Derive the solution of the characteristic equation, Electrical Engineering

For a vehicle suspension, a basic two-degree-of-freedom "quarter-car" model would be slightly more complicated than the spring-mass-damper system I chose to study in Figure 1. The input would be the road profile xin(t).

990_circuit.png

A step input would correspond to a sudden bump on the road with a height of 1 meter, which would be unrealistic, but we could simply look at the response to a sudden bump with a height of 10 cm for instance. The responses x1(t) and x2(t) would be proportional to what we would observe for a real step input. 

Note that Kt represents the stiffness of the tire while Ks and Cs represent the stiffness and the damping coefficient of the suspension, respectively. Mu is the unsprung mass, and Ms is the sprung mass (where the driver sits).

22_circuit1.png

 

(1a) Derive the equation of motion for the 2nd order mechanical system shown in Figure 1. Show your solutions in function of the damping ration ς and the natural frequency ωn after expressing the damping ratio ς and the natural frequency ωn as a function of M, C and K.  

 

Figure 1: Mechanical system with mass M, damping constant C, and spring constant K. The system input is the displacement F(t). The system output is the displacement x(t).

(1b) The system input is the force F(t) and the system output is the displacement x(t). Derive the transfer function of the system, i.e. X(s) / F(s) as a function of ς and ωn:     

 

(1c) What does a step input mean physically? Now, assume that M = 1 kg. Calculate the extra mass that would needed to be suddenly added to M  at  t = 0 to yield a response equivalent to a unit step response. 

 

(1d) Derive the solution of the characteristic equation of the system ODE for this system, i.e. find the open-loop poles of this system as a function of ς and ωn. You should have three different solutions based on the value of ς: 

   Case 1:  0 < ς < 1 (the system is called underdamped)   

   Case 2:  ς = 1 (the system is called critically damped)  

   Case 3:  ς > 1 (the system is called overdamped)  

 

(1e) Assuming M = 1 kg, choose a value for K and a value for C and that they yield an underdamped response. Do not make it too close to a critically damped response (pick  0.1 < ς < 0.4 ). Using Matlab, plot the step response of this underdamped system.   

Keep the same values for M (1 kg) and K, and modify the value of C such that it yields a critically damped response. Using Matlab, plot the step response of this critically damped system.  

Keep the same values for M (1 kg) and K, and modify the value of C such that it yields an overdamped response. Using Matlab, plot the step response of this overdamped system. 

Show the numerical values of your poles for each of the three answers.

(1f) Build a Simulink model of the system and simulate the same three step responses of the system than in question 1.e. Note: Do not use the state-space matrices A, B, C, and D. Build a Simulink model that directly represents the equation of motion (you should have M, C and K in your Simulink diagram).

Posted Date: 3/19/2013 2:08:51 AM | Location : United States







Related Discussions:- Derive the solution of the characteristic equation, Assignment Help, Ask Question on Derive the solution of the characteristic equation, Get Answer, Expert's Help, Derive the solution of the characteristic equation Discussions

Write discussion on Derive the solution of the characteristic equation
Your posts are moderated
Related Questions
Hybrid Model for Two-port Network:-             V 1 = h 11 i 1 + h 12 V 2                                                                             I 2 = h 1 i 1

Q. Show Spectral Analysis and Signal Bandwidth? Spectral analysis is based on the fact that a sinusoidal waveform is completely characterized by three quantities: amplitude, ph

Q. What is Session Layer? Session layer is used to allow users to identify themselves when waiting access to the network. This is concerned with setting up and sustaining an op

Explain the construction of a MOSFET. Draw the symbols and diagrams of both P-channel and N-channel MOSFET. As JFET and MOSFET also have source, drain and gate. Its gate is ins

Subtract Registers and Borrow from Accumulator The  contents of the  register  and borrow  flag  are subtracted from   the contents of the accumulator and the  result is  store

Q Determine the effective input noise temperature of a long piece of waveguide (that connects an antenna to a receiver) with a loss of 3.4 dB at 12 GHz and a physical temperature o

For the network shown in Figure, find the current delivered by the 10-V source and the voltage across the 3- resistor by means of mesh-current analysis.

who to write resume for job

Some wire of cross-sectional area 1mm 2 has a resistance of 20Ω. Determine: a)  The resistance of a wire of the same length and material if the cross-sectional area is 4 mm 2 .

Modern complex systems are normally constructed by interconnecting sub-units. Therefore the systems engineer is often not too concerned about the details of what is inside these