(دکتر حشمدار) Dr.Hashamdar دکتر حمیدرضا حشمدار ——– Dr.Hamidreza Hashamdar
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AN EFFICIENT METHOD FOR NONLINEAR DYNAMIC ANALYSIS OF 3D SPACE STRUCTURES
Prof.Dr.Hashamdar
The equation of dynamic motion
The equation of dynamic motion for a system can be written as:
where M= mass determinant
C (t) = Damping matrix K (t) = stiffness matrix
X = Displacement vector = Velocity vector
= Acceleration vector P (t) = Load vector, t= time
The solution of equation cannot be expressed in the functional form and it is necessary to plot or tabulate to solution curve point by point, beginning at (t0 , x0) and then at selected intervals of t, usually equally spaced, until the solution has been extended to cover the required range. Thus the solutions of the nonlinear equations require a step-by-step approach and are normally based on the use of the interpolation or the finite difference equations. The independent variable t is divided into equal intervals , over the range of the desired solution. Thus the variables after n and (n+1) intervals are given by tn= n. , and tn+1=(n+1) respectively. At time tn it is assumed that the values of all the parameters as well as the values for same parameters at all previous intervals (n-1), (n-2),…..,2,1 are known. At time tn+1 it is assumed that the values of the variable parameters are not known and the purpose of the analysis is to find the value of xn+1 and its derivatives which satisfy.
In the following sections, the equations of motion for single degree of freedom system and multi degree of freedom system will be discussed.
A formal assessment of nonlinear dynamic response methods
The linear acceleration method
This method is based on the step-by-step time integration of the equations of motion. A method is linear change of acceleration during each time step is assumed. Equilibrium of the dynamic forces is established at the beginning and at the end of each time interval. The nonlinear nature of a structure is in accordance with the deformed state at the beginning of each time increment.
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