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# Engineering science lesson 3

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### Engineering science lesson 3

1. 1. Chapter 2- Dynamic Engineering Systems 2.1 Uniform acceleration • linear and angular acceleration • Newton’s laws of motion • mass, moment of inertia and radius of gyration of rotating components • combined linear and angular motion • effects of friction 2.2 Energy transfer • gravitational potential energy • linear and angular kinetic energy • strain energy • principle of conservation of energy • work-energy transfer in systems with combine linear and angular motion • effects of impact loading 2.3 Oscillating mechanical systems • simple harmonic motion • linear and transverse systems; • qualitative description of the effects of forcing and damping
2. 2. Review Hook’s law relaxed position FX = 0 x x=0 relaxed position FX = -kx > 0 x x<0 x=0
3. 3. Simple Harmonic Motion (SHM)• Definition: The motion that occurs when an object is accelerated towards a mid- point. The size of the acceleration is dependent upon the distance of the object from the mid-point. Very common type of motion eg. sea waves, pendulums, springs
4. 4. Simple harmonic motion X=0 X=A; v=0; a=-amax X=0; v=-vmax; a=0 F=-kx F=ma X=-A; v=0; a=amax ma = -kx a= -kx/m X=0; v=vmax; a=-0 x = -ma/k X=A; v=0; a=-amax X=-A X=A
5. 5. Properties
6. 6. SHM plotted on a paper
7. 7. Basic definitions
8. 8. Transverse and longitudinal waves
9. 9. Relation between SHM & Circular motion
10. 10. Simple Harmonic Motionx(t) = [A]cos(ωt)v(t) = -[Aω]sin(ωt)a(t) = -[Aω2]cos(ωt) Maximum value xmax = A Angular frequency vmax = Aω ω = 2πf = 2π/T amax = Aω2
11. 11. Pendulum• A mass, called a bob, suspended from a fixed point so that it can swing in an arc determined by its momentum and the force of gravity.• The length of a pendulum is the distance from the point of suspension to the center of -Lmg sinθ =Iα gravity of the bob. α = -(Lmg/I) θ and α= - ω 2 θ Lmg ∴ ω= I
12. 12. • A clock has a pendulum that performs one full swing every 1.0 sec. The object at the end of the string weights 10.0 N. What is the length of the pendulum?
13. 13. Damped oscillations• Only ideal systems oscillate indefinitely• In real systems, friction retards the motion• Friction reduces the total energy of the system and the oscillation is said to be damped• Example: Shock absorber: With a low viscosity fluid, the vibrating motion is preserved, but the amplitude of vibration decreases: This is known as underdamped oscillation
14. 14. More Types of Damping• With a higher viscosity, the object returns rapidly to equilibrium after it is released and does not oscillate – The system is said to be critically damped• With an even higher viscosity, the piston returns to equilibrium without passing through the equilibrium position, but the time required is longer – This is said to be over damped Plot a : under damped Plot b : critically damped Plot c : over damped
15. 15. Combination of springs
16. 16. Questions1. A mass spring system has m=5kg and k=2000N/m, A=50cm. Find the velocity and acceleration when x=30cm.2. A mass spring system has m=5kg and k=1600N/m, A=2m. Find 1. Total energy 2. Kinetic energy to potential energy ratio when x=1m3. A k=800N/m spring is stretched by 200N force on a horizontal surface with a 4kg object connected to the one end of it and released. Find the 1. Amplitude 2. Maximum acceleration and velocity 3. Find the velocity when x=30cm and x=-30cm
17. 17. • An object performs simple harmonic motion, its position given by x(t)=20cos(31.4t) in cm. Find the – Amplitude – Frequency and period of oscillation• An object performs SHM, its position is given by x(t)=50cos(20πt) in cm. Find the – Maximum value of displacement, velocity and acceleration – Number of oscillations in 5 sec.• An object makes 10 complete oscillations in 5 seconds between two points 40cm apart. – Write an expressions for displacement and velocity – Find the position and speed at t=0.125s• An oscillations having an amplitude of 10cm and a period of 4s starts from Xmax at t=0 – Find x at t=1s – Find x at t=0.5s