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# Integration Methods

A set of coupled differential and algebraic equations define the equations of motion of a SolidWorks Motion model. A numerical solution to these equations is obtained by integrating the differential equations while satisfying algebraic constraint equations at every time step.

A set of differential equations is numerically stiff when there is a wide spread between high and low frequency eigenvalues, while the high-frequency eigenvalues are overdamped. The speed of solution of the equations of motion depends on the numerical stiffness of the equations. The stiffer the equations, the slower the solution.

A stiff integration method is an efficient computational method for solving stiff systems. Numerically stiff differential equations require stiff integration methods to compute the solutions efficiently because other types of methods for solving differential equations perform poorly and are too slow.

The SolidWorks Motion solver offers three stiff integration methods for computing motion:
• The GSTIFF integration method developed by C. W. Gear is a variable order, variable step size integration method. It is the default method used by the SolidWorks Motion solver. The GSTIFF method is a fast and accurate method for computing displacements for a wide range of motion analysis problems.
• WSTIFF is another variable order, variable step size stiff integrator.
GSTIFF and WSTIFF are similar in formulation and behavior. Both use a backwards difference formulation. They differ in that the GSTIFF coefficients are calculated assuming a constant step size, whereas WSTIFF coefficients are a function of the step size. If the step size changes suddenly during integration, GSTIFF introduces a small error, while WSTIFF can handle step size changes without loss of accuracy. Sudden step size changes occur whenever there are discontinuous forces, discontinuous motions or abrupt events such as contact in the model.
• SI2_GSTIFF, a Stabilized Index-2 method, is a modification of the GSTIFF method. This integration method provides better error control over the velocity and acceleration terms in the equations of motion. Provided the motion is sufficiently smooth, SI2_GSTIFF velocity and acceleration results are more accurate than those computed with GSTIFF or WSTIFF, even for motions with high frequency oscillations. SI2_GSTIFF is also more accurate with smaller step sizes, but is significantly slower.

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