Principle of Virtual work - Beams

  Principle of Virtual work - Beams 

Principle of virtual work applied for beams - How to find reactions explained. Simply supported beam with two point loads.

Watch it on YouTube: https://youtu.be/RztUxwYikq0

In the below picture shown a simply supported beam with two point loads, load W1 is acting at a distance a from support A and load W2 is acting at a distance (a+b) from support A. Length of the beam is L. 

In the below pictures described the procedure to calculate reactions at supports A and D using the principle of virtual work.

Principle of Virtual work - Beams

 Principle of Virtual work - Beams 

Principle of virtual work applied for beams - How to find reactions explained. Simply supported beam with a point load.

Watch it on YouTube: https://youtu.be/aiTVHK_AtcQ 

In the below picture shown a simply supported beam with a point load at a distance of a from support A, length of the beam is L. 

In the below pictures described the procedure to calculate reactions at supports A and C using the principle of virtual work.

Principle of Virtual work - Beams

 Principle of Virtual work - Beams 

Principle of virtual work applied for beams - How to find reactions explained. Simply supported beam with a point load.

Watch it on YouTube: https://youtu.be/LdR2DjcmfXo

In the below picture shown a simply supported beam with a point load at the center of the beam, length of the beam is L. 

In the below pictures described the procedure to calculate reactions at supports A and C using the principle of virtual work.

Cam with oscillating flat faced follower - Acceleration equations

 Cam with oscillating flat faced follower - Acceleration equations

Cam with oscillating flat faced follower - Acceleration equations derived using analytical method.

Check this page for Position and Displacement equations: https://www.kinematics-mechanisms.com/2022/06/cam-with-oscillating-flat-faced.html

Check this page for velocity equations: https://www.kinematics-mechanisms.com/2022/06/cam-with-oscillating-flat-faced_22.html

To find acceleration, differentiate velocity equations with respect to time once and follow the procedure described below.

Cam with oscillating flat faced follower - Displacement and velocity equations

 Cam with oscillating flat faced follower - Displacement and velocity equations

Cam with oscillating flat faced follower - Displacement and velocity equations derived using analytical method.

Check this page for Position and Displacement equations: https://www.kinematics-mechanisms.com/2022/06/cam-with-oscillating-flat-faced.html

To find velocity, differentiate displacement equations with respect to time once and follow the procedure described below.

Cam with oscillating flat faced follower - position and displacement equations

 Cam with oscillating flat faced follower - position and displacement equations

Cam with oscillating flat faced follower - Position and displacement equations derived using analytical method.

In the below picture shown is a cam with oscillating flat faced follower.

The above cam follower is converted to an equivalent mechanism, in which crank length is L2, angle between crank and horizontal axis is theta 2. Coupler link length is L3,  and the angle is theta 3, fixed link length is L1. angle between follower link and horizontal is theta 4. Angle between follower link and coupler is 90 degrees.

In the below pictures described the procedure to calculate various angles and length L4 for any given crank angle theta 2.

In the below picture described the procedure to calculate position of the point C on the mechanism at any given crank angle theta 2.

Scotch yoke | Variation of scotch yoke | Displacement, velocity and acceleration equations

 Variation of scotch yoke - Displacement, velocity and acceleration equations

Variation of scotch yoke mechanism displacement, velocity and acceleration equations derived using analytical method.

In the below picture shown is a variation of the scotch yoke mechanism, in which crank radius is r and radius of the sliding block is R. Omega is angular velocity of the crank and alpha is angular acceleration of the crank.

When crank rotates by an angle theta in counter clock wise direction, the plunger displaces by a distance S, in the below pictures described how to calculate plunger displacement, plunger velocity and plunger acceleration.

To find velocity of the plunger, differentiate displacement equation S with respect to time once and to find acceleration of the plunger differentiate displacement equation S twice with respect to time.

Scotch yoke | Variation of scotch yoke | Displacement, velocity and acceleration equations

 Variation of scotch yoke - Displacement, velocity and acceleration equations

Variation of scotch yoke mechanism displacement, velocity and acceleration equations derived using analytical method.

In the below picture shown is a variation of the scotch yoke mechanism, in which crank radius is r and radius of the sliding block is R. Omega is angular velocity of the crank and alpha is angular acceleration of the crank.

When crank rotates by an angle theta in counter clock wise direction, the plunger displaces by a distance S, in the below pictures described how to calculate plunger displacement, plunger velocity and plunger acceleration.

To find velocity of the plunger, differentiate displacement equation S with respect to time once and to find acceleration of the plunger differentiate displacement equation S twice with respect to time.

Whitworth Quick return mechanism | Time ratio

 Whitworth Quick return mechanism | Time ratio

Whitworth Quick return mechanism, Time ratio equation derived using analytical method.

In the below picture shown is Whitworth Quick return mechanism. This mechanism is mostly used in shaping and slotting machines. In this mechanism, the link AB is fixed whose length is L1. The driving crank AC (length L2) rotates at a uniform angular speed. The slider attached to the crank pin at C slides along the slotted bar CD, which oscillates at a pivoted point B. The connecting rod DE carries the ram at E to which a cutting tool is fixed. The motion of the tool is constrained along the line EB produced, i.e. along a line passing through B and perpendicular to AB.

The forward or cutting stroke occurs when the crank rotates from the position AC1 to AC2 (or through an angle theta) in the clockwise direction. The return stroke occurs when the crank rotates from the position AC2 to AC1 (or through angle beta) in the clockwise direction.

In the below pictures described the procedure to calculate theta, beta and time ratio (Time of forward stroke to time of return stroke)

Crank and slotted lever Quick return mechanism | Time ratio

 Crank and slotted lever Quick return mechanism | Time ratio

Crank and slotted lever Quick return mechanism, Time ratio equation derived using analytical method.

In the below picture shown a Crank and slotted lever mechanism. This mechanism is mostly used in shaping machines, slotting machines and in rotary internal combustion engines. 

The link AC is fixed and its length is L1, Link CB whose length is L2 is a crank, which rotates 360 degrees about point C with uniform angular velocity. A sliding block attached to the crank pin at B slides along the slotted bar AD and thus causes AD to oscillate about the pivoted point A. Length of link AD is L3. Link DE whose length is L4, transmits motion from link AD to the ram which carries the tool and reciprocates along the line of stroke.

The forward or cutting stroke occurs when the crank rotates from the position CB1 to CB2 (or through an angle theta) in the clockwise direction. The return stroke occurs when the crank rotates from the position CB2 to CB1 (or through angle beta) in the clockwise direction.

In the below pictures described the procedure to calculate theta, beta and time ratio (Time of forward stroke to time of return stroke)

Four bar mechanism | Grashof's Law | Variations of four bar mechanism

 Four bar mechanism | Grashof's Law | Variations of four bar mechanism

Watch it on YouTube: https://youtu.be/34FFTF4zEZs