Time History - CAESAR II - Help

CAESAR II Users Guide (2019 Service Pack 1)

Language
English
Product
CAESAR II
Search by Category
Help
CAESAR II Version
11.0 (2019)

Time history analysis is a more accurate, more computationally intensive analytical method than response spectrum analysis. It is best suited to impulse loadings or other transient loadings where the profile is known. This method of analysis involves the actual solution of the dynamic equation of motion throughout the duration of the applied load and subsequent system vibration, providing a true simulation of the system response.

As noted in Harmonic Analysis, the dynamic equation of motion for a system is

This differential equation cannot be solved explicitly but may be integrated using numeric techniques by slicing the duration of the load into many small time steps. Assuming that the change in acceleration between time slices is linear, the system accelerations, velocities, displacements, and corresponding reactions, internal forces, and stresses are calculated at successive time-steps.

Because the total response of a system is equivalent to the sum of the responses of its individual modes of vibration, the above equation can be simplified assuming that the damping matrix C is orthogonal. Use the transformation x = FX, to be expressed in modal coordinates:

Where:

= acceleration vector (in modal coordinates), as a function of time

C´ = diagonal damping matrix, where entry C´i = wi ci

wi = angular frequency of mode i

ci = ratio of damping to critical damping for mode i

(t) = velocity vector (in modal coordinates), as a function of time

x(t) = displacement vector (in modal coordinates), as a function of time

W = diagonal stiffness matrix, where entry Wi = wi2

This transformation represents N uncoupled second order differential equations, where N is the number of modes of vibration extracted. N can then be integrated and summed, using the in-phase, algebraic summation method to give the total system response. CAESAR II uses the Wilson q method (an extension of the Newmark method) to integrate the equations of motion, providing an unconditionally stable algorithm regardless of time step size chosen.

Only one dynamic load can be defined for a time history analysis. This dynamic load case can be used in as many static/dynamic combination load cases as necessary. The single load case may consist of multiple force profiles applied to the system simultaneously or sequentially. Each force versus time profile is entered as a spectrum with an ordinate of Force (in current units) and a range of Time (in milliseconds). The profiles are defined by entering the time and force coordinates of the corner points defining the profile.

A time can only be entered once. A time with zero force outside of the defined profile need not be entered explicitly.

For example, the profiles shown in the following figure are entered as:

Time (MS)

Force

Time (MS)

Force

0.0

0.0

20.0

1000.0

10.0

300.0

60.0

1000.0

20.0

1000.0

30.0

0.0

The load profiles are linked with force sets (indicating magnitude, direction, and location of the applied load) in the shock case. The magnitude of the applied load is determined by the product of the profile force, the force set magnitude, and the scale in the shock case.

You can enter only forces, not moments or restraint displacements, in the time history load profile. Model moments using force couples and simulate restraint displacements by entering forces equal to the displacement multiplied by the restraint stiffness in the direction of the displacement.

Process output from a Time History analysis in three ways:

  • Use the output processor to review the natural frequencies, mode shapes, participation factors, included mass/force, displacements, and restraint loads, forces, or stresses in report form. CAESAR II’s implementation of time history analysis provides two types of results. One results case contains the maximum individual components (such as axial stress, X-displacement, and MZ reaction) of the system response, along with the time at which it occurred. Several results cases represent the actual system response at specific times. Dynamic results also show the largest modal contributor, along with the mode and transient load responsible for that contribution.

  • Animate the shock displacement for the transient load cases. During animation, the displacements, forces, moments, stresses, and other data associated with individual elements are displayed at every time step and for the dynamic load alone, or for any of the static/dynamic combinations.

  • Animate the individual mode shapes included in the time history response.