Mechanical System Fundamentals Slides

7/21/2019 Mechanical System Fundamentals Slides

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Mechanical System Fundamentals K. Craig 1

Mechanical System Fundamentals


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Mechanical Building Blocks

Motion and force are concepts used to describe the behavior of

engineering systems that employ mechanical components.

It is hard to imagine any engineering system that does not havemechanical components.

Motion is a term used to describe the movement of a point

relative to another and it is described using the terms distance,

velocity, and acceleration. The three are related by

differentiation and integration. For rectilinear (straight-line)

motion we can use scalars and write:2

1

2

1

t

2 1

t

t

2 1

t

dxv x x v dt

dt

dva v v a dtdt

= =

= =


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Shown is a symbolic or circuit diagram of a mechanical

component whose ends are undergoing translationalmovement.

Since velocity is a relative term, the above figure implies the

existence of a reference that is fixed. Shown below is the

same figure with a fixed reference.

21 2 1

21 2 1

21 2 1

x x x

v v v

a a a

=

= =


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Relative motion between the ends of a mechanical component

cant exist without a force being present. The force has both amagnitude and a sign.

V2 > V1

V2 < V1

V2 = V1


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If a force F is applied over an incremental distance ds in the

direction of the force, an amount of work dWis done equal to:

We can write an expression for dWin terms of velocity of a

point moving in the direction of the force. The amount of

work done in an interval from t1 to t2 can be obtained by

integration.

Power is the rate at which work is performed.

dW Fds=

dsdW F dt Fvdt

dt= =

2

1

t

t

W Fv dt=

dWP Fv

dt= =


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The three basic mechanical building block elements are:

Spring (elastic) element

Damper (frictional) element Mass (inertia) element

There are both translational and rotational versions of these

basic building blocks. These are passive (non-energy producing) devices

Driving Inputs

force and motion sources which cause elements to respond


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Each of the elements has one of two possible energybehaviors:

stores all the energy supplied to it

dissipates all energy into heat by some kind offrictional effect

Spring stores energy as potential energy

Mass stores energy as kinetic energy

Damper dissipates energy into heat

The Dynamic Response of each element is important.


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Spring Element A spring is a fundamental mechanical component found

intentionally or unintentionally in almost every mechanical

system.

Real-world spring is neither pure nor ideal

Real-world spring has inertia and friction

Pure spring has only elasticity - it is a mathematical model,

not a real device

Some dynamic operation requires that spring inertia and/or

damping not be neglected

Ideal spring: linearNonlinear behavior may often be preferable and give

significant performance advantages


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Device can be pure without being ideal (e.g., nonlinear

spring with no inertia or damping)

Device can be ideal without being pure (e.g., device which

exhibits both linear springiness and linear damping)

Pure and ideal spring element:

Ks = spring stiffness (N/m or N-m/rad) 1/Ks = Cs = compliance (softness parameter)

( )( )

s 1 2 s

s 1 2 s

f K x x K x

T K K

= =

= =

s

s

x C f

C T

=

=

Ksx f f x

Cs


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Energy stored in a spring

Dynamic Response: response to an input is

instantaneous. Real springs will not behave exactly like the pure/ideal

element.

2 2

s ss

C f K x

E 2 2= =

21 21

21

F kx k v dt

1 dFv

k dt

= =

=


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Spring Element

( ) ( )

( )0

s

2 2x

s 0 s 0s

0

Differential Work Done

f dx K x dx

Total Work DoneK x C f

K x dx2 2

= =

= = =


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Linearization

for a

Nonlinear Spring

( ) ( )

( )

0 0

0

22

0

0 0 2

x x x x

0 0

x x

x xdf d f y f (x ) x x

dx 2!dx

dfy y x x

dx

= =

=

= + + +

+

( )0

0 0

x x

dfy y x x

dx

y Kx

=

+

=


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Real Springs

nonlinearity of the

force/deflection curve

noncoincidence of the

loading and unloading

curves (The 2nd Law of

Thermodynamics

guarantees that the area

under the loading f vs. xcurve must be greater

than that under the

unloading f vs. x curve.It is impossible to recover

100% of the energy put

into any system.)


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Several Types of Practical

Springs: coil spring

hydraulic (oil) spring

cantilever beam spring

pneumatic (air) spring

clamped-end beam spring

ring spring

rubber spring (shock mount)

tension rod spring

torsion bar spring


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Spring-like Effects in

Unfamiliar Forms

aerodynamic spring

gravity spring (pendulum)

gravity spring (liquid

column)

buoyancy spring magnetic spring

electrostatic spring

centrifugal spring


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Damper Element A damper is a mechanical component often found in

engineering systems.

A pure damper dissipates all the energy supplied to it,i.e., converts the mechanical energy to thermal energy.

Various physical mechanisms, usually associated with

some form of friction, can provide this dissipative

action, e.g.,

Coulomb (dry friction) damping

Material (solid) damping Viscous damping


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Shown is a typical mechanical viscous damper. If the mass

and springiness of the piston and cylinder are small, then theforce will be a function of the relative velocity between the

piston and the cylinder.


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Shown is a symbolic (circuit) diagram of a viscous damper

along with a graphical representation.

Pure / ideal damper element provides viscous friction.

All mechanical elements are defined in terms of their

force/motion relation. (Electrical elements are defined in

terms of their voltage/current relations.)


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Pure / Ideal Damper

Damper force or torque is directly proportional to therelative velocity of its two ends.

Forces on the two ends of the damper are exactly equal

and opposite at all times (just like a spring); puresprings and dampers have no mass or inertia. This is

NOT true for real springs and dampers.

Units forB to preserve physical meaning: N/(m/sec)

(N-m)/(rad/sec)

( )2 1

2 1 21

dx dx

F B B v v Bvdt dt

= = =


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Operational Transfer Function

We assume the initial conditions are zero.

( )

22

2

2

dx d xDx D x

dt dt

x x(x)dt x dt dt

D D

Differential

Operator

Notation

f BDx

T BD

=

=

( ) ( )

( ) ( )

f TD BD D BDx

x 1 1D D

f BD T BD


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Damper element dissipates into heat all mechanical

energy supplied to it.

Force applied to damper causes a velocity in same

direction.

Power input to the device is positive since the force

and velocity have the same sign.

It is impossible for the applied force and resulting

velocity to have opposite signs.

Thus, a damper can never supply power to anotherdevice; Power is always positive.

( ) ( )

2dx dx

Power force velocity f Bdt dt

= =


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A spring absorbs power and stores energy as a forceis applied to it, but if the force is gradually relaxed

back to zero, the external force and the velocity now

have opposite signs, showing that the spring isdelivering power.

Total Energy Dissipated

( ) ( )2

dx dxP dt B dt B dx f dx

dt dt

= = =


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Damper Element


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Real Dampers A damper element is used to model a device designed

into a system (e.g., automotive shock absorbers) or for

unavoidable parasitic effects (e.g., air drag). To be an energy-dissipating effect, a device must exert

a force opposite to the velocity; power is always

negative when the force and velocity have oppositedirections.

Lets consider examples of real intentional dampers.


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Viscous (Piston/Cylinder) Damper

A relative velocity between thecylinder and piston forces the

viscous oil through the clearance

space h, shearing the fluid and

creating a damping force.

22 22 2 1

2 13

2

6 L h R R B R R hhh 2

R2

=

= fluid viscosity


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Simple Shear Damper

AndViscosity Definition

fluid viscosity

shearing stress F / A

velocity gradient V / t

=

2AF V

tF 2A

BV t

=

= =


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Examples

of

Rotary Dampers

3D LB 4t

=

4

0DB16t

=


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Commercial Air Damper

laminar flow

linear damping

turbulent flownonlinear damping

(Data taken with valve shut)

Air Damper

much lower viscosity less temperature dependent

no leakage or sealing problem


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Eddy-Current Damper

Motion of the conducting

cup in the magnetic field

generates a voltage in thecup.

A current is generated in

the cups circular path.

A current-carrying

conductor in a magnetic

field experiences a force

proportional to the current. The result is a force

proportional to and

opposing the velocity. The dissipated energy

shows up as I2R heating of

the cup.


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Temperature Sensitivity

OfDamping Methods


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Other Examples

of

Damper Forms


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The damper element can also be used to represent

unavoidableparasitic energy dissipation effects in mechanical

systems.

Frictional effects in moving parts of machines

Fluid drag on vehicles (cars, ships, aircraft, etc.)

Windage losses of rotors in machines

Hysteresis losses associated with cyclic stresses inmaterials

Structural damping due to riveted joints, welds, etc.

Air damping of vibrating structural shapes


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Hydraulic Motor Friction

and its Components


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Mass or Inertia Element

All real mechanical components used in engineeringsystems have mass. It is frequently possible to treat a

component as if all its mass were concentrated at a single

point called the center of gravity.

Point 1 is either fixed or has constant velocity.


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A designer rarely inserts a component for the purpose of

adding inertia; the mass or inertia element often represents an

undesirable effect which is unavoidable since all materialshave mass.

There are some applications in which mass itself serves a

useful function, e.g., accelerometers and flywheels.


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Useful Applicationsof

Inertia

Flywheels are used as

energy-storage devices or as

a means of smoothing outspeed fluctuations in engines

or other machines.

Accelerometer


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Newtons Law defines the behavior of mass elements

and refers basically to an idealized point mass:

The concept of rigid body is introduced to deal withpractical situations. For pure translatory motion, every

point in a rigid body has identical motion.

Real physical bodies never display ideal rigid behaviorwhen being accelerated.

The pure / ideal inertia element is a model, not a real

object.

( ) ( )forces mass acceleration=


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Rigid and FlexibleBodies:

Definitions and Behavior


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Inertia Element

Real inertias may beimpure (have some

springiness and friction)

but are very close toideal.

( ) ( )2 2x 1 1

D D

f MD T JD

= =

Inertia Element stores

energy as kinetic energy:

2 2Mv J or

2 2


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Force and Motion Sources

The ultimate driving agency of any mechanical

system is always a force not a motion; force causes

acceleration, acceleration does not cause force.

Motion does not occur without a force occurring

first. At the input of a system, what is known, force or

motion? If motion is known, then this motion was

caused by some (perhaps unknown) force andpostulating a problem with a motion input is

acceptable.


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There are only two classes of forces:

Forces associated with physical contact between two

bodies

Action-at-a-distance forces, i.e., gravitational, magnetic,and electrostatic forces.

There are no other kinds of forces! (Inertia force is a

fictitious force.) The choice of an input form to be applied to a system

requires careful consideration, just as the choice of a

suitable model to represent a component or system.

Here is an example of a force source and a motion

sources.


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Force and Motion Inputs

acting on a

Multistory Building


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Summary


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Summary


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Mechanical System Fundamentals K Craig 45

Summary

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Mechanical System Fundamentals Slides