Let's solve the problem step by step: --- ### **Step 1: Given Data** - **Vehicle speed, \( v \) = 100 km/h = \( \frac{100 \times 1000}{60 \times 60} = 27.78 \) m/s** - **Spatial period of road roughness, \( L = 4 \) m** - **Mass when fully loaded, \( m_{\text{full}} = 1200 \) kg** - **Mass when empty, \( m_{\text{empty}} = 400 \) kg** - **Effective spring constant, \( k = 400 \) kN/m = \( 400,000 \) N/m** - **Damping ratio when fully loaded, \( \zeta_{\text{full}} = 0.4 \)** - **Damping ratio when empty: Not given; assume same unless otherwise noted** --- ### **Step 2: Excitation Frequency** The road profile is a sinusoid with spatial period \( L \), and the vehicle moves at speed \( v \), so the excitation frequency is: \[ f = \frac{v}{L} \] \[ f = \frac{27.78}{4} = 6.945 \text{ Hz} \] Angular excitation frequency: \[ \omega = 2\pi f = 2\pi \times 6.945 = 43.64 \text{ rad/s} \] --- ### **Step 3: Natural Frequency of the Vehicle** \[ \omega_n = \sqrt{\frac{k}{m}} \] #### **(a) Fully Loaded** \[ \omega_{n,\text{full}} = \sqrt{\frac{400,000}{1200}} = \sqrt{333.33} = 18.26 \text{ rad/s} \] #### **(b) Empty** \[ \omega_{n,\text{empty}} = \sqrt{\frac{400,000}{400}} = \sqrt{1000} = 31.62 \text{ rad/s} \] --- ### **Step 4: Frequency Ratio (\( r \))** \[ r = \frac{\omega}{\omega_n} \] #### **(a) Fully Loaded** \[ r_{\text{full}} = \frac{43.64}{18.26} = 2.39 \] #### **(b) Empty** \[ r_{\text{empty}} = \frac{43.64}{31.62} = 1.38 \] --- ### **Step 5: Magnification Factor (for Absolute Motion)** The steady-state magnification factor for absolute motion (displacement of mass relative to the base) is: \[ M = \sqrt{ \frac{1 + (2 \zeta r)^2 }{ (1 - r^2)^2 + (2 \zeta r)^2 } } \] Typically, for **absolute motion**: \[ M_{\text{abs}} = \sqrt{ \frac{1 + (2\zeta r)^2}{(1 - r^2)^2 + (2\zeta r)^2} } \] #### **(a) Fully Loaded ( \( \zeta = 0.4 \) )** \[ M_{\text{abs,full}} = \sqrt{ \frac{1 + [2 \times 0.4 \times 2.39]^2}{[1 - (2.39)^2]^2 + [2 \times 0.4 \times 2.39]^2} } \] Calculate numerator: \[ 2 \times 0.4 \times 2.39 = 1.912 \] \[ (1.912)^2 = 3.658 \] \[ \text{Numerator} = 1 + 3.658 = 4.658 \] Calculate denominator: \[ 1 - (2.39)^2 = 1 - 5.712 = -4.712 \] \[ (-4.712)^2 = 22.205 \] \[ \text{Denominator} = 22.205 + 3.658 = 25.863 \] \[ M_{\text{abs,full}} = \sqrt{ \frac{4.658}{25.863} } = \sqrt{0.1802} = 0.424 \] --- #### **(b) Empty (Assume \( \zeta = 0.4 \), unless otherwise specified)** \[ M_{\text{abs,empty}} = \sqrt{ \frac{1 + [2 \times 0.4 \times 1.38]^2}{[1 - (1.38)^2]^2 + [2 \times 0.4 \times 1.38]^2} } \] \[ 2 \times 0.4 \times 1.38 = 1.104 \] \[ (1.104)^2 = 1.219 \] \[ \text{Numerator} = 1 + 1.219 = 2.219 \] \[ 1 - (1.38)^2 = 1 - 1.9044 = -0.9044 \] \[ (-0.9044)^2 = 0.818 \] \[ \text{Denominator} = 0.818 + 1.219 = 2.037 \] \[ M_{\text{abs,empty}} = \sqrt{ \frac{2.219}{2.037} } = \sqrt{1.0895} = 1.044 \] --- ## **Final Answers:** ### **Steady-state magnification factor for the vehicle's absolute motion:** - **Fully loaded:** \( \boxed{0.424} \) - **Empty:** \( \boxed{1.044} \) --- **If you need the answer for a different value of damping ratio when empty, please specify.**