MT-1620 Fall 2002 Forced vibration The homogeneous solution is still valid, but must add a particular solution The simplest case here is a constant load with time Figure 20.2 Representation of constant applied load with time (think of the load applied suddenly step function at t= O) The governing equation is mq kq= F The particular solution has no time dependence since the force has no time dependence Paul A Lagace @2001 Unit 20-6
MIT - 16.20 Fall, 2002 Forced Vibration The homogeneous solution is still valid, but must add a particular solution The simplest case here is a constant load with time… Figure 20.2 Representation of constant applied load with time (think of the load applied suddenly ⇒ step function at t = 0) The governing equation is: m q˙˙ + k q = F0 The particular solution has no time dependence since the force has no time dependence: F0 qparticular = k Paul A. Lagace © 2001 Unit 20 - 6
MT-1620 al.2002 Now use the homogeneous solution with this to get the total solution qt)=C snot C2cosot k The Initial Conditions are q 0)=0 F6 q(0)=0→C1=0 So the final solution is F6 q coS ot k with Plotting this Paul A Lagace @2001 Unit 20-7
qt MIT - 16.20 Fall, 2002 Now use the homogeneous solution with this to get the total solution: F () = C1 sinω t + C2 cosω t + 0 k The Initial Conditions are: @ t = 0 q = 0 q˙ = 0 q () 0 = 0 ⇒ C = − F0 2 k q ˙() 0 = 0 ⇒ C1 = 0 So the final solution is: F q = 0 (1 − cosω t) k with ω = k m Plotting this: Paul A. Lagace © 2001 Unit 20 - 7
MT-1620 Fall 2002 Figure 20.3 Ideal dynamic response of single spring-mass system to constant force Dynamic response Static response 七 Note that Dynamic response= 2 X static response dynamic magnification factor'-will be larger when considering stresses over their static values Know this doesn't really happen(i. e response does not continue forever) What has been left out? DAMPING Paul A Lagace @2001 Unit 20-8
MIT - 16.20 Fall, 2002 Figure 20.3 Ideal dynamic response of single spring-mass system to constant force Dynamic response Static response Note that: Dynamic response = 2 x static response “dynamic magnification factor” - will be larger when considering stresses over their static values Know this doesn’t really happen (i.e. response does not continue forever) What has been left out? DAMPING Paul A. Lagace © 2001 Unit 20 - 8
MT-1620 al.2002 Actual behavior would be Figure 20.4 Actual dynamic response with damping of single spring mass system to constant force rate of damping dependent on magnitude of damper(c) Have considered a simple case. But, in general forces are not simple steps. Consider the next level The Unit Impulse Paul A Lagace @2001 Unit 20-9
MIT - 16.20 Fall, 2002 Actual behavior would be… Figure 20.4 Actual dynamic response (with damping) of single springmass system to constant force rate of damping dependent on magnitude of damper (c) Have considered a simple case. But, in general forces are not simple steps. Consider the next “level”… The Unit Impulse Paul A. Lagace © 2001 Unit 20 - 9