Non-uniform damping

Introduction

Damping is present in all oscillatory systems and is a measure of a vibration structure to dissipate energy.

The dynamic tools of SCIA Engineer take account of damping. Sometimes it is defined directly by the damping ratio, sometimes by a damping spectrum.  Please consult the specific chapters for detail.  In general we can say that by default a mean value of damping is taken for the complete system for a mode.

With non-uniform damping, the user is allowed to define damping on the members and on the supports. The program computes a mean value for each mode based on energy considerations.

The non-uniform damping is supported for:

Harmonic load

Seismic load

Theoretical background

SCIA Engineersupports two types of damping:

• Material damping ξ is defined as a percentage of critical damping (logarithmic decrement, another quantity frequently used to defined damping, is defined by )
• Geometrical damping from soil, defined by a coefficient at the level of the supports.

Both types of damping are taken into account by an equivalent damping ratio. For each mode, it is computed as following:

(7.1)

where

ξ_j  is the damping ratio in node j

k_j is the stiffness in node j

Φⁱ_j  is the modal displacement at node j for mode i

ω_i  is the frequency of mode i

C_s is the damping constant of support s (kg/s)

Φⁱ_s  is the modal displacement at support node s for mode i

α is a user defined parameter (>0, default =0.5)

Not: At supports, damping is defined separately for the 3 global directions X, Y and Z

Proof 

The equivalent modal damping is obtained by ponderation of the potential energies used by the mode in part j of the structure where the damping is .

Case 1: material damping (members)

where  [K] and [M] are, respectively, the stiffness matrix and the mass matrix.

With m_jthe mass at node j

We assume that

In case of the modal displacement is normed by the mass

then the mean internal modal damping of mode I is given by:

For practical use, we take the same value of damping for all the elements,

We obtain

Case 2: geometrical damping (supports)

For a simple oscillator, the damping is defined by:

where c is the coefficient of damping

By analogy with material damping, we can state:

 

We obtain in total:

The SEPTEM recommends the value β=0.5 (hence α=1)

Practical use

Damping on members is linked to the material.

Sometimes the user want one value assigned to the whole structure

Sometimes the user have mixed structures

The user is able to overrule easily the damping for some members

The user is able to easily define and correct the damping for the whole structure

Damping at supports is only valid for:

Flexible nodal support

Support orientated in global direction XYZ only

Damping groups

The user defines damping groups (similar to mass groups). In order to activate non-uniform damping:

The functionality “Non-proportional damping” is enabled in Project settings

A damping group is defined and linked to a mass combination

A mass combination is defined and linked to a dynamic load case (harmonic or seismic)

Two types of default damping data are available for members (1D/2D). The user is allowed to select the proper default for each damping group:

One value for the whole structure, the “Global default”

One value per material used in the project, the “Material default”

Default damping settings are stored in the set-up of the damping service.

In case of global damping, one value is displayed {default 5%}

In case of material defaults:
The damping is defined in the properties of materials in the material library of SCIA Engineer.

The set-up of the damping service also contains

Max modal damping: one value {used to limit the calculated damping per mode; default 30% }

Alfa {factor for supports; >0; default 1}

Add data for damping on members and supports

In each damping group, the user can assign damping on the elements (similar to additional mass on members in mass groups).  Damping can be added on:

a 1D member

a 2D member

a nodal support

Following data is available:

In case of 1D and 2D members, there are 3 possibilities:

Logarithmic decrement: one value {default = 0.001}

Relative damping: one value {default =0.001}

Rayleigh damping: two values { Alfa and beta; default both = 0}

In case of nodal support: 3 values, one value per global direction X, Y and Z {C value; default = 0}

Note that it is not possible to add support dampers

• if the support in not flexible
• if the support is rotated

In this case, a message is displayed and input is cancelled

In case the user adds damping in a specific group, then the default is overruled for that member in that group.