Question

Derive the equation of motion of the system shown in Figure Q5b, using the following methods: (0) Newton's second law of motion. (4 marks) D'Alembert's principle. (3 marks) (iii) Principle of conservation of energy. (5 marks) ki k2 000 m Figure Q5b

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Answer 1

A spring mass arrangement and its force block diagram are shown in the given figure: X k, m m kx kx (1) Write the equation of motion for the given system according to the Newton's second law is given as follows: F(t)=-kx-kx = më më +(kı +kz) x = 0 Thus, the equation of motion for the given system according to the Newton's second law is mä + (ka + ky ) x = 0.. (ii) Write the equation of motion for the given system according to the D'Alembert's principle is given as follows: F(t)- mx = 0 -kx-kzx- mä = 0 më +(k, +k, )x=0 Thus, the equation of motion for the given system according to the D'Alembert's principle is mä + k + kz) x = 0..

Calculate the kinetic energy (T) of given system as follows: T = mi? Calculate the strain energy (U) of the given system as follows: U kx+-k Calculate the total energy (TE) of the given system which is conserve for the system as follows: 1 TE kır? mi? = 0 Differentiate above equation with respect to t as follows: d (T+U) = 0 dt më +(k, +k)x=0 Thus, the motion equation for the given system according to the principle of conservation of energy is më + (k + ks)x = 0.