机械振动学非定常瞬态振动.pptVIP

Arbitrary Force The force can be regarded to be made up of a series of impulses of varying magnitude. At time τ, the force F(τ) acts on the system for a short period of time Δτ, the impulse acting at t = τ is given by F(τ) Δτ. At any time t, the elapsed time since the impulse is t- τ, so the response of the system at t due to this impulse alone is given as The total response at time t can be found by summing all the response due to the impulses acting at all times τ. Letting Δτ→0, and replacing the summation by integration, one obtains Arbitrary Force Duhamel integral Note above equation doesn’t consider the effect of the initial conditions. Example – Compacting Machine with Cam F(t) x(t) m k/2 k/2 c Cam Platform Material being compacted Follower Motion of Cam F(t) 0 t 1 δF F(t)=δFt Find: Response of the system Solution: the response can be obtained by evaluating Duhamel integral with F(t)=δFt m =5kg; ωn=20 rad/s; ξ = 0.01; δF0 = 1000; Example – Compacting Machine with Cam F(t) Piston Cylinder Platform x(t) m k/2 k/2 c Example – Compacting Machine Material being compacted A compacting machine, modeled as a SDOF system, is shown in left figure. The force acting on the mass m (m includes the masses of the piston, the platform and the material being compacted) due to a sudden application of the pressure can be idealized as a step force, shown as figure below. Determine the response of the system. F(t) F0 0 t Example – Compacting Machine Given: Compacting machine subjected to a step force. Find: Response of the system Solution: Since the compacting machine is modeled as a SDOF system, the problem is to find the response of a damped SDOF system subjected to a step force. By noting F(t) = F0. Then the response can be calculated as Example – Compacting Machine m =5kg; ωn=20 rad/s; ξ = 0.05 ; F0 = 1000; F0/k Problem: why step force cannot produce steady-state vibration for damped system? Example – Compacting Machine If damp ratio is zero, then m =5kg; ωn