User:Yongtao-Li/NM.HW5.4

=Problem 5.4 a Mass-Spring-Damper with Gaussian noise and Cauchy noise=

GIVEN
An ideal mass-spring-damper system with mass m, spring constant k and viscous damper of damping coefficient c is subject to an oscillatory force u. This can be illustrated in the following image.



FIND
1)Derive equations of motion in terms of d, c, k, m, u.

2)Let $$ {\mathbf{x}} = \left\{ {\begin{array}{ccccccccccccccc} d \\  {\dot d} \end{array}} \right\} = \left\{ {\begin{array}{ccccccccccccccc}   \\ \end{array}} \right\}$$. Find $$\left( {{\mathbf{F}},{\mathbf{G}}} \right) $$

3)Find $${c_{cr}}$$ in terms of k, m st. this system is critically damped.

4)Let k=1, m=1/2, $$ {{\mathbf{x}}_0} = {\left[ {0.8, - 0.4} \right]^T} $$

a)For u=0, plot $${{\mathbf{x}}_k}$$ for $$c = \frac{1}{2}{c_{cr}},{c_{cr}},\frac{3}{2}{c_{cr}}$$.

b)For u=0.5 Gaussian noise and $$c = \frac{3}{2}{c_{cr}}$$, plot $${{\mathbf{x}}_k}$$.

c)For u=0.5 Cauchy noise and $$c = \frac{3}{2}{c_{cr}}$$, plot $${{\mathbf{x}}_k}$$.

SOLUTION
1) Since the system is subject to an oscillatory force generated from the spring, we have this oscillatory force

and also a damping force from the damper:

Since we can use [http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton.27s_second_law Newton's second law

Therefore we can derive equations of motion as following:

2) From Eq. 5.4.4, we can solve for $${\ddot d}$$ as following:

Therefore we can rewrite $${\dot x}$$ in matrix form

If we make a discretization as

and use forward Euler method

We can have

3) Since we can rewrite Eq. 5.4.4 as

where $$\omega_0 = \sqrt{ k \over m } $$, $$\zeta = { c \over 2 \sqrt{m k} }$$

When $$\zeta = 1$$, the system is said to be critically damped. A critically damped system converges to zero faster than any other and without oscillating. Therefore we have $${c_cr}$$ indicated in Eq. 5.4.12 and can look for damping for more explanations.

4) a) The following matlab codes are used to solve the problem without any noise, and the damping coefficients can be chose to create different cases with under-damping, critical damping or over-damping.



Comment:

Different c values cause different damping cases. From the pic above, we can find out a critically damped system converges to zero faster than any other and without oscillating.

b) The following matlab codes are used to solve the problem with the Gaussian Random Noise and over-damping case.



Comment:

When the system has Gaussian noise, the convergence can be archived, however the final result is not exactly zero since there is still noise all the time and the final result oscillated around zero.

c) The following matlab codes are used to solve the problem with the Cauchy Noise and over-damping case.



Comment:

When the system has Cauchy noise, the same result was generated as Gaussian noise. And the oscillation around the zero is much larger than the case with Gaussian noise.