Since you will be using a guide, then my thoughts are the following. Assume at some point the rod forms an angle $\phi$.
General Idea
Since you are pushing up the platform the downward component of the force is equal to the reaction on the pivot below the platform $R_{1y}$.
this will create a horizontal component on the rod which will need to satisfy the following equation
$$\tan\phi = \frac{R_{1y}}{H}$$
Where $H$ is the horizontal component of the force. ($H= \frac{R_{1y}}{\tan\phi}$). Notice that the force reduces as $\phi$ approaches $90\deg$
So, the force H is the forces needed to overcome the weight. The only tricky part is that you need to estimate the angle $\phi$
calculate $\phi$
In order to calculate $\phi$ for an angle $\theta$ that the platform pivots from the horizontal you'd need (if you need a sketch drop me a comment and I'll sketch it for you tomorrow):
- $H_{Total}$: The total height between the platform top pivot and the pivot of the vehicle (I assume it is 11+24=35 in).
- $H_{Platform}$: The platform height (11 in)
- $W_{Platform}$: The platform width (12 in)
- $L_{rod}$: The length of the rod. (35 in)
Given the above and theta you can calculate $H_{Pl,\theta}$, which is the vertical distance between the top pivot of the platform and the pivot below the platform. For me the simplest way to calculate this is using a rotation matrix (quantity $\color{red}{y_\theta}$ is $H_{Pl,\theta}$).
$$\begin{bmatrix}x_\theta\\ \color{red}{y_\theta} \\0 \end{bmatrix} =
\begin{bmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta & \cos\theta & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}\frac{W_{Platform}}{2}\\ -H_{Platform}\\0 \end{bmatrix}
$$
This reduces to :
$$H_{Pl, \theta} = \frac{W_{Platform}}{2} \sin\theta - H_{Platform}\cos\theta$$
Note: $H_{Pl, \theta}$ should be negative for angles less that 60 deg.
Then the angle $\phi$ (as a function of \theta) is given by:
$$\phi = asin\left(\frac{H_{total}+H_{Pl, \theta}}{L_{rod}}\right)$$
$$\phi(\theta) = asin\left(\frac{H_{total}+\frac{W_{Platform}}{2} \sin\theta - H_{Platform}\cos\theta }{L_{rod}}\right)$$
Therefore you can now plot the force for all $\theta$ angles between 0 and 60 $\deg$ that is the limit of your movement.
Calculate $R_{1y}$
The following is the "free body diagram" of the top platform.
- Point 1: is the pivot below the platform
- Point 2: is the pivot at the top of the platform
The following equations describe the "balance" of the system:
Balance on x axis
$$\sum F_x =0 \rightarrow R_{2x} - R_{1x}=0$$
Balance on y axis
$$\sum F_y =0 \rightarrow R_{2y} + R_{1y} - W=0$$
Moment around point 2
$$\sum M_2 =0 \rightarrow y_\theta R_{1x} + x_\theta R_{1y} - x_w W=0$$
relation between $R_{1x}, R_{2x}$
$$\tan\phi = \frac{R_{1y}}{R_{1x}}$$
$$\sin\theta = \frac{x_w}{H_{platform}}$$
5 equations, with 5unkwowns ($R_{1x},R_{1y},R_{2x},R_{2y}, x_w$). They can be reduced to the following three:
$$\begin{cases}
R_{2x} - \frac{R_{1y}}{\tan\phi}=0\\
R_{2y} + R_{1y} - W=0\\
- y_\theta \frac{R_{1y}}{\tan\phi} + x_\theta R_{1y} - \sin\theta H_{platform} W=0
\end{cases}
$$
NOTE: $y_\theta$ will have negative values.
Calculation of cart position $x_{cart}(\theta)$
Since you now have a way of calculating the position of the pivot underneath the platform $(x_\theta, y_\theta)$, and the angle $\phi$, you can easily estimate the position of the bottom pivot by vector calculus. In order to calculate
$$x_{cart}(\theta) = x_\theta - L_{Rod} \cdot \cos\phi$$
Then you can plot the force with respect to $x_{cart}(\theta)$.
What I got is the following:
Additional points
There is also an additional force component (dynamic) which has to do with the centrifugal force that you need to overcome. The magnitude of the centrifugal force would be approximately 1.5[lbf], which would make its effect negligible (compared to the 385[lbf] of the mass).
Although, I am not entirely convinced that you don't need to consider acceleration in general (you have a constant angular velocity, and therefore you need to accelerate and decelerate the cart and the mass of 385lb is quite high)
Python code
# %%
import numpy as np
import matplotlib.pyplot as plt
# %%
H_tot = 35
H_plat= 11
W_plat= 12
L_rod = 35
# %%
theta = np.radians(90)
r_th = lambda theta: np.array(( (np.cos(theta), -np.sin(theta), 0),
(np.sin(theta), np.cos(theta),0) ,
(0,0,1)
))
# %%
v = np.array((W_plat/2, -H_plat,0)).T
# %%
def get_pivot1_coords(theta):
coords = r_th(theta).dot(v)
return coords[0:2]
# %% [markdown]
# # plot x,y theta
# xy contains two column with the coordinates of the pivot at the bottom of the platform
# %%
thetas= np.linspace(0,np.pi/3,60)
xy_raw = r_th(thetas).dot(v)[:2]
xy = np.vstack((xy_raw[0],xy_raw[1])).T
# %%
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(xy[:,0], xy[:,1])
ax.set_aspect('equal')
ax.set_title('Trajectory of the pivot 1 ( bottom of the platform)')
# %% [markdown]
# # calculate angle $\phi$ wrt $\theta$
# %%
def calc_phi(theta):
xy1 = get_pivot1_coords(theta)
phi = np.arcsin((H_tot+ xy1[1])/L_rod)
return phi
# %%
phis = []
for theta in thetas:
# print(theta)
phis.append(calc_phi(theta))
phis = np.array(phis)
# %%
plt.figure()
plt.plot(thetas, phis)
plt.xlabel('$\\theta$ [rad]')
plt.ylabel('$\\phi$ [rad]')
plt.title('angle $\\phi$ as a function of $\\theta$')
# %% [markdown]
# # calculate R1y
# %%
W = 385
def calc_R1(theta):
''' returns R1 (x, y) for a given theta
'''
xy1 = get_pivot1_coords(theta)
phi = calc_phi(theta)
R1y=(np.sin(theta)*H_plat*W)/(-xy1[1]/np.tan(phi) + xy1[0])
R1x= R1y/np.tan(phi)
return [R1x, R1y]
# %% Calculate R1s for all theta angles
R1 = []
for theta in thetas:
# print(theta)
R1.append(calc_R1(theta))
R1 = np.array(R1)
x_cart = xy[:,0]-np.cos(phis)*L_rod
# %%
plt.figure()
plt.plot(thetas*180/np.pi, R1[:,0], '.')
plt.xlabel('$\\theta$ [rad]')
plt.ylabel('$R_{1x}$ [rad]')
plt.title('$R_{1x}$ w.r.t. $\\theta$')
# %%
plt.figure()
plt.plot(xy[:,0]-np.cos(phis)*L_rod,phis*180/np.pi)
plt.xlabel('$x_{cart}$ [in]')
plt.ylabel('$\\phi$ [deg]')
plt.grid()
plt.title('$\\phi$ w.r.t. $x_{cart}$')
plt.figure()
plt.plot(x_cart, R1[:,0],label='$R_{1x}$')
plt.plot(x_cart, R1[:,1], label='$R_{1y}$')
plt.xlabel('$x_{cart} [in]$')
plt.ylabel('$Force$ [lbf]')
plt.legend()
plt.title('x and y components for R w.r.t. $\\theta$')
# %%
plt.figure(figsize=(10,8))
plt.plot(x_cart, R1[:,0], label='$R_{1x}$')
plt.xlabel('$x_{cart}$ [in]')
plt.ylabel('$R_{1x}$ [lbf]')
plt.title('x components for R w.r.t. $x_{cart}$')
plt.legend()
plt.show()
# %%
