SSRAO

Explanation:
Natural frequency wn = sqrt(k/m) = sqrt(100/1) = sqrt(100) = 10 rad/s. This is the fundamental formula for undamped free vibration of a SDOF system.

Explanation:
wn = sqrt(k/m) = sqrt(400/4) = sqrt(100) = 10 rad/s. fn = wn/(2*pi) = 10/(2*pi) approx 1.59 Hz. Option A: 10/(2*pi) is correct.

Explanation:
For springs in parallel, keq = k1 + k2 = 200 + 300 = 500 N/m. In parallel arrangement, both springs undergo the same displacement, so forces add: F = k1*x + k2*x = (k1+k2)*x.

Explanation:
For springs in series: 1/keq = 1/k1 + 1/k2 = 1/200 + 1/300 = 5/600. Therefore keq = 600/5 = 120 N/m. In series, same force acts through both springs but displacements add.

Explanation:
Critical damping cc = 2*sqrt(k*m) = 2*sqrt(100*1) = 20 N.s/m. Damping ratio zeta = c/cc = 4/20 = 0.2. Since zeta < 1, the system is underdamped.

Explanation:
wd = wn*sqrt(1-zeta^2) = 10*sqrt(1-0.36) = 10*sqrt(0.64) = 10*0.8 = 8 rad/s. The damped natural frequency is always less than the undamped natural frequency for underdamped systems.

Explanation:
Logarithmic decrement delta = ln(x1/x2) = ln(10/5) = ln(2) approx 0.693. It represents the natural logarithm of the ratio of successive amplitudes in underdamped free vibration.

Explanation:
delta = 2*pi*zeta/sqrt(1-zeta^2). Given delta = 0.4*pi: 0.2 = zeta/sqrt(1-zeta^2) => 0.04*(1-zeta^2) = zeta^2 => zeta^2 = 0.04/1.04 approx 0.0385 => zeta approx 0.196 approx 0.2.

Explanation:
Static deflection dst = mg/k => k/m = g/dst. wn = sqrt(g/dst) = sqrt(9.81/0.025) = sqrt(392.4) approx 19.8 approx 20 rad/s. Key result: wn = sqrt(g/dst), independent of mass.

Explanation:
wn = sqrt(g/dst) = sqrt(9810/4.9) = sqrt(2002) approx 44.7 rad/s. fn = wn/(2*pi) = 44.7/6.283 approx 7.12 Hz. Using dst = 4.9 mm = 0.0049 m.

Explanation:
Since 0 < zeta = 0.3 < 1, the system is underdamped. The response x(t) = X*e^(-zeta*wn*t)*cos(wd*t - phi) is oscillatory with exponentially decreasing amplitude. The envelope decays as e^(-zeta*wn*t).

Explanation:
wn = sqrt(k/m) = sqrt(200/2) = 10 rad/s. Critical damping cc = 2*m*wn = 2*2*10 = 40 N.s/m. Alternatively, cc = 2*sqrt(k*m) = 2*sqrt(200*2) = 2*20 = 40 N.s/m.

Explanation:
Since zeta = 1.5 > 1, the system is overdamped. The response is non-oscillatory and returns to equilibrium without oscillation, but more slowly than a critically damped system. Solution involves two real negative exponential terms.

Explanation:
For torsional vibration: wn = sqrt(kt/I) = sqrt(1000/0.1) = sqrt(10000) = 100 rad/s. The formula is analogous to translational vibration with torsional stiffness replacing linear stiffness and moment of inertia replacing mass.

Explanation:
In Rayleigh's method, at maximum displacement all energy is potential (strain energy), and at equilibrium all energy is kinetic. Setting maximum KE = maximum PE gives the natural frequency. This is valid for conservative (undamped) systems.

Explanation:
For a cantilever shaft with end mass, the equivalent mass = md + ms/3. This accounts for the distributed inertia of the shaft. The factor 1/3 comes from the mode shape integration for a uniform beam in bending.

Explanation:
For a simple pendulum: wn = sqrt(g/L) = sqrt(9.81/1) = sqrt(9.81) approx 3.13 rad/s. The period T = 2*pi/wn approx 2.007 s approx 2 s, which is the well-known result for a 1-m pendulum.

Explanation:
For a compound pendulum: wn = sqrt(mgh/I) where I is the moment of inertia about the pivot point. This reduces to sqrt(g/L) for a simple pendulum where I = mL^2 and h = L.

Explanation:
A rigid body in 3D space has 6 degrees of freedom: 3 translational (x, y, z) and 3 rotational (rotation about x, y, z axes). This is fundamental to vibration analysis of multi-body systems.

Explanation:
For a cantilever beam with end load P, deflection delta = PL^3/(3EI). Therefore keq = P/delta = 3EI/L^3. This is the standard result from beam bending theory used in vibration analysis.

Explanation:
For a simply supported beam with central load P, deflection at center delta = PL^3/(48EI). Therefore keq = P/delta = 48EI/L^3. This is the standard result used when modeling beams as equivalent springs.

Explanation:
Amplitude A = sqrt(x0^2 + (v0/wn)^2) = sqrt(0.02^2 + (0.1/5)^2) = sqrt(0.0004 + 0.0004) = sqrt(0.0008) = 0.02*sqrt(2) approx 0.0283 approx 0.028 m.

Explanation:
Torsional stiffness kt = GJ/L where J = pi*d^4/32 for solid circular shaft. Therefore kt = G*(pi*d^4/32)/L = pi*G*d^4/(32L). Both expressions A and B are equivalent and correct.

Explanation:
In a two-rotor system, the node is the point of zero amplitude. For free vibration, I1*L1 = I2*L2 where L1 and L2 are distances from each rotor to the node. This ensures the angular momentum is conserved.

Explanation:
Dunkerley's empirical formula: 1/wn^2 approx 1/w1^2 + 1/w2^2 + ... where w1, w2, ... are the natural frequencies when each mass acts alone. This method gives a lower bound estimate of the fundamental natural frequency.

Explanation:
Rayleigh's method always gives an upper bound for the fundamental natural frequency. Any assumed mode shape (other than the exact one) introduces additional constraints, increasing the apparent stiffness and hence the frequency estimate.

Explanation:
In viscous damping, the damping force Fd = c*x_dot, where c is the viscous damping coefficient and x_dot is the velocity. This linear relationship makes the equations of motion linear and analytically tractable.

Explanation:
In Coulomb damping, the amplitude decreases by a constant amount 4*mu*m*g/k per cycle (linearly). This contrasts with viscous damping where amplitude decreases exponentially. Motion stops when restoring force equals friction force.

Explanation:
In hysteretic damping, energy dissipated per cycle Ed = pi*h*k*X^2 where h is the hysteretic damping factor, k is stiffness, and X is amplitude. The energy loss is proportional to the square of amplitude, independent of frequency.

Explanation:
For Coulomb damping, ceq = 4F/(pi*w*X). Derived by equating energy dissipated per cycle: Coulomb energy = 4FX, viscous energy = pi*ceq*w*X^2. Setting equal: 4FX = pi*ceq*w*X^2 => ceq = 4F/(pi*w*X).

Explanation:
For undamped free vibration: m*x_ddot + k*x = 0. This is a second-order linear ODE with solution x(t) = A*cos(wn*t) + B*sin(wn*t) where wn = sqrt(k/m). Constants A and B are determined from initial conditions.

Explanation:
With x0 = 0 and v0 > 0: x(t) = (v0/wn)*sin(wn*t). Comparing with A*sin(wn*t + phi): A = v0/wn = 0.5/10 = 0.05 m, phi = 0 degrees. The motion starts from zero displacement with maximum velocity.

Explanation:
For a vertical spring-mass system, the equation of motion about the static equilibrium position is m*x_ddot + k*x = 0, giving wn = sqrt(k/m). The gravity term cancels with the static spring force, so gravity does NOT affect the natural frequency.

Explanation:
Natural frequency is the frequency at which a system vibrates when disturbed from equilibrium and left to vibrate freely without any external forcing. It depends only on system parameters (mass and stiffness).

Explanation:
Time period T = 1/fn = 1/2 = 0.5 s. The period is the time for one complete oscillation. Alternatively, T = 2*pi/wn = 2*pi/(2*pi*2) = 1/2 = 0.5 s.

Explanation:
Logarithmic decrement delta = ln(x1/x2), therefore x1/x2 = e^delta. The amplitude ratio between successive cycles is e^delta where delta = 2*pi*zeta/sqrt(1-zeta^2). For small damping, delta approx 2*pi*zeta.

Explanation:
For n cycles: x1/x(n+1) = e^(n*delta). Therefore n*delta = ln(x1/x(n+1)), giving delta = (1/n)*ln(x1/x(n+1)). This formula is used when measuring over multiple cycles to improve accuracy of damping measurement.

Explanation:
Exact: delta = 2*pi*zeta/sqrt(1-zeta^2). For small zeta, sqrt(1-zeta^2) approx 1, so delta approx 2*pi*zeta. This approximation is valid for zeta < 0.2. For example, if delta = 0.2, then zeta approx 0.2/(2*pi) approx 0.032.

Explanation:
Stiffness of helical spring k = G*d^4/(64*N*R^3) = G*d^4/(8*N*D^3) where D = 2R is the mean coil diameter. This is derived from the torsion of the wire and the geometry of the coil.

Explanation:
The critical speed (whirling speed) of a rotating shaft equals the natural frequency of its transverse (lateral) vibration. At this speed, resonance occurs and deflections become theoretically infinite (in undamped case). Nc = (30/pi)*sqrt(g/dst) rpm.

Explanation:
Nc = (30/pi)*sqrt(g/dst) = (30/pi)*sqrt(9810/1) = (30/pi)*99.05 approx 946 approx 950 rpm. The formula uses dst in mm: Nc = (30/pi)*sqrt(g/dst_m) where dst_m is in meters.

Explanation:
Transmissibility TR < 1 (isolation) occurs when w/wn > sqrt(2) for undamped systems. For w/wn < sqrt(2), TR > 1 (amplification). At w/wn = sqrt(2), TR = 1. For isolation, the natural frequency should be much lower than the excitation frequency.

Explanation:
When a dynamic vibration absorber is tuned (wa = w), the amplitude of the main mass becomes zero. The absorber mass vibrates and exerts a force equal and opposite to the excitation force on the main mass, effectively canceling the vibration.

Explanation:
When a spring is attached at the center of a beam in parallel (both supporting the same load), they act in parallel. The equivalent stiffness keq = kb + ks. Both the beam and spring deflect by the same amount under the applied load.

Explanation:
For a U-tube manometer with total liquid length L, the restoring force for displacement x is 2*rho*A*g*x. Mass = rho*A*L. wn = sqrt(2g/L). This is analogous to a spring-mass system with k = 2*rho*A*g and m = rho*A*L.

Explanation:
For vertical oscillation of a floating body: restoring force = rho*A*g*x (buoyancy change). Equation of motion: m*x_ddot + rho*A*g*x = 0. wn = sqrt(rho*A*g/m). This is analogous to a spring-mass system with k = rho*A*g.

Explanation:
Using energy equivalence: KE of flywheel = (1/2)*I*omega^2 = (1/2)*I*(x_dot/r)^2 = (1/2)*(I/r^2)*x_dot^2. Therefore equivalent mass meq = I/r^2. The natural frequency wn = sqrt(k/meq) = sqrt(k*r^2/I).

Explanation:
For a cantilever beam, the fundamental frequency: wn = (1.875)^2*sqrt(EI/(m_bar*L^4)) = 3.516*sqrt(EI/(m_bar*L^4)). The value 1.875 is the first root of the frequency equation cos(beta*L)*cosh(beta*L) = -1.

Explanation:
For simply supported beam: wn = (n*pi/L)^2*sqrt(EI/m_bar) for nth mode. For n=1: wn = (pi/L)^2*sqrt(EI/m_bar) = pi^2/L^2*sqrt(EI/m_bar) = pi^2*sqrt(EI/(m_bar*L^4)). Both expressions A and B are equivalent and correct.

Explanation:
A seismometer measures the relative displacement between the seismic mass and the instrument case (base). When w/wn >> 1, the relative displacement approx absolute displacement of the base. This is the principle of displacement measurement.

Explanation:
An accelerometer operates when w/wn << 1 (low frequency ratio). In this range, the relative displacement of the seismic mass is proportional to the acceleration of the base: Z approx (w^2/wn^2)*Y. The instrument must have high natural frequency.

Explanation:
Stiffness influence coefficient kij = force required at coordinate i to maintain equilibrium when unit displacement is given at coordinate j (all other coordinates fixed). The stiffness matrix [K] is symmetric: kij = kji.

Explanation:
Flexibility influence coefficient aij = displacement at coordinate i due to unit force applied at coordinate j (all other forces zero). The flexibility matrix [A] = [K]^(-1). By Maxwell's reciprocal theorem, aij = aji.

Explanation:
Mode shapes {phi}r and {phi}s are orthogonal with respect to both mass and stiffness matrices: {phi}r^T*[M]*{phi}s = 0 and {phi}r^T*[K]*{phi}s = 0 for r not equal to s. This orthogonality property is used to decouple the equations of motion in modal analysis.

Explanation:
For two equal masses connected by three equal springs (fixed-mass-spring-mass-spring-fixed): The characteristic equation gives w^2 = k/m and w^2 = 3k/m. First mode: both masses move in same direction. Second mode: masses move in opposite directions.

Explanation:
When referring to shaft 1: angular velocity of shaft 2 = w1/n. KE of gear 2 = (1/2)*I2*(w1/n)^2 = (1/2)*(I2/n^2)*w1^2. Therefore equivalent inertia at shaft 1: Ieq = I1 + I2/n^2. The natural frequency wn = sqrt(kt/Ieq).

Explanation:
Adding a vibration absorber converts the SDOF system to a 2-DOF system with two natural frequencies. Both new natural frequencies are different from the original wn = sqrt(k/m). The absorber is tuned so that wa = sqrt(ka/ma) = w (excitation frequency).

Explanation:
The node in torsional vibration is the point of zero angular displacement (amplitude = 0). At the node, the shaft does not twist. The two rotors vibrate in opposite phase. The node location depends on the ratio of the two moments of inertia.

Explanation:
At resonance (w = wn), the amplitude of an undamped system theoretically becomes infinite. In practice, some damping always exists, limiting the amplitude. The dynamic magnification factor DMF = 1/|1-(w/wn)^2| -> infinity as w -> wn.

Explanation:
Forced vibration occurs when an external periodic force is applied to the system. Free vibration occurs when the system is disturbed from equilibrium and left to vibrate without external force. Underdamped, overdamped, and critically damped are all types of free vibration.

Explanation:
wn = sqrt(g/dst) = sqrt(9810/5) = sqrt(1962) = 44.3 rad/s. fn = wn/(2*pi) = 44.3/6.283 = 7.05 Hz. Note: the mass does not appear explicitly because wn = sqrt(g/dst) is independent of mass.

Explanation:
When liquid is displaced by x from equilibrium, one side rises by x and other falls by x, creating a height difference of 2x. Restoring force = weight of excess liquid = rho*A*(2x)*g = 2*rho*A*g*x. This gives wn = sqrt(2g/L).

Explanation:
Transmissibility TR = sqrt[(1+(2*zeta*r)^2)/((1-r^2)^2+(2*zeta*r)^2)] where r = w/wn is the frequency ratio and zeta is the damping ratio. For r > sqrt(2), TR < 1 (isolation). For r < sqrt(2), TR > 1 (amplification).

Explanation:
For a fixed-fixed beam with central point load P, deflection at center delta = PL^3/(192EI). Therefore keq = P/delta = 192EI/L^3. This is stiffer than a simply supported beam (48EI/L^3) due to the fixed end conditions.

Explanation:
Holzer's method is a tabular method used to find natural frequencies of multi-rotor torsional systems. It is a trial-and-error method where a frequency is assumed and the residual torque at the free end is calculated. The correct frequency gives zero residual torque.

Explanation:
The matrix iteration (power) method using the dynamical matrix [D] = [K]^(-1)*[M] converges to the lowest natural frequency (fundamental mode). To find higher modes, the mode shapes must be swept out using orthogonality conditions.

Explanation:
For a taut string: wave speed c = sqrt(T/m_bar). Fundamental frequency f1 = c/(2L) = sqrt(T/m_bar)/(2L). The nth natural frequency fn = n*f1 = n*sqrt(T/m_bar)/(2L). This is the basis of stringed musical instruments.

Explanation:
For an undamped system, there is no energy dissipation. The total mechanical energy E = KE + PE = constant. Energy oscillates between kinetic (maximum at equilibrium) and potential (maximum at extreme positions), but the total remains constant.

Explanation:
The amplitude envelope is A*e^(-zeta*wn*t). For amplitude to decay to A/e: e^(-zeta*wn*t) = e^(-1) => zeta*wn*t = 1 => t = 1/(zeta*wn). This time constant tau = 1/(zeta*wn) characterizes the rate of decay of free vibration.

Explanation:
The general solution is x(t) = A*cos(wn*t + phi). Amplitude A = sqrt(x0^2 + (v0/wn)^2). This is the maximum displacement from equilibrium. The phase phi = arctan(-v0/(wn*x0)).

Explanation:
For shafts in series (same torque, angles add): 1/keq = 1/kt1 + 1/kt2 => keq = kt1*kt2/(kt1+kt2). This is analogous to springs in series. The equivalent stiffness is less than either individual stiffness.

Explanation:
For undamped free vibration: x = A*cos(wn*t + phi), x_dot = -A*wn*sin(wn*t + phi). Eliminating t: x^2/A^2 + x_dot^2/(A*wn)^2 = 1, which is an ellipse. For damped vibration, the trajectory is a converging spiral.

Explanation:
Increasing the mass ratio mu = ma/m widens the frequency range over which the absorber is effective (the frequency band where the main mass amplitude is acceptably small). However, a larger absorber mass increases the total system weight.

Explanation:
For a non-uniform shaft, the equivalent torsional stiffness is found by integration: 1/keq = integral from 0 to L of dx/(G*J(x)) where J(x) is the polar moment of inertia at position x. This accounts for the varying stiffness along the shaft length.

Explanation:
Natural frequencies of plates depend on: material properties (E, rho, nu), geometry (length a, width b, thickness h), and boundary conditions. For simply supported rectangular plate: wmn = pi^2*[(m/a)^2+(n/b)^2]*sqrt(D/(rho*h)) where D = E*h^3/[12*(1-nu^2)].

Explanation:
The logarithmic decrement method measures successive amplitudes from free vibration decay: delta = (1/n)*ln(x1/x(n+1)), then zeta = delta/sqrt(4*pi^2+delta^2). This is the most direct method for free vibration. The half-power bandwidth method is used for forced vibration.

Explanation:
For a semi-elliptic leaf spring: k = 8*E*n*b*t^3/(3*L^3) where L is the half-length, n is number of leaves, b is leaf width, t is leaf thickness. This is derived from beam bending theory treating the spring as a series of cantilever beams.

Explanation:
A critically damped system (zeta = 1) returns to equilibrium in the shortest possible time without oscillating. The response is x(t) = (A + B*t)*e^(-wn*t). This is the boundary between underdamped (oscillatory) and overdamped (non-oscillatory) behavior.

Explanation:
For a massless cantilever with tip mass m: keq = 3EI/L^3 (from delta = PL^3/3EI). Natural frequency wn = sqrt(keq/m) = sqrt(3EI/(m*L^3)). This is the standard result for a cantilever beam with concentrated end mass.

Explanation:
For undamped system: TR = 1/|1-(w/wn)^2| = 1/|1-4| = 1/3. At w/wn = 2, the transmissibility is 1/3, meaning only 1/3 of the force is transmitted to the foundation. This demonstrates effective isolation when w/wn > sqrt(2).