Question

  

Create a simulink model of the nonlinear equation (1) and obtain a plot from 0 to 3 seconds of the

pendulum problem. Be sure to include a scope box so the displacement plot can be viewed.

d2q/dt2 + (g/L)*sinq = 0

q = angle of displacement of the pendulum [rad]

g = gravitational constant 9.81 [m/sec2 ]

L = pendulum length [m]

      The sinq term introduces the nonlinearity. This makes the solution difficult. To simplify the problem,       equation is linearized by using the approximation that sin q = q. This is a valid approximation for small angles of q only. Run

and create a scope plot for displacement vs time for approximately 5 cycles of a 0.5 m long pendulum that has a starting angular displacement of 10 degrees and an initial angular velocity of – 10 m/sec. Note use the linearized equation.

  

Answer 1

TO REALIZE THE DIFFERENTIAL EQUATION IN LINEAR FORM , CONNECT THE BLOCKS AS PER THE SHOWN BELOW

STEP01 Understand that every time integrator is applied the differntial order decreases by one hence we neeed q (displacement in rad) vs time graph so we need to get q(rad) as output and connect it to scope .

There are two gains which you are seeing as feedback first gain block is having value zero beacuse we do not have any dq/dt term is orignial equation.

second term of gain denoted as k in above figure is (g/L). so just click it and put in block value space as 9.81/0.5 .

where g=9.81m/sec

and L=0.5m. given

step(02) there is intail angular velocity dq/dt is given =-10m/sec so to enter that click on first integrator (1/s) block and enter data as shown below.

Step(03) similarly for displacement intial value click on second (1/s) block and enter as shown below ...

please note enter the data in radians for intail value of q

q in degree is 10

q in rad=10*pi/180

          =0.17453

Step(04)

we need to run for five cycles.

FOR PENDULUM time period

T=2*pi *underroot(L/g)

T=1.4185    sec             put L=0.5 meter and g=9.81m/sec2

for time period of five cycle =5*T=5*1.4185=7.09 sec

Hence run the simulation for approx 8 sec

and click on scope to see output dispalcement vs time graph

Note : if you face any problem segrading steps .feel free to ask in comment section.