An interactive introduction to MATLAB

78 additional exercises

10. 2D plotting
    The vibrations of a helicopter due to the periodic force applied by the
    rotation of the rotor can be modelled by a frictionless spring-mass-damper
    system subjected to an external periodic force as shown in Figure 20.
    The positionx(t)of the mass is given by the equation:

m

F(t)

c

k

x

Figure 20: Modelling helicopter rotor vibrations with spring-mass-damper system

x(t) =

2f 0

ωn^2 −ω 2 sin

(

ωn−ω

2 t

)

sin

(

ωn−ω

2 t

)

,

whereF(t) =F 0 sin(ωt), andf 0 =Fm^0 ,ωis the frequency of the applied

force, and ωn is the natural frequency of the helicopter. When the

value ofωis close to the value ofωn the vibration consists of fast

oscillation with slowly changing amplitude called beat. UseFm^0 =12 N/kg,

ωn=10 rad/s, andω=12 rad/sto plotx(t)as a function of t for

06 t 6 10 s.

11. 2D plotting
    The ideal gas equation states thatPVRT=n, wherePis the pressure,V
    is the volume,Tis the temperature,R=0.08206(L atm)/(mole K)is
    the gas constant, andnis the number of moles. For one mole (n= 1 )
    the quantityPVRT is a constant equal to 1 at all pressures. Real gases,
    especially at high pressures, deviate from this behaviour. Their response
    can be modelled with the van der Waals equation:

P=VnRT−nb−n

(^2) a
V^2 ,
whereaandbare material constants. Consider 1 mole (n= 1 ) of nitrogen
gas atT =300 K(a=1.39 L^2 atm/mole^2 , andb=0.0391 L/mole).
Use the van der Waals equation to calculatePas a function ofVfor
0.08 6 V 6 6 L, using increments of0.02 L. At each value ofVcalculate
the value ofPVRT and make a plot ofPVRT versusP. Does the response of
nitrogen agree with the ideal gas equation?