Courses

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.
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

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 − rG 

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

 

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!
Download EduRev app here for Civil Engineering (CE) preparation.

Up next

Author
Constructing Careers
Views
170
Rating
4.9
Description
Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev notes for Civil Engineering (CE) is made by best teachers who have written some of the best books of Civil Engineering (CE). It has gotten 170 views and also has 4.9 rating. You can download Free Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev pdf from EduRev by using search above. You can also find Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev ppt and other Civil Engineering (CE) slides as well. If you want Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev Tests & Videos, you can search for the same too. Civil Engineering (CE) Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev Summary and Exercise are very important for perfect preparation. You can see some Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev sample questions with examples at the bottom of this page. Complete Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev chapter (including extra questions, long questions, short questions, mcq) can be found on EduRev, you can check out Civil Engineering (CE) lecture & lessons summary in the same course for Civil Engineering (CE) Syllabus. EduRev is like a wikipedia just for education and the Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev images and diagram are even better than Byjus! Do check out the sample questions of Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev for Civil Engineering (CE), the answers and examples explain the meaning of chapter in the best manner. This is your solution of Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev search giving you solved answers for the same. To Study Summary of Equations of Motion for Rigid Bodies (Part - 1) Civil Engineering (CE) Notes | EduRev for Civil Engineering (CE) this is your one stop solution.

Related Searches

Viva Questions

,

Previous Year Questions with Solutions

,

study material

,

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

,

Important questions

,

Extra Questions

,

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

,

MCQs

,

Summary

,

Exam

,

video lectures

,

practice quizzes

,

Free

,

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

,

Semester Notes

,

Sample Paper

,

ppt

,

shortcuts and tricks

,

pdf

,

Objective type Questions

,

past year papers

,

mock tests for examination

;
Follow Us
© EduRev