A general vibrating model of Such a model, that is

shown in Figure below, includes the body bounce $x,$ body roll $,$ body pitch $,$ wheels hop

$x1,x2,x3,$ and $x4$ and independent road excitations $y1,y2,y3,$ and $y4.$ A full car vibrating

model has seven DOF with the following equations of motion.

$m_f{x}_1^{}-c_f((x^{})-x_1^{}+b_1({}^{})-a_1({}^{}))-k_f(x-x_1+b_1-a_1)$

$-k_R\frac{1}{w}(-\frac{x_1-x_2}{w})+k_{t_f}(x_1-y_1)=0$

$m_f{x}_2^{}-c_f((x^{})-x_2^{}-b_2({}^{})-a_1({}^{}))-k_f(x-x_2-b_2-a_1)$

$+k_R\frac{1}{w}(-\frac{x_1-x_2}{w})+k_{t_f}(x_2-y_2)=0$

$m_r{\stackrel{}{x}}_3-c_r((x^{})-x_3^{}-b_1({}^{})+a_2({}^{}))$

$-k_r(x-x_3-b_1+a_2)+k_{t_r}(x_3-y_3)=0$

$m_r{\stackrel{}{x}}_4-c_r((x^{})-x_4^{}+b_2({}^{})+a_2({}^{}))$

$-k_r(x-x_4+b_2+a_2)+k_{t_r}(x_4-y_4)=0$

Consider a vehicle with the following characteristics.

$m=840kg,m_f=53kg,m_r=76kg$

$I_x=820kg{m}^2,I_y=1100kg{m}^2$

$a_1=1\mathrm{.}4m,a_2=1\mathrm{.}47m$

$b_1=0\mathrm{.}7m,b_2=0\mathrm{.}75m$

$k_f=10000\frac{N}{m},k_r=13000\frac{N}{m}$

$k_{t_f}=k_{t_r}=200000\frac{N}{m},k_R=10000N\frac{m}{r}ad$

Determine;

$1-$ Natural frequencies and mode shapes of the full car model by using related

commands in MATLAB script file. Plot all mode shapes.

$2-$ Suppose that the front left wheel encounters a road bump when the vehicle velocity

is $1\frac{m}{s}.$ Solve the differential equations for this kind of road excitation and plot all

states variation versus time. Use related codes in MATLAB script file. The road

bump can be approximated as half circle with $10cm$ radius.

Put all the scripts in one YourNameLastname.m format file and upload to the Moodle

webpage before due time. There is no need to prepare a report, just upload your

MATLAB script file. The evaluation is based on the results that your code gives when

it is running.