JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III Year DYNAMICS OF MACHINERY Previous Question Paper (March - 2021 - R18)

To determine the input torque T on link AB for static equilibrium, we use the principle of virtual work or the method of superposition of forces. Given the configuration, we first determine the instantaneous centers of rotation or use the velocity ratio method to relate the input torque to the output force.

Let the angular velocity of the input link AB be ω_AB. By the principle of virtual work, the sum of virtual work done by external forces and torques must be zero: T × δθ_AB + F × δs_DC = 0, where δs_DC is the virtual displacement of the point of application of force F on link DC.

1. Kinematic Analysis: Determine the velocity of point E (where force F acts) relative to the fixed link AD. Using the velocity polygon or instantaneous center method, find the ratio of the velocity of point E (v_E) to the angular velocity of link AB (ω_AB).

2. Force Analysis: The force F = 1500 N acts on link DC. The virtual work done by this force is F × v_E × cos(φ), where φ is the angle between the force vector and the direction of velocity v_E.

3. Torque Calculation: The equilibrium condition is T = -(F × v_E × cos(φ)) / ω_AB. Given the dimensions AD=500mm, AB=400mm, BC=1000mm, DC=750mm, and DE=350mm, construct the configuration diagram to find the specific angles and position of the instantaneous center of link DC relative to the fixed frame AD.

4. Result: By calculating the mechanical advantage (v_E / (ω_AB × AB)), the required input torque T is obtained by multiplying the force component by the effective lever arm length relative to the input crank AB. For the specified geometry, the torque T is calculated as the product of the force F and the perpendicular distance from the pivot A to the line of action of the force, adjusted by the transmission ratio of the linkage mechanism.

When a two-wheeled vehicle like a motorcycle takes a turn, it must lean inward to maintain stability. This lean is necessary to balance the centrifugal force acting outward at the center of gravity with the inward component of the normal reaction force.

Consider a vehicle of mass m moving at velocity v around a curve of radius R. The centrifugal force F_c = mv²/R acts horizontally outward at the center of gravity (CG). The weight mg acts vertically downward at the CG. To prevent overturning, the vehicle leans at an angle θ (angle of heel) with the vertical.

Taking moments about the point of contact with the ground, the condition for stability is that the moment of the centrifugal force must be balanced by the moment of the weight:

Where h is the height of the CG from the ground. Simplifying this equation:

This angle θ is the required angle of banking or lean. If the vehicle leans at an angle greater than this, it tends to fall inward; if less, it tends to skid outward. Stability is further influenced by the gyroscopic effect of the rotating wheels. The angular momentum of the wheels creates a gyroscopic couple when the vehicle steers, which acts to either assist or resist the leaning motion depending on the direction of rotation relative to the turn. For a standard motorcycle, the gyroscopic couple C = Iωω_p (where I is the moment of inertia of the wheels, ω is the angular velocity of the wheels, and ω_p is the angular velocity of precession) acts to maintain the vehicle's upright orientation or assist in the turn depending on the wheel rotation direction.