MM401 Lecture Slides (Set 1)
Assignment 2
Two Parts
1. 2DOF mass-spring-damper system.
Forced vi
ation. Phasor analysis
using Excel
2. Modelling of DC electric motor
using Simulink and Matlab.
Submission
• Submit Simulink model and Matlab
code (together in a zipped folder).
• Submit Excel workbook.
• Submit report as PDF file.
Part 1
• Forced analysis of 2DOF system with
• Two input sinusoidal forces with
same frequencies but different
phases
• Damping
• Modify Phasor Analysis workbook to
analyse system.
Part 1
Diagram shows system, where
?1 = 1 kN/m
?1 = 200 kg/s
?2 = 2 kN/m f1(t)
Part 1
?
?2 = 50 kg/s
?1 = 55 kg
?2 = 10 kg
?1= 4 sin(?? + 60°) N
?2 ? = 2sin ?? − 155° N
• Adapt Phasor Analysis workbook (checking it carefully
first) to:
• Allow all model parameters to be entered and changed
m 2
m 1
k 2
k 1
c 2 f 2 ( t )
c 1
Part 1
• For the parameters given, calculate and plot amplitudes
and phases of the displacements of both masses
against
? showing the frequency response clearly
• For the parameters given, calculate and plot the
amplitude of the force in spring ?2 against ?. Be careful!
• Label all plots and include units.
• In report:
Part 1
• Draw FBDs
• Write equations of motion
• For specified parameter values, give mass,
damping, and stiffness matrices (i.e. with
numbers)
• Specify clearly how you calculated the
amplitude of the force in the spring.
• Include plots
Part 1
• In report:
• You will analyse the difference the phasor analysis makes
to the results. To do that, include equivalent plots for the
situation where the two forces are in antiphase (i.e. where
one force amplitude is negative), and where damping is
zero. That system could be analysed without using phasor
analysis. Compare the two sets of results.
• From the results, discuss the conditions under which a
phasor analysis of the system is required for accurate
esults (in a situation where you’re just looking for rough
Part 1
estimates of the vi
ation amplitudes and the maximum
spring force).
Part 2
• Small
ushed DC electric motor:
• One cu
ent through rotating armature, ??,
and one through stator coil, ??
Part 2
?? = ??
Part 2
?? = ???
where ? = ?? ?? .
The value of ?
depends on the coil
dimensions.
? ?? is the magnetic
flux.
Motor properties:
Part 2
?? = 40Ω
?? = 5mH
? = 24gcm2
? = 1× 10−6 Nms
ad
Assume that ?? = 24 V giving ? = 0.058 N ∙ m/A.
• Create state-variable Simulink model
for the system
Part 2
• The load torque is ?|?|?, where, for
initial tests, ? = 1 × 10−6 N ∙ m ∙ s2.
• For all tests, start model with system
turned on at ? = 0, and run for 200 s
• You will vary the motor speed in two ways:
• By setting ?? equal to a constant value
etween 0 an 24V
• Using Pulse Width Modulation (PWM).
Part 2
• Use Pulse Generator block for PWM, with
frequency of 490Hz (as supplied by
Arduino) and amplitude of 24 V. Vary the
pulse width from 0% to 100%
• Measure the motor speed over time, and
find the steady-state speed
• Measure total heat generated in armature
coil (i.e. 0200׬??2????).
Part 2
• Test 1:
• Use Matlab code to run model with
constant armature voltage, and plot both
steady-state speed and total heat
generated against ?? for ?? from 0 to 24 V
• Also, plot ?? and ? against ? for ?? = 12V
and ?? = 24V (don’t need to do that with
same code).
Part 2
• Test 2:
• Use Matlab code to run model with PWM,
and plot both steady-state speed and total
heat generated against pulse width for
pulse width from 0 to 100%
• Also, plot ?? and ? against ? for pulse width
equalling 50% and equalling 100% (don’t
need to do that with same code).
Part 2
• Test 3:
• Create a linearized incremental version of
the model with constant ??, with the
operating point at the steady-state speed
• Plot ?? and ? against ? for ?? = 12V and ??
= 24V (don’t need to do that with same
code).
• Test 4:
Part 2
• Change the nonlinear model with constant
?? by adding a PID controller block (with
default settings) to control speed. Set
desired ? to 100 rad/s. Note that ?? must
e between 0 and 24V (you can use a
Saturation block)
• Plot ??, ?? and ? against ?.
• Test 5:
Part 2
• Change model so desired ? is reduced to 10
ad/s, and so that load changes over time with
? ? = 1 × 10−4 1 + sin ? N ∙ m ∙ s2
• Plot ??, ?? and ? against ?.
• Your report should contain:
• Images of the Simulink models and any Matlab code
used
Part 2
• The state variable equations for both the linearized and
nonlinear models, and the derivation of formulas for the
operating point parameters
• Details of the time-stepping methods used and
justification
• A discussion of time stepping for the different tests • The
equested results, with all plots properly labelled • A
discussion of the results from the tests.
Submission
– Submit report as a PDF file.
– Submit your Excel workbook