---
title: "Damping matrix"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Damping_matrix"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:46:56.497350+00:00"
instance: "kb-cron"
---

In applied mathematics, a damping matrix is a matrix corresponding to any of certain systems of linear ordinary differential equations.  A damping matrix is defined as follows.  If the system has n degrees of freedom un and is under application of m damping forces. Each force can be expressed as follows:

  
    
      
        
          f
          
            D
            i
          
        
        =
        
          c
          
            i
            1
          
        
        
          
            
              
                u
                
                  1
                
              
              ˙
            
          
        
        +
        
          c
          
            i
            2
          
        
        
          
            
              
                u
                
                  2
                
              
              ˙
            
          
        
        +
        ⋯
        +
        
          c
          
            i
            n
          
        
        
          
            
              
                u
                
                  n
                
              
              ˙
            
          
        
        =
        
          ∑
          
            j
            =
            1
          
          
            n
          
        
        
          c
          
            i
            ,
            j
          
        
        
          
            
              
                u
                
                  j
                
              
              ˙
            
          
        
      
    
    {\displaystyle f_{Di}=c_{i1}{\dot {u_{1}}}+c_{i2}{\dot {u_{2}}}+\cdots +c_{in}{\dot {u_{n}}}=\sum _{j=1}^{n}c_{i,j}{\dot {u_{j}}}}
  

It yields in matrix form;

  
    
      
        
          F
          
            D
          
        
        =
        C
        
          
            
              U
              ˙
            
          
        
      
    
    {\displaystyle F_{D}=C{\dot {U}}}
  

where C is the damping matrix composed by the damping coefficients:

  
    
      
        C
        =
        (
        
          c
          
            i
            ,
            j
          
        
        
          )
          
            1
            ≤
            i
            ≤
            n
            ,
            1
            ≤
            j
            ≤
            m
          
        
      
    
    {\displaystyle C=(c_{i,j})_{1\leq i\leq n,1\leq j\leq m}}