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

## 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.
All you need of Civil Engineering (CE) at this link: Civil Engineering (CE)

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 • The total force acting on the body • The total linear impulse exerted by forces during the time interval • The total moment (including torques) acting on the body • The tot al angular impulse exerted on the body during the time interval • The rate of work done by forces and torques acting on the body • The total work done by forces and torques on the body during the time interval 6.6.2 Inertial properties

• The total mass is • The position of the center of mass is • • The mass moment of inertia about the center of mass where d = r − rG

For a 2D body with mass per unit area µ we use • The total mass is • The position of the center of mass is • The mass moment of inertia about the center of mass is 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) • The spin tensor is related to R by  • Rotation through an angle θ about an axis parallel to a unit vector  n = nx i +ny j + nzk is • The angular velocity vector ω = ω x i +ωy j+ ω zk is related to W by • 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 6.6.5 Conservation laws

• Linear momentum  • Angular momentum • Work-Power - Kinetic Energy relation • Energy equation for a conservative system d/dt (T+V) =0  T0+ V0= T1+V1

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

## Introduction to Dynamics and Vibrations- Notes, Videos, MCQs

20 videos|53 docs

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;