Theory of Machines and Vibrations

Vibration and machine dynamics

Vibration and machine dynamics deals with the oscillation of machine elements and the forces they transmit. The core relations are the equation of motion of a single degree of freedom system, m\ddot{x}+c\dot{x}+kx=F(t), the natural frequency \omega_n=\sqrt{k/m}, and the damping ratio \zeta=c/c_c, where the critical damping is c_c=2m\omega_n. Springs in parallel add their stiffnesses, so the equivalent stiffness is the sum of the individual stiffnesses. A cantilever under a transverse tip load has stiffness 3EI/L^3, so a cantilever in parallel with a spring gives \omega_n=\sqrt{(3EI/L^3+K)/m} when the masses of the beam and spring are negligible.

A reciprocating mechanism produces a primary inertia force that rotates at the crank speed. The secondary force, which arises from the finite connecting-rod length, has amplitude m\omega^2R^2/L and varies at twice the crank speed. It is therefore equivalent to the primary force of a virtual crank of radius R^2/(4L) rotating at twice the engine speed.

A spinning rotor whose axis is forced to precess develops a gyroscopic couple C=I\omega_s\omega_p\sin\theta, where \theta is the angle between the spin axis and the precession axis. The couple therefore vanishes when the two axes are parallel.

For multi-degree-of-freedom systems the equations of motion follow from Lagrange's equation with L=T-V. Non-conservative forces such as an applied external force do not appear in L and enter only as generalized forces on the right-hand side. The kinetic energy of a uniform rod hinged to a moving cart contains the translational energy of its centre of mass, the rotation about its centre with I_G=mb^2/12, and a cross term from the horizontal velocity of the centre of mass.

A rigid body oscillating about a pivot behaves as a physical pendulum with period T=2\pi\sqrt{I_P/(mgd)}, where d is the distance from the pivot to the centre of mass. The moment of inertia about the pivot is obtained from the parallel axis theorem, I_P=I_G+md^2.

For free viscous vibration, successive peaks decrease by the factor e^{-\delta}. The logarithmic decrement over n cycles is \delta=\frac{1}{n}\ln(x_0/x_n), and the damping ratio is \zeta=\delta/\sqrt{4\pi^2+\delta^2}. For small damping \delta\approx 2\pi\zeta, so the next peak follows directly from a given peak.

The damped natural frequency is \omega_d=\omega_n\sqrt{1-\zeta^2}. If the damping ratio is set equal to the ratio of the damped to the undamped frequency, the condition \zeta=\sqrt{1-\zeta^2} gives \zeta=1/\sqrt{2}.

For a forced system, the steady-state amplitude is largest at a frequency ratio r=\sqrt{1-2\zeta^2}. This peak frequency lies below \omega_d, which in turn lies below \omega_n, so the ordering \omega_p<\omega_d<\omega_n holds for any positive damping.

The transmissibility of a mount is T=\sqrt{\frac{1+(2\zeta r)^2}{(1-r^2)^2+(2\zeta r)^2}}. For frequency ratios above \sqrt{2}, a softer mount and lighter damping both reduce the force reaching the foundation, so ranking isolators requires computing r for each stiffness.

When a rope passes over a pulley without slip, the effective inertia is J+mr^2 and the effective stiffness is kr^2, so \omega_n=\sqrt{kr^2/(J+mr^2)}.

The responses of a forced system differ qualitatively. With positive damping the transient dies out and the envelope settles. Negative damping makes the amplitude grow exponentially. Undamped exact resonance gives linear growth with time, and undamped near-resonance excitation produces beats.

For a rotating lever carrying several springs, the equivalent torsional stiffness about the pivot is K_t=\sum k_i a_i^2, the sum of each stiffness times the square of its lever arm. The natural frequency is then \sqrt{K_t/I}.

Belt drive and gear train speed ratio

A belt drive that does not slip transmits speed inversely to the pulley diameters, so N_1D_1=N_2D_2. In a compound arrangement, the speed ratio of the whole train is the product of the belt ratio and the gear ratios.

Gears fixed to the same shaft share one angular speed. The overall speed ratio of a gear train is the product of the tooth ratios over each mesh, Z_{driver}/Z_{driven}. Each external mesh reverses the sense of rotation, so an odd number of external meshes reverses the output, an even number preserves it, and an internal mesh does not reverse it.

An idler gear meshes between two gears and carries no power off its own shaft. Its tooth number cancels from the speed ratio, and it only changes the direction of rotation.

In an epicyclic train the arm carries the planet gears, and the Willis relation expresses the speed of any gear relative to the arm. For a sun gear and an internal ring meshing through external planets, \frac{\omega_{ring}-\omega_{arm}}{\omega_{sun}-\omega_{arm}}=-\frac{Z_{sun}}{Z_{ring}}.

The tooth count of an internal ring equals the sun tooth count plus twice the planet tooth count, Z_{ring}=Z_{sun}+2Z_{planet}. The module cancels in every tooth ratio, so it does not affect the speed ratio.

Instantaneous centre of rotation

The instantaneous centre of rotation of two bodies in relative motion is the point, common to both bodies at that instant, about which one body rotates relative to the other. A mechanism with n links has n(n-1)/2 such centres.

The Aronhold-Kennedy theorem states that the three instantaneous centres of any three bodies in relative motion lie on one straight line. This theorem is used to locate an unknown centre from known ones.

In pure rolling without slip, the contact point is an instantaneous centre, so the velocity of the rolling body at that point is zero.

The velocity of a point equals the angular velocity multiplied by its distance from the instantaneous centre. For a four-bar mechanism, the ratio of two link angular velocities is therefore the inverse ratio of the distances of the corresponding centres.

Flywheel energy fluctuation

A flywheel stores kinetic energy when the driving torque exceeds the resisting torque and releases it when the resisting torque exceeds the driving torque. The energy fluctuation is the integral of the excess torque over the cycle, \Delta E=\int (T-T_{mean})\,d\theta, taken over the interval where the integrand is positive.

The energy fluctuation is related to the moment of inertia by \Delta E=I\omega_m^2 C_s, where \omega_m is the mean angular speed and C_s is the coefficient of fluctuation of speed. The coefficient is defined as C_s=(\omega_{max}-\omega_{min})/\omega_m.

A permitted band of plus or minus a percentage of the mean speed corresponds to a coefficient of fluctuation equal to twice that percentage, expressed as a fraction. The mean speed must be in rad/s in the formula, so a speed in rpm is converted by \omega=2\pi N/60.

When \Delta E and C_s are both specified, the moment of inertia follows directly from I=\Delta E/(\omega_m^2 C_s), even if the torque is given as a harmonic function.

Logarithmic decrement damping

The logarithmic decrement \delta=\ln(x_1/x_2) is the natural logarithm of the ratio of successive amplitudes on the same side of equilibrium. For viscous damping it satisfies \delta=\frac{2\pi\zeta}{\sqrt{1-\zeta^2}}.

Inverting the relation gives \zeta=\delta/\sqrt{4\pi^2+\delta^2}. The damping coefficient is then c=\zeta c_c=2m\zeta\omega_n.

The natural circular frequency must be used in c=2m\zeta\omega_n. A frequency given in hertz must first be converted by \omega_n=2\pi f.

The damping ratio is also \zeta=c/c_c. When \zeta is small, the logarithmic decrement is approximately 2\pi\zeta.

Quick return mechanism ratio

The quick return ratio Q is the ratio of the time of the cutting (forward) stroke to the time of the return stroke. Since the crank angle is proportional to time, Q=\frac{360^\circ-\alpha}{\alpha}, where \alpha is the crank angle swept during the return stroke.

In a Whitworth mechanism, the crank radius r and the distance d between the two fixed centres satisfy \sin\beta=r/d, where the slotted lever swings through 2\beta. The forward crank angle is 180^\circ+2\beta and the return angle is 180^\circ-2\beta, so Q=\frac{180^\circ+2\beta}{180^\circ-2\beta}.