ME- 495
Mechanical and Thermal
Systems Lab
Fall 2011
Chapter 5: MEASURING SYSTEM RESPONSE
Professor: Sam Kassegne, PhD, PE
Signal
Response of Measurand?
RESPONSE
• System Response: an evaluation of the systems
ability to faithfully sense, transmit and present all
pertinent information included in the measurand
and exclude all else:
• Key response characteristics/components
are:
–
–
–
–
Amplitude response
Frequency response
Phase response
Slew Rate
COMPONENTS OF SYSTEM RESPONSE
1) Amplitude response:
• ability to treat all input
amplitudes uniformly
– Overdriving – exceeding
an amplifiers ability to
maintain consistent
proportional output
– Gain = Amplification =
So/Si
– Smin<Si<Smax
Overloaded in this range.
COMPONENTS OF SYSTEM RESPONSE
2) Frequency
Response
• ability to measure
all frequency
components
proportionally
• Attenuation: loss of
signal frequencies
over a specific
range
Attenuated in this range.
COMPONENTS OF SYSTEM RESPONSE
3) Phase Response
• amplifiers ability to
maintain the phase
relationships in a
complex wave.
• This is usually
important for
complex waves
unlike amplitude
and frequency
responses which
are important for
all types of input
wave forms. Why?
COMPONENTS OF SYSTEM RESPONSE
• 4) Delay/Rise time:
time delay between
start of step but
before proper output
magnitude is
reached.
• Slew rate:
maximum rate of
change that the
system can handle
(de/dt) (i.e. for
example 25
volts/microsecond)
Dynamic Characteristics of Simplified Mechanical Systems
Generalized Equation of Motion for a Spring Mass Damper
System(1-axis)
• F(t) = general excitation force
•  = fundamental circular forcing frequency
1   d 2s    ds 
F(t )  m 2        Ks
g c   dt    dt 
Ao 
F( t ) 
  Cn cos(nt  n )
2 n 1
C n  An  Bn
2
tan 
Bn
An
2
(I)FIRST ORDER SYSTEM
(I.A) Step Forced
If mass = 0, we get a first-order system.
E.g. Temperature sensing systems
• F(t)=0 for t<0
• F(t) = F0 for t >= 0
– t=time, k=deflection constant
– s=displacement, =damping coefficient
– Fo=amplitude of input force
•
This can be reduced to the general form:
(after integration over time and simplification)
–
–
–
–
ds
F( t )    ks
dt
t
P  P  PA  P e 
P=magnitude of any first order system at time t
P=limiting magnitude of the process as t 
PA=initial magnitude of process at t=0
 = time constant = /k
The above equation could be used to define processes such as a heated/cooled bulk or
mass, such as temperature sensor subjected to a step-temperature change, simple
capacitive-resistive or inductive-resistive circuits, and the decay of a radioactive source.
Figure (a) depicts progressive process
Figure (b) depicts decaying process
(I)FIRST ORDER SYSTEM
(I.B) Harmonically Excited
ds
F( t )    ks
dt 1
P
G (f ) 
d
Ps

F(t) = Fo cost
1  ( 2f) 2
Pd  m axim u m_ am pl itu de_ of _ th e_ pe riodic_ dyn am ic_ proce ss
Ps  stati c_ displace mne t _ of _ F 
Fo
k
1
T

  ti m e_ con stant 
k
ph ase_ re spon se_ be com e s: _ (f )  tan1 ( 2f)
ide al ly_ G  1 _ an d_   0 _& _ work s_ as _ 2f   1
f  fre qu e n cy
( Hz) 
PHASE LAG
First Order System – Harmonically Excited –
Example
Temperature Probe Example
TEMPERATURE PROBE EXAMPLE - Continued
TEMPERATURE PROBE EXAMPLE - Continued
(II) SECOND ORDER SYSTEM
(II.A) Step Input
1
F (t ) 
gc
  d 2 s 
 ds 


m


   Ks
  2 
 dt 
  dt 
• Step input
– F=0 when t<0
– F=Fo when t>0
• Underdamped Eq:
OVERDAMPED SECOND ORDER SYSTEM
• =  / C >1
This represents
both damped and
under-damped
cases.
(II) SECOND ORDER SYSTEM
(II.B) Harmonically Excited
F(t) = Fo cost
Second Order System – Harmonically Excited
Example
Microphone Example
MICROPHONE EXAMPLE
MICROPHONE EXAMPLE