Simply Supported Beam with Springs and Mass

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☕ https://buymeacoffee.com/pankajkporwal ☕ • • 1. Equivalent Stiffness of Springs in Parallel and Series • How to find equivalent stiffness of springs in series ? • How to find equivalent stiffness of springs in parallel ? • When two springs are in series ? • When two springs are in parallel ? • What are the differences in springs in series and in parallel ? • Parallel System • If two springs are connected such that under the action of external load they have same displacement and different forces then they are said to be in parallel. • Equivalent stiffness is sum of stiffnesses of the parts. keq = k1 + k2 • Series System • If two springs are connected such that under the action of external load they have different displacements and same force then they are said to be in parallel. • (1/keq) = (1/k1) + (1/k2) • keq = (k1 k2)/(k1 + k2) • 2. Equivalent stiffness of springs with mass in middle • How to find equivalent stiffness of springs in series ? • How to find equivalent stiffness of springs in parallel ? • When two springs are in series ? • When two springs are in parallel ? • What are the differences in springs in series and in parallel ? • Parallel System • If two springs are connected such that under the action of external load they have same displacement and different forces then they are said to be in parallel. • Equivalent stiffness is sum of stiffnesses of the parts. keq = k1 + k2 • Series System • If two springs are connected such that under the action of external load they have different displacements and same force then they are said to be in parallel. • (1/keq) = (1/k1) + (1/k2) • keq = (k1 k2)/(k1 + k2) • 3. Stiffness of cantilever and simply supported beam • How to find stiffness of cantilever beam ? • How to find stiffness of simply supported beam ? • What is stiffness of beam in vibration problems ? • What role do beams play in vibration problems ? • Stiffness of Cantilever and Simply Supported Beams • In Strength of Materials (or Mechanics of Solids ) and Structural Analysis we deal with deformable elements and members. We can related the force and displacement at the point of application of force with spring equation F = k x. • For cantilever beam – displacement of free end due to concentrated force F ( perpendicular to longitudinal axis) at the free end is given by • y = FL^3/ ( 3EI ) • or F = (3EI/L^3) y • so k = 3EI/L^3 • For simply supported beam – displacement of mid span due to concentrated force F ( perpendicular to longitudinal axis) at the mid span is given by • y = FL^3 / ( 3EI ) • or F = (48EI/L^3) y • so k = 48EI/L^3 • • 4. Stiffness of Beam vs Bar • What is stiffness of a bar ? • What is the stiffness of a cantilever beam ? • When should we use k = 3EI/L^3 and when should we use k = AE/L ? • This video explains the difference between stiffness of a beam ( cantilever ) and a bar. In other words when should we use k = 3EI/L^3 and when should we use k = AE/L • Beam is a structural member which is subjected to force perpendicular it its longitudinal axis. • For cantilever beam – displacement of free end due to concentrated force F ( perpendicular to longitudinal axis) at the free end is given by • y = FL^3/ ( 3EI ) • or F = (3EI/L^3) y • so k = 3EI/L^3 • Bar is a structural member which is subjected to force along its axis. • For a bar – displacement of free end along its length due to concentrated force F (along the axis of the bar ) is given by • x = FL / AE • or F = (AE/L) x • so k = AE/L • In above • E = Young’s Modulus of Elasticity • I = Second moment of area • L = Length of the beam or bar • A = Area of cross section of bar • 5. Stiffness of cantilever beam and spring combination • 6. Stiffness of simply supported beam and spring combination • 7. Equivalent stiffness of beam and spring combination Complete • Equivalent Stiffness of Spring and Beam in Parallel and Series • Parallel System ( Spring and Beam ) ( Beam and Spring ) • If two springs (or one spring and one beam) are connected such that under the action of external load they have same displacement and different forces then they are said to be in parallel. • Equivalent stiffness is sum of stiffnesses of the parts. keq = k1 + k2 • Series System ( Spring and Beam ) or ( Beam and Spring ) • If two springs (or one spring and one beam) are connected such that under the action of external load they have different displacements and same force then they are said to be in parallel. • (1/keq) = (1/k1) + (1/k2) • keq = (k1 k2)/(k1 + k2)

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