Section B Formula sheet for questions 4-6. mx + cx + kx = F sin cot Χ _ 2πζω, 275 8=In = = X₂ @d 1-5² 2π = 0,₁ √1-5² @d *+250x+x=0 F X = [(k_o²m)² + (oxc)² ]/² CxXxX (k − w²m) tano = Where m= mass (kg) c= damping coefficient (Ns/m) k = spring stiffness (N/m) F = harmonic force (N) w = harmonic frequency (rad/s) 8 = logarithmic decrement Wn ural frequency (rad/s) @d damping frequency (rad/s) =damping ratio Td = damping period (s) *1/x₂= ratio of successive peaks X = amplitude of vibration Ø=phase angle (0) =nat- = Question 5 A machine, subject to a harmonic force of 500N with 0.5Hz frequency, is modelled as a mass of 800kg supported by a rigid spring of stiffness 20kN/m and dashpot of damping coefficient 6400Ns/m. a. Derive the equation of motion of the vibrating system and calculate its natural fre- quency and damping ratio. [8 marks] b. Draw the vector diagram of all the forces acting in the system, showing the phase re- lationships between them. [4 marks]
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