Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev

Introduction to Dynamics and Vibrations- Notes, Videos, MCQs

Civil Engineering (CE) : Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev

The document Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Introduction to Dynamics and Vibrations- Notes, Videos, MCQs.
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In this section, we collect together all the important formulas from the preceding sections, and summarize the equations that we use to analyze motion of a rigid body.

We consider motion of a rigid body that has mass density ρ during some time interval t0 < t < t1 , and define the following quantities:

6.6.1 Forces, torques, impulse, work, power

Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev

  • The total force acting on the body  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • The total linear impulse exerted by forces during the time interval  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • The total moment (including torques) acting on the body  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • The tot al angular impulse exerted on the body during the time interval  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • The rate of work done by forces and torques acting on the body  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • The total work done by forces and torques on the body during the time interval  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev

6.6.2 Inertial properties

  • The total mass is  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • The position of the center of mass is  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
    • Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • The mass moment of inertia about the center of mass

Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev

where d = r − rG

For a 2D body with mass per unit area µ we use

Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev

  • The total mass is  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • The position of the center of mass is  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • The mass moment of inertia about the center of mass is  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev

where d = r − r

6.6.3 Describing motion

  • The rotation tensor (matrix) maps the vector connecting two points in a solid before it moves to its position after motion rB − rA = R(pB − pA)

Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev

  • The spin tensor is related to R by   Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRevSummary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • Rotation through an angle θ about an axis parallel to a unit vector  n = nx i +ny j + nzk is

Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev  

  • The angular velocity vector ω = ω x i +ωy j+ ω zk is related to W by

Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev  

  • The angular acceleration vector is α = dω/dt
  • The velocities of two points A and B in a rotating rigid body are related by v B −v A = ω ×(rB − rA )
  • The accelerations of A and B are related by aB − aA = α ×(rB−rA) + ω×(vB − vA) = α ×(rB−rA) + ω× ω×(rB−rA)]

6.6.4 Momentum and Energy 

  • The total linear momentum is p = MvG
  • The angular momentum (about the origin) is h =rG × MvG+ IG ω
  • The total kinetic energy is T= 1/2 MvG ·vG +1/2 ω·IGω

For 2D planar problems, we know ω = ωzk .   In this case, we can use

  • The total linear momentum is p = MvG
  • The total angular momentum (about the origin) is h =rG × MvG+IGzzω z k
  • The total kinetic energy is  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev

6.6.5 Conservation laws 

  • Linear momentum  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRevSummary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • Angular momentum  Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • Work-Power - Kinetic Energy relation   Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev
  • Energy equation for a conservative system d/dt (T+V) =0  T0+ V0= T1+V1

 

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