- spectral amplitude decreases as damping ratio increases
- spectrum becomes less jagged with increasing damping
