Dynamics Assignment-

Q1. The schematic of an electric vehicle is shown in Figure 1a. It is given that k₁ = 12500 N/m, k₂ = 10⁵ N/m, m₁ = 450 kg, m₂ = 24 kg, I =96 kgm², a = 0.7 m, b = 0.8 m.

a) First, neglecting the mass of the wheels and taking the equivalent spring stiffness k_e at both ends (See Figure 1b) write down the potential and kinetic energy terms in terms of y_G, θ and proceed to derive Lagrange's equations of motion. You need to first show that as k_e = 1/(1/k₁+1/k₂).

b) Arrange the equations in matrix form [K]{q} - ω²[M]{q} = {0} where {q} = .

c) Using the MATLAB code given in Appendix 1, find the natural frequencies and modes. You will need to change the definition of elements of the matrices K and M.

d) Now considering the mass of the wheels, write down the potential and kinetic energy expressions in terms of all four degrees of freedom y_L, y_R, y_G, θ and proceed to derive the equations of motion using Lagrange's formula. 

e) Arrange the equations in matrix form [K]{q} - ω²[M]{q} = {0} where {q} = .

f) Modify the MATLAB program given to find the natural frequencies and modes of the vehicle.

g) Obtain the solution to the four dof problem, for the case of m₂ = 0 and compare with results in part (b).

h) Comment on the effect of neglecting the mass of the wheels.

Q2. Two engineers (fresh graduates) Amy and Barry disagree on how to model a framework that supports a sensitive equipment which needs to be protected against vibration. They are aware that if the framework has a natural frequency close to the frequency of any disturbance, large amplitude oscillations may damage the equipment. The framework consists of two 500 mm long vertical steel (elastic modulus E = 210 GPa, density ρ = 7800 kg/m³) beams of 24 mm x 10 mm rectangular cross section, connected to a rigid platform of 30 kg mass which supports the instrument of negligible mass.  See Figure 2a.

Amy suggests modelling the beams as mass-less, fixed at the bottom, constrained against rotation at the top but able to translate, and providing a stiffness 12 EI/L³ (neglecting any gravity effect) and treating the floor as a translating mass half of which is carried by each beam. See Figure 2b for Amy's model. Barry agrees that the mass of the beams are negligible compared to the platform but argues that for this reason, the platform should be treated as fixed and that the natural frequency of the system can be found by treating the beam as clamped at both ends which gives the first natural frequency parameter λ as 4.73 (readily obtainable from textbooks) where λ = L(ρAω²/(EI))^(1/4) where A is the cross sectional area of the beam and ω is the natural frequency. See Figure 2c for Barry's model. Both agree that the top of the beam cannot rotate.

a) From first principles show that the exact frequency equation for a beam fixed at one end and carrying a translating mass at the other end is given by: sin(λ)cosh(λ) + cos(λ)sinh(λ) + rλ(cosh(λ)cos(λ)-1) = 0, where r is the ratio of the attached sliding end mass to the mass of the beam.

b) Find the first three roots of the above equation using a MATLAB program and solve the problem to compare with Amy's and Barry's models and comment on your findings. Consider the vibration of the framework in its plane (see possible modes in Figures 2b and 2c). The bending of the beam will then be about the weaker axis. This is needed to choose the correct breadth and depth in the formula for the second moment of area of a rectangle.  A MATLAB code that can be used to find the roots of a frequency equation for a different set of boundary conditions is provided to you. (Appendix 2). You will need to modify this. The statements of the frequency equation that needs to be modified are highlighted.

c) Explain why both agree that the top of the beam cannot rotate.

Attachment:- Appendixs.rar