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Hydraulic Press: Analyze Coupled Dynamics

Use a linear hydraulic actuator model to connect stiffness, damping and stability.

All worksheets
5 questions0m 0s

Prerequisites

Dimensional analysis, conservation laws, differential equations and elementary fluid mechanics.

Learn the skill

For positive constants C, m, A, b, L, let C dp/dt = Q - A v - Lp and m dv/dt = A p - b v, with zero external load. Here p is differential pressure, L is leakage conductance and b is viscous damping. Small signals, constant coefficients and no saturation are assumed. Use diagrams and calculations only; never build or operate a press for this activity.

Worked example

Eliminating p gives Cm d^2v/dt^2 + (Cb + Lm) dv/dt + (Lb + A^2)v = A Q. With L = 0, C = 10^-10 m^3/Pa, m = 100 kg, A = 0.001 m^2 and b = 1000 N s/m, omega_n = 10 rad/s and damping ratio = 0.5.

A two-equation linear model couples pressure rate and piston acceleration. Pressure drives motion, piston motion consumes flow, and leakage and damping can remove energy from the perturbation.
A two-equation linear model couples pressure rate and piston acceleration. Pressure drives motion, piston motion consumes flow, and leakage and damping can remove energy from the perturbation.
Question 1 With L = 0, what is effective hydraulic stiffness A^2/C?
Question 2 For the example, what is natural angular frequency sqrt(k/m)?
Question 3 What is damping ratio b/(2 sqrt(mk)) for the example?
Question 4 With L = 0 and constant Q, what is steady velocity?
Question 5 If b = 0 and L = 0, what does this ideal unforced linear model predict?

Further inquiry

Eliminate pressure from the coupled equations and derive the transfer function V(s)/Q(s). Use the characteristic coefficients to test asymptotic stability, including the zero-damping and zero-leakage limit. Derive a perturbation-energy balance and explain why saturation, nonlinear friction and distributed wave propagation lie outside this model.

Review criteria

  • State units, signs, initial conditions and all simplifying assumptions.
  • Check the derived result against conservation and at least one limiting case.
  • Explain model limitations without treating a classroom calculation as a machine design.