7.1  Particle kinetics

The method for solving problems in kinetics is an extension of statics and the methods used in the corresponding sections. Kinetics is the study of the relationship between forces and accelerations, and a starting point is Newton’s equation of motion

\[ \sum {\mathbb F}_i =m\;\mathbb a \]

where \(\mathbb a\) is the center of mass (COM) of a body (or particle). For completeness, we shall here state the complete generalized equations of motion extended by Euler.

\[ \left\lbrace \begin{array}{ll} \mathbb F=\sum {\mathbb F}_i =m\;\ddot{\mathbb r} & \textrm{Translation}\\ \mathbb M=\sum {\mathbb M}_i +{\mathbb M}_e =\sum {\mathbb r}_i \times {\mathbb F}_i +{\mathbb M}_e =\bar{I} \ddot{\theta} & \textrm{Rotation} \end{array}\right. \]

where \(\bar{I}\) is the moment of inertia for the body at the center of mass, otherwise known as the rotational inertia of a rigid body. We will give a proper introduction to this quantity in the section on rigid body kinetics. Note that the moment of inertia is given by the geometric relation

\[ \bar{I} =\int_V \rho |\bar{\mathbb R} |^2 \;dV \]

where \(\bar{\mathbb R}\) is the vector from the center of rotation (which in this case is the COM) to each point in the body.

Notice that in this general formulation we have \({\left\lbrack F_x ,F_y ,F_z \right\rbrack }^T =m\ddot{\mathbb r}\) and \({\left\lbrack M_x ,M_y ,M_z \right\rbrack }^T =\bar{I} \ddot{\theta}\), thus six equations (ODEs), where for each component we derive these equations by using a free body diagram, exactly as in statics. In order to get a specific solution to the system of equations, we need corresponding initial conditions and/or boundary conditions.

Suffice to say for now, if the body has a near-zero size, then \(\bar{\mathbb R}\to 0\) and we can assume particle dynamics and disregard the rotational part of the equation of motion, resulting in only three degrees of freedom instead of six. Similarly, if the rotational acceleration \(\ddot{\theta}\) is sufficiently low, we can assume particle dynamics. We will revisit the generalized equations of motion in rigid body kinetics.

Note that Newton’s second law of motion does not explicitly relate force to acceleration, but rather to momentum, paraphrased: “The time rate of change of a body’s momentum equals the force acting upon it,” i.e.,

\[ \mathbb F=\dfrac{d}{dt}\left(m\left(t\right)\;\dot{\mathbb r} \left(t\right)\right)\iff \mathbb F=\dot{m} \dot{\mathbb r} +m\ddot{\mathbb r} \]

Thus, we can create models where the body has a change of mass, e.g., rockets. When a body has constant mass, the expression reduces to what we are familiar with.

The solution of the equations of motion is typically done by solving a differential equation using either a symbolic or numerical solver. In traditional courses on classical mechanics, countless chapters are spent on deriving various analytical methods and formulas for simplifying problems or solving problems specifically designed to be solved using simple hand calculations. We shall here instead focus on solving problems completely and explicitly visualizing the whole motion.

# Define symbolic variables
mu_k, m, g, v_0, P, theta, N, a_x, t = sp.symbols('mu_k m g v_0 P theta N a_x t')
x = sp.Function('x')(t)

# Define vectors for the forces and acceleration
F_ext = P * sp.Matrix([sp.cos(theta), -sp.sin(theta)])
F_N = sp.Matrix([0, N])
F_W = sp.Matrix([0, -m*g])
F_f = sp.Matrix([-mu_k*N, 0])
a = sp.Matrix([a_x, 0])

# Newton's second law: sum of forces = m*a
ekv = sp.Eq(F_ext + F_N + F_W + F_f, m*a)
ekv

sol = sp.solve(ekv, (a_x, N), dict=True)[0]
a_x_expr = sol[a_x]
N_expr = sol[N]

# Define the piecewise constant force P(t)
P_const = sp.Piecewise((4000, t < 0.3), (0, t > 0.3))

f_P = sp.lambdify(t, P_const, 'numpy')
t_vals = np.linspace(0, 2, 1000)
P_vals = f_P(t_vals)
#plt.figure(figsize=(8, 5))
plt.plot(t_vals, P_vals, linewidth=3, color='blue')
plt.xlabel('$t$ [s]'); plt.ylabel('$P(t)$ [N]'); plt.title('Piecewise constant load')
plt.show()