Equations of motion, block-diagram models, and numerical simulation of physical systems.
These problems start from a physical system and end with a model you can simulate. You choose the coordinates, write the energy or the circuit laws, and turn that into a differential equation. The solution then shows how the equation is built in a block diagram and what the response looks like.
Mechanical systems is the first chapter: rigid bodies, constraints, and Lagrange's equation. Later chapters can sit beside it. Each card is one assignment, numbered from 1, so a new problem is another card on this page and another file in the same folder.
The calculus these solutions use is in Differential Calculus and Integrals. The same rigid-body motions, without the simulation step, are practiced in Physics.
Problem 1
HardMechanical systems
A planetary gear train lies in a vertical plane. Gear 2 (radius $r_2$, weight $G_2$) rolls without slipping inside the fixed gear 1 (radius $r_1$) and is driven by the crank $OA$ (weight $G$), which is acted upon by a couple of moment $M_0$. Define a generalized coordinate and derive the equation of motion.
A circular cylinder of mass $m$ and radius $r$ rolls without slipping inside a circular groove of radius $R$ cut into a cart of mass $M$. The cart is attached to fixed walls by two springs of equal stiffness $k$. Derive the equations of motion with Lagrange's equations.
Three massless beams of length $2l$, each pinned at one end and carrying a point mass $m$ at the other, are linked by four identical springs of stiffness $k$. The motion stays in a vertical plane. Derive the equations of motion with Lagrange's equations.
A homogeneous cylinder of mass $M$ and radius $2a$ hangs from a spring of stiffness $k_1$ and can rotate freely about its centre. A mass $m$ hangs from the outer rim through a spring of stiffness $k_2$, and a mass $2m$ hangs from the inner drum of radius $a$. Find the equations of motion.
A light rod of length $L$ hangs from a fixed pivot and carries a mass $m$ at its end. A horizontal spring of stiffness $k$ joins the rod to a wall at a distance $a$ below the pivot. In equilibrium the rod is vertical and the spring is unstretched. Determine the equation of motion, including the small-angle case.
Four masses slide without friction. Mass $m_1$ is tied to the left wall by spring $k_1$ and damper $B_1$. Masses $m_2$ and $m_3$ are chained to $m_1$ through springs $k_2$ and $k_3$. Mass $m_4$ is linked to $m_1$ by spring $k_4$ and to the right wall by damper $B_2$. Determine the equations of motion.
A block of mass $M$ lies on a smooth horizontal plane, tied to a wall by a spring of stiffness $k_1$ and driven by $F(t) = F_0\sin\omega t$. A load of mass $m$ rides on the block's smooth top, tied to the block by a spring of stiffness $k_2$. Find the equations of motion, including the small-oscillation case.
The rotor–gearbox–motor assembly of total mass $m$ is attached to the helicopter fuselage of mass $M$ by a spring of stiffness $k_1$ and a damper with damping coefficient $B_1$. The rotor is subjected to the disturbing force $F(t) = F_0\sin\omega t$. Find the equations of oscillation of the system.
A homogeneous cylinder of mass $m_2$ and radius $R$ turns on a fixed frictionless axle. A rope that does not slip is anchored to the floor through a spring of stiffness $k_2$, and its other end carries a spring of stiffness $k_1$ from which a mass $m_1$ hangs. Find the equations of motion.
Three identical pendulums hang side by side. Springs connect neighboring pendulums at mid-length and are unstressed when the pendulums hang straight down. Find the equations of motion, and the equations for small oscillations.
A double pendulum has rod lengths $L_1$ and $L_2$ and masses $m_1$ and $m_2$. The rods are rigid and massless, and the joints are frictionless. The coordinates are the absolute angles $\theta_1$ and $\theta_2$ of the two rods, measured from the vertical. Find the equations of motion for arbitrary oscillations and for small oscillations.
A cart of mass $M$ moves without friction and is held between two walls by springs of stiffness $k$. A massless rod of length $a$, pivoted on the cart, carries a point mass $m$. The coordinates are the cart displacement $x$ and the rod angle $\theta$ from the vertical. Find the equations of motion, including small oscillations.
A circular cylinder of radius $r$ and mass $m$ rolls without slipping inside a semicircular groove of radius $R$. The groove, of mass $M$, rests on a spring of stiffness $k$ and slides vertically without friction. The coordinates are the groove displacement $y$ and the angle $\theta$ of the cylinder from the vertical. Find the equations of motion, including small oscillations.
A rigid beam of negligible weight and length $2l$ turns about an axle at its centre. Balls of equal mass $m$ sit at each end, and a third ball of mass $m$ hangs between two springs on the right. Three springs of stiffness $k$ support the system, and a torque $\tau=\tau_0\sin\omega t$ drives the beam. Find the equations of motion.
Two equal levers of length $l$ and mass $M$ turn about fixed joints. Point masses $m=M/6$ sit at the left ends, and three springs of stiffness $k$ connect the levers to each other and to the ceiling and the floor. The springs are vertical and unstretched when both levers are horizontal. Find the equations of motion.
Two masses $m_1$ and $m_2$ sit on a rigid beam of negligible mass and length $3L$. Springs of stiffness $k_1$ and $k_2$ and a damper $B$ support the beam, and a force $F(t)=F_0\cos\omega t$ acts on $m_1$. Find the equations of motion.
For an incremental converter with simple excitation, the inductance depends on the air gap as $L(\delta)=\mu_0 N^2 s/\delta$. The armature of mass $m$ is held by a spring $k$ with viscous friction $B$, and a voltage $u(t)$ feeds the coil. Find the dynamic equations of motion.
With no charge on the plates, a mass $m$ rests against a stop. A charge $Q$ is placed on a parallel-plate capacitor of area $S$ and initial gap $d$, and the stop is removed. The movable plate is held by a spring $k$, and the capacitor is then disconnected so $Q$ stays constant. Find the equation of motion.
A mass $m$ carries one plate of a capacitor and the iron core of a coil, so $C(x)=C_0 d_0/(d_0+x)$ and $L(x)=L_0(1-x/x_0)$. A spring $k$ and a damper $B$ hold the mass, and a voltage $U$ feeds the series circuit. Find the equations of motion.
A condenser microphone is a series circuit of a coil $L$, a resistor $R$ and a parallel-plate capacitor. Two springs of combined stiffness $k$ join the plates, a constant voltage $E$ feeds the circuit, and a force $f(t)=F_0\sin\omega t$ acts on the movable plate of mass $m$. At equilibrium the capacitance is $C_0$ and the plate spacing is $a$. Find the equations of motion.
A mass $M$ hangs on a spring $k$ and a damper $B$ between the plates of a capacitor. The charge depends on the voltage and the position, $q=a\dot\lambda^{1/3}+b\dot\lambda(x-c)^4$. A current source and a resistor feed the capacitor, and gravity acts on the mass. When $x=c$ the mass is centred and the spring is relaxed. Find the equations of motion.