1. MESB 374 System Modeling and Analysis Translational Mechanical System

2. Translational Mechanical Systems
Basic (Idealized) Modeling Elements
Interconnection Relationships -Physical Laws
Derive Equation of Motion (EOM) - SDOF
Energy Transfer
Series and Parallel Connections
Derive Equation of Motion (EOM) - MDOF
In this and next class, we will talk about the modeling and analysis of translational mechanical systems. The basic topics are listed here. By the way, section 3.1 of the textbook gives us a good review of basic concepts of mass, weight, force and systems of units. I wish that you may read it.
At first, we would like to introduce some basic, idealized modeling elements. It should be kept in mind that we may use bunches of these elements to model a single object in realty. For example, a series of mass, spring and damper may be used to model a single flexible beam.
Then we will discuss the physical laws used to describe the interconnection relationship . By using these physic laws, for example Newton’s second law, we may obtain the equation of motion of system To learn it step by step, we firstly discuss systems with single degree of freedom. System with multiple degree freedom will be discussed later. Also, we will discuss the topic of energy transfer and how to simply to model in case of series or parallel connections.

3. Key Concepts to Remember
Three primary elements of interest
Mass ( inertia ) M
Stiffness ( spring ) K
Dissipation ( damper ) B
Usually we deal with “equivalent” M, K, B
Distributed mass  lumped mass
Lumped parameters
Mass maintains motion
Stiffness restores motion
Damping eliminates motion
Throughout this course, we will always three basic elements, inertia element, or mass, spring element which is used to model the stiffness of a object, element dissipating energy or damper. Usually we use M to denote mass, K to denote stiffness, B to denote damping. For system with single degree of freedom, M, K, B are scalars. For system with multiple degree of freedom, they are matrices.
As we discussed in the first lecture, all the system in realty are distributed system. If we want to model them in lumped models, we need to find the equivalent mass, stiffness and damping.
Now let us look at the physical meaning of these three elements. At first, what is mass? How will mass affect the motion? How can we measure the mass? Mass maintains the motion. It means that if there is no external force acting on an object, then it will maintain its current situation. If it moves, then it keep on moving at the same speed. If it original stops, then it will never move until a force acts on it. Stiffness can restore the motion. It is related to potential energy. Damping can eliminate the motion, which means to resist the motion of a object. Does it mean that damping eliminate energy? No. we all know that energy can only be changed from one form to another and never be eliminated. In our case, damping usually change the energy to heat
(Kinetic Energy)
(Potential Energy)
(Eliminate Energy ? )
(Absorb Energy )

4. Variables x : displacement [m] v : velocity [m/sec]
a : acceleration [m/sec2]
f : force [N]
p : power [Nm/sec]
w : work ( energy ) [Nm]
1 [Nm] = 1 [J] (Joule)
Now let’s look at some basic variables used to describing motion.
Displacement is usually defined with respect to a reference frame. Velocity is used to describe how fast the displacement changes with respect to time. Mathematically, it is time derivative of displacement. Acceleration is used to describe how fast velocity changes. Hence, it is Vectors, scalars

5. Basic (Idealized) Modeling Elements
Spring
Stiffness Element
Reality
1/3 of the spring mass may be considered into the lumped model.
In large displacement operation springs are nonlinear. K x2
Linear spring
 nonlinear spring
 broken spring !!
Idealization
Massless
No Damping
Linear
Stores Energy
Hard Spring
Potential Energy
Soft Spring

6. Basic (Idealized) Modeling Elements
Damper
Friction Element
Mass
Inertia Element x2 x1 fD B x f1 f2 f3 M
Dissipate Energy fD
Stores Kinetic Energy

7. Interconnection Laws Newton’s Second Law Newton’s Third Law
Lumped Model of a Flexible Beam
K,M
Newton’s Third Law
Action & Reaction Forces x x M K M K
Massless spring
E.O.M.

8. Modeling Steps
Understand System Function, Define Problem, and Identify Input/Output Variables
Draw Simplified Schematics Using Basic Elements
Develop Mathematical Model (Diff. Eq.)
Identify reference point and positive direction.
Draw Free-Body-Diagram (FBD) for each basic element.
Write Elemental Equations as well as Interconnecting Equations by applying physical laws. (Check: # eq = # unk)
Combine Equations by eliminating intermediate variables.
Validate Model by Comparing Simulation Results with Physical Measurements

9. Vertical Single Degree of Freedom (SDOF) System
Define Problem
The motion of the object g
Input
Output xs
Develop Mathematical Model (Diff. Eq.)
Identify reference point and positive direction. x
B,K,M f K B M g
Draw Free-Body-Diagram (FBD) M M x f
Write Elemental Equations
From the undeformed position
From the deformed (static equilibrium) position
Validate Model by Comparing Simulation Results with Physical Measurement

10. Energy dissipated by damper
Energy Distribution
EOM of a simple Mass-Spring-Damper System
We want to look at the energy distribution of the system. How should we start ?
Multiply the above equation by the velocity term v : Ü What have we done ?
Integrate the second equation w.r.t. time: Ü What are we doing now ? x K M B f
Change of kinetic energy
Energy dissipated by damper
Change of potential energy

11. Example -- SDOF Suspension (Example)
Simplified Schematic (neglecting tire model)
Suspension System
Minimize the effect of the surface roughness of the road on the drivers comfort.
From the “absolute zero”
From the path
From nominal position g M x K B x x p

12. Series Connection Û Springs in Series K1 K2 x1 x2 fS KEQ K1 K2 x1 x2xj fS

13. Series Connection Û Dampers in Series x1 x2 x1 fD x2 fD fD fD B1 B2
BEQ

14. Parallel Connection Û Springs in Parallel x1 x2 fS K1 K2 x1 x2 fS fS
KEQ

15. Parallel Connection
Dampers in Parallel x1 x2 fD B1 B2 x2 fD x1
BEQ Û

16. Horizontal Two Degree of Freedom (TDOF) System
Absolute coordinates
FBD
Newton’s law

17. Horizontal Two Degree of Freedom (TDOF) System
Static coupling
Absolute coordinates
Relative coordinates
Dynamic coupling

18. Two DOF System – Matrix Form of EOM
Input vector
Absolute coordinates
Output vector
Mass matrix
Damping matrix
Relative coordinates
Stiffness matrix
SYMMETRIC
NON-SYMMETRIC

19. MDOF Suspension Suspension System
Simplified Schematic (with tire model)
Suspension System x p
TRY THIS

20. MDOF Suspension Suspension System
Simplified Schematic (with tire model)
Suspension System
Assume ref. is when springs are
Deflected by weights
Wheel
Reference
Car body
Suspension
Tire
Road

21. Example -- MDOF Suspension
Draw FBD
Apply Interconnection Laws M2
FS2 x2
FD2 M1
FS1 x1
FD1
FS2
FD2

22. Example -- MDOF Suspension
Matrix Form
Mass matrix
Damping matrix
Stiffness matrix
Input Vector