# When to Use Dynamic Analysis

Static studies assume that loads are constant or applied very slowly until they reach their full values. Because of this assumption, the velocity and acceleration of each particle of the model is assumed to be zero. As a result, static studies neglect inertial and damping forces.

For many practical cases, loads are not applied slowly or they change with time or frequency. For such cases, use a dynamic study. Generally if the frequency of a load is larger than 1/3 of the lowest (fundamental) frequency, a dynamic study should be used.

Linear dynamic studies are based on frequency studies. The software calculates the response of the model by accumulating the contribution of each mode to the loading environment. In most cases, only the lower modes contribute significantly to the response. The contribution of a mode depends on the load’s frequency content, magnitude, direction, duration, and location.

Objectives of a dynamic analysis include:

• Design structural and mechanical systems to perform without failure in dynamic environments.

• Modify system's characteristics (i.e., geometry, damping mechanisms, material properties, etc.) to reduce vibration effects.

## Equations of Motion

### Single Degree of Freedom (SDOF) Systems

Consider the simple mass-spring system. The mass (m) is subjected to a force F(t) in the u direction as, a function of time. The mass is allowed to move in the u direction only and hence this is a Single-Degree-Of-Freedom (SDOF) system. The motion is resisted by a spring of stiffness (k)

.

Writing Newton's Second Law (force equals mass times acceleration) for this system at time (t) results in:

F(t)-ku(t) = mu..(t)

or:

mu..(t) + ku(t) = F(t)

where:

u..(t) is the acceleration of the mass at time (t) and it is equal to the second derivative of u with respect to time

k = is the stiffness of the spring

Theoretically, if the mass is displaced and released, it will continue to vibrate with the same amplitude forever. In practice, the mass vibrates with progressively smaller amplitudes until it comes to rest. This phenomenon is called damping and it is caused by loss of energy through friction and other effects. Damping is a complex phenomenon. For the purpose of this discussion, assume that the damping force is proportional to the velocity. This type of damping is called viscous damping.

Considering damping, the above equation becomes:

mu..(t) + cu.(t) + ku(t) = F(t)

where:

u.(t) is the velocity of the mass at time (t), and it is equal to the first derivative of u with respect to time

Note: In static studies, velocity and acceleration are so small that they can be neglected and F and u are not functions of time. The above equation reduces to: F=ku.

### Multi Degree of Freedom (MDOF) Systems

For a Multi-Degree-of-Freedom (MDOF) system, m, c, and k become matrices instead of single values and the equations of motion are expressed as:

, where

[M]: mass matrix

[K] : stiffness matrix

[C] : damping matrix

{u(t)}: displacement vector at time t (displacement components of every node)

acceleration vector at time t (acceleration components of every node)

velocity vector at time t (velocity components of every node)

{f(t)}: time varying load vector (force components of every node)

### Related Topics

Linear Static Versus Linear Dynamic Analysis

Loads and Result Options for Dynamic Analysis

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