Engine Resonance Speed Calculation

Understanding the resonance speed of an automotive engine supported by springs is crucial for preventing excessive vibrations and ensuring the longevity of the system. Resonance occurs when the engine's operating frequency matches its natural frequency, leading to large amplitude oscillations. This solution details the steps to calculate the engine speed in revolutions per minute (rpm) at which resonance is likely to occur.

Engine System Components and Data

First, let's identify the given parameters for the automotive engine and its supporting springs. These values are essential for determining the system's dynamic characteristics.

• Mass of the automotive engine ($\text{m}$): 240 kg
• Number of front springs: 2
• Stiffness of each front spring ($\text{k}_{\text{front}}$): 16 MN/m
• Number of rear springs: 2
• Stiffness of each rear spring ($\text{k}_{\text{rear}}$): 32 MN/m

It's important to convert the stiffness values from meganewtons per meter (MN/m) to newtons per meter (N/m) for consistency in calculations:

• $\text{k}_{\text{front}} = 16 \text{ MN/m} = 16 \times 10^6 \text{ N/m}$
• $\text{k}_{\text{rear}} = 32 \text{ MN/m} = 32 \times 10^6 \text{ N/m}$

Total Stiffness Calculation

Since the four springs collectively support the entire engine, their stiffnesses add up in parallel. This combined stiffness is known as the equivalent stiffness ($\text{k}_{\text{eq}}$) of the system.

The total stiffness from the two front springs is:

$\text{k}_{\text{total, front}} = 2 \times \text{k}_{\text{front}} = 2 \times (16 \times 10^6 \text{ N/m}) = 32 \times 10^6 \text{ N/m}$

The total stiffness from the two rear springs is:

$\text{k}_{\text{total, rear}} = 2 \times \text{k}_{\text{rear}} = 2 \times (32 \times 10^6 \text{ N/m}) = 64 \times 10^6 \text{ N/m}$

Therefore, the total equivalent stiffness ($\text{k}_{\text{eq}}$) for the entire system is the sum of the stiffnesses from all springs:

$\text{k}_{\text{eq}} = \text{k}_{\text{total, front}} + \text{k}_{\text{total, rear}}$

$\text{k}_{\text{eq}} = (32 \times 10^6 \text{ N/m}) + (64 \times 10^6 \text{ N/m})$

$\text{k}_{\text{eq}} = 96 \times 10^6 \text{ N/m}$

Natural Frequency Determination

The natural frequency ($\omega_{\text{n}}$) of a spring-mass system is the frequency at which it oscillates freely without any external forces. It is calculated using the formula:

$\omega_{\text{n}} = \sqrt{\frac{\text{k}_{\text{eq}}}{\text{m}}}$

Where:

• $\omega_{\text{n}}$ is the natural frequency in radians per second (rad/s)
• $\text{k}_{\text{eq}}$ is the equivalent stiffness in N/m
• $\text{m}$ is the mass in kg

Plugging in the calculated values:

$\omega_{\text{n}} = \sqrt{\frac{96 \times 10^6 \text{ N/m}}{240 \text{ kg}}}$

$\omega_{\text{n}} = \sqrt{400 \times 10^3 \text{ rad}^2/\text{s}^2}$

$\omega_{\text{n}} = \sqrt{400000 \text{ rad}^2/\text{s}^2}$

$\omega_{\text{n}} \approx 632.455 \text{ rad/s}$

Resonance Speed in RPM

Resonance occurs when the engine's operating speed (frequency) matches its natural frequency. To express this in revolutions per minute (rpm), we first convert the natural frequency from radians per second to Hertz (cycles per second), and then to rpm.

The relationship between angular frequency ($\omega_{\text{n}}$) and frequency in Hertz ($\text{f}_{\text{n}}$) is:

$\text{f}_{\text{n}} = \frac{\omega_{\text{n}}}{2\pi}$

$\text{f}_{\text{n}} = \frac{632.455 \text{ rad/s}}{2 \times 3.14159}$

$\text{f}_{\text{n}} \approx \frac{632.455}{6.28318} \text{ Hz}$

$\text{f}_{\text{n}} \approx 100.658 \text{ Hz}$

Finally, to convert the frequency from Hertz (cycles per second) to revolutions per minute (rpm), we multiply by 60:

Engine Speed (RPM) = $\text{f}_{\text{n}} \times 60$

Engine Speed (RPM) = $100.658 \text{ Hz} \times 60 \text{ s/min}$

Engine Speed (RPM) $\approx 6039.48 \text{ rpm}$

Rounding this value to the nearest whole number gives approximately 6039 rpm. Comparing this with the given options:

The calculated engine speed at which resonance is likely to occur is approximately 6039.48 rpm, which is closest to 6040 rpm.