import matplotlib
if not hasattr(matplotlib.RcParams, "_get"):
    matplotlib.RcParams._get = dict.get

Potential energy conceptual

Click –> Live Code on the top right corner of this screen to investigate some potential energy

import micropip
await micropip.install("ipympl")

import numpy as np
import sympy as sym
import matplotlib.pylab as plt
from ipywidgets import widgets, interact

%matplotlib widget

F = 10
EI = 20e3
L = 3

# Polynomial trial function
x, C1, C2, C3, C4, F, u0= sym.symbols('x, C1, C2, C3, C4, F, u0')
w = C1*x**3+C2*x**2+C3*x + C4
eq1 = sym.Eq(w.subs(x,0),0)
eq2 = sym.Eq(w.diff(x).subs(x,0),0)
eq3 = sym.Eq((-w.diff(x,2)*EI).subs(x,L),0)
eq4 = sym.Eq(w.subs(x,L),u0)
sol = sym.solve((eq1, eq2,eq3,eq4),(C1,C2,C3,C4))
w_sol = sym.nsimplify(w.subs(sol))
w_numpy_polynomial = sym.lambdify((u0,x),w_sol)

M_sol_polynomial = sym.nsimplify(EI * w_sol.diff(x,2))

Ev_polynomial = sym.integrate(sym.nsimplify(M_sol_polynomial**2/(EI*2)),(x,0,3))

Ev_numpy_polynomial = sym.lambdify(u0,Ev_polynomial)

V_polynomial = Ev_polynomial - 10 * u0
V_numpy_polynomial = sym.lambdify(u0,V_polynomial)

# Cosine trial function
w_sol = u0*sym.sin(x/12*2*sym.pi-sym.pi/2)+u0
w_numpy_sin = sym.lambdify((u0,x),w_sol)

M_sol_sin = sym.nsimplify(EI * w_sol.diff(x,2))

Ev_sin = sym.integrate(sym.nsimplify(M_sol_sin**2/(EI*2)),(x,0,3))

Ev_numpy_sin = sym.lambdify(u0,Ev_sin)

V_sin = Ev_sin - 10 * u0
V_numpy_sin = sym.lambdify(u0,V_sin)

# Sin trial function
w_sol = 2*u0*(sym.cosh(x/sym.pi))-2*u0
w_numpy_cosh = sym.lambdify((u0,x),w_sol)

M_sol_cosh = sym.nsimplify(EI * w_sol.diff(x,2))

Ev_cosh = sym.integrate(sym.nsimplify(M_sol_cosh**2/(EI*2)),(x,0,3))

Ev_numpy_cosh = sym.lambdify(u0,Ev_cosh)

V_cosh = Ev_cosh - 10 * u0
V_numpy_cosh = sym.lambdify(u0,V_cosh)

fig, axs = plt.subplots(2, 2, figsize=(10, 6))
fig.canvas.toolbar_visible = False
def func(u,trial_function):
    axs[0,0].clear()
    axs[0,1].clear()
    axs[1,0].clear()
    axs[1,1].clear()

    x = np.linspace(0,3,100)
    if trial_function == 'Polynomial':
        w_numpy = w_numpy_polynomial
        Ev_numpy = Ev_numpy_polynomial
        V_numpy = V_numpy_polynomial
    elif trial_function == 'cos':
        w_numpy = w_numpy_sin
        Ev_numpy = Ev_numpy_sin
        V_numpy = V_numpy_sin
    elif trial_function == 'cosh':
        w_numpy = w_numpy_cosh
        Ev_numpy = Ev_numpy_cosh
        V_numpy = V_numpy_cosh
    axs[0,0].plot(x,w_numpy(u,x))
    axs[0,0].set_xlim([-0.2,3.2])
    axs[0,0].set_ylim([-0.003,0.01])
    axs[0,0].invert_yaxis()
    axs[0,0].annotate(text='', xy=(3,w_numpy(u,3)), xytext=(3,0), arrowprops=dict(arrowstyle='fancy'))
    axs[0,0].text(3.1,w_numpy(u,3),'$u_0$')
    axs[0,0].axis('off')
    axs[0,1].axis('off')
    axs[0,1].set_xlim([-0.2,3.2])
    axs[0,1].set_ylim([-0.003,0.01])
    axs[0,1].annotate(text='', xy=(1.5,w_numpy(u,3)), xytext=(1.5,w_numpy(u,3)-0.003), arrowprops=dict(arrowstyle='simple'))
    axs[0,1].invert_yaxis()
    x_axis= ['$E_v$','$A_F$','$E_v - A_F$']
    y_axis = [Ev_numpy(u),w_numpy(u,3)*10,Ev_numpy(u)-w_numpy(u,3)*10]
    axs[1,0].bar(x_axis,y_axis,color=('blue','green','orange'))
    axs[1,0].set_ylim([-0.03,0.1])
    axs[1,0].set_yticklabels([])
    axs[1,0].set_yticks([])
    
    u_range=np.linspace(-0.0015,0.01,100)
    axs[1,1].plot(u_range,Ev_numpy(u_range),label='$E_v$',color='blue')
    axs[1,1].plot(u_range,w_numpy(u_range,3)*10,label='$F_A$',color='green')
    axs[1,1].plot(u_range,V_numpy(u_range),label='$E_v - A_F$',color='orange')
    axs[1,1].legend()
    axs[1,1].plot(u,Ev_numpy(u),'o',color='blue')
    axs[1,1].plot(u,w_numpy(u,3)*10,'o',color='green')
    axs[1,1].plot(u,Ev_numpy(u)-w_numpy(u,3)*10,'o',color='orange')
    axs[1,1].set_ylim([-0.03,0.1])
    axs[1,1].set_xlim([-0.0015,0.01])
    axs[1,1].set_xlabel('$u_0$')
    axs[1,1].set_xticks([])
    axs[1,1].set_xticklabels([])
    axs[1,1].set_yticks([])
    plt.draw()

interact(func, u = widgets.FloatSlider(min=-0.001, max=0.01, value=0, step=0.0002, description="Displacement u_0",readout_format='.4f',style= {'description_width': '180px'},layout = {'width': '400px'}),
        trial_function = widgets.ToggleButtons(options=['Polynomial', 'cos', 'cosh']));