Parameter ID of a SpringMassDamper System

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Parameter ID of a Spring-Mass-Damper System
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Car 1
Car 2
Car 3
car 3
car 2
car 1
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• operational concept
• we input a torque at the motor which exerts a
  force on the car
• the position of the car is measured by the
  encoder
• shutdown safety switches stop unstable operation

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position encoder
shutdown switch
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• Where do we start?
• learn about the encoders (one for each car)
• the encoder somehow measures the position of
  the car

? ? Will learn next week.
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scale (for human use)
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• for our problem we will most likely
• slide the block to the zero position by hand
• then enter a command on the PC that tells it we
  are at the zero position

scale (for human use)
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• What do we want to do with this?
• model it
• obtain m1, m2, m3 car masses k1, k2, k3,
  k4 spring constants L01, L02, L03, L04 spring
  free lengths c damper value
• write the governing equations of motion of the
  system in terms of these parameters
• create a Simulink model

Car 1
Car 2
Car 3
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• What do we want to do with this?
• model it (cont)
• design a controller
• control it
• implement the controller
• determine value for input force at each instant
  of time in order to position the cars as specified

Car 1
Car 2
Car 3
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• we have a way to measure the position of each car
• our only input to the system is a force applied
  to the first car
• what do we want to control?

the position of the cars
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• lets start with a one car system
• m1 mass of the car
• k1, L01 spring constant and free length
• c damping coefficient

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• How to determine thevalues for the constantsm1,
  k1, L01, and c ?
• Lets write the response of the system
• if c0 and at t0, x1xstart, and (0) 0
• Also we will apply no external force
• i.e. f(t) 0
• equation of motion ?

How to determine the values for the constants m1,
k1, L01, and c?
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How to determine the values for the constants m1,
k1, L01, and c?
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• suppose
• m1 2 kg 2 N sec2/m k1 3 N/cm 300
  N/m L01 6 cm 0.06 m
• xstart 10 cm 0.1 m

How to determine the values for the constants m1,
k1, L01, and c?
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• x1(t) 0.06 0.04 cos(12.2474 t)

How to determine the values for the constants m1,
k1, L01, and c?
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• in general

How to determine the values for the constants m1,
k1, L01, and c?
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How to determine the values for the constants m1,
k1, L01, and c?
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• another way to represent sinusoids is by using
  Eulers formula
• ej?t cos (?t) j sin (?t)
• since cos(a) cos(-a) and sin(a) sin(-a)

How to determine the values for the constants m1,
k1, L01, and c?
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• so, we can disconnect the damper, pull the car to
  some starting value and record the motion
• the frequency will equal

How to determine the values for the constants m1,
k1, L01, and c?
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• but this gives us the value of
• how to get m1 and k1?

How to determine the values for the constants m1,
k1, L01, and c?
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• let m1 be the mass of the car with no extra
  weights on it
• pull car to starting point and determinefrequenc
  y of vibrations,
• add a known weight, ?m, to car and repeat

How to determine the values for the constants m1,
k1, L01, and c?
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• we have two equations in two unknowns
• ?12 m1 k1
• ?22 (m1?m) k1

How to determine the values for the constants m1,
k1, L01, and c?
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• use a large ?m so ?1 and ?2 will differ enough so
  that any error in measuring the frequencies will
  not significantly impact the calculations

How to determine the values for the constants m1,
k1, L01, and c?
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• but in reality, even with the damper
  disconnected, there will be some damping due to
  friction

How to determine the values for the constants m1,
k1, L01, and c?
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• with friction
• equation of motion

How to determine the values for the constants m1,
k1, L01, and c?
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• with friction
• equation of motion

due to friction
How to determine the values for the constants m1,
k1, L01, and c?
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How to determine the values for the constants m1,
k1, L01, and c?
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• suppose
• m1 2 kg 2 N sec2/m k1 3 N/cm 300
  N/m L01 6 cm 0.06 m
• c 2.5 N sec/m
• xstart 10 cm 0.1 m

How to determine the values for the constants m1,
k1, L01, and c?
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c25
c250
c2.5
How to determine the values for the constants m1,
k1, L01, and c?
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How to determine the values for the constants m1,
k1, L01, and c?
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• suppose the damping is small (since its due to
  only friction)
• c2 4 m1 k1 will be negative

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• lets define two new terms
• now the term can be written as

units of rad/sec
dimensionless
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where
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• now if c2 lt (4 m1 k1) then lt 1

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c2 lt (4 m1 k1), ? lt1

• another way to represent sinusoids is by using
  Eulers formula
• ej?t cos (?t) j sin (?t)

How to determine the values for the constants m1,
k1, L01, and c?
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c2 lt (4 m1 k1), ? lt1

• but for small c, c2 4 m1k1 will be negative

How to determine the values for the constants m1,
k1, L01, and c?
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c2 lt (4 m1 k1), ? lt1
How to determine the values for the constants m1,
k1, L01, and c?
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c2 lt (4 m1 k1), ? lt1
How to determine the values for the constants m1,
k1, L01, and c?
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c2 lt (4 m1 k1), ? lt1

• could leave result as is, but will make one last
  change
• suppose we have
• A1 cos(?t) A2 sin(?t)
• this can be written as

call this sin(f)
call this cos(f)
f is uniquely determined
How to determine the values for the constants m1,
k1, L01, and c?
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c2 lt (4 m1 k1), ? lt1

• now have
• where

How to determine the values for the constants m1,
k1, L01, and c?
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c2 lt (4 m1 k1), ? lt1

• applying to our case

or
where
How to determine the values for the constants m1,
k1, L01, and c?
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c2 lt (4 m1 k1), ? lt1

• we now have the time response when there is small
  damping (c2 4 m1 k1 lt 0)

Solution Case Underdamped
How to determine the values for the constants m1,
k1, L01, and c?
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• in lab we may get a time response like the
  following

How to determine the values for the constants m1,
k1, L01, and c?
How to determine the values for the constants m1,
k1, L01, and c?
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• we can measure the frequency that will give us

How to determine the values for the constants m1,
k1, L01, and c?
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How to determine the values for the constants m1,
k1, L01, and c?
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• let t1 be the time when the first peak occurs
• let tn be the time when the nth peak occurs
• at t1 and tn

1
1
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• taking the natural log of each equation gives

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• subtracting the second from the first gives

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• but tn t1

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• L01 can be obtained from the data as the
  steadystate position
• (x1(t1)-L01) and (x1(tn)-L01) can be measured
• solve for ?

How to determine the values for the constants m1,
k1, L01, and c?
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• measure frequency of oscillation, ?d, solve for
  ?n since ? is now known
• add a known mass ?m to the car and obtain ?2 and
  then ?n2

How to determine the values for the constants m1,
k1, L01, and c?
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Use the definition of the natural frequency term
to write
Lastly, determine c for the two cases from
How to determine the values for the constants m1,
k1, L01, and c?
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