The document discusses the Newmark method for solving initial value problems, particularly in the context of second-order differential equations. It highlights the limitations of methods like Runge-Kutta in dealing with ill-conditioned system matrices and introduces the Newmark method as a solution. The method involves evaluating acceleration, velocity, and displacement, with specific parameters that must adhere to constraints.

1. Newmark Method for Initial Value Problems
Mohammad Tawfik
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Newmark Method for Initial
Value Problems
Mohammad Tawfik

2. Newmark Method for Initial Value Problems
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Newmark method
• In many cases, the reduction of the second
order differential equation into a set of first order
systems results in ill-conditioned system
matrix!
• In such cases, the Runge-Kutta method fails to
converge.
• Many other methods were proposed, in this
lecture we will introduce the Nemark Method

3. Newmark Method for Initial Value Problems
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Newmark method
• The Newmark method handles problems
in the form of:
• Where [M] is the mass matrix, [C] is the
damping matrix, [K] is the stiffness matrix,
and {f(t)} is the generalized excitation
(force) term.
           tfxKxCxM  

4. Newmark Method for Initial Value Problems
Mohammad Tawfik
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Algorithm
1. Evaluate the acceleration of the system
using

5. Newmark Method for Initial Value Problems
Mohammad Tawfik
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http://WikiCourses.WikiSpaces.com
2. Then the velocity and displacement may
be evaluated using

6. Newmark Method for Initial Value Problems
Mohammad Tawfik
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http://WikiCourses.WikiSpaces.com
• In the above algorithm, δ and μ are
parameters that have to obey the
constraints