Question 13.4.3: Determine the mode shapes and mode frequencies of the vehicl......

The mode ratio can be found from either (13.4.5) or (13.4.6). Choosing the former, we obtain

\left(m s^2+k_1+k_2\right) A_1+\left(k_2 L_2-k_1 L_1\right) A_2=0              (13.4.5)

\left(k_2 L_2-k_1 L_1\right) A_1+\left(I_G s^2+k_1 L_1^2+k_2 L_2^2\right) A_2=0            (13.4.6)

\frac{A_1}{A_2}=\frac{k_1 L_1  –  k_2 L_2}{m s^2  +   k_1  +  k_2}                  (1)

The mode ratio A_1/A_2 can be thought of as the ratio of the amplitudes of x and θ in that mode. From Figure 13.4.3a we find that

and for small angles θ,

D \approx \frac{x}{\theta}=\frac{A_1}{A_2}            (2)

The distance D locates a point called a “node” or “motion center” at which no motion occurs (that is, a passenger located at a node would not move if the vehicle were moving in the corresponding mode). Thus, there are two nodes, one for each mode.
Equation (1) shows that A_1, the amplitude of x, will be zero if

k_1 L_1-k_2 L_2=0              (3)

In this case, equation (2) shows that D = 0. Thus, no coupling exists between the bounce motion x and the pitch motion θ, and the node for each mode is at the mass center. As we will discuss shortly, this condition will result in poor ride quality. Note also, that if equation (3) is satisfied, the characteristic equation (13.4.7) can be factored as follows:

\left(m s^2+k_1+k_2\right)\left(I_G s^2+k_1 L_1^2+k_2 L_2^2\right)+\left(k_2 L_2-k_1 L_1\right)^2=0              (13.4.7)

or

m s^2+k_1+k_2=0              (4)

I_G s^2+k_1 L_1^2+k_2 L_2^2=0            (5)

Each of these equations has a pair of imaginary roots. Thus both modes are oscillatory, and the modal frequencies are

\omega_1=\sqrt{\frac{k_1  +  k_2}{m}}

\omega_2=\sqrt{\frac{k_1 L_1^2  +  k_2 L_2^2}{I_G}}