Question

A system made up of a mass (m), attached to a spring of stiffness k [N/m] will oscillate to a specific amplitude (A) which will depend on an external force (F) and initial conditions. If all the variables involved are given in Table 1, formulate the necessary Pi groups to describe this behavior. Make sure you write the Pi groups using the parameters involved Parameter Variable Variable Units Amplitude A т Mass m kg Spring k N/m constant External F N Force Frequency rad/S w

Answer 1

Given variables and their Dimensions are m -> as following: Amplitude, a [MOLTO] Maes, [ML To Het External force, F - [MLT 2 Frequency [ru T']. spring contant, un W - Total no of variables sen Number of fundamental variables 3-M . No. of Pe terms med will be (nom) - 5-3 = 2. Two terms, P, and Pa are the required Pe terme dech are dimensionled. The no ff repeating terms will be m=3 which are A m, w we have da [LT] L = A M = m . (Muto W = [More] T= win 3.

we have ky [mot? kx ['L T2 will be dimencionley. P k[(m") (A) (-1)] k Pi = is one Pi- term. mo2 k Pi = mot We also have, F - [MLT 2 Fx [l[ ] will be dimensionless Per & [(m5" (as" (con in Pa P₂= F mAw² is another sitem. are The required se groups P1 = Ps k and moz mA Co* P = f (P.).