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MECH 364MECHANICAL MECHANICAL VIBRATIONSVIBRATIONS
Clarence W. de Silva, Ph.D., D.Eng. (hc), P.Eng. Professor of Mechanical EngineeringThe University of British Columbia
e-mail: [email protected]:// www.mech.ubc.ca/~ial
C.W. de Silva
Presentation Part 3
PlanPlan
Time Response AnalysisTime Response AnalysisTo Study:• Element Behavior• Naturally Oscillating Systems• Undamped Oscillator• Damped Oscillator• Free Response• Forced Response
MECH 364 Road Map
Time Response AnalysisTime Response Analysis• Time Domain:
– Independent variable is time t– System is represented by differential equations
• Free Response:– No external forcing input– Response is due to an initial-condition excitation– Exhibits “natural” response– E.g.: system shut-down conditions– Useful in measuring natural frequency and damping ratio
• Forced Response:– Response is due to a forcing excitation– E.g.: system start-up and operating conditions– Initial-condition effects are not present (or, decayed due to
damping)– Useful in determining resonant conditions
Element BehaviorElement Behavior
Mechanical ElementsMechanical ElementsMass/Inertia Element
Spring/Flexibility Element
k
b
Damper/Dissipation Element
Mass (Inertia) Element
Constitutive Equation (Newton’s 2nd Law): m dv
dtf=
; m = mass (inertia) Power = fv = rate of change of energy è
E fv dt m dvdt
v dt mv dv= = = ∫∫∫ è Energy E mv=12
2 (Kinetic Energy)
Note: Energy storage element
Integrate constitutive equation è v t vm
f dtt
( ) ( )= +−
−∫0 1
0
Set t = +0 è v v( ) ( )0 0+ −= unless force is infinite. Note: 0− denotes instant just before t = 0 and 0+ denotes instant just after t = 0. Observations:
1. Velocity (across variable) represents the state of an inertia element è “A=Type Element” Notes: 1. Velocity at any t is completely determined from initial velocity and the
applied force; 2. Energy of inertia element is represented by v alone. 2. Velocity across an inertia element cannot change instantaneously unless infinite
force/torque applied. 3. A finite force cannot cause an infinite acceleration in inertia element. A finite
instantaneous change (step) in velocity needs an infinite force è v is a natural output (or state) variable and f is a natural input variable for inertia element.
Spring (Flexibility) Element
Constitutive Equation (Hook’s Law): dfdt
kv= ; k = stiffness Note: Differentiated version of familiar force-deflection Hooke’s law in order to use velocity (as for inertia element)
Energy E fv dt fk
dfdt
dtk
f df= = = ∫∫∫1 1
è Energy E fk
=12
2
(Elastic potential energy)
f t f k v dtt
( ) ( )= +−
−∫00
è f f( ) ( )0 0+ −= unless an infinite velocity is applied
Note: Energy storage element Observations: 1. Force (through variable) represents state of spring element è “T-Type Element” Justified because: 1. Spring force of a spring at time t is completely
determined from initial force and applied velocity; 2. Spring energy is represented by f alone.
2. Force through stiffness element cannot change instantaneously unless an infinite velocity is applied to it.
3. Force f is a natural output (state) variable and v is a natural input variable for spring.
Note: Displacement x may be used in place of force f in the above discussion.
Damping (Dissipation) Element
Constitutive Equation: f bv= b = damping constant (damping coefficient); for viscous damping
Observations:
1. This is an energy dissipating element (D-Type Element) 2. Either f or v may represent the state 3. No new state variable is defined by this element.
Gravitation Potential EnergyGravitation Potential EnergyWork done by external force in raising an object against gravity
Lumped mass m is raised to height y by a constant external force f(Note: f = mg in order to avoid acceleration). Work done by the external force è Energy:
Gravitational Potential Energy: PE = mgy
E fdy mgdy mgy= = =∫ ∫
m
y
f = mg
mg
GroundReference
Constitutive (Physical) Equations
Constitutive Relations for:
System Energy Storage Elements Energy Dissipating Elements
Type A-Type (Across) Element
T-Type (Through) Element
D-Type (Dissipative) Element
Translatory- Mechanical v = velocity f = force
Mass m dv
dtf=
(Newton’s 2nd Law) m = mass
Spring dfdt
kv=
(Hooke’s Law) k = stiffness
Viscous Damper f bv=
b = damping constant
Electrical v = voltage i = current
Capacitor C dv
dti=
C = capacitance
Inductor L di
dtv=
L = inductance
Resistor Ri v= R = resistance
Thermal T = temperature difference Q = heat transfer rate
Thermal Capacitor C dT
dtQt =
Ct = thermal capacitance
None Thermal Resistor R Q Tt =
Rt = thermal resistance
Fluid P = pressure
difference Q = volume flow rate
Fluid Capacitor C dP
dtQf =
Cf = fluid capacitance
Fluid Inertor I dQ
dtPf =
If = inertance
Fluid Resistor R Q Pf =
Rf = fluid resistance
Naturally Oscillating SystemsNaturally Oscillating Systems
km
x
(a)
(d)
l
mg
θ
(b)
KJ θ
(c)x
m
k1 k2k = k1+ k2
y
yh
l
Mass density = ρ
A(e)
(f)i
vC
VL
L
C+ -
Six Examples of Single-D.O.F Oscillatory Systems: (a) Translatory, (b) Rotatory, (c) Flexural, (d) Pendulous, (e) Liquid slosh, and (f) Electrical.
UndampedUndamped OscillatorOscillator
Mass-Spring System(Simplified model of a rail car impacting against a snubber)
Conservation of Energy:Differentiate: Generally at all tè
è
Natural frequency:
UndampedUndamped OscillatorOscillator
2 21 12 2mx kx constant+ =&
mxx kxx&&& &+ = 0&x ≠ 0 &&x k
mx+ = 0
&&x xn+ =ω 2 0ω n
km=
k
m
x
General Solution of Equation of Motion:Proof: Satisfies the equation; Has two unknowns (A and )Initial Conditions:A = amplitude; = phase angle
Free Free UndampedUndamped ResponseResponsesin( )nx A tω φ= +
φ(0) sin ; (0) coso o nx x A x v Aφ ω φ= = = =&
φA x
vo
o
n
= +22
2ω
φω
= −tan 1 n o
o
xv
Natural Frequencies of Natural Frequencies of Six Types of Systems Six Types of Systems
Approaches for Equations of MotionApproaches for Equations of Motion
Example: Conveyor System OscillationsExample: Conveyor System Oscillations
Tracked Conveyor System
Free-Body Diagram
( )1 22 0Jm x k k xr
+ + + =
&&
( )1 2 2/nJk k mr
ω = + +
k k keq = +1 2m m Jreq = + 2
( )K r k keq = +21 2J r m Jeq = +2
Heavy SpringHeavy Spring
Distributed Parameter (Continuous) System: A Heavy Spring
Note: Mass and flexibility are distributed throughout the spring
(not at a few discrete points)
Lumped Model of a Heavy Spring
ms = mass of spring; k = stiffness of spring; l = length of spring One end fixed and the other end moving at velocity v
Kinetic Energy Equivalence
Local speed of element δx = vlx . Element mass = x
lms δ è
Element kinetic energy KE = 2)(21 v
lxx
lms δ
As δx → dx, Total KE = 321
21)(
21 2
0
23
2
0
2 vmdxxlvmv
lxdx
lm s
ls
ls ∫∫ ==
è Equivalent lumped mass concentrated at free end = ×31 spring mass
Assumption: Conditions are uniform along the spring.
Lumped-Parameter Approximation of a Heavy Spring
Note: Natural frequency equivalence may give a different approximate model
Free (Unforced) Response of Free (Unforced) Response of Damped OscillatorDamped Oscillator
Natural Frequency:
Damped Oscillator (Free)Damped Oscillator (Free)
kx
b
k
m
m
mx bx kx&& &+ + = 0
&& &x x xn n+ + =2 02ζω ω
ω nkm=
2ζω nbm
=
ζ = 12
bkmDamping Ratio:
Damping constantdamping ratioDamping constant for critically damped conditions
ζ = =Note:
Steps:1. Substitute into equation of motion (i.e., seek) solution of the
form:2. Solve the resulting characteristic equation:
to give roots:
3. General response:
4. Determine the unknowns using initial conditions (ICs)
Free Response DeterminationFree Response Determination
Three Cases of Roots:1. Real and unequal è Overdamped (Nonoscillatory)2. Real and equal è Critically damped (Nonoscillatory)3. Complex conjugate è Underdamped (Oscillatory)
x Ce t= λ
λ ζω λ ω2 22 0+ + =n n
λ ζω ζ ω= − ± −n n2 1 1 2 and λ λ=
1 21 2
t tx C e C eλ λ= + λ λ1 2≠
x C e C tet t= +1 2λ λ λ λ λ1 2= =
Free Response ResultsFree Response Results
Typical Time ResponsesTypical Time Responses
0 0.2 0.4 0.6 0.8 1
-0.4
-0.2
0
0.2
0.4
0.6Responsex (m)
ζ = 0.5
Time t (s)
(a)
0 0.2 0.4 0.6 0.8 1
-0.4
-0.2
0
0.2
0.4
0.6Responsex (m) ζ = 1.0
Time t (s)
ζ = 2.0ζ = 4.0
(b)
Period:
Logarithmic Decrement Method ofLogarithmic Decrement Method ofDamping MeasurementDamping Measurement
Decay Ratio During r Periods:
Td
=2πω
( )
sin( )( )( ) sin[ ( ) ]
n
n
td
t rTd
Ae tx tx t rT Ae t rT
ζω
ζω
ω φω φ
−
− +
+=
+ + +
Note:2
2
1( )
nn
n
rt
rTt rT
e e ee
π ζζω
ζζωζω
−−
− += = = 2 and d rT rω π=2
2 21
n nd
T π πω ω
ω ζ= =
−
2
2exp1
i
i r
A rA
π ζ
ζ+
=
− Using peaks è
è
2
Logarithmic decrement per cycle:
1 2ln1
i
i r
Ar A
πζδ
ζ+
= =
− ( )2
1 = Logarithmic decrement per radian21 2 /
δζ
ππ δ= ≈
+è
Dependence of Free Response Dependence of Free Response (Stability) on Pole Location(Stability) on Pole Location
A and B are stable; C is marginally stable; D and E are unstable
Im
Re
s-Plane
(Eigenvalue Plane)
E
E
A
A
DB
C
C
Forced Response of Forced Response of Damped OscillatorDamped Oscillator
Equation of Motion:
Forced OscillatorForced Oscillator
è
Massm
Viscous Damperb
Springk
y
f(t)
( )my by ky f t+ + =&& &
&& & ( )y y y u tn n n+ + =2 2 2ζω ω ω
Note: In control systems nomenclature, u = input; y = output
Normalized force u(t) = f(t)/k (has units of displacement)
Forced Response ComponentsForced Response Components
Particular Solution Method1. Add a suitable particular solution (a solution that satisfies the forced
equation) to the homogeneous solution (general solution of free response)2. Determine the unknown constants (in the homogeneous solution) by
using ICs
Convolution Integral Method1. Determine the impulse response h(t)—the response to a unit impulse
(assumes zero ICs)2. Use convolution integral to determine the zero-IC response:
3. Add the zero-input response (i.e., IC response)
Laplace Transform Method1. In the equation of motion represent time-derivative by Laplace variable
s and corresponding ICs2. Express the forcing function (input) by its Laplace transform 3. Express the response in terms of s (algebraic manipulations)4. Using Laplace tables determine the inverse Laplace
Methods of Forced Response DeterminationMethods of Forced Response Determination
0 0( ) ( ) ( ) ( ) ( )y t h t u d h u t dτ τ τ τ τ τ
∞ ∞= − = −∫ ∫
Particular Solutions for Useful Input Functions
Useful Results on Useful Results on Forced Response Forced Response