page_title: Impulse vector
section_title: Property 1: Resultant of two impulse vectors.
hierarchical_section_title: Impulse vector / Properties / Property 1: Resultant of two impulse vectors.
caption_attribution_description: English: CGKang Image
context_page_description: An impulse vector is a mathematical tool to graphically design and analyze input shapers that could suppress residual vibration. The impulse vector can be applied for both undamped and underdamped systems, and for both positive and negative impulses in a unified way. The impulse vector makes it easy to obtain impulse time and magnitude of the input shaper graphically.
  A vector concept for an input shaper was first introduced by W. Singhose for undamped systems with positive impulses, and an impulse vector was first introduced by C.-G. Kang to generalize Singhose idea to underdamped systems with positive and negative impulses.
context_section_description: The impulse response of a second-order system corresponding to the resultant of two impulse vectors is same as the time response of the system with a two-impulse input corresponding to two impulse vectors after the final impulse time regardless of whether the system is undamped or underdamped.
caption: Two representations 
    
      
        
          
            
              I
            
            
              R
              1
            
          
          ,
          
            
              I
            
            
              R
              2
            
          
        
      
      {\displaystyle \mathbf {I} _{R1},\mathbf {I} _{R2}}
    
   of the resultant of two impulse vectors 
    
      
        
          
            
              I
            
            
              1
            
          
          ,
          
            
              I
            
            
              2
            
          
        
      
      {\displaystyle \mathbf {I} _{1},\mathbf {I} _{2}}
    
   and the corresponding impulse responses.
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