Case studies and course project

This course has developed the core of classical mechanics: kinematics with constant and changing acceleration, Newton's laws and their application to springs, drag, curves and accelerating frames, work, energy and stability, momentum and collisions, rotation, equilibrium and elasticity, gravitation and orbits, and oscillations and waves. Real engineering problems rarely sit inside one lesson. This lesson works through three case studies that combine several: stopping a car safely with a highway crash cushion, sizing the drive of a passenger elevator, and estimating the power, loads and vibration of a large wind turbine. Each starts with a physical question, chooses the principles that answer it most simply, and checks the result. The lesson then sets out the course project and looks ahead.

Case study 1: a highway crash cushion

Crash cushions (impact attenuators) protect vehicles that leave the road in front of rigid hazards such as bridge piers and the ends of concrete barriers. Their job is to bring a car to rest over a controlled distance, keeping the deceleration of its occupants survivable. Energy and momentum give the essential numbers.

Different cushions use different physics. Crushable and hydraulic units absorb energy by deformation or by pushing fluid through orifices (work against a force). Arrays of sand-filled barrels instead transfer momentum to the sand they hit: each barrel the car strikes takes up some of its momentum, so the car slows progressively, an application of conservation of momentum in a sequence of inelastic collisions. Because kinetic energy grows as v², a car at 120 km/h carries 44% more energy than at 100 km/h, so a cushion rated for the lower speed may be overwhelmed at the higher one.

Case study 2: sizing an elevator drive

A traction elevator hangs the car and a counterweight on opposite ends of steel ropes passing over a driven sheave. The counterweight typically equals the car mass plus about half the rated load, so the motor never has to lift the whole load, only the imbalance.

The rope tension on the car side during upward acceleration is 2200 × (9.81 + 1.0) = 23.8 kN. Shared among six ropes, each carries about 4 kN, and elevator codes require the ropes' breaking strength to exceed the working load by a large safety factor, around ten or more, to allow for wear, fatigue over millions of bending cycles over the sheave, and emergency stops (Lesson 9). The same analysis, with different numbers, applies to mine hoists, cranes and funiculars.

Case study 3: a wind turbine

A modern wind turbine combines almost every topic of this course: the kinetic energy of moving air, rotational motion of the rotor, large centripetal forces in the blades, and vibration of a tall, slender tower.

The rotor turns slowly but its blade tips move fast. Turbines are designed for a tip speed ratio (tip speed divided by wind speed) of about 7 to 9. At a ratio of 8 in a 10 m/s wind, the tip moves at 80 m/s, so the rotor turns at \(\omega = 80/60 = 1.33\) rad/s, about 13 revolutions per minute (Lesson 8). Each blade, perhaps 20 t with its center of mass 20 m from the hub, then pulls on the hub with a centripetal force of \(m\omega^2r = 20\,000 \times 1.78 \times 20 = 710\) kN (Lesson 4), on top of its weight and the aerodynamic loads, all cycling once per revolution: fatigue over 20 to 30 years and some 10⁸ load cycles governs the blade root design.

Finally, vibration. The rotor excites the tower at its rotation frequency (called 1P, here 0.21 Hz) and at the blade-passing frequency (3P, 0.64 Hz for three blades). The tower's first natural frequency (Lesson 11) must avoid both to prevent resonance. Most large turbines use a "soft-stiff" tower, with a natural frequency between 1P and 3P, around 0.3 to 0.4 Hz, which is lighter and cheaper than a tower stiff enough to stay above 3P. Variable-speed turbines must avoid lingering at speeds where 1P or 3P crosses the tower frequency.

The method behind the case studies

Each case study followed the same steps:

• State the question in physical terms: a stopping distance, a power, a force, a frequency.
• Choose the simplest principle that answers it: energy when speeds and distances matter, momentum for impacts and flows, Newton's second law for forces and accelerations, natural frequencies for vibration.
• Make the idealizations explicit: constant deceleration, rigid car, uniform wind, and decide whether each is conservative.
• Check the answer two ways where possible (energy and kinematics gave the same cushion force) and against experience.
• Interpret the result as a design decision with a margin, and note what a more detailed analysis would add.

The course project

Choose one of the following problems, or propose your own of similar scope with your instructor's approval, and prepare a report with your physical model, calculations, a measurement or simulation where possible, checks and conclusions.

• Project A: Vehicle performance. For a real car, bicycle or electric scooter, estimate drag and rolling resistance from published data or a coast-down test, find the power needed at several speeds, the energy per kilometer and the effect of a hill, and compare with stated range or fuel consumption.
• Project B: Measuring g and damping. Build a pendulum or mass-spring system, measure its period for several lengths or masses, determine g or k with an uncertainty estimate, and measure how quickly the amplitude decays.
• Project C: Structure and stability. Analyze a crane, a ladder, a cantilevered sign or a tower: find support reactions, the tipping or slipping condition, member stresses and the deformation under load, and propose a safety margin.
• Project D: Energy systems. Compare two ways of storing or delivering energy (for example, pumped hydro and batteries for a community, or a flywheel and a battery for a bus), using energy, power, efficiency and mass, and recommend one with reasons.

Every report should show free-body diagrams where forces are involved, state assumptions and their effect, check units and orders of magnitude, compare at least one result with an independent estimate or measurement, and express conclusions in engineering terms.

Looking ahead

Mechanics is the foundation for much of engineering. Statics and dynamics (ENGG-130 and later dynamics courses) develop the methods of this course into systematic tools for structures and machines; strength of materials extends stress and strain to beams, shafts and columns; fluid mechanics applies Newton's laws and energy to flowing water and air; vibrations and control engineering build on oscillations; and PHYS-131 turns from mechanical forces to electric and magnetic ones, using the same ideas of fields, potential energy and conservation laws.