#### Continuum Mechanics

According to the Cayley-Hamilton theorem, a second-order tensor should satisfy its own characteristic equation, thus

Using the results of Problem 1 with A = F (the deformation gradient), we find that

Let A be a given general second-order tensor and consider that we need to satisfy V.A.v>0

The behavior of an isotropic non-linear material is described by Green elastic expression and where a, b, c are constants. Please ( a ) derive the stress-strain relations for this material; ( b ) show that the constitutive equations are reduced to those of the isotropic linear elastic material for c = 0; ( c ) find the elastic moduli E and for this case.

The equilibrium equation for the linear elastic one-dimensional rod (as shown in Fig. 1), with body force neglected, is d/dx EA du/dx= 0 Please solve u1 at x=1 by Galerkin’s approach.

Assume is a symmetric second order tensor, please prove that satisfy the equilibrium equation when , Fi=0

The stress tensor at a point under the working condition is given by

Please calculate or explain that (a) EijkErjk = ? (b)EijkUjUk=?

Assuming that =find Kp and Ka such that both of the closed loop poles (roots of the characteristic equation) are at -= m = 1 = 1 and that g = 10 (instead of 9.81, to make the numbers nice),mp1.

d. A slightly more complicated controller has the form D(s) = Kp + Kas. Derive the closed-looptransfer function when this controller is in use.

c. Is there a value of K, that can make the system stable? Why or why not? (While we have not yetdiscussed controller design or stability much, you should be able to answer this based on thematerials covered already.)

Example 2.7 in the 7th edition of the FPE textbook describes a model for an inverted cart-and-pendulum system that is similar to the Segway scooter. Let's take a more detailed look at what is neededto balance an inverted pendulum or Segway. a. First, start with the equations of motion for this system that are given in the book (equation 2.30): \begin{array}{l} \left(I+m_{p} l^{2}\right) \ddot{\theta}^{\prime}-m_{p} g l \theta^{\prime}=m_{p} l \ddot{x} \\ \left(m_{t}+m_{p}\right) \ddot{x}+b \dot{x}-m_{p} l \ddot{\theta}^{\prime}=u \end{array} Derive the transfer function from the cart input U(s)to the angle O'(s).

f. Using Matlab, plot the unit step response of your closed loop system with these gains to a stepchange in reference input. Use zero initial conditions. (Submit a printout of your Matlab code andyour plot.) What happened? Did you get the response you want?

b. Now assume you have a feedback loop closed around the system with a controller D(s) as in thefollowing block diagram: Derive the closed loop transfer function from reference 0,(s) to pendulum angle O'(s), assumingthat the controller is a simple gain, D(s) = Kp. You should also assume that b 0 (makes thingscleaner for now).%3D