Questions for practice- MIN-511B (Modeling and Simulation)
1. An excitation pulse is concentrated on a one-degree-of-freedom system. The
system is initially at rest in equilibrium. The pulse is half sinusoidal wave (see
adjacent figure). Calculate the response of the system using Duhamel’s integral.
(tip: Product-to-Sum Formula sin(u)sin(v)=(1/2)[cos(u-v)-cos(u+v)])
2. Determine the Fourier series presentation for the excitation in figure 2. . Plot
the result using proper amount terms.
3. A spring-mass system initially at rest is subjected to a triangular impulse as
shown below. Determine the displacement, x, of the mass at a time t=ζ/2 where
ζ=natural period in seconds. Use convolution (Duhamel’s integral), and be sure
to show all calculus and algebraic steps.
Figure 3
4. Find the response of a string of length l, fixed at x = 0 and x = l, under the
action of the harmonic force f (x, t) = fo(x)𝑒 𝑖𝜔𝑡 , where ω is the forcing frequency.
Assume the initial displacement and velocity of the string to be zero.
5. Using Fourier integral formula, prove that

2 ( 2  2) cos x
e cos x  
d
0
2  4
x
6. A load of roosters 𝑚𝑟 is dropped on the floor of a truck bed. Assuming the
roosters do not move and that the truck bed is modelled as a spring-mass-damper
system (of values k, m, and c, respectively).The load is modelled as a force F(t)
= 𝑚𝑟 g applied to the spring-mass-damper system, as illustrated in the following
figure. This allows a crude analysis of the response of the truck’s suspension.
First assume that the trucks damper is broken, how does the maximum dynamic
displacement compare to the static displacement. What would happen to the
maximum displacement if the damper was repaired on the truck?
Figure 4: Dump truck being loaded with roosters showing (a) roosters going into the truck bed;
and (b) the single-degree-of-freedom vibration model.
7. In testing, a hammer is used to excite a 1-DOF system with an impact (i.e.
impulse), however, the hammer ascendingly impacts the system twice. The first
impact has a force of 0.2 N, while the second has a force of 0.1 N and happens
0.1 seconds after the first impact. Plot the response for the double impact. The
system has the parameters m = 1 kg, c = 0.5 kg/s, k = 4 N/m.
8. Considering the forced system:
Figure 5: A spring-dashpot-mass model of a 1-DOF system with external excitation.
Set the forcing function to be Fo sin(ωt) and calculate the transfer function.
9. Considering the following system:
Figure 6: 2-DOF system with two masses and two independent confidante systems x1 and x2.
Calculate response for the system if m1=9kg, m2=1kg, k1 = 24N/m, and k2 =
3N/m with the initial conditions x10 = 1mm, v10 = 0mm/s, x20 = 0mm, and v20
= 0mm/s.
10.
Plot the input and output response with the help of Matlab.
11. Find the Fourier transform of following functions:
12. The problem is to find the steady state response y(t) of a spring/mass/damper
system modelled by
Where F(t) is the periodic square wave function shown in the diagram
Figure 7
13. write the following functions in terms of unit step function(s). sketch each
waveform.
(a) A 12-v source is switched on at t = 4s.
(b)
Assume a >0.
(c) one cycle of a square wave, f(0)=4, amplitude = 4, period =2 seconds.
(d) The unit Ramp function (i.e. f(t) = t for t > 0).
(e)
14. A spring-mass system with mass 2, damping 4, and spring constant 10 is
subject to a hammer blow at time t = 0. The blow imparts a total impulse of 1 to
the system, which as initially at rest. Find the response of the system using
Laplace transformation.
15. In the mechanical system shown in Figure 8, the spring is unstreched for all t
< 0. The applied force fa(t) is zero fort < 0, but has a constant value of A for all t
> 0. Find an expression for the displacementx for all t > 0.
Figure 8: First-order mechanical system.
16. For the translational system shown in Figure 9, find unit step response yu(t)
and unit impulse response h(t) when the output is the velocity v(t).
Figure 9: Translational mechanical system.
17. Find the unit step response for a second order system described by input –
output equation.
Plot the response for several values of the damping ratio ζ, with ωn held constant.
18. Use laplace transformation to find the unit step response of marginally stable
system described by
19. Derive the transform of the triangular pulse shown in Figure 10, defined by
the equation
Figure 10: Triangular pulse.
Hint: Any pulse that consists of straight lines can be decomposed; into a sum of step
functions and ramp functions.
20. Solve the following initial value problems using properties of dirac delta and
laplace transformations:
(a)
(b)
21. The following differential equation is the equation of motion for an ideal
spring-mass system with damping and an external force F(t). find the transfer
functions using laplace transformations
Figure 11: Spring-mass system with damping.
22. Find the Poles and Zeros of the following transfer functions:
(a)
(d)
(b)
(e)
(c)
23. Compute the N-point DFT (Discrete fourier transform) of
(a) x(n)=3δ(n)
(b) x(n)=7(n−n0)
24. Calculate the four-point DFT of the aperiodic sequence x[k] of length N =
4, which is defined as follows:
25. Calculate the inverse DFT of